Jan 6, 2012 - rate 1.0E-4 (by initial market share controlled and β2). . . . . . . 101 vii ..... This version of the investment game has very similar properties of the ...... why there is less challenging to the project leader or changing the project ...... 하다 정체시에는 250% 정도로 요금을 올리는 것이 적절한 정책이라는 해석을.
Ph.D Dissertation in Economics
Three Essays in the Computational and Experimental Economics
by
Cho, Nam-Un
Department of Economics Graduate School Korea University January 2012
박 만 섭 교수지도 박사학위논문
Three Essays in Computational and Experimental Economics
이 논문을 경제학박사 학위논문으로 제출함.
2010년 1월 6일
고려대학교 대학원 경제학과 조
남
운
조남운의 경제학박사 학위논문 심사를 완료함.
2012년 1월 6일
위원장
박 만 섭
(인)
위
원
안 도 경
(인)
위
원
이 우 진
(인)
위
원
최 정 규
(인)
위
원
한 치 록
(인)
부모 , 주 , 이연, 그리고
을 나누었거나 나눌 모든 사람들에게
Three Essays in the Computational and Experimental Economics Namun, Cho
Contents Contents
i
List of Tables
iii
List of Figures
v
1 Information under Interdependent Expectations
1
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Theoretic Approach . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.4
Agent-based Simulation . . . . . . . . . . . . . . . . . . . . . .
43
1.5
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .
77
1.A Specifications of Experiments . . . . . . . . . . . . . . . . . . .
78
1.B Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
1.C Korean Abstract . . . . . . . . . . . . . . . . . . . . . . . . . .
82
2 What makes open source software development sustainable? 83 2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
2.2
The structure of Agent-Based-Simulation Model
. . . . . . . .
88
2.3
The result of the simulation . . . . . . . . . . . . . . . . . . . .
94
2.4
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . .
96
2.A Appendix: The comparison of simulation results . . . . . . . . . 100 i
2.B Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.C Korean Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3 민간자본 고속도로의 최적 요금 검토
109
3.1
서론 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2
이원화된 고속도로 요금 체계와 이용 현황 분석 . . . . . . . . . 111
3.3
이원화된 고속도로에서의 PoA 산출 . . . . . . . . . . . . . . . . 117
3.4
이질성 가정하에서 PoA 산출 : 시뮬레이션 . . . . . . . . . . . . 124
3.5
결론 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.A 부록 : 통행량의 추정 . . . . . . . . . . . . . . . . . . . . . . . . 137 3.B Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.C 영문초록 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
ii
List of Tables 1.1
Payoff matrix of 2-person binary choice investment game . . . . . .
7
1.2
Best response of agent i in 2-person investment game . . . . . . . .
9
1.3
Optimal strategy when ηi ≈ 0
. . . . . . . . . . . . . . . . . . . .
20
1.4
Parameter setting for Experiment . . . . . . . . . . . . . . . . . . .
23
1.5
Reward System: Overview . . . . . . . . . . . . . . . . . . . . . . .
28
1.6
Estimation results : random-effects GLS regression of ∆βit and ˜ t−1 with different information group . . . . . . . . . . . . . . . . R ˜t − R ˜ e between informed and uninformed groups Result of t-test: R
31
with 3/3 comprehension score . . . . . . . . . . . . . . . . . . . . .
32
1.7 1.8
it
Variance comparison test of Var(∆βit ) between informed and uninformed groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Summary for Crisis State . . . . . . . . . . . . . . . . . . . . . . . ˜ t of the groups regrouped by the crisis experience. . . . 1.10 Average R
35
1.11 Overview: Main Parameter Setting . . . . . . . . . . . . . . . . . .
44
1.12 Payoff Comparison: ADD vs. PROP . . . . . . . . . . . . . . . . . ˜ t after 1000 round in multiple strategy set 1.13 Standard deviation of R ˜ t in the different comparison 1.14 Comparison of standard deviation of R
60
regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
1.15 Groups Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
2.1
91
1.9
Happiness matrix of each agents in the static environment . . . . . iii
39
66
2.2
Skill distribution based on Sourceforge.net . . . . . . . . . . . . . .
93
3.1
천안–논산 구간 고속도로 상세 현황 . . . . . . . . . . . . . . . . . . 112
3.2
천안–논산 구간 고속도로 요금 (2008년 10월 현재)
3.3
정체 비용을 제외한 천안–논산 구간 고속도로 이용 비용 (승용차 기준)113
3.4
천안–논산 구간 민자고속도로 이용현황 (차량수/시간) . . . . . . . . 114
3.5
부산–대구 구간 고속도로 상세 현황 . . . . . . . . . . . . . . . . . . 115
3.6
부산–대구 구간 고속도로 요금 (2008년 10월 현재)
3.7
정체 비용을 제외한 대구–부산 구간 고속도로 이용 비용 (승용차 기준)116
3.8
대구–부산간 민자고속도로 이용현황 . . . . . . . . . . . . . . . . . 116
3.9
업무통행 시간가치 (2008년 기준) . . . . . . . . . . . . . . . . . . . 121
. . . . . . . . . 113
. . . . . . . . . 115
3.10 VDF 함수의 도로 차선 수 (lk ) 에 따른 계수값 . . . . . . . . . . . . 122 3.11 2008년 제품별 주유소 가격 . . . . . . . . . . . . . . . . . . . . . . 122 3.12 천안–논산 고속도로 요금체계 (승용차 기준) . . . . . . . . . . . . . 122 3.13 대구–부산 (양산) 고속도로 요금체계 (승용차 기준) . . . . . . . . . . 122 3.14 동질성 가정하에서 수학적으로 도출된 민자도로 최적 가격정책 . . . 124 3.15 구간별 추정 상시 교통량 . . . . . . . . . . . . . . . . . . . . . . . . 125 3.16 시뮬레이션 개관 (F는 False, T는 True를 의미) . . . . . . . . . . . 126 3.17 평가기준별 민자 고속도로 최적요금 (1종 기준) . . . . . . . . . . . . 135 3.18 천안–논산 간 고속도로의 구간별 요금과 동일 요율 적용시 도로요금 137
iv
List of Figures 1.1
Mixed strategy Nash equilibria in the 2-person binary investment
1.2
11
1.3
game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¯ = 0.8 . . . . . . . . . . . Best Response of 2-person Game when R ¯ = 0.4 . . . . . . . . . . . Best Response of 2-person Game when R
1.4
Graph of ϵ, β2 , β1NE − β1∗ . . . . . . . . . . . . . . . . . . . . . . . .
14
1.5
The rate of return π1 of each investment strategy with regard to x2 at the local crisis case
1.6
. . . . . . . . . . . . . . . . . . . . . . .
11
15
Replicator dynamics with Non-investment strategy in the local crisis condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7
9
16
Screen capture of intro web page for investment game experiment conducted at fall, 2011 . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.8
Flowchart of experiment procedure: client part . . . . . . . . . . .
25
1.9
Flowchart of experiment procedure: server part . . . . . . . . . . .
25
1.10 Screen capture of investment game web page . . . . . . . . . . ˜ t with different information type and unwaged game . . . . . 1.11 R 1.12 First degree optimal strategy of ∆βit considering β¯t−1 . . . . . ˜ t−1 . . . . . . . . . . . . . . . . 1.13 Correlation between ∆βit and R
. .
26
. .
29
. .
30
. .
30
˜ t−1 by information and comprehension 1.14 Scatter graph of ∆βit and R score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
31
˜ t and R ˜ e , grouped by compre1.15 Box graph of difference between R it hension score. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ e , grouped by information and com1.16 Scatter graph of ∆βit and R it
32
prehension score . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
1.17 Average payoff of last round, partitioned by max round per session and crash condition . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ t grouped by experience . . . . . . . . . . . . . . . 1.18 Box graph of R ˜ 1 between experienced and unexperienced groups. 1.19 Box graph of R ˜ t of the groups classified by 1.20 Average and standard deviation of R
34
the crisis experience . . . . . . . . . . . . . . . . . . . . . . . . . .
38
1.21 Box graph of Wi at last round classified by reward system . . . . ˜ t classified by the size of re1.22 Average and standard deviation of R
40
ward: extra credit ratio . . . . . . . . . . . . . . . . . . . . . . . . ˜ t classified by N . . . . . . . 1.23 Average and standard deviation of R ˜ t by controlled factors . . . . 1.24 Average and standard deviation of R
40 42
1.25 5 × 5 torus lattice and an agent A’s neighbors with boundary one .
44
1.26 Flow Chart of Simulation Process . . . . . . . . . . . . . . . . . . . ˜ t by payoff schemes, with no mimic and no mutation . 1.27 Graph of R
46
36 37
41
59
1.28 Histograms of µi (mean) after 2500 rounds with additive payoff regime, mimic process, κ = 0.0441, δ = 0.1, fixed CON strategy with same agent set. . . . . . . . . . . . . . . . . . . . . . . . . . . ˜ t by δ, κ with fixed CON strategy . . . . . . . . . . . . 1.29 Graph of R ˜ t with mimic and mutation . . . . . . . . . . . . . . . . 1.30 Graph of R ˜ t by payoff and information condition with fixed mem1.31 Graph of R
62 63 65
ory length, global neighbor, full round comparison . . . . . . . . . ˜ t , one iteration for closer view . . . . . . . . . . . . . . 1.32 Graph of R ˜ t . All graphs with same parameters are merged to one. 1.33 Graph of R
67
1.34 Histogram of wi2500 by each iteration . . . . . . . . . . . . . . . . .
69
1.35 Kernel density of all iteraions with same condition . . . . . . . . .
69
vi
67 68
1.36 Graph of relative wealth of richest agent to total wealth in each iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
1.37 Graph of occupy rate of strategies at each iterations and different conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
1.38 Box graph of Wi2500 and ln Wi2500 . . . . . . . . . . . . . . . . . . .
72
1.39 Bar graph of average Wi2500 and ln Wi2500 . . . . . . . . . . . . . . 72 ˜ t in the proportional payoff regime . . . . . 73 1.40 Aggregated graph of R ˜ t in each iterations with 500 rounds of comparison period 74 1.41 Graph of R 1.42 Occupy rate of strategies in the 500 rounds of comparison period . ˜ t , Proportional payoff condition. Comparison Period: 1.43 Graph of R
75
1000 rounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
2.1
The movement of heterogeneous skilled OSS users without mutation (by initial market share controlled). . . . . . . . . . . . . . . .
2.2
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled). . . . . . . . . . . .
2.3
96
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-5 (by initial market share controlled). . . . . . . . . . . .
2.4
95
97
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 70% (by skill level controlled). 97
2.5
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 and OSS initial market share 30% (by skill level controlled). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6
98
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.7
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
2.8
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled and β2). . . . . . . 101 vii
2.9
The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 90% (by skill level controlled). 101
2.10 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 80% (by skill level controlled). 102 2.11 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 70% (by skill level controlled). 102 2.12 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 60% (by skill level controlled). 103 2.13 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 50% (by skill level controlled). 103 2.14 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 40% (by skill level controlled). 104 2.15 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 30% (by skill level controlled). 104 2.16 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 20% (by skill level controlled). 105 2.17 The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 10% (by skill level controlled). 105 3.1
천안–논산 고속도로 망의 구조 . . . . . . . . . . . . . . . . . . . . . 112
3.2
대구–부산 고속도로 망의 구조 . . . . . . . . . . . . . . . . . . . . . 115
3.3
천안 논산 고속도로의 차량흐름, 가격정책별 PoA 분포 . . . . . . . 123
3.4
이질성 정도에 따른 시뮬레이션 결과 개관 . . . . . . . . . . . . . . 127
3.5
CN5,DB5를 30×4000 pcu 으로 1회 시뮬레이션했을 때의 PoA 그래프128
3.6
천안–논산간 고속도로의 0–15,000pcu 구간에서의 PoA . . . . . . . 129
3.7
천안–논산 구간 고속도로의 500–3,000pcu 구간에서의 PoA . . . . . 130
3.8
대구–부산 구간 고속도로의 0–15,000pcu 구간에서의 PoA
3.9
이론적으로 계산한 천안–논산 구간의 PoA . . . . . . . . . . . . . . 132
. . . . . 131
3.10 수정된 PoA로 시뮬레이션 1–5 재계산한 결과 개관 . . . . . . . . . 133 viii
3.11 CN5: 수정된 PoA로 재계산 . . . . . . . . . . . . . . . . . . . . . . 134 3.12 DB5: 수정된 PoA로 재계산 (10회 계산 후 평균값) . . . . . . . . . 135
ix
Chapter 1
Information under Interdependent Expectations Abstract The origin of fundamental uncertainty lies in the interdependent expectations. In this paper, simple investment game with proportional payoff regime is considered to analyse the effect of additional information. In this game, agents make portfolio of risky asset and risk free asset. Risk free asset gives fixed rate of return regardless of macroeconomic state but the rate of return of risky asset is dependent on macroeconomic state: crisis state and normal state. This macroeconomic state is determined by the weighted average of individual portfolio and wealth. The crisis threshold is common knowledge. In this game, the Nash equilibrium portfolio is exactly the crisis threshold level. But because of the loss from crisis state and uncertainty about other agents’ strategies, Nash equilibrium portfolio lies safer area than theoretical Nash equilibrium. The web based, extra credit rewarded experiments for this investment game was conducted with undergraduate students in Korea University. They used their information to maximize their own wealth, but the positive effect to the rates of return of informed groups were not prominent. In the
1
agent-based simulation for this investment game was done and one more condition was considered: additional payoff regime to compare the result of traditional game theory. The result was that information showed positive effect in the additional payoff regime. However in the proportional payoff regime, additional information showed no clear contribution to the rate of return and clear negative effect to the fluctuation of the macroeconomic state. This suggests that there may exist some condition in which true additional information have negative effect. Keywords: Interdependent Expectation, Information, Evolutionary Game Theory, Economic Experiment, Proportional Payoff Regime, Agentbased simulation JEL Classification Numbers: C73, C91, G02, G11, G17
1.1
Introduction
Economic agents make their decisions based on their expectation of other agents’ behavior. Yet, the behavior of each of the other agent depends on his or her expectation of all other agents’ behavior which are dependent on their expectation about the behavior of others. This interdependent expectation is one of the main questions for the complexity of economic phenomenon. This issue arose in the early twentieth century. Keynes (1936) pointed out the well-known story of ‘Beauty Contest’ to show selecting the prettiest face is different from selecting the one which would be regarded as the prettiest by others. Suppose B is the set of selectable faces and Ei (B) is the face regarded as the prettiest by ith competitor among N competitors. To win this contest, the ith competitor must choose the face which would be considered as the prettiest by the others. This face can be expressed as Ei Ej∈U (B). U is the set of all competitors. But it is not the end of the solution. If it is concerned that other competitors would reason exactly the same as ith competitor, i’s expectation must be like this: Ei Ej∈U Ek∈U (B). And this expectation would 2
be taken by the others, so the expectation would be the endless expectations of others’ expectations. This endless expectation is caused by the fact that the expectation to win the contest depends on the others’ expectations. This condition can be stated as ‘Interdependent Expectation(henceforth IE)’. The circumstance like IE condition can easily be observed in economic reality. In reality, many economic signals come from aggregated decisions based on the IE. Arthur (1995) suggested the model of financial market in which the price was determined by the individual investment decision. Then he showed that stable equilibrium could not be achieved if the assumptions of homogeneous agents and unbiased expectations on which the rational expectation theory was based were relaxed. Arthur (1994) also suggested the so-called ‘Bar Problem’ that all players wanted to go to the uncrowded bar. This also can be the example of the IE situation. To solve this model, he made a set of inductive reasoning pattern from the observations and made the decision rule which have the property of correcting their own strategy by their payoff. Although there is no stochastic process in this model, the result shows the noisy pattern around the critical threshold. Nagel (1995) designed a numerical version of Keynesian Beauty Contest game to test the depth of reasoning in the IE situation and conducted an experiment. In this experiment, most participants made their decision from at most three or infinity degree of consideration. Above researches consider the fundamental uncertainty – or Keynesian uncertainty – arising from the fact that decision making process determined by the expectation of others who decide from the expectation of others. This paper is similar to them, but it focuses on another issue: the information of systemic risk and payoff scheme. To do so, this study develops a simple investment game in which the macro state is determined by collective decisions of participants and conducts a web-based experiment with under3
graduate students in the Korea University. And the agent-based simulation was performed to replicate and analyze the effect of information in the IE environment. The organization of this paper is following: Section 1.2 sets up the simple investment game and considers the stochastic dynamics theoretically. Section 1.3 describes the result of experiment and exports the mechanical patterns of participants. Section 1.4 presents the dynamics of the strategies extracted from section 1.3 by agent-based simulation of the investment game. Section 1.5 provides concluding remarks.
1.2
Theoretic Approach
Model This paper is based on a kind of N -person investment game. In period t, ith participant of this game can invest his endowment Wit in the two types of asset. In fact, the number of investment category can be three for the reason that they can choose not to invest any of these assets. These choices are called ‘Risk Free Asset(WSit )’, ‘Risky Asset(WRit )’, and ‘Not-Invested Asset(WN it )’. If the agent i decides to invest the endowment at the rate of αit , βit , γit ∈ [0, 1] then this can be expressed as Equation (1.1).
Wit = (αit + βit + γit )Wit = WSit + WRit + WN it ,
i = 1, · · · , N
(1.1)
Categories of Investment Assets Risk Free Asset
In period t, if agent i decides to invest WSit in Risk Free
Asset, the agent receives (1 + rS )WSit at the beginning of the next period.
4
Risky Asset
In period t, if agent i invests WRit in Risky Asset, the agent
receives a return dependent on the macrostate of the economy: normal and crisis state. This state is determined by total risky asset rate and the given ¯ ∈ (0, 1). If total risky asset rate β¯t which is the ratio of total risk threshold R risky asset invested by all agents and total asset invested by all agents is higher ¯ the macro state of this economy becomes crisis state and all of the than R, risky asset disappear. Otherwise the state is normal and receive higher return at the rate of rR than that of the risk free asset. Let ∆WRi,t+1 be a return from riksy asset, then
∆WRi,t+1 = rRt WRit Ψ(β¯t ≤ Rt ) =
0,
if β¯t > Rt ∈ [0, 1]
rRt WRit ,
otherwise
∑N Rt = ∑N i
i
WRit
(WSit + WRit )
rRt > rSt
(1.2)
(1.3)
(1.4)
Where Ψ is the logical function that is 1 if the input expression is true and is 0 otherwise. Not-Invested Asset
If agent i decides not to invest certain amount WN it ,
the non-invested endowment WN it is just carried over to the next period. If the agent is rational, this amount should be zero. i.e., strategies in which WN it > 0 is dominated strongly except for the case of relative payoff maximization.1 This property can be used for testing the rationality of participants. WN it = Wit − (WSit + WRit ) = γit Wit
(1.5)
1 If the agents’ objects are to maximize their ‘relative’ payoffs, strategy with WN it > 0 can be used to induce crisis state. It is because safe strategy has external effect of making macro state more stable. If one is able to induce crisis state by no investment and not by safe investment, non-investment strategy can be optimal strategy. This issue is discussed in the Subsection 1.2.
5
Individual Payoff Let Sit be agent i’s investment vector consisted of asset ratios: (αit , βit , γit ) (αit , βit , γit ),
αit + βit + γit = 1,
αit , βit , γit ∈ [0, 1]
(1.6)
Let ∆WSi,t+1 be a return from risk free asset. Then Wi,t+1 , the wealth of an agent i at period t + 1 can be derived as Equation (1.7) with respect to Sit , individual portfolio:
Wi,t+1 = Wit + ∆WSi,t+1 + ∆WRi,t+1 ¯ = γit Wit + αit (1 + rSt )Wit + βit (1 + rRt )Wit Ψ(Rt ≤ R) γit = 1 − (αit + βit )
(1.7)
(1.8)
Then agent i decides Si∗ to maximize his or her wealth.2 Sit∗ = arg max Wi,t+1 Si
Overall Processes
s.t.
αit + βit ≤ 1
(1.9)
Overall process in period t takes the following steps:
Step 1: Each agent decides his or her investment portfolio Step 2: Macroeconomic status is set by the result of agents’ aggregated investment decisions Step 3: Return is distributed to the agents according to each portfolio and macroeconomic status 2 If the object of the agent is not to maximize absolute wealth but to maximize relative wealth, the strategy of no-investment γit > 0 should be no longer considered as strictly dominated. This issue will be considered at the later section.
6
Equilibrium Analysis In this game, the crucial factor that each agent must consider is the fact that ¯ If Rt does not exceed the macroeconomic state whether or not Rt exceeds R. ¯ regardless of agent i’s portfolio, the best strategy of agent i is to invest R all endowment to the risky asset: Sit = (0, 1, 0), the same representation for βi = 1. For the same reason, if β¯t exceeds Rt for agent i’s all feasible portfolios, the best strategy is αit = 1. However, these situations would not be general if it is taken into account that all other agents would decide their strategies in the same way. Typically, the resulting strategy is not stable but oscillating around critical level as Arthur (1994), Nagel (1995) and many other literatures. Equilibrium in 2-person Game In the beginning, consider the investment game in which two participants are ¯ = 0.8. And assume endowed with same endowment. Let the crisis threshold R additionally that agents can choose from two investment portfolio (αit , βit ): (0, 1) or (1, 0). In this 2-person binary choice investment game, the payoff table is described in Table (1.1). i.e., if all agent choose the risky strategy – Sit = (0, 1)
∀i –, the macroeconomic state goes into the crisis state and the
rate of return is −100%. Otherwise, agents would receive the positive return by their portfolio.
Saf e1 Risky1
Saf e2 (rS W1 , rS W2 ) (rR W1 , rS W2 )
Risky2 (rS W1 , rR W2 ) (−W1 , −W2 )
Table 1.1: Payoff matrix of 2-person binary choice investment game. agent i’s strategy Saf ei and Riskyi means (αi , βi ) = (1, 0) and (0, 1) strategy each. This version of the investment game has very similar properties of the Hawk-Dove game and the Chicken game.3 If the payoff matrix of row player 3
But the difference from these games would become more apparent if it is considered
7
is set to Equation (1.10), elements of this matrix for the hawk-dove game and the chicken game are c > a > b > d, while c > a = b > d in the investment game. a b , A= c d
c>a=b>d
Pure Strategy Nash Equilibrium
(1.10)
The pure strategy Nash equilibria are
the same as Hawk-Dove game: (Saf e1 , Risky2 ), (Risky1 , Saf e2 ). The best strategy is choosing opposite strategy to the opponent’s. Mixed Strategy Nash Equilibria In this paragraph, the mixed strategy is considered: Si = (α ˆ i , βˆi ), α ˆ i + βˆi = 1. In this case, α ˆ , βˆ is the probability of choose risk free or risky strategy each. Then payoff of agent i can be expressed as Equation (1.11).4
A1 =
rS rR
T πi (Si ) = Si Ai S−i , rS r , A2 = S −1 rS
Si = (ˆ αi , βˆi ) rR , 1 > rR > r S > 0 −1
(1.11)
From the Equation (1.11), Equation (1.12) is derived. Table (1.2) is the best response of agent i. πi (βˆi ) = βˆi (−rS + (1 − βˆ−i )rR − βˆ−i ) + rS
(1.12)
Therefore, the mixed strategy Nash equilibria are: {(βˆ1∗ , βˆ2∗ )} = {(1 − (1 + rS )/(1 + rR ), 1 − (1 + rS )/(1 + rR )), (1, 0), (0, 1)}. On the other hand, in the multiplicity environment. The payoff of investment game is determined by the whole agents globally while that of hawk-dove game is determined by the strategy of opponent agent locally. 4 It is worth to remark that α, ˆ βˆ is different to the portfolio α, β
8
βˆ−i
Mixed Strategy βˆi∗ = 1 ∗ βˆi ∈ [0, 1] βˆ∗ = 0
0 ≤ βˆ−i < 1 − (1 + rS )(1 + rR ) βˆ−i = 1 − (1 + rS )/(1 + rR ) 1 − (1 + rS )/(1 + rR ) < βˆ−i ≤ 0
i
Table 1.2: Best response of agent i in 2-person investment game
1
β2
1-rS/rR
0 0
1-rS/rR
1
β1 Figure 1.1: Mixed strategy Nash equilibria in the 2-person binary investment game. Dashed line is for participants 1, and solid for participant 2.
more general expression of the mixed strategy Nash equilibria from the payoff matrix(Equation (1.10)) can be expressed as Equation (1.13). ( ) b−d b−d ∗ ∗ (β1 , β2 ) = 1 − ,1 − , (1, 0), (0, 1) c−a+b−d c−a+b−d Portfolio Strategies
(1.13)
Now turn to the case of portfolio investment: each
agent is able to invest in some portfolio of risk free asset and risky asset. Let βi be the risky asset investment ratio of agent i, then γit > 0 is strictly dominated
9
strategy when absolute wealth maximization is each agent’s object.5 It leads to γit = 0 ∀i, and the π1 which is the payoff of agent 1 can be expressed as Equation (1.14).6
π1 (β1 , β2 ) =(1 − β1 )rS W1 (∑ ) βi Wi i ¯ + rR β1 W1 Ψ ∑ ≤R i Wi ) (∑ βi Wi i ¯ >R − β1 W1 Ψ ∑ i Wi
(1.14)
In this case best response of each agent’s strategy can be described like Figure 1.2.7 The best response of agent 1 is line ABC, and that of agent 2 is line BCD. Therefore the Nash equilibrium of this 2-person investment game can be described by the line BC. This equilibrium means that the best strategy of each agent is to raise βi as high as not going into crisis state. ¯ ∈ [.5, 1], but when R ¯ < .5 the This type of response curve is valid when R shape of response curve is like Figure (1.3). However, the Nash equilibrium is also line BC where two response curves are overlapped. AB, CD means that the best strategy is to invest all endowment into the risk free asset in the crisis state. When N – the number of agents under the assumption of same endowment – is higher than 3, the Nash equilibrium is the hyperplane which satisfies 5 When each agent’s object has some property of relative wealth maximization, γit > 0 could be no longer strictly dominated. This problem would be discussed later in Subsection (1.2). 6 Hence, time index t is omitted for simplicity of equations. When the endowment is equal with each other, the average investment ratio of risky asset can be reduced to the equation below: ∑2 ∑2 βi Wi i βi ¯ = ≤R ∑i 2 2 W i i 7
The range of β1 for getting higher return at the riskier state is not feasible. β1 < −
rR − rS ¯ (2R − β2 ) < 0 rS + 1
10
¯ = 0.8 Figure 1.2: Best Response of 2-person Game when R
¯ = 0.4 Figure 1.3: Best Response of 2-person Game when R
11
¯ But when R ¯ ∈ [0, 1/N ], any agent can cause the crisis state and β¯t = R. ¯ ∈ [(N − 1)/N, 1], any agent can guarantee normal state. In the 2when R person game, cases of Figure (1.2),(1.3) come under the former and latter case each. If N exceeds 3, no single agent can cause or prevent the crisis ¯ ∈ [1/N, (N − 1)/N ]. The problem raised by the fact that single state when R agent or small group can determine the macro state becomes trivial since [1/N, (N − 1)/N ] ∼ (0, 1) at high N .8 Trembling hand Nash Equilibrium
Taking into account the possibility
of slight mistakes, the Nash equilibrium shifts to the safer area(Selten (1975)). If there is small probability ϵ > 0 to choose the strategy of no intent, i.e., random portfolio of X ∼ U (0, 1) with the probability of ϵ, expected payoff of agent 1 is expressed as Equation (1.15).9
π1 (β1 , β2 ) = (1 − β1 )rS W1 +
[ ] ¯ − β1 )rR − (1 − 2R ¯ + β1 ) β1 W1 (1.15) (1 − ϵ)rR β1 W1 + ϵ (2R
The second derivative of this expected payoff is always negative and the point which satisfies first order condition of β1 which maximize E(π1 ) is best response of agent 1. Such β1∗ can be derived as: β1∗
1 = 2ϵ
(
1 + rS 1− 1 + rR
)
¯−1 +R 2
(1.16)
8
But if some agents have high wealth, this problem can reoccur. This problem will be discussed in Section 1.4. 9 This can be derived as below: [∫ ¯ ] ∫ 1 2R−β1 π1 = (1 − β1 )rS W1 + (1 − ϵ)rR β1 W1 + ϵ rR β1 W1 dX2 − β1 W1 dX2 0
¯ 2R−β 1
¯ − β2 must be satisfied since the strategy that causes crisis state when other agent β1 ≤ 2R make no mistake is strictly dominated.
12
The feasible condition which β1∗ ∈ [0, 1] is Equation (1.17). RHS of Equation (1.17) is positive and smaller than 1 for all possible parameters. Therefore if ϵ ≤ β1∗ , the best response is not changed but if ϵ > βi∗ it changes into ¯ − 1, β ∗ ) ≤ β NE = 2R ¯ − β2 .10 max(2R 1
1
) ( 1 1 + rS ϵ> 1− ¯ 1 + rR 3 − 2R
(1.17)
ϵ can be understood as the uncertainty of others’ strategy. Existing Nash equilibrium becomes more risky when it is more uncertain. It means the best response is formed more inner area as uncertainty is increasing. This relation is described in Figure (1.4). In this figure, the vertical axis represents the distance between the existing Nash equilibrium and the current best response. This is ˆ − 1 can be stated as the fallback strategy an increasing function on ϵ. β1 = 2R of agent 1. Replicator Dynamics Replicator Dynamics in Local Conditions
The game in which only
mixed strategy is available – i.e., portfolio is not allowed – and the crisis state is locally determined is equivalent with traditional Hawk-Dove game or Chicken game. The term ‘Local’ means that the crisis state is not determined by all, but by the game partner. Let x1 be the frequency of the agents whose 10
The more strict Nash equilibrium β1NE is: ¯ − β2 ) β1NE = min(1, 2R
However, the condition such that β1NE = 1 is available only if opponent player uses fallback strategy S2 = (1, 0) which guarantee normal macrostate. This means that function min can ¯ > 1 − 1/N . be omitted when there is no fallback strategy such that N > 3 and R ¯ > 1/2, β1∗ > β1NE can be feasible and the range On the other hand, β1∗ ≤ β1NE is true if R of ϵ which makes this condition be feasible is: ) ( ) ( 1 1 + rS 1 1 + rS · < ϵ < 1 − · ¯ 1− ¯ 1 + rR 1 + rR 3 − 2R 2R + 1 However, such β1∗ is dominated strategy since this causes crisis state if opponent doesn’t make mistake.
13
Figure 1.4: The graph of ϵ, β2 , β1NE − β1∗ . Parameters are set as the experiment parameters.
strategy si = (1, 0) = S and x2 be the frequency of the agents whose strategy si = (0, 1) = R. In this situation, the rate of return of each strategy πS , πR with regard to x2 can be derived like below:
πS (x2 ) = (1 − x2 )rS + x2 rS = rS πR (x2 ) = (1 − x2 )rR − x2 = rR − (1 + rR )x2
(1.18)
Suppose the reproduction mechanism is set to that of (Bowles, 2004, p.79). i.e., the growth rate of xi has positive relation with the payoff of each strategy added by basic reproduction rate φ. The x∗1 and x∗2 , population ratio of each strategy in the stationary state, follow Equation (1.19) and are evolutionary stable. 1 + rS 1 + rR 1 + rS x∗2 = 1 − 1 + rR x∗1 =
14
(1.19)
Figure 1.5: The rate of return π1 of each investment strategy with regard to x2 at the local crisis case
In spite of considering the strategy of no investment, this result is not changed. In this case, A˜ – the payoff matrix of agent i – can be expressed as Equation (1.20). Each of the first, second, third row of this matrix means risk free, risky, not-invested asset each.
rS
rS
rS
A˜1 = rR −1 −1 0 0 0
(1.20)
The population state shown in the Equation (1.19) is evolutionary stable. Replicator Dynamics in the Global Condition
In this paragraph the
replicator dynamics is considered when the crisis state is determined globally. The term ‘Global’ means that macro state is determined by all the partici-
15
Figure 1.6: The replicator dynamics with Non-investment strategy in the local crisis condition. The arrowed point is the only point which is evolutionary stable. Graph is drawn from Dynamo version 1.1: Sandholm et al. (2011)
pants. Each strategy with regard to x2 can be expressed as Equation (1.21).11
πS (x2 ) = rS ¯ − Ψ(x2 > R) ¯ = πR (x2 ) = rR Ψ(x2 ≤ R)
rR ,
¯ if x2 ≤ R
−1,
¯ if x2 > R
(1.21)
Considering the replicator dynamics in which the population with each strategy increased by the payoff comes from each strategy, replicator equation 11
˜ t , the weighed average of other Unlike Hawk-Dove game, the payoff is determined by R agents’ strategy and their wealth in this investment game. The cause of this difference comes from the property that critical condition is determined globally. This type of game has the character of public good game. The reason why the investment game with two pairwise participants has similar property of hawk-dove game is that total participants are only two agents. Therefore the difference between global and local condition can be ruled out.
16
can be defined as below:
x˙ i = [Average Payoff of Strategy i] − [Average Payoff of All Strategy] (1.22) xi It is assumed for simplicity that the endowment is same for all agents. 12 ¯ i.e., the case which current ratio of Then the condition of crisis is x2 > R, risky asset investors exceeds the threshold of crisis.
¯ − (rR − rS )Ψ(x2 ≤ R)] ¯ x˙ 1 = x1 x2 [(1 + rS )Ψ(x2 > R) ¯ + (rR − rS )Ψ(x2 ≤ R)] ¯ x˙ 2 = x1 x2 [−(1 + rS )Ψ(x2 > R)
(1.23)
¯ However, this result can be The result is that x2 oscillates around R. adjusted if it is taken into account that the assumption of the same endowment is thrown out for the reason that the endowments of risky asset investors are gone at the crisis state. If the switch of strategy is not available, only investors who invested in risk free asset would be remained after first crisis state. But this is not evolutionary stable if the strategy invasion or mutation is considered.13 In this case the ratio of risky asset investors would start to ¯ even if the endowment is not increase again. Therefore x2 oscillates around R same if the strategy switching or invasion and mutation is enabled. It means ¯ the ratio of risky asset investor x2 would increase and otherwise x2 if x2 < R, would decrease. Meaning of not investing Although the strategy of not investing cannot be rationalized in the sense of maximizing absolute payoff, this strategy has some interesting characteristics.14 Suppose that current total risky asset ratio 12 This assumption is valid in additional payoff regime and not in proportional payoff regime because it neglects the wealth effect. This issue will be discussed in Section 1.4. 13 The effect of mutation process will be discussed in Section 1.4. 14 See the footnote 1.
17
¯ – the threshold of the crisis state. In this case, any agent can is exactly R trigger crisis state with very small loss if the agent invest very small asset to risky asset and keep the rest of the endowment not invested. Then the agent’s risky asset ratio becomes 100% and the total risky asset ratio exceeds the threshold of crisis state. This strategy does not maximize the agent’s absolute payoff, but that can be considered rationally if the agent’s object is to maximize the relative payoff. This strategy is strictly inferior when the individual object is to maximize absolute payoff, but it is not the case when the object is to maximize relative payoff. Equilibrium in three or more Person Investment Game In this subsection, investment game with portfolio strategy and N ≥ 3 is considered. The fallback strategy which can trigger or prevent crisis state always exists when N = 2, but when N ≥ 3, such βi doesn’t exist if 1/N ≤ ¯ ≤ 1 − 1/N , i.e., the only possible fallback strategy is to set βi = 0 – not R ¯ ∈ [1/N, 1 − 1/N ].15 In this case, to invest in the risky asset at all – when R ˜ t = R. ¯ Nash equilibrium is the hyperplane satisfying R Crisis probability η can be defined as Equation (1.24). Then ηi , the partial derivative of η of βi would be Equation (1.26). ηi has the meaning that the influence to the η by the strategy of agent i. ηi is always positive for η is increasing function of βi . ˜ t > R), ¯ η ≡ Pr(R
˜t ≡ R
N ∑ βi Wit i
Wt
,
Wt ≡
N ∑
Wit
i
˜t ∂R Wit = ∂βi Wt ηi = 15
∂η > 0∀i ∂βi
¯ would be general case since limN →∞ [1/N, 1 − 1/N ] ∼ (0, 1) Such R
18
(1.24)
(1.25)
(1.26)
The maximization problem of an agent i can be expressed as: max πi (βi , β−i ) = (1 − βi )rS Wi − ηβi Wi + (1 − η)rR βi Wi βi
(1.27)
The sign of the second order derivative of πi is dependent on the relation with ηi , ηii and βi , the first and second order partial derivative of η of βi .16 ∂ 2 πi < 0, ηii = ∂βi2 > 0,
if ηii ≥ −2ηi /βi ,
β ̸= 0
(1.28)
otherwise
The first case in which there exists internal solution of Equation (1.28), βi∗ satisfying the first order condition can be expressed as Equation (1.29).17 If βi∗ ∈ [0, 1], this is the best strategy for agent i (internal solution). Otherwise, 0 or 1 would be the best solution (external solution). βi∗
1 = ηi
[(
1 + rS 1− 1 + rR
)
] −η
(1.29)
η for internal solution must satisfy the condition as Equation (1.30). η ≤1−
1 + rS ≤ η + ηi 1 + rR
(1.30)
1+rS by ηi . ηi which Equation (1.30) means η must be slightly smaller than 1− 1+r R ∑ determines the size of interval becomes larger if Wi / i Wi is large enough or ˜ t and R ¯ is close to each other. This implies that ηi is not small if macro state R 16
This disequation can be derived by: ∂πi = Wi − (1 + rR )Wi (η + ηi βi ) ∂βi ∂ 2 πi = −Wi (1 + rR )(2ηi + ηii βi ) ∂βi2
17
The more strict expression is η = η|βi =βi∗ , ηi = ηi |βi =βi∗ because η is a function of βi . However ∑ the influence of βi for η is proportional to the ratio of the Wi to entire wealth: Wi / i Wi . This means that as N is larger, the relation between βi and η become smaller. Therefore this analysis is valid with sufficiently large N . And it is worth to mention that this is valid only in additional payoff regime. This issue will be discussed in Section 1.4.
19
βi∗
η ≤1− 1
1+rS 1+rR
η >1− 0
1+rS 1+rR
Table 1.3: Optimal strategy when ηi ≈ 0 is sufficiently close to the Nash equilibrium.18 ηi for internal solution would decrease if N increases or individual wealth ratio become smaller. In the case of ηi ≈ 0, i.e., N is sufficiently large or individual wealth ratio is sufficiently small, the optimal strategy can be arranged as Table (1.3). In short, individual optimal strategy becomes extreme – invest all asset to risky asset or risk free asset – as individual influence to macro state being smaller. 1+rS . This threshold value The η which become the threshold of decision is 1− 1+r R
is about 19% with the parameter used by the experiment(Table (1.4)). i.e., when N is sufficiently large, the optimal strategy is to invest all asset to risky asset if η < 19% or to risk free asset if η ≤ 19%.
Influence of Information The optimal βi (Equation (1.29)) regarding the probability of small mistake (Subsection 1.2) becomes even smaller if agents are risk averse. This implies that the investment to risky asset become less attractive in a more uncertain situation. In the experiment of Section 1.3 and simulation in Section 1.4, additionally ˜ t−1 , the total risky asset ratio in previous provided information at period t is R ˜ e , future total period. Participants would use this information to expect R t e ˜ ¯ risky asset ratio. If Rt is very close to R, ηt would rise and agents adjust ˜ t ≪ R, ¯ βi would be adjusted each βi to lower level. On the other hand, if R to a higher level. This can be called first degree strategy. Let this strategy 18
In this case, η would also become large. The elasticity of η to βi get higher as macro state approaching to Nash equilibrium. In Section 1.4, η is defined deductively as ratio of crisis state and normal state. But for the exact analysis, the function η must be defined strictly. This question remain on the agenda for future research.
20
be Si1 . But another strategy can be chosen considering others would behave 1 . This is the second degree strategy. By such way, S k , the strategy of as S−i i
agent i with kth degree strategy, can be defined.19 For this reason, measuring the actual thinking depth is an important issue. Nagel (1995) measured a degree of considering others’ strategy by experiment of the numerical version for Keynesian ‘Beauty Contest’ guessing game. It was the result that the deepest level of the participants who choose strategies based on pure kth degree expectation was at most three or infinity. It is worth to note that the strategic depth had positive relation with the agent’s payoff from the experiment.(Camerer et al. (2004)) As discussed before, the strategy would be formed by the crisis condition. And this condition is determined by theoretical Nash equilibrium state: ˜t = R ¯ and the uncertainty. From this uncertainty, agents would have internal R ˜t = R ¯ − δi . δi is the gap of agent i concerning uncertainty decision criteria: R of the crisis state.20 In the experiment of Section 1.3, there are two groups: one is informed explicitly about previous total risky asset ratio, and the other has no explicit information. It can be expected that uninformed group would face more uncertain environment and their strategy would be more careful: this will make their δi be higher. But it does not mean that uninformed group has absolutely no information. They could have implicit information from previous macroeconomic state.21 For example, if previous state was not in crisis, this should ˜ t−1 ≤ R. ¯ So, if the uninformed group’s macro state is normal for mean that R the several periods, the first degree strategy would be raising their individual risky asset ratio βi . 19
Cognitive Hierarchy Model of Camerer et al. (2004) can be one of kth degree strategy approach model. This is also called as P-CH(Poisson-Cognitive Hierarchy) model from the assumption that the distribution of lower degree strategy follows Poisson distribution. 20 In Section 1.4, this would be considered as ‘reserved total riksy asset ratio’. 21 This reasoning is not valid only if they invest all their wealth to risk free asset.
21
Figure 1.7: Screen capture of intro web page for investment game experiment conducted at fall, 2011
Response to the Risk Another factor to be considered is δi : agent i’s risk averseness. The size of δi is not affected only by the environmental uncertainty, but also by agent i’s risk averseness. Therefore δi would have positive relation with both global uncertainty and agent i’s risk averseness.22 The learning effect should also be examined. Participants could adjust their strategy as their knowledge is increased. And also the experience of crisis can effect their strategies.
1.3
Experiment
In this section, the behavioral difference under different information condition is investigated by experiment of Web-based investment game. 22 For this reason, the survey for measuring risk averseness was conducted after 2011, the result has not been fully investigated yet.
22
Parameter
Value
R Wi0 rS rR (rR − rS )/(rR + 1)
0.8 10,000ECU 0.05 0.3 0.19
Table 1.4: Parameter setting for Experiment
Experimental Design and Hypotheses Participants connect to the web server for the experiment with their own USERID/PASSWORD and make investment decisions. There are two different game by the endowment regime. In some experiments, endowment(Wi0 ) was provided only in the first round – henceforth ‘unwaged’ group – and the other experiments, same endowment was provided at each round – henceforth ‘waged’ group. The value of main parameters are set as the Table (1.4). The term ECU(Experimental Currency Unit) is used to express the currency unit in this experiment. ˜ t−1 , risky asset ratio in the previous period is proTo informed groups, R vided and to uninformed groups, any explicit information is not provided. All the other settings are the same. These groups has exactly the same rules but the global condition is independent to each other. i.e., global variables such ˜ t is calculated for each group independently. as R The strategies of participants can be classified by the expected global macroeconomic condition. First, agent i will invest all their asset to risky asset ˜ e for R ˜ t would not exceed R ¯ with certainty. Second, if agent i’s expectation R it
˜e agent i will invest all their asset to risk free asset if agent i’s expectation R it ˜ t would exceed R ¯ with certainty. Third, they will make some portfolio of for R assets and adjust their portfolio between periods if the macroeconomic state is uncertain.
23
Knowledge about previous risky asset ratio means that participants of informed groups could expect current macro state better than uninformed groups. It may lead to more aggressive investment. Or, agents may learn over time how to avoid crisis state. On the other hand, the members of the uninformed groups might make investment decisions in a ad hoc or highly safe manner. Also they can learn over time how to avoid crisis state by trial and error. Experimental Procedure At the first round, all participants are endowed with same wealth: 10, 000ECU . In unwaged game, 10, 000ECU is provided only at the first round. And in waged game, 10, 000ECU is provided at each round. This may affect to the attitude to risk, because one can no longer make decision in unwaged game when hi or she is totally bankrupt. Participants log in to the investement game web page with their USERID/PASSWORD once per weekday and make decisions for the investment game.23 They decide their portfolio for risk free asset, risky asset, and not invested asset. The web server stopped at scheduled times ˜ t and assign subjects’ payoff.24 to calculate R After the session ends, participants are regrouped and information condition is reset. Participants of some groups are asked survey questions such as individual risk averseness, their strategies and comprehension for the rules of this game.25 In these experiments, focused variables are the adjustment behavior of βi , 23
The address of web page for this expriment is http://econ.korea.ac.kr/~hokyoung/ investmentgame. 24 This server update was conducted at 2-4 PM in 2009 experiments, 10-12 PM in 2010, 11-12 PM in 2011 Spring, 0-1 AM in 2011 Fall. 25 Surveys for measuring risk averseness were conducted after 2011 experiments. The questions for measuring risk averseness referred to those of Holt and Laury (2002) and Deck et al. (2010). Also the comprehension test of this game, expectation of β¯t at each round and survey about individual strategy were checked in these experiments.
24
Log in Write Portfolio to the SERVER Set this Client to have invested
Has invested this turn?
Y
N
Log out
Print Current Wealth (and Information if informed) Input Investment Decision
Y
Compatibility Check N
Figure 1.8: Flowchart of experiment procedure: client part
Close Participants log in
Gather Portfolio information from Database Sum W_S and W_R and Residual
N
Is Total risky asset exceeds Crisis Threshold?
Y Set All W_R to ZERO Calculate Returns and Set
Open Participants log in
Figure 1.9: Flowchart of experiment procedure: server part
25
Figure 1.10: Screen capture of investment game web page
˜ t−1 , expectation of R ˜ e , participant ratio. Controlled variables response to the R it are information condition, N , payoff, experience, endowment type and length of session.26 Considerations for External Factors Generally, economic experiments are conducted in the strictly controlled lab and subjects are paid in cash. But this experiments have difference with general economic experiments. For this reason, these differences must be considered. Absence of Strict Control Standard economic experiments take relatively short time and is strictly controlled. However, experiments for investment game were conducted for from 1 to 5 weeks with loosly controlled environ26
See appendix for more informations.
26
ment. The main problem of these experiments is that some participants could not participate some rounds and this was not controllable. To 2010 experiments, average participate rate was 79.4% and this means that participants had not logged in average once in five rounds. In addition, the probability of ad hoc decision by misunderstanding may be higher than the experiments with controlled environment. All these factors are probable to lead to more noisy results than those of ordinary experiments.27 The only solution of these problems is sufficiently large observations. For these reasons, the size of groups is controlled and indirect measure for γi – ratio of not invested asset – and direct comprehension question was provided.28 Another problem is that participants have chance to discuss about their strategies. They can talk about their individual strategy and uninformed group could infer the macro economic state of other informed group. For these reasons, groups were divided more than two and shuffled randomly after each session and the information about detail of the group composition was not provided. Reward In these experiments, it could be another problem that the reward is not cash but absolute or relative additional credit. Camerer and Hogarth (1999) surveyed 74 economic experiments by the amount of pecuniary incentive and the result was that as cash incentive being higher, participants consider their strategy more actively to get the reward, but the average result is not so different. Issac et al. (1994) compared results of cash rewarded public good game experiments with that of extra credit rewarded ones. The 27 The issues raised by web based experiment is discussed in Rubinstein (2007), Suri and Watts (2011). 28 As formerly discussed in previous Subsection 1.2, positive γi can be reasonable if agent’s object is ‘relatively’ high payoff. The payoff of these experiments were relative credit, but the payoff was given not by group, but by overall groups with same experiment process. Therefore, the individual effect from the aggressive strategy with no investment is negative. By the 2010 experiment, the ratio of relatively high γi strategy was less than 1%.
27
Reward type x%RP x%RG x%AG x%RG+y%RP
Reward x%, relative extra credit only by participation x%, relative extra credit only by asset of last round x%, additional extra credit only by asset of last round relative extra credit, x% by asset of last round and y% by participation
Table 1.5: Reward System: Overview
result was that the cash reward group shows 10%point higher contribution than the extra credit reward group. Bosch-Domènech et al. (2002) conducted the guessing game under various environments such as labs, lectures, assignments, internet newsgroups, newspaper contests. They found some differences between groups but participants behaved in qualitatively similar ways. To investigate the effects of rewards, experiments with several reward structures were conducted. The specifications of reward types are described in Table (1.5).
Experiment Results ˜ t – total risky asset ratio of the group – approach to 0.8 and then Generally, R oscillate below 0.8. Observed facts are described in subsequent subsections. Effects of Information Observation 1 Participants used their information progressively. Difference can be inspected between informed and uninformed groups. First of all, informed groups adjust their βi to avoid crisis state. This adjust˜ t in the right panel of Figure (1.11). ment can be seen as the oscillation of R On the other hand, uninformed groups raise βi gradually as left panel of Fig˜ t is more spread. ure (1.11) and the distribution of R ˜ t−1 – provided risk asset Figure (1.13) shows the correlation between R ratio of previous round – and ∆βit – adjustment amount of βit . Theoreti-
28
Informed
.6 .5 .3
.4
Risky Asset Ratio
.7
.8
Uninformed
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Round
Graphs by info
˜ t with different information type and unwaged game Figure 1.11: R
cally, the first degree optimal strategy can be stated as Equation (1.31) and Figure (1.12). ∆βit > 0,
˜ t−1 < Reserved risk level if R
∆βit < 0,
otherwise
(1.31)
Inverse correlation of informed groups is relatively stronger than that of uninformed groups at the 99% confidence level. ˜ t−1 + ui ∆βit = A + AIN F O Ii + (B + BIN F O Ii )R 1, if agent i belongs to informed group Ii = 0, otherwise
(1.32)
˜ t−1 goes closer to This means more agents doesn’t increase their βit when R ¯ when they are informed. This suggest that many participants’ strategy are R close to the first degree strategy.
This relation becomes more prominent if 29
+1
∆βit
First Degree Optimal Strategy
0 First Degree Optimal Strategy
-1 0
˜ t−1 R
Reserved Risk Level
1
Informed
-.5
0
.5
1
Uninformed
-1
Adjustment Amount of Individual Risk Asset Ratio
Figure 1.12: First degree optimal strategy of ∆βit considering β¯t−1
.2
.4
.6
.8
1 .2
.4
.6
.8
Total Risk Asset Ratio Adjustment Amount
Fitted values
Graphs by info
˜ t−1 Figure 1.13: Correlation between ∆βit and R
30
1
Variable A AIN F O B BIN F O
Coefficient 0.158∗∗∗ 0.241∗∗∗ -0.251∗∗∗ -0.372∗∗∗
(Std. Err.) (0.034) (0.050) (0.053) (0.079)
No Info, 1/3 Correct
No Info, 2/3 Correct
No Info, 3/3 Correct
Info, 0/3 Correct
Info, 1/3 Correct
Info, 2/3 Correct
Info, 3/3 Correct
-.5
0
.5
1
-1
-.5
0
.5
1
No Info, 0/3 Correct
-1
Adjustment Amount of Individual Risk Asset Ratio
˜ t−1 Table 1.6: Estimation results : random-effects GLS regression of ∆βit and R with different information group
.4
.6
.8
1 .4
.6
.8
1 .4
.6
.8
1 .4
.6
.8
1
Total Risk Asset Ratio Adjustment Amount
Fitted values
Graphs by info and comprehension
˜ t−1 . Row classifies information conFigure 1.14: Scatter graph of ∆βit and R dition and column classifies comprehension score.
the comprehension level is considered. Participants of waged groups provided 3 comprehension questions about this game at the end of each session. The ˜ t−1 is most prominent when degree of comprenegative relation of ∆βit and R hension is highest and additional information is provided(as in Figure (1.14)). ˜ e more accurately than uninformed Observation 2 Informed group expect R t group.
31
1 Realized RA ratio - Expected RA ratio 0 .5 -.5 No info
0
Info
No info
Info
No info
Info
No info
Info
1 2 3 Nonresponse excluded, Group by Comprehension
˜ t and R ˜ e , grouped by compreFigure 1.15: Box graph of difference between R it hension score.
˜t − R ˜ e between informed and uninformed groups Table 1.7: Result of t-test: R it with 3/3 comprehension score
˜ te at round t was asked after the 2011 experiments. Expectation about R ˜ t and R ˜ e . This reveals the fact Figure (1.15) shows the difference between R it
˜ t more accurately(Table (1.7)). Especially that informed group expect global R expectations of uninformed participants whose comprehension scores are 0 or 1 are inaccurate and positively biased. 32
No Info, 2/3 Correct
No Info, 3/3 Correct
Info, 0/3 Correct
Info, 1/3 Correct
Info, 2/3 Correct
Info, 3/3 Correct
-.5
0
.5
1
-1
-.5
0
.5
1
No Info, 1/3 Correct
-1
Adjustment Amount of Individual Risk Asset Ratio
No Info, 0/3 Correct
0
.5
1 0
.5
1 0
.5
1 0
.5
1
Expectation of Total Risk Asset Ratio Adjustment Amount
Fitted values
Graphs by info and comprehension
˜ e , grouped by information and comFigure 1.16: Scatter graph of ∆βit and R it prehension score
Observation 3 Participants make decision by provided information rather than expectation. ˜ t−1 as one can see Other than negative correlation between ∆βit and R ˜ e as Figin Figure (1.14), any correlation was not observed in ∆βit and R it ure (1.16). This result suggests that participants make decision not by their uncertain expectation but by relatively certain provided information. This also suggests that partipants behave in some adaptive manner. Observation 4 Information seems to have no obvious effect for higher payoff. Figure (1.17) shows final payoff of each session. Average wealth of informed groups was higher than that of uninformed groups when max round per session was 10, 15 and when max round per session was 5, uninformed groups took higher payoff. 33
Average Payoff at Last Round
29073.1
No Info
Info
Max Round:15 230920 196387
0
20380.2
mean of wealth 100000 200000 300000
Max Round:10***
0
12666.3
mean of wealth 10,000 20,000 30,000
13848.8
0
mean of wealth 5,000 10,000 15,000
All Sessions Max Round: 5***
No Info
Info
No Info
Info
29073.1
No Info
Info
mean of wealth 100000 200000 300000
Max Round:10 27818.6
Max Round:15 256845
260870
No Info
Info
0
mean of wealth 10,000 20,000 30,000
14643.8
0
15125.8
0
mean of wealth 5,000 10,000 15,000
Uncrashed Sessions Only Max Round: 5**
No Info
Info
Figure 1.17: Average payoff of last round, partitioned by max round per session and crash condition. Asterisked graphs mean that it is able to reject null hypothesis(equal mean).(∗ ∗ ∗:1%, ∗∗:5%, ∗:10% confidence level)
The striking fact is that payoff of informed groups is similar to that of uninformed groups but standard deviation of informed groups is higher if crashed sessions are excluded. This means that payoff is dominated by crash. Observation 5 Average fluctuations of informed groups are higher than those of uninformed groups As can be seen from Table (1.8), variance of ∆βit in the informed groups is significantly higher than that of ∆βit in the uninformed groups. Observation 6 Information has positive effect in avoiding crisis state. Another role of the information is risk hedging, i.e., avoiding crisis state. It is expected that informed groups would use their additional information to avoid crisis state. Table 1.9 shows the groups experienced crisis state. 34
Table 1.8: Variance comparison test of Var(∆βit ) between informed and uninformed groups. group
session
crashed round
max_round
info
β¯
wage
208 211 215 301 302 302 308 310 310 310 312
1 1 1 2 2 2 2 2 2 2 2
9 3 11 1∗ 1∗ 4 1∗ 2 4 14 12
10 5 15 5 5 5 10 15 15 15 15
0 1 0 0 1 1 0 0 0 0 1
0.815 0.804 0.843 0.804 0.838 0.832 0.824 0.884 0.808 0.801 0.808
0 0 1 0 0 0 0 1 1 1 1
Table 1.9: Summary for Crisis State
Above result shows that informed groups experience crisis state less than uninformed groups. In the 28 games, crisis state occurred 11 rounds out of 260 rounds. 4 of them were informed groups and 7 of them were uninformed groups.29 It is worth to note that 3 of the crisis state occurred in the first round of the second session. This seems to have relation with learning effect and this effect would be discussed in later subsection. Also these cases has no meaning with information because there was no information to both informed and uninformed groups at the first round. If these case were excluded, 3 of 8 29
Among the crashed round, some of them were in the same session. Total number of sessions was 28 and number of crashed sessions was 8.
35
1 Individual Risk Asset Ratio .4 .6 .8 .2
Unexperienced
Experienced
˜ t grouped by experience Figure 1.18: Box graph of R
crisis occurred among the informed groups, 5 of 8 crisis occurred among the uninformed groups. However, these observations for the case of crisis state were too few to be said that informed groups have critical advantage to avoid crisis state. Learning Effect ˜t. Observation 7 Experienced groups starts with higher R Figure 1.18 compares the investment pattern between experienced and ˜ 1 , the ratio of risky asset invested unexperienced groups. In Figure (1.19), R in the first round of experienced groups is much higher than unexperienced groups. Observation 8 Return rates of experienced groups are higher than those of unexperienced groups. 36
Max Round: 10
Max Round: 15
.7 .6 .5 .4
Individual Risk Asset Ratio at 1 Round
.8
Max Round: 5
Unexperienced Experienced Unexperienced Experienced Unexperienced Experienced Graphs by maxr
˜ 1 between experienced and unexperienced groups. Figure 1.19: Box graph of R
Return rates of experienced groups were higher than that of unexperienced groups if cases in which it were crashed at the first round were excluded. This suggests that participants had learned something from the previous session ˜ t was very close to R. ¯ Especially, at and it leaded to higher payoff. But R ˜ t was very close to the crisis state. But the first round of second session, R the experience with the crisis of first round could also be another kind of experience, but these could not investigated in these experiments.30 Observation 9 Experience have negative effect to avoid crisis state. As one can see in Table (1.9), crisis states occurred 4 times in the second session and 2 times in the first session. Because these observed cases were too few, this fact 10 needs further observations. 30 Only two groups were proceeded to third session to test the first round effect and total number of rounds of the session were just 3. It was too limited to conclude, but the rate of return was highest and standard deviation was low.
37
˜ t of the groups classified by Figure 1.20: Average and standard deviation of R the crisis experience
Observation 10 It seems that experience has more influence to the rate of return than information. In Figure (1.20), investment strategy is investigated to see whether there is different strategy pattern from agents who has experience of crisis state. From the left side, each group represents: groups with no crisis, no crisis and excluded in current session, no crisis and informed, and groups with experienced crisis before. There are no evident difference. This leads to the fact 11 Observation 11 Crisis experience has no clear influence on individual strategy. To confirm this, two groups were shuffled by the crisis experience without any notice and played extra three round with information provided. The result
38
Round Experienced Crisis Not Experienced Crisis
1 0.7 0.7077
2 0.708 0.712
3 0.733 0.763
˜ t of the groups regrouped by the crisis experience. Table 1.10: Average R was shown in the Table (1.10). There were no clear difference.31 Reward Effect ˜ t than Observation 12 Groups rewarded by investment result shows higher R those rewarded only by participation. To consider the effect of reward system discussed in the Subsection 1.3, the correlation between the reward system and strategy is investigated. The result is Figure (1.21). Term ‘Alt’ means the groups of first round crisis were excluded. The groups of which reward is proportional to only participation ˜ t and high standard deviation. This result is shows relatively low average R same to that of Camerer and Hogarth (1999). This result suggests that the participants act in looser manner under participation based reward system. Observation 13 The amount of reward seems to have positive relation with ˜t. the standard deviation of average R The amount of reward means the ratio of extra credit of this game to the total credit. The result is Figure (1.22). Weak positive relation with payoff size is shown. 31
Very interesting thing was that groups – although number of participants was about 50 each – showed very similar pattern. But this is not clear because of too few observation. This result may have some relation with the guessing game of which object was guessing ¯ closest R.
39
40,000 Wealth at Last Round 10,000 20,000 30,000 0 by Result, Additional
by Participation, Relative
by Result, Relative
Figure 1.21: Box graph of Wi at last round classified by reward system
˜ t classified by the size of Figure 1.22: Average and standard deviation of R reward: extra credit ratio
40
˜ t classified by N Figure 1.23: Average and standard deviation of R
Another Factors ˜ t classified by Figure (1.23) shows the average and standard deviation of R the group size N . Average is similar except size 60, but this case is biased: reward systems of these groups were participation based. Standard deviation has negative relation with N . In Figure (1.24), all observed facts mentioned above are summarised.
Conclusion Contrary to the expectation, there was no clear evidence from the experiments that ‘additional’ information contribute to raise rate of return of the agents. More surprising result was that the rates of return of uninformed groups were similar to that of informed groups. Although the agents in the informed groups used their information more actively but the difference was not evident and ˜ t in informed groups was observed. only higher fluctuation of R These results suggest that even if true ‘additional’ information is used actively, it may not lead to higher rate of return and could lead to more unstable macro state under some interdependent expectation environment.
41
˜ t by controlled factors Figure 1.24: Average and standard deviation of R This suggestion is hypothetical by now and more strict analysis is needed.32 In the next section, more strict analysis using evolutionary game theory and agent-based simulation would be provided. 32 In fall 2011, more careful experiment was conducted but the result is not fully discovered. To sum up, uninformed and experienced groups showed significantly more crisis states than others. Other findings including this observation are planned to be considered in forthcoming research.
42
1.4
Agent-based Simulation
The subject of this section is to analyze investment game by agent-based simulation. Also formal analysis would be provided under simple assumptions.
Model description This model description follows the ODD rule suggested by Grimm et al. (2006).33 Purpose The purpose of this model is to analyze the influence of the provided information in the investment game environment. State variables and scales This model is composed of two types of object: Agent and Space. Agent is a micro object identified by the following state variables: id, strategy, wealth, location, memory. At the starting point, agents are created with random statevariables. They make their investment decisions by their own strategy and payoffs given by their investment decisions. Space is a macro object to aggregate individual investment decisions and assign payoffs to agents. As simulation goes on, agents chosen randomly at given mimic rate evaluate their neighbors’ strategies and mimic the strategy of the most successful neighbor. Also some agents are mutated and change their strategy randomly. All agents are located in a two-dimensional torus lattice. The structure of this space is square lattices where each vertex is connected by an edge to its four closest neighbors and vertices at the conners are connected to vertices 33
For more technical information, refer to the Java API document for this agent-based model at http://econ.korea.ac.kr/~hokyoung/riskInfo/doc/.
43
Parameter
Notation
Basic Value
Maximum Rounds Lattice Size Number of Agents id Maximum Memory Neighbor Search Bound Comparing Period for Mimic Mimic Rate Mutation Rate Adjust Gap Confidence Level Reserved Risky Asset Ratio Initial Endowment Wage per Round Return (Risky Asset) Return (Risk free Asset) Crisis Threshold
T √ √ N× N N i m √i κ ∗ N /2 − 1 t−τ δ ζ ui , di ci Li wi0 wi0 rR Wit−1 rS Wit−1 ¯ R
Alternative Value
100×100 10000 [1, N ] ∈ N 50 10 ∞ 0.01 0.001 [−0.1, 0.1] [0, 0.1] [0.75, 0.8] 100 100 {0.3, −1} × Wit−1 (PROP) 0.05 × Wit−1 (PROP) 0.8
[0, 50] ∈ N 12 25, 500, 1000 0, 0.02 0
{30, −100}(ADD) 5 (ADD)
Table 1.11: Overview: Main Parameter Setting
N
N
N
N
N N N
N
N
N
N
A
N N N
N N
A
Figure 1.25: 5 × 5 torus lattice and an agent A’s neighbors with boundary one. Left: local search regime, Right: global search regime
located at opposite conner. Agents get neighbors locally or globally in this spatial structure as Figure (1.25).34 34
If agents get some neighbors globally, spatial structure has no meaning because getting neighbor is the only process which have relation to the spartial structure of Space. Network structure of agents is not considered in this paper.
44
Process overview and scheduling This model proceeds by rounds. Figure (1.26) is the flow chart of this process. In the beginning of round t, previous information about total risky asset ˜ t−1 ) is provided to each agent if additional information is available. ratio(R The contents can be limited as the given information condition. On the other hand, previous information whether macroeconomic state was in crisis or not is provided regardless of information condition because all agents are able to know this from the rate of return of their portfolio. This information is accumulated at the degree of their memory length(mi ). Based on their information and strategy, investment decision of each agents is determined.35 After individual decision making process, all of the assets are aggregated and macro state is determined. Return rate of the risky asset rR is set by this macro state. By individual portfolio βit and rate of return, current individual wealth Wit is determined and then it is assigned to each agent. After wealth distribution, agents chosen arbitrarily compare them with their neighbors and mimic the strategy of most successful neighbor. And at given rate, small number of agents goes into mutation process and reset their strategy randomly. Design concepts Emergence
In the early rounds, extremely safe strategies are weeded out ¯ the and total risky asset ratio approaches to the Nash equilibrium level: R, crisis threshold and the first crisis state occurs. After this crisis, extremely risky strategies are weeded out and total risky asset ratio goes to lower level ¯ And approaches to the R ¯ again. This shows oscillating pattern around than R. the Nash equilibrium level, but the pattern can be different with different set of strategies, payoff regime, mimic rule, mutation rate, and information set. 35
For more detailed information, refer to the Subsection 1.4.
45
Start
Initialization Space,Agent,Strategy
All Agents:
Wealth Distribution Process
Information Gathering Investment Decision Wealth Distribution Process
Sum Risky Asset Sum Safe Asset
Agents: Wealth Distribution
Crisis?
Some Agents:
Yes
Mimic Most Successful Neighbor
No
Strategy Mutation
Return rate = Normal rate
Final round?
Wealth(t) ← Wealth(t-1)*Return rate
Some Agents:
No
Return rate = Crisis rate
All Agents: Yes
End
Figure 1.26: Flow Chart of Simulation Process
There are some inferior strategies in all rounds, but there is no dominant strategy which overwhelm others in the complex strategy setting. Adaptation Among all agents, arbitrarily chosen agents at the rate of δ compare their rate of return with their neighbors and mimic the strategy of the most successful one. The evaluation rule is simple: most successful one is who shows the highest rate of return during predetermined rounds. Fitness
Measurement of fitness is clear: the wealth of agent Wit . 46
Prediction
Each agent has their own history of previous macroeconomic
status. They predict future macroeconomic state by their own reasoning scheme and macroeconomic history. Reasoning scheme and the memory length mi are set when strategy parameters are initialized. Sensing
An agent can observe his or her neighbors’ wealth and strategy.
This setting is different from that of experiments discussed in the previous section. Interaction There are two interaction channel among agents. One is the influence of aggregation of agents’ decisions to the macroeconomic state. Agents with risky portfolio would increase the probability of crisis state and agents with safe portfolio would decrease it. This means that risky strategy has negative externality and safe strategy has positive externality.36 Another interaction channel is evaluation process among neighbors.37 By mimic process, more successful strategy would be prevalent and the influence of this strategy would increase. But the successful property can disappear if this strategy is too dominant to be profitable. Stochasticity There is no stochasticity in individual decision making. Stochasticity involves only in the initialization of strategy parameters and selecting agents to mimic or mutate. Influence of stochasticity would be tested by restricting mimic rate, mutation rate, and number of strategy set. 36 This implication leads to another strategy set that is not discussed here yet: Punishment strategy. For this strategy, no investment strategy must be introduced and agents’ object must be maximizing their relative wealth. This would raise very interesting issues but for now it is leaved for future research. 37 There are two neighbor selection regime: local and global. In the local regime, neighbors of an agent are in some boundary whose center is the agent. On the other hand, in the global regime, neighbors of an agent are chosen randomly.
47
Collectives Agents are not grouped explicitly, but they can be said to be grouped implicitly. It is the reason that if agent i and agent j have the same strategy, they will make the same portfolio and this means they behave identically. Observation At each simulation, this paper has focus on these variables: ˜ t ), Strategy ratio among agents, wealth ratio of the total risky asset ratio(R agent who has the highest wealth and her strategy. And the individual state variables of all agents at last round was observed to analyze wealth distribution. Initialization All agents are placed randomly on the lattice and endowed with same wealth. Endowment rule is distinguished by STOCK and FLOW in this model. In STOCK endowment rule, wealth is provided only in the first round but in FLOW endowment rule, wealth is provided in every round. STOCK rule is more strict than FLOW rule because once bankrupt agent cannot make any decision to the final round in STOCK endowment rule. Input In this model, agent set is provided externally to remove the unintentional effect from agents’ initial conditions. The external agent set is made from a random seed. Submodels Investment Rule The rule of this investment game is discussed in Section 1.2. The agents’ investment strategy is discussed in Subsection 1.4.
48
Literature Review Hawk-Dove Game and Financial Crisis The attempt to link the financial crisis and the Hawk-Dove game can be found in Hanauske et al. (2010). In Hanauske et al. (2010), hawk strategy is risky investment strategy and dove is risk free investment strategy. But the hawk-dove game in Hanauske et al. (2010) is played by two paired players, the investment game in this paper is played by a large number of players. This means payoffs are determined by their aggregated strategy. Minority Game
Games with payoff condition driven by aggregated indi-
vidual strategies is analyzed by many scholars, including Challet et al. (2005), Arthur (1999) and Challet et al. (2000).38 The investment game dealt in this ¯ is paper also has similar properties to minority game. The crisis threshold R the critical condition to determine payoff. This threshold condition can be re˜t > R ¯ means garded as a kind of minority criteria in terms of minority game. R ˜ t , weighted average of risky asset ratio exceeds R ¯ and agents who invest at R lower risky asset ratio than others’ are relatively smaller than others. In this situation, low risk strategies can yield higher return than high risk strategies. ˜ t ≤ R, ¯ high risk takers can earn more return. This On the other hand, when R payoff structure has similarity to minority game. Threshold Structure
¯ is the most important factor The crisis threshold R
in the investment game. One kind of the game which has threshold for payoff is the threshold public good game(TPGG). Bach et al. (2006) designed 3-person TPGG that contribution is effective only when quantity of total contribution exceed a given threshold. The result in that paper was that benefit and cost of the contribution was crucial to induce the contribution to public good. 38 Especially, Challet et al. (2005) gives detailed information about literatures for minority game.
49
The investment game in this paper also has properties of public good game. ¯ in the sense Individual risky asset ratio βit has positive externality if βit < R that it would decrease Rt so that this can reduce the chance of crisis state. ¯ it ) if Wit ≈ 0. On the Individual cost of this strategy is approximately Wit (R−β Wt ¯ has negative externality.39 other hand, individual investment decision βit > R Depth of Reasoning
Bosch-Domènech et al. (2002) designed a numerical
version of Keynesian Beauty Contest game to measure the depth of reasoning and measured experiments with it. The result was that only 1, 2, 3, and infinity depth of reasoning could be inspected. Camerer et al. (2004) suggested Poisson Cognitive Hierarchy model which has more explanation power than traditional Nash equilibria approach in one-shot game experiment in a large number of experimental literatures. In that paper, average depth of reasoning was about 1.5. In this investment game, at most two depths of reasoning can be distinguished by inverse strategy. Inverse strategy is the strategy which take more risk when the future macro state is estimated to be risky. More generally speaking, the inverse strategies can be the result of even depths of reasoning and the direct strategies can be the result of odd depths of reasoning. Information Analysis
To analyze the information in game model, Challet
et al. (2000) made simple market model and conducted computer simulation. There were two types of agents: producers and speculators. Producers who use fixed strategy also generate market information and speculators who use flexible strategy consume such information. The result was that agents with limited rationality or information power could only achieve marginal efficiency. Also these agents could achieve equilibrium approximately by inductive behaviors. Challet et al. (2000) also considered interesting topics of information: although 39
It is worth to mention that punishment is available if agent i’s another strategy γit – deciding γit Wit to be carried over to next round(t + 1) – is introduced.
50
certain information is meaningless in some situation, when sufficiently many agents believe it and react to the information, self-fulfilling process would make the meaningless information to have some meaning. But speculators consume this information to seek additional profit and profitability of the information is diminishing so that speculators exit from this market and then the information can have meaning again. Different Points of the Investment Game
The investment game has
several different points from games discussed in above literatures. First, minority game has similar to this game in the sense that macro state is dependent on the aggregated individual strategy but the payoff scheme is different. Payoff scheme of the minority games is additive but that of the investment game is proportional. This means that total payoff of minority game can grow arithmetically but that of investment game can grow exponentially. As one can see in Bouchaud and Mezard (2000), exponential growth overwhelms any arithmetic growth. Second, this article has some similarity to Bouchaud and Mezard (2000) which deals with proportional payoff scheme but in Bouchaud and Mezard (2000), the payoff is only the result of simple stochastic process. There is no room for individual strategies. The investment game model excludes any randomness in all strategies to eliminate the possibility of noise based on noise in strategies. Third, most literatures treat combination of at most three-player game in which the payoff is determined by local partners but this investment game is one n-player game in which the payoff is determined by all agents like minority game.40 To compare with the result of similar literatures such as Arthur (1999), additive payoff version of this game is considered in Subsection 1.4. 40
Traditional local game version of this investment game is discussed in Subsection 1.2.
51
Analytic Approaches Strategies In this subsection, agents’ strategies are defined formally. Cognition from Information
Every agent i = 1, 2, · · · , N has his or her
own information Iit−1 at round t. The quantity of information is limited to individual memory length mit . If the additional information of history of total IN F O and if not, I N OIN F O . Informarisky asset ratio is given, Iit−1 = Iit−1 it−1 = Iit−1
tion from crisis history is common knowledge even in the uninformed situation because everyone can know the crisis state in past round by observing their return. IN F O Iit−1 ∈
t−1 { ∏
ˆτ R
τ =t−m ¯i
}
t−1 { } ∏ ˜τ R
×
(1.33)
τ =t−m ¯i t−1 { } ∏ ˆτ R
N OIN F O Iit−1 ∈
(1.34)
τ =t−m ¯i
ˆ t ∈ {0, 1} is binary variable for macroeconomic state at round t and R ˜t ∈ R [0, 1] is real number variable for total risky asset ratio at round t. ∑N ˜t ≡ R
i
βit Wit ∈ [0, 1] ∑N
(1.35)
i
¯ is the threshold of macroeconomic state. R 1, if R ˜t > R ¯ ˆ Rt ≡ 0, otherwise
(1.36)
At the starting point of round t, every agent expects the macroeconomic state after individual investment from their information set Iit−1 . An agent i would ˜ t |Iit−1 ) > R. ¯ Otherwise, expect that it would be crisis state at round t if Ei (R ˜ t |Iit−1 ) ≤ R. ¯ Then binary function normal state would be expected: Ei (R 52
e ∈ {0, 1} which shows agent i’s expectation whether it would be in crisis or ηit
not at round t can be defined as: e ηit (Iit−1 ) ≡ Ei (Rˆt |Iit−1 )
(1.37)
There are five different expectation scheme. ˜ t−1 exceeds Li , INFO Agent i expect crisis at the beginning of round t if R the individual criterion for determine the macroeconomic risk. 1, if R ˜ t−1 > Li e ηit (Iit−1 ) = 0, otherwise
(1.38)
CHARTINFOTS Agent i expect crisis at the beginning of round t if ex˜ te derived from linear time trend {R ˜ t−m ˜ ¯ pected R ¯ , · · · , Rt−1 } exceeds R i
e ηit (Iit−1 )
=
1,
˜ e ≡ a∗ t + b∗ if R it i i
0,
otherwise
(1.39)
∑τ =t−1 ∑ ∑ ˜ ˜ Rτ τ τ =t−m ¯ i τ Rτ − ∑ 2 ∑τ 2 τ = m ¯ i τ τ − ( τ τ) ∑ ∑ ˜ ∑ 2 ∑ ˜ ( τ τ ) τ Rτ − τ τ τ τ R τ ∗ ∑ ∑ bi = m ¯ i τ τ 2 − ( τ τ )2 a∗i
m ¯i
CHARTINFO Agent i expect crisis state at the beginning of round t if the t-test of which the null hypothesis is that average risk level of recent mit ¯ rounds exceeds R. 1, if H0 : R ¯τ ¯ cannot be rejected >R τ =t−m ¯ i ,··· ,t−1 e (1.40) ηit (Iit−1 ) = 0, otherwise
53
CHART Agent i expect crisis state at the beginning of round t if the frequency of crisis state during recent mit rounds exceeds ci , the individual confidence level of risk. e ηit (Iit−1 )
=
1,
if
0,
otherwise
Λt−mit ,t−1 m ¯i
> c¯i ∈ [0, 1]
(1.41)
ˆ τ = 1|τ = t − m Λt−mit ,t−1 ≡ n({R ¯ i , · · · , t − 1}) MARKOV Agent i expect crisis state at the beginning of round t if the ( ) ˆt) > R ¯ coming from the every possible probability of crisis state Prit−1 Ei (R macroeconomic state transition frequency derived from recent mi rounds exceeds ci , the individual confidence level for determining macroeconomic state.
ηit (Iit−1 ) =
1,
( ) ˆ t |Iit−1 ) > R ¯ if Prit−1 Ei (R
0,
otherwise
(1.42)
( ) ˆτ = 1 ∧ R ˆ τ −1 = R ˆ t−1 ) n( R ˆ t |Iit−1 ) ≡ Prit−1 Ei (R ˆτ = R ˆ t−1 ) n(R
τ =t−m ¯ i ,··· ,t−1
Based on above expectation, agents make investment decisions. These can be regarded as functions of information φi . φi : Ii ∈ Rnmi 7−→ βi ∈ R1 ∈ [0, 1],
n=
1, 2,
if agents are uninformed if agents are informed (1.43)
Fixed Strategy with No Information(CON)
An agent i who uses the
strategy CON make investment decision from predetermined fixed risk asset ratio µi . This is the simplest strategy. In this case, the depth of reasoning is zero. βit = φi (Iit−1 ) = µi ∈ [0, 1] 54
(1.44)
Constant Strategy based on Information(CON****)
An agent i who
uses these strategy adjust the level of µi dependent on the expectation of macroeconomic state as described in Paragraph 1.4: CONINFO, CONCHARTINFOTS, CONCHARTINFO, CONMARKOV, and CONCHART strategy. ui , di ∈ [−0.1, 0.1] is adjustment interval.41 µit−1 + ui , if Ei (R ˆ t |Iit−1 ) = 1 µit = µit−1 − di , otherwise Flip Strategy based on Expectation(FLIP****)
(1.45)
An agent i who uses
this strategy set has two kind of individual variable: hi ∈ [0, 1] and li ∈ [0, 1]. The agent set βit to one of these values based on the expectation of macroeconomic state. When crisis state is expected, agent i set βit to li , otherwise to hi . By the expectation method of agent i, there are five strategy: FLIPINFO, FLIPCHARTINFOTS, FLIPCHARTINFO, FLIPMARKOV, FLIPCHART.42
βit = φi (Iit−1 ) =
h ¯ i ∈ [0, 1],
e =1 if ηit
¯li ∈ [0, 1],
otherwise
(1.46)
Analytic Approach Case: All Fixed CON Strategy and Proportional Payoff Scheme In this subsection, the simplest case that all strategy is fixed CON strategy is considered. It is assumed for analytical simplicity that there is no mimic process and no mutation process. And endowment is given in the flow manner: w0 is given in the beginning of every round. The parameters for the strategy 41 If ui , di is positive, degree of reasoning is odd. If ui , di is negative, degree of reasoning is even. If ui , di is zero, these strategies are equivalent as simple CON strategy. if ui di < 0, µi is adjusted to 0 or 1 regardless of expectation. 42 The depth of reasoning also can be considered here. In the case of hi > li , the degree is odd. And if hi < li , the degree is even. If hi = li , this strategy is equivalent to simple CON strategy.
55
µi is generated by uniform distribution between 0 and 1. Agents’ index i is numbered in the ascending order of µi . µi ∼ i.i.d.U (0, 1)
(1.47)
∀i < j
(1.48)
µi < µj
βit , agent i’s individual risk asset ratio at round t, is constant µi regardless of round. Therefore WRit , agent i’s investment to risk asset at round t can be stated like equation below:
WRit = βit Wit = µi Wit
(1.49)
WRt , total investment to risk asset at round t, is weighted average of µi and Wit−1 .
WRt =
N ∑
WRit =
i
N ∑
µi Wit
(1.50)
i
On the other hand, rit , agent i’s rate of return at round t is ¯ i0 + (1 + rit−1 )Wit−1 Wit = W
(1.51)
ˆ it )(rR − rS )µi − R ˆ it (1 + rS )¯ rit = rS + (1 − R µi
(1.52)
Wt , total summation of the asset at the beginning of round t, is determined by aggregation of individual investment decisions and the macroeconomic state ˆ t−1 at past round t − 1. Let W ¯ 0 ≡ Nw R ¯0 , then Wt can be stated like equation below:43 ¯ 0 + (1 + rS )Wt−1 + ψt−1 WRt−1 Wt = W 43
(1.53)
rS is rate of return from invested risk free asset, rR is rate of return from invested risky asset.
56
ψt−1 is the rate of return from risky asset invested in round t − 1. ψt−1 can ˆ t−1 . be defined by macroeconomic state R ˆ t−1 )(rR − rS ) − R ˆ t−1 (1 + rS ) ψt−1 ≡ (1 − R
(1.54)
Let ρt be the ratio of Wt and Wt−1 . If Equation (1.53) is divided by Wt−1 , ρt can be derived like equation below:44 ρt ≡
¯0 Wt W ˜ t−1 = + (1 + rS ) + ψt−1 R Wt−1 Wt−1
(1.55)
Then ∆WRt can be derived.
∆WRt =
N ∑
µi ∆Wit =
N ∑
i
¯ i0 + rit−1 Wit−1 ) µi (W
(from Equation 1.52)
i
1 ¯ = W 0 + rS Wt−1 2 ∑ ˆ t−1 ) + (1 − R µ ¯i (rS + (rR − rS )µi ) Wit−1 ˆ t−1 +R
∑
i
µ ¯i (rS − (1 + rS )µi ) Wit−1
i
∑ 1 ¯ = W µ2i Wit−1 0 + rS WRt−1 + ψt−1 2
(1.56)
i
By dividing both side by Wt−1 , equations below can be derived. ¯0 1 W WRt − WRt−1 Wt ˜ t−1 + ψt−1 = + rS R Wt Wt−1 2 Wt−1
∑
2 i µi Wit−1
Wt−1
(1.57)
˜t. Above equation can be rewritten by R
˜t = R 44
¯0 1 W 2 Wt−1
˜ t−1 + (rR − rS ) + (1 + rS )R
˜ t−1 = By definition, R
ρt WRt−1 Wt−1
57
∑
µ2i Wit−1 Wit−1
i
ˆ t−1 = 0) (1.58) (When R
˜t = R
¯0 1 W 2 Wt−1
˜ t−1 − (1 + rS ) + (1 + rS )R
∑
µ2i Wit−1 Wit−1
i
ρt
ˆ t−1 = 1) (1.59) (When R
Combine above equations and Equation (1.55), then
˜t = R
¯ ∑ 2 1 W0 ˜ t−1 +(rR −rS ) i µi Wit−1 +(1+rS )R 2 W Wit−1 t−1 , ¯ W 0 +1+r +(r −r )R ˜ t−1 S R S Wt−1
∑
¯ 1 W 0 +(1+r )R ˜ t−1 −(1+rS ) S 2 Wt−1 ¯ W0 ˜ +1+r −(1+r )R S
Wt−1
S
2 i µi Wit−1 Wit−1
,
˜ t−1 ≤ R ¯ if R (1.60) ˜ t−1 > R ¯ if R
t−1
0 For a sufficiently large round t, Wt−1 ≫ W0 and this leads to WWt−1 ≈ 0. ∑ 2 2 ˜ ˜ Let Σt−1 = i µi Wit−1 /Wit−1 then Σt−1 > R t−1 and the sign of ∆Rt can be
derived.45 ˜ t oscillate around R ¯ permanently. If we take parameters from This means R ˜ t is Figure (1.27). Table 1.11, The graph of R
˜t = ∆R
˜2 (rR −rS )(Σt−1 −Rt−1 ) > 0, ˜ 1+r +(r −r )R S
R
S
t−1
˜2 Rt−1 −Σt−1 < 0, ˜ 1−R t−1
45
˜ t−1 ≤ R ¯ if R ˜ t−1 > R ¯ if R
Proof of the disequation. ∑ ∑ ∑ 2 − i µ2i Wit i Wit ˜ t2 − Σt = ( i µi Wit ) ∑ R ( i Wit )2 ∑ ∑ 2( i̸=j µi µj Wit Wjt − i̸=j µ2i Wit Wjt ) ∑ = ( i Wit )2 ∑ 2 i̸=j µi Wit Wjt (µj − µi ) ∑ = ( i Wit )2 ∑ ∑ 2( i>j µi Wit Wjt (µj − µi ) + j>i µi Wit Wjt (µj − µi )) ∑ = ( i Wit )2 ∑ 2 i>j Wit Wjt (µj − µi )2 ∑ =− R ¯ if R
59
(1.62)
rR (crisis) rR (normal) rS WRit + WSit
Proportional Payoff(PROP) rR WRit −WRit rS WSit Wt−1
Additional Payoff(ADD) ¯ Rit rR W ¯ −WRit ¯ Sit rS W ¯ W
Table 1.12: Payoff Comparison: ADD vs. PROP
Wt can be derived by summation Wit through rounds of τ = 1, · · · , t. Λt ∈ N is the number of crisis state from 0 to t round.
¯ rS t + 1 W ¯ (rR − rS )(t − Λt−1 ) − 1 W ¯ rS Λt−1 Wit = W0 + W 2 2
Λt ≡ ∑
t ∑
ˆτ R
(1.63)
(1.64)
τ =0
¯ = 1W ¯ regardless rounds because In this payoff scheme, WRt is i µi W 2 ¯ . For this reason, R ˜ t is 1/2 always as Figure (1.27). WRit ≡ µi W ˜ tADD R
∑ ¯i µi W 1 ∑ ≡ ∑i ¯ = µi = µ ¯ N i Wi i
(1.65)
The cause of this difference is followed from the fact that total assets invested in each round is constant regardless of agents’ own wealth. However, ˜ t , the weighted average of βit and Wit , in the proportional payoff scheme, R can be changed by more the individual wealth even if there is not any mimic or mutation process. Introducing Mimic Process
Let’s introduce the mimic process to above
procedure. δ is defined as the rate of mimickers at each round and κ as rate ˜ t starts to change by the distribution of neighbors of each mimicker. Then R transition of µit . And let U be the set of all agents and Θx (U ) be the set of
60
˜ t oscillate around R ¯ through the mimic index of sample of U at rate x. Then, R process.
µjt =
max(µkt−1 − 0, µjt−1 ) ∈ [µjt−1 , max(µi )] ∀i,
¯ if µ ¯t−1 ≤ R
min(µkt−1 + 0, µjt−1 ) ∈ [min(µi ), µjt−1 ] ∀i,
¯ if µ ¯t−1 > R
∀j ̸= k
s.t.
j ∈ Θδ (U ),
(1.66)
k ∈ Θκ (U )
In any state, an agent i who find the neighbor j with higher rate of return adjust µit to µjt . This leads to overlap of strategies. Total number of strategies is decreasing function of t. Decreasing can occur when an agent is a unique user of a certain strategy and the agent could find more successful neighbor.46 In the normal state, agents with high µ earn more wealth and individual µi tend to increase. And in the crisis state, agents with low µ earn more wealth and individual µi tend to decrease. Assume there are only three strategies µa < µb < µc and Let a, b, c be the ratio of agents using respective strategy. In each round, δN agents compare κN neighbors. If the normal state persists, c will increase and a will decrease with positive probability in every mimic process. b will increase with probability aδ((a + b)κN − aκN ) and decrease with probability bδ(1 − (a + b)κN ) in every mimic process. But because there exist κN such that make b decrease, b would decrease if κN is sufficiently large.47 If crisis state persists, agents with low µ earn more wealth and µi 46
Increasing can occur when mutation process occur to an agent with overlapped strategy. However there is no mutation process in this case. 47 Proof. Let r be ab . Then r ≤ 0. The increasing probability F can be defined by F =A−B A = aδ((a + ar)κN − aκN )B = bδ(1 − (a + ar)κN ) [ ( )κN ( )κN ] 1 1 κN κN F = aa (1 + r) 1+r−r − a(1 + r) 1+r When r = 0, then F = 0. And the derivative of F with respect to r is [ ] d F = 1 + (1 + r)−κN −1 κN + a−κN (rκN − (1 + r)) dr
61
400 400
Density
30
600
400 0
0
10
20
20
40
30
600
10
20
20
40
Density
10
20
20
40
30
600
10
20
20
40
30
40
Mutation
60
NO Mutation
0
.5
1
0
.5
1
0
.5
1
0
.5
1
0
.5
1
0
.5
mean
1 0
.5
1 0
.5
1 0
.5
1 0
.5
1
mean
Graphs by itr
Graphs by itr
Figure 1.28: Histograms of µi (mean) after 2500 rounds with additive payoff regime, mimic process, κ = 0.0441, δ = 0.1, fixed CON strategy with same agent set.
will decrease and mimic process makes lowest strategy increase. Therefore pure mimic process with sufficiently large κ has tendency to make strategy extreme as Figure (1.28). But it is feasible that strategy set converge to only one strategy if all agents using second large strategy moves to largest strategy set in certain state. In that case, there are no change in strategy. For sufficiently large t, assume that strategy set is only 0 and 1. µi ∈ {0, 1}
∀i
(1.67)
Then ˜ t = n1 R
(1.68)
¯ and decreases otherwise. This mean n1 Therefore n1 increases when n1 ≤ R ¯ The frequency has positive relation with δ, the rate of oscillate around R. This means F ′ > 0 if κN > 1 + 1r . Therefore F > 0 if κN > 1 + 1r . QED.
62
.8 .6 rr .4 .2 0
0
.2
.4
rr
.6
.8
1
delta=0.02, kappa=0.0441
1
delta=0.01 kappa=0.441
0
500
1000 1500 round
2000
2500
0
1000 1500 round
2000
2500
.8 .6 rr .4 .2 0
0
.2
.4
rr
.6
.8
1
delta=0.02, kappa=0.0625
1
delta=0.01, kappa=0.0625
500
0
500
1000 1500 round
2000
2500
0
500
1000 1500 round
2000
2500
˜ t by δ, κ with fixed CON strategy. Same pool of agents Figure 1.29: Graph of R is used. mimic process. This relation can be seen in Figure (1.29). Introducing Mutation process
Mutation process can increase the num-
ber of strategies. With combination to the mimic process, clustering of strategies is observed as Figure (1.29). Case: Proportional Payoff Scheme with Mimic Process In this subsection, mimic process is introduced at the case discussed in Subsection 1.4. Mimickers are chosen at the rate of δ, and they compare the return rate of investment of the neighbors who are chosen at the rate of κ. If there are more successful neighbor, then the agent mimic the neighbor’s strategy. The ˜ t is not difference with additive payoff scheme is arisen from the fact that R the average of µit , but the weighted average of µit and Wit (Equation (1.50)).
63
As Equation (1.52), the rate of return from each strategy is dependent on the macroeconomic state. Then πiτ,t , the rate of return from τ to t round, can be expressed as the product of (1 + rit ) for sufficient large τ . Λ is defined as the number of crisis state from τ to t round. 1 + πit = (1 + rS + (rR − rS )µi )t−τ −Λ (1 + rS − (1 + rS )µi )Λ Λτ,t ≡
t ∑
(1.69)
ˆs R
s=τ
There exist
µ∗i
such that makes πit max. By taking the derivative of both side
with respect to µi , µ∗i is: µ∗i = 1 − Λτ,t t−τ
1 + rR Λτ,t rR − rS t − τ
(1.70)
is ητ,t , the probability of crisis state during s ∈ [τ, t] round.
Now consider the strategy of an agent i at the macro state is normal: ˜ t ≤ R. ¯ In the case of R ˜ t < µi < R, ¯ πit exceeds average rate of return π R ˜t . ˜ t > R, ¯ πit exceeds average rate of return if µi < R. ˜ And in the crisis state: R ˜ t , strategy µit can earn more if |µit − R| ¯ < |R ˜ t − R|. ¯ This Therefore for any R ¯ is the optimal portfolio µ∗ for risky asset and individual µi converge means R i
¯ Then the amplitude of vibration of R ˜ t converges to 0 in the long run. to R. Combining this result and Equation (1.70) leads to the ητ,t , the rate of crisis state in the long run. ¯ rR − rS ητ,t = (1 − R) 1 + rR
(1.71)
Case: Introducing Mutation Process ˜ t from regular oscillation or convergence to R ¯ as Mutation process prevent R Figure 1.30.
64
PROP
rr .5
.5
.6
.6
rr
.7
.7
.8
.8
ADD
0
500
1000 1500 round
2000
2500
0
500
1000 1500 round
2000
2500
˜ t with mimic and mutation Figure 1.30: Graph of R
Proportional Payoff Scheme and Information ˜ t oscillate around R ¯ and the amplitude of As the results discussed above, R this oscillation is persistent in additive payoff scheme. This oscillation is based on the fact that the strategies of agents cannot converge to one. On the other ˜ t also oscillate around R ¯ in the proportional payoff scheme, but the hand, R amplitude converges to zero because the strategies of agents is gathered to one. Even if other strategies are introduced, the main property of this invest¯ would not be disappeared but more ment game that the oscillation around R complicated pattern would emerge. Now the strategies discussed in Subsection 1.4 is introduced and the result is analyzed by computer simulation.
65
Payoff Scheme ADD ADD PROP PROP
Information Condition INFO NO INFO INFO NO INFO
˜t Stdev of R 0.260014 0.0645506 0.280344 0.0045959
˜ t after 1000 round in multiple strategy set Table 1.13: Standard deviation of R
Simulation Results Information and Payoff Scheme In this subsection, the results of simulations with various strategy set are discussed. All simulations are conducted in the condition of two kind of payoff scheme: ADD, PROP, and two kind of information regime: No Info, Info. Same agent set is provided in each case to eliminate the result originated from different randomness of agent initialization. ˜t Risky Asset Ratio R
˜ t is presented The fluctuation of risky asset ratio R ˜ t is oscillating around R ¯ = 0.8 in all case as Figure (1.31) and Figure (1.32). R but the oscillation pattern is different in both payoff schemes. In proportional payoff scheme, the amplitude of oscillation is less than that in additional payoff scheme but is not converged to zero. This result is similar to the results from typical minority games except for the center of the oscillation. This difference comes from the asymmetric character of the payoff. The standard deviation ˜ t is given in the Table (1.13). In the additional payoff scheme, simulations of R with informed condition shows more stable pattern, but in the proportional payoff scheme, simulations with uninformed condition shows more stable pattern. Wealth Distribution
The wealth distribution at last round is presented
in the Figure (1.34) and Figure (1.35). Kernel density plot in the additional 66
.5 1 0 0
.5
1 0
.5
1 0 .5 1 0 .5 0
Risky Asset Ratio
.5
1
ADD, Info
1
ADD, No Info
0
2000
0 3000
2000 1000
0 3000
2000 1000
0 3000
0 3000
2000 1000
0 3000
2000 1000
PROP, No Info
PROP, Info
3000
.5 1 0 .5 1 0 .5 0
.5
1 0
.5
1 0
.5
1
round
0
Risky Asset Ratio
2000 1000
round
1
1000
0
2000 1000
0 3000
2000 1000
0 3000
2000 1000
0 3000
2000 1000
0 3000
2000 1000
round
0 3000
2000 1000
3000
round
˜ t by payoff and information condition with fixed memFigure 1.31: Graph of R ory length, global neighbor, full round comparison. 11th graph is overlapped graph of all other iterations.
ADD, No Info
PROP, Info
PROP, No Info
.4 1 .8 .6 .4
Risky Asset Ratio
.6
.8
1
ADD, Info
0
1000
2000
0 3000
1000
2000
round
˜ t , one iteration for closer view Figure 1.32: Graph of R
67
3000
ADD, No Info
PROP, Info
PROP, No Info
1 .8 .4
.6
Risky Asset Ratio
.4
.6
.8
1
ADD, Info
0
1000
2000
0 3000
1000
2000
3000
round
˜ t . All graphs with same parameters are merged to one. Figure 1.33: Graph of R payoff scheme is similar to the normal density plot but not exactly same.48 On the other hand, kernel density plot of individual log wealth in the proportional payoff scheme has horizontal tail in the left. This means agents with low income distributed by power law.49 Figure (1.36) shows the relative wealth of richest agent. In the additional payoff scheme, it shows descending trend. However, opposite trend is observed in the proportional payoff scheme. This means that macroeconomic state is determined by the strategy of a few rich agents. 48
If the return rate rit follows identical and independent stochastic process of same pdf, ∑ then Wit = t rit w ¯0 will be normally distributed by central limit theorem. But the reverse is not true. This result suggest the result can be expressed by certain pdf, but this is not verified yet. 49 Clauset et al. (2009) pointed out that the income distribution in the real world looks like log-normal in the body with Pareto-style upper tail. Wealth distribution in this investment game has Pareto-style lower tail.
68
ADD, Info
2.3e-04 0
Density
0
2.4e-04 0
1.6e-04 0 0
1.9e-04
Density
0
2.2e-04
2.4e-04
ADD, No Info
511724.88 530531.88
511724.88 530531.88
510886.63 530531.88
510886.63 527992.06
wealth
PROP, No Info
PROP, Info
527992.06
.0174 0 0
.02
Density
0
.0185 .021 0 .0215 0 0
Density
510886.63 527992.06
wealth
.0194
511724.88
4.831411
4.831411 396.1482
4.831411 404.0536
6.575668 404.0536
6.575668 403.7787
log_wealth
6.575668 395.4165
420.503
log_wealth
Figure 1.34: Histogram of wi2500 by each iteration. Last(11th) one is the aggregated histogram.
.00005 0
0
.00005
Density .00015 .0001 .0002
ADD, Info
Density .00015 .0001 .0002
ADD, No Info
510000
515000
520000 wealth
525000
530000
510000
kernel = epanechnikov, bandwidth = 222.6461
515000
525000
530000
.015 .01 .005 0
0
.005
.01
Density
.015
.02
PROP, Info
.02
PROP, No Info
Density
520000 wealth
kernel = epanechnikov, bandwidth = 172.7696
0
100
200 log_wealth
300
400
kernel = epanechnikov, bandwidth = 4.3961
0
100
200 log_wealth
300
400
kernel = epanechnikov, bandwidth = 7.1927
Figure 1.35: Kernel density of all iterations with same condition. Dashed line shows normal density with average and variance of each condition.
69
0 3000
2000 1000
0 3000
round
.000102.000106 .000102.000106 .000102.000106 .0001 .000104.0001 .000104.0001 .000104
.000102 .000106 .000102 .000106 .000102 .000106 .0001 .000104 .0001 .000104 .0001 .000104
2000 1000
2000 1000
ADD, Info
0
2000 1000
3000
0 3000
2000 1000
0 3000
2000 1000
3000
round
PROP, No Info
.04
.05
.06
.08 0
.1 0
.02
.05
.04
.06
.1 0
.08 0
.02
.05
.04
.06
.1
.08
PROP, Info
0
.02
Richest Agent's Wealth / Total Wealth
0
0
Richest Agent's Wealth / Total Wealth
ADD, No Info
0
2000 1000
0 3000
2000 1000
0 3000
round
0
2000 1000
2000 1000
3000
0 3000
2000 1000
0 3000
2000 1000
3000
round
Figure 1.36: Graph of relative wealth of richest agent to total wealth in each iteration Strategy Dynamics
Figure (1.37) shows the occupy rate of strategies.
There are no dominant strategies in the additional payoff regime. But there are some dominant strategies in the proportional payoff regime. CON strategy is the most dominant strategy in the no information condition and CON strategy and FLIPCHARTINFOTS strategy are dominant in the information condition. CON strategy is degree zero strategy that doesn’t consider others’ strategy. On the other hand, degree of FLIPCHARTINFOTS strategy is more than one. The origin of fluctuation observed in the proportional payoff scheme and information condition lies in this strategy dynamics. Payoff in the last round
Figure (1.38) and Figure (1.39) show the last
payoff in different conditions. In spite of the different fluctuation of macro state, the last payoff is almost same in the different information condition. In proportional payoff scheme, sum of log wealth in the uninformed environment 70
.5 1 0 0
.5
1 0
.5
1 0 .5 1 0 0
.5
Occupy Rate
.5
1
ADD, Info
1
ADD, No Info
1000 2000 30000
1000 2000 3000
0
1000 2000 3000
PROP, No Info
PROP, Info .5 1 0 .5 1 0 0
0
.5
1 0
.5
1 0
.5
1
round
.5
Occupy Rate
1000 2000 30000
round
1
0
0
1000 2000 30000
1000 2000 3000
0
1000 2000 30000
round
1000 2000 3000
round
Figure 1.37: Graph of occupy rate of strategies at each iterations and different conditions is higher than that in the informed, but the sum of wealth in the uninformed environment is lower than that in the informed. Finite Comparison Period The comparison period to assess the agents’ investment performance was current wealth. This is equivalent to return rate from the first round to current round because agents have same endowment at the first round. But it is more realistic that comparison period for determine the performance of each agent is fixed. This idea is similar to that of the rolling window regression in time series analysis. 71
PROP
100
515000
200
520000
wealth
log_wealth
300
525000
400
530000
ADD
No Info
Info
No Info
excludes outside values
Info
excludes outside values
0
0
100000
100
mean of wealth 200000 300000
mean of log_wealth 200
400000
300
500000
Figure 1.38: Box graph of Wi2500 and ln Wi2500 .
No Info
Info
No Info
ADD
Info
PROP
Figure 1.39: Bar graph of average Wi2500 and ln Wi2500
72
1
.5 0
rr
1
0
0
1000
2000
30000
1000
2000
3000
round Graphs by info
˜ t in the proportional payoff regime. 0 is Figure 1.40: Aggregated graph of R uninformed game and 1 is informed game
Comparison Period: 25 rounds
Figure (1.40) shows the extreme fluctu-
ation regardless of information condition. Comparison Period: 500 rounds
Figure (1.41) shows the fluctuation of
macro state in the 500 rounds of comparison period. Fluctuation is decreased in the additional payoff scheme. But in the proportional payoff and uninformed ˜ t is observed as Table (1.14). But the case large increase of fluctuation of R result remain unchanged. The reason of this result can be explained by the fact that the dominance of CON strategy is weaken in the uninformed game as one can be seen is Figure (1.42). Comparison Period: 1000 rounds
As comparison period being longer,
the result shows similar pattern to the simple wealth comparison condition discussed in Subsection 1.4. Theoretically, these two condition is equivalent
73
.5 1 0 0
.5
1 0
.5
1 0 .5 1 0 .5 0
Risky Asset Ratio
.5
1
ADD, Info
1
ADD, No Info
0
2000
0 3000
2000 1000
0 3000
2000 1000
0 3000
0 3000
2000 1000
0 3000
2000 1000
PROP, No Info
PROP, Info
3000
.5 1 0 .5 1 0 .5 0
.5
1 0
.5
1 0
.5
1
round
0
Risky Asset Ratio
2000 1000
round
1
1000
0
2000 1000
0 3000
2000 1000
0 3000
2000 1000
0 3000
2000 1000
round
0 3000
2000 1000
0 3000
2000 1000
3000
round
˜ t in each iterations with 500 rounds of comparison Figure 1.41: Graph of R period. 11th graph is aggregated graph of all iterations with same condition. Agent set is same as the case of full round comparison.
Payoff Scheme Information Condition Comparison Period ADD INFO ADD NO INFO PROP INFO PROP NO INFO
˜t Standard deviation of R Full rounds 500 rounds 0.260014 0.0165341 0.0645506 0.0444451 0.0280344 0.0226837 0.0045959 0.0153402
˜ t in the different comparison Table 1.14: Comparison of standard deviation of R regime
74
.5 1 0 0
.5
1 0
.5
1 0 .5 1 0 0
.5
Occupy Rate
.5
1
ADD, Info
1
ADD, No Info
1000 2000 30000
1000 2000 3000
0
1000 2000 3000
PROP, No Info
PROP, Info .5 1 0 .5 1 0 0
0
.5
1 0
.5
1 0
.5
1
round
.5
Occupy Rate
1000 2000 30000
round
1
0
0
1000 2000 30000
1000 2000 3000
0
round
1000 2000 30000
1000 2000 3000
round
Figure 1.42: Occupy rate of strategies in the 500 rounds of comparison period
to first 1000 round. However the fluctuation is higher than simple wealth comparison condition.
Conclusion In this computer simulation of simple investment game, agents use their information to maximize their payoff but the macroeconomic state shows more fluctuation as the strategies being introduced. There exists dominant strategy in the simple strategy set, but dominant strategy disappears in the multiple strategy environment.50 Instead of single dominant strategy, a few strategies get dominant status in the rotating manner. This leads to fluctuation of macro state and this fluctuation feedback to the strategy via information. For this reason, environment with more accurate information can lead to more unsta50 Under no mutation condition, some strategy can be the only one strategy but this is not evolutionary stable because this strategy disappears with mutation. Mutation process acts as the filter for evolutionary stable strategy(ESS).
75
.5 1 0 .5
Risky Asset Ratio
0
.5
1 0
.5 0
.5
1 0
Risky Asset Ratio
1 0
.5
1
PROP, Info
1
PROP, No Info
0
1000 2000 30000
1000 2000 3000
0
1000 2000 30000
round
1000 2000 3000
round
Graphs by itr
Graphs by itr
˜ t , Proportional payoff condition. Comparison Period: Figure 1.43: Graph of R 1000 rounds ble state in this investment game. In the traditional additive payoff scheme, agents in the more informed environment shows more stable macro state. However, different result was observed in the proportional payoff scheme: agents in the more informed environment shows less stable macro state. The wealth is concentrated in the hand of a few richest agents and the macro state was determined by them. But in the uninformed game the strategy was more stable than in the informed game. Therefore, under the environment of interdependent expectation, information, which can make micro agents to avoid risk or earn more payoff, can have negative effect which makes macro state more unstable even though the information is correct.
76
1.5
Concluding Remarks
Under interdependent expectations, there is no ‘true’ information. This fact leads to the fundamental uncertainty. Economic agents tries to maximize their wealth with their information, but this behavior can cause the macro economic fluctuation. Even if certain information is fully artificial, the information can be realized when significant agents believe it. Economic phenomena is said to be highly complex. Interdependent expetations may be placed in this complexity. In this paper, simple investment game with proportional payoff regime was considered. This regime has some interesing properties such as making macro state to Nash equilibrium even if there are no profit seeking effort. In this regime, more successful agents have higher influence to the macro state. And the exponential character of this payoff regime leads to the concentration of wealth. Under the additional payoff regime and the interdependent expectation environment, additional information is successful both for wealth maximization and macroeconomic stablility. However, under the proportional payoff regime and the interdependent expectation environment, additional information is used to microeconomic optimization and shows some contribution to microeconomic success, but this also caused macroeconomic fluctuation. These result suggests that it is probable in certain interdependent expectation environment that information can have negative effect even if the information is accurate.
77
Group
Class
Term
S#
Exp
R#
Reward
Info
N
Lecturer
1-1 2-1 2 4 5 p2-1 p2-2 p1-1′ p1-2′ pY pN p3 p4 1-1 2-1 3 4 5 p1-1 p1-2 p2-1′ p2-2′
PE2 PE2 PE1 OT PE PE2 PE2 PE2 PE2 PE2 PE2 PE1 PCT PE2 PE2 PE2 OT PE PE2 PE2 PE2 PE2
09f 09f 09f 09f 09f 10s 10s 10s 10s 10s 10s 10s 10s 09f 09f 09f 09f 09f 10s 10s 10s 10s
1 2 1 1 2 1 1 2 2 3 3 1 1 2 1 1 1 1 1 1 2 2
F T F F T F F T T T T F F T F F F F F F T T
5 5 5 5 5 10 10 10 10 3 3 5 5 5 5 5 5 5 10 10 10 10
0.5%RG 0.5%RG 5%RP 3%AG 5%RG 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP 5%RP 3%AG 0.5%RG 0.5%RG 5%RP 3%AG 5%AG 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP 2.5%RG+2.5%RP
T T T T T T T T T T T T T F F F F F F F F F
55 58 54 65 7 26 25 25 25 51 50 17 13 55 58 86 65 7 25 25 26 25
CHO CHO LEE AHN CHO CHO CHO CHO CHO CHO CHO LEE AHN CHO CHO LEE AHN CHO CHO CHO CHO CHO
Table 1.15: Groups Overview
1.A
Specifications of Experiments
• Experiment URL: http://econ.korea.ac.kr/~hokyoung/investmentgame/ • Experiment Manual(Korean): http://econ.korea.ac.kr/~hokyoung/ investmentgame.tmp/manual/Instruction_Manual(KOR).pdf
78
1.B
Bibliography
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theoretical biology, 238(2):426–434. Bosch-Domènech, A., Montalvo, J. G., Nagel, R., and Satorra, A. (2002). One, Two, (Three), Infinity, ... : Newspaper and Lab Beauty-Contest Experiments. The American Economic Review, 92(5):1687–1701. Bouchaud, J. and Mezard, M. (2000). Wealth condensation in a simple model of economy. Physica a-Statistical Mechanics and Its Applications, 282:536– 545. Bowles, S. (2004).
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Challet,
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Minority Games: Interacting Agents in Financial Markets (Oxford Finance). Oxford University Press, USA. Clauset, A., Shalizi, C. R., and Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, pages 661–703. Deck, C., Lee, J., and Reyes, J. (2010). Risk Taking Behavior: An Experimental Analysis of Individuals and Pairs. comp.uark.edu. Eisert, J., Wilkens, M., and Lewenstein, M. (1999). Quantum Games and Quantum Strategies. Physical Review Letters, 83(15):3077–3080. Grimm, V., Berger, U., Bastiansen, F., Eliassen, S., Ginot, V., Giske, J., GossCustard, J., Grand, T., Heinz, S., and Huse, G. (2006). A standard protocol for describing individual-based and agent-based models. Ecological Modelling, 198(1-2):115–126. Hanauske, M., Kunz, J., Bernius, S., and Koenig, W. (2010). Doves and hawks in economics revisited: An evolutionary quantum game theory based analysis of financial crises. Physica a-Statistical Mechanics and Its Applications, 389(21):5084–5102. Holt, C. A. and Laury, S. K. (2002). Risk Aversion and Incentive Effects. The American Economic Review, 92(5):1644–1655. Issac, R. M., Walker, J. M., and Williams, A. W. (1994). Group size and the voluntary provision of public goods. Journal of Public Economics, 54:1–36. Keynes, J. M. (1936). The general theory of employment, interest and money. Harcourt, Brace. Nagel, R. (1995). Unraveling in guessing games: An experimental study. The American Economic Review, 85(5):1313–1326. 80
Rubinstein, A. (2007). Instinctive and Cognitive Reasoning: A Study of Response Times. The Economic Journal. Sandholm, W., Dokumaci, E., and Franchetti, F. (2011). Dynamo: Diagrams for Evolutionary Game Dynamics, version 1.1. Selten, R. (1975). Reexamination of the perfectness concept for equilibrium points in extensive games. International journal of game theory. Suri, S. and Watts, D. (2011). Cooperation and Contagion in Web-Based, Networked Public Goods Experiments. PLoS ONE.
81
1.C
Korean Abstract
상호의존적 기대환경 하에서의 선택은 근원적 불확실성을 내포한다. 이러한 환 경에서 추가적인 정확한 정보가 가지는 의미를 분석하고자 간단한 투자 게임을 설계하였다. 이 게임에서는 참가자들이 위험자산과 안전 자산에 대한 포트폴 리오를 만들고, 위험자산의 비중이 일정 수준을 넘으면 파산상태가 되어 위험 자산에 투자한 자산을 모두 잃게 되지만, 그렇지 않을 경우에는 안전 자산보다 더 높은 수익을 보장한다. 이 위험자산의 비중은 각 개인의 위험자산 투자율과 보유한 개별 자산의 가중평균이 되며, 내쉬 균형은 평균 위험자산 투자율이 정확 히 파산의 기준이 되는 수준일 때가 된다. 하지만 위험상태가 되면 손실이 크기 때문에 이러한 위험을 감안한 내쉬균형은 그보다 좀 더 안전한 영역에 위치하게 된다. 고려대에 재학중인 학부생 500여명을 대상으로 웹 기반 투자게임을 실시 하고 그룹간 정보의 조건을 통제하였다. 그 결과 참가자들은 자신에게 주어진 정보를 최대한 활용하려 하지만 최종 투자수익 큰 차이를 보이지 않고 다만 위험자산 투자율의 변동성을 키우는 결과만 가져왔다. 한편으로는 행위자 기반 모형을 구성하여 각 행위자들이 자신의 정보를 통해 투자 위험을 판단하고 그에 따라 포트폴리오를 조정하는 전략을 구사토록 하여 시뮬레이션을 수행했다. 그 결과 기존 문헌에서 주로 다루는 단리적인 보상 구조에서는 추가적 정보가 거시 상태의 안정성 면에서나 최종 수익의 측면에서 모두 긍정적인 결과를 이끌어 냈으나, 본 논문에서 고려하고 있는 복리적인 보상 구조에서는 추가적 정보가 최종수익에서는 미세하게 긍정적인 결과를 이끌어내지만 한편으로 거시 상태의 변동성을 크게 증가시키는 긍정적 효과와 부정적 효과가 동시에 발생했다. 이는 상호의존적 기대상황에서 정확한 더 많은 정보가 오히려 역효과를 초래할 수 있는 조건이 존재할 수 있음을 시사한다. 핵심어 : 상호의존적 기대, 정보, 진화게임이론, 실험경제학, 복리적 수익 구 조, 행위자 기반 시뮬레이션 JEL Classification Numbers: C73, C91, G02, G11, G17
82
Chapter 2
What makes open source software development sustainable?: Agent based model on two conjectures Abstract Open source software(OSS) has no explicit pecuniary incentive. OSS project has the free riding problem but there is no way to punishment. However many OSS projects are deveoloped and survived. How can OSS compete against Proprietary Software(PS) and be survived? Many researchers have tried to explain this puzzle. This paper have focus on the group selection originated from the competition between OSS and PS. For this, a simple agent-based model is suggested. In this model, all agents can choose among OSS using for free, OSS developing, and PS using with endogeneously determined price. The object of agents is to maximize their own measurable happiness from using S/W or enjoying leisure time. Also the heterogeniety of the developing power is consid-
83
ered. In this model, initial market share of OSS and the probability to take part in the developing OSS is important factor for the survive of OSS, and heterogeneity of developing power showed no clear role in the survival of OSS. Keywords: Open Source Software, Heterogeneity, Agent-based model JEL Classification Numbers: L17, L86
2.1
Introduction
Recently, open source softwares (henceforth, OSS) have showed the remarkable successes in software industry. Along with famous success of Linux among operating systems and of Firefox in web browsers, many other OSS projects have showed the remarkable successes in the software industry. Gimp (graphic tools) and R (statistical packages) are also well known products of OSS development. Besides daily-used softwares, OSS is very common in professional areas: database program such as MySQL and scripting languages such as PHP, Perl, Lua, Python, Ruby and Tcl. It shows that OSS developments are active in various popularly used softwares and compete with the proprietary softwares. The functions of each OSS are different but they share a common one that their developments are basically based on the unconstrained access to the source codes. Source codes are human readable computer language. They are converted into executable files by compilers, then those complied files make up software applications. In OSS development, the originators who write the source code for the first time freely distribute their efforts mostly with certain licenses1 . The main 1
There are many kinds of OSS licenses, such as GPL, LGPL, and MPL. You can find them on the Open Source Initiative web page. The existence and the characteristics of the licenses is the one of the crucial factor of the OSS’s success. This topic, however, is beyond our paper.
84
characteristic of OSS licenses is that they allow modification and redistribution of the source code without paying to the originator. This is a very big different feature from the traditional commercial software development. The source code of proprietary softwares (henceforth, PS), such as Windows, Internet Explorer, and Photoshop, is not publicly revealed. And most of them charge the monetary cost to their users while OSS is free to use. OSS development has raised three key economic problems: motivation, development process and governance, and competitive dynamics (Bonaccorsi and Rossi, 2003; von Krogh and von Hippel, 2006). Motivational question is related with why developers voluntarily contribute their source codes without direct or explicit reward. Development process is closely related with organizational problem: how do the dispersed programmers around world effectively coordinate with each other while producing millions of lines of code? The last question, competitive dynamics is about popularity of OSS in the competitive environment with PS. They are closely related because a large number of well-organized voluntary developers will contribute to improve the quality of the software and the software which has the higher performance will have more chance to survive in the market. Most economists feel convenient to argue that OSS developers are affected by extrinsic motivations. Extrinsic motivations are often related to the direct benefit from solving the problem (scratching a itch) or from signaling effects as a good developer in the related job market (Lerner and Tirole, 2002). This homo-economicus explanation, however, misses some important points in the OSS phenomena. There will be no direct self-interested relation between scratching a itch and public distribution of the source code. Also if a certain OSS has the popular gain, the career of its developer could have the good reputation in the labor market, however, it is excessive to say that the originator starts the project to increase their income even when its success is uncertain. Also extrinsic motiva85
tions cannot explain why people devote their time to non-essential dirty-hands works such as making help files or providing local language supplements and why there is less challenging to the project leader or changing the project as a personal one (forking) (Hertel et al., 2003). With this line of criticisms, some studies have stressed on the intrinsic motivations, such as hobby culture, generalized reciprocity, and gift-giving (Lakhani and Wolf, 2005). A few surveys showed that the OSS developers’ motivation is usually combination of extrinsic and intrinsic one (Lakhani and Wolf, 2005; Spiller and Wichmann, 2002). Two motivational arguments lead to the two innovation models: private and collective model. Private model is related with the extrinsic motivation. The innovators are pursuing the benefit from the innovation. However, this model gives very limited explanation to the OSS development process. OSS developers give their efforts, even they do not have exact private returns because of free revealing of their innovations. In contrast with the private innovation model, collective model tries to explain voluntary innovators’ role in the projects. As Johnson (2002) and von Hippel and von Krogh (2003) point out that production of OSS could be understood as the voluntary provision of public good. In that case, we should consider free-riding problem which will affect the competitiveness with PS. The software’s performance will be stagnated or declined if there are only free-riding users in the OSS community without increasing numbers of developers. In that case, OSS could lose the user base. However, against the collective model’s expectation, the OSS innovation process is not hindered by free riders (von Hippel and von Krogh, 2003). However, if we admit the various motivations, the problem lies in the governance of the development process and its competitiveness. This paper will focus on the behavior of the OSS developers rather than their motivation. Our primary concern is about innovators’ behavior and innovation process: 86
why and how volunteer’s innovation development process can be sustained? Regardless of each developers’ motivations, in the OSS project group, each of them pursues the common goal to make a software which has a specific function. However, within the project there is a free-riding problem. The developers bear the developing costs to make solutions but the users don’t. In that sense, developers are altruists. The voluntary developers’ good will, however, would not seem likely to survive in any selection process which favors traits with higher payoffs. It means that the free-riding users have the reproductive advantage over the developers. Because the free-riding users will have more benefit than the developers, all the people will be users after a long time. If we assume that the softwares’ quality depends on the number of developers, the OSS development’s main problem is how to keep up voluntary developers to compete with PS. There are some researches on this topic. Johnson (2002) shows that the developers will participate only if the benefit-cost ratio is higher than a certain threshold. Bessen (2005) makes a model in which free riding is a less concerning problem if the OSS is the first product in the market. Nevertheless, these models are lacking to analyze the dynamic market competition. To analyze the demand side’s non-linear dynamics, network effects from path-dependent and lock-in effects (Arthur, 1989; David, 1985) can be used. In this regard, Bonaccorsi and Rossi (2003) run simulations on their own OSS adoption function. Bitzer (2004) and Leoncini et al. (2008) consider heterogeneity and competition in the market. Dejean (2008) deals with freeriding and heterogeneity on making information in the online communities. Baldwin and Clark (2006) argues that the modularity of OSS increases the developer’s incentive to join and remain in OSS development and decreases the amount of free-riding. Those previous studies separately or partly together examine free-riding, competition, and heterogeneity in OSS developments. However, this paper 87
tries to combine those problems. To figure out the friendly environment for OSS developers with free-riding and market competition, we use the agentbased simulation. In section 2.2, we describe the OSS favorable scenario and the basic structure of simulation. In section 2.3, the simulation results is presented and interpreted. In section 2.4, as a conclusion, we sum up the implications and suggest the future research topic.
2.2
The structure of Agent-Based-Simulation Model
Basic ideas and its implication We can imagine the world of which population consists of people who troubles with the same problems in using computer. For example, in Korea, Mac OS X and Linux users have the problem to type the Hangul, Korea’s native alphabet, into the software applications. The company, Apple have not fully developed the proper solution for Koreans yet. In this situation, a natural language input problem is solved by the OSS project, the Baram and the Nabi2 . In our model, all the agent should have their own solutions and they are categorized as four kinds by using or developing PS and OSS. They are not exclusive to each other. For instance, a agent can be a PS developer and also OSS user. Every agents should spend 24-hours in a day3 . We assume that the agents allocate 8-hours for sleeping, 8-hours for working, and 8-hours for leisure. OSS developers divide the 8-hours leisure time into OSS development (dt) and leisure (lt). 2
Each projects homepages are: http://kldp.net/projects/baram-kim/src and http: //kldp.net/projects/nabi 3 This time period is not actually matched to the time in real life.
88
Agents’ objective is to increase their happiness (or satisfaction). Softwares’ good performance (A) and consumption in leisure time (L) will increase each agent’s happiness (H), but the switching cost (SC) will decrease it.
Hi = Ai + Li − SCi
(2.1)
For the simplicity, we assume that softwares’ performance depends on the total developing time. There is no governance problem in the model. The proprietary software company would hire the developers (npd ) and order them to make softwares for 8 hours in everyday. The OSS community would assign the task to the voluntary developers (nod ) but the developing time depends on their freewill. Developers has different developing skill levels (βi ) and become the members of developers’ pool in each cases (D(P S), D(OSS)).
Ai = log Asw ∑npd i∈D(P S) βi dti , Asw = ∑ nod βi dti i∈D(OSS)
(2.2) (P S)
(2.3)
(OSS)
The consumption in leisure time has the positive relationship with happiness. We assume that the agents consume every income and the consumption per hour increases happiness. In the case of the proprietary software users, they should pay the cost to use it. It will decrease their fixed wage income (Ii ), then happiness. For the software’s price (Psw ), we assume that the PS firm would take the mark-up (µ) pricing strategy. As pay-per-service, the firm will charge the cost in certain used periods .
89
(
Ii − Psw E(lti − dti )
Li = α log ∑npd (1+µ) i∈D(P S) Ii E(D) P = 0 (OSS)
) × (lti − dti ) (P S)
(2.4)
(2.5)
When the agents change their software from the proprietary to the OSS, or vice versa, they will have switching cost (Shapiro and Varian, 1999). The longer agents use (ut) or develop (dt) the specific software, the more they have specific knowledge (k(t)) about it. Also they could feel difficulty (h(t)) to use the other software. This cost will be lower as it has been developed to have more compatibility. In many cases softwares’ interoperability is crucial in switching cost, however, our simple model doesn’t consider it. If the OSS users try to switch to use proprietary software, then they should pay the price for it. In the other case, the proprietary switchers will not pay for the OSS. However, the proprietary software’s price is reflected in the disposable income, so also we don’t include it in switching cost.
SCi = k(t) − h(t) = utsw + dtsw − log Asw
(2.6)
If we assume the static environment where the performance of softwares is same and there is no switching cost, we can draw the happiness matrix as table 2.1. In any cases, OSS user would have more happiness, every agents want to be a OSS user and no one wants to be a OSS developer. If everyone becomes a OSS user, OSS development stops. It becomes orphanized. If there are more OSS developers than free-riders, the performance of OSS will be enhanced and be more attractive. However, if there are more free-riding OSS users, it becomes less attractive and OSS users will choose PS. Because the PS firm will continue the development and the OSS’s development stops, 90
happiness
(
Ii − Pps 8
P Suser
A + 8 × α log
OSSuser
( ) Ii A + 8 × α log 8 (
OSSdeveloper
A + (8 − dti ) × α log
)
Ii 8 − dti
)
Table 2.1: Happiness matrix of each agents in the static environment
which leads to the different performance. Agents may choose it for the better performance while they should pay for the PS. We call it as a performance effect. In the model, otherwise, the price of PS could be lower if there are more PS users. Also if there are less PS users, the price of PS could be higher. We call it as a price effect. So, if the the number of OSS developers decreases, there will be negative performance effect, then the number of PS user increases as the number of OSS user decreases, there will be positive price effect. In the long run, there will be no OSS users and developers in the population. Although it seems that the fall of OSS is more plausible, we can think that still there is a chance if there are large number of high-skilled developers in the population. Because he or she could make the software without loss of their leisure time. Or OSS users just change their minds to devote their time in the OSS development, we call it as a mutation rate. If there is positive performance effect, OSS becomes more attractive, then Some PS users change to OSS users. As the number of PS users decreases, there is negative price effect and PS becomes less attractive.
91
Parameters and variables of the agent-based simulation For the simulation, we fix some variables: the number of agents4 are 1,000; wage per hour is 10; initial employment of PS firm is 30 (3%) which draws from employment rate5 ; initial OSS community member6 is 10. We control the three variables: software development skill distribution, mutation rate, and the initial market share of OSS. As explained in previous section, developers who has the higher skills could use less time than lower skilled developers. Also, the higher mutation rate could give more possibility to sustain development. It is very hard to know exact software development skill distribution. As a proxy measure, we use the SourceForge.net Database. It has a question about the software development skill, its value is skill_level_id in the database. Table 2.2 shows the distribution in selected times. With given average distribution of skills, we normalized the skill level (β1) and introduced arbitrarily skill level (β2). There is no learning: the skill level is not changed through the simulation. In addition to the SourceForge.net Database, for comparison we include the flat skill (β0 = 1) and big leap skill level (β2 = 1, 10, 100, 1000, 10000) among agents. The mutation rate is a certain probability of voluntary participation in OSS development, in this model, we compare 1.0 × 10−4 per using OSS and 1.0 × 10−5 . The initial market share of OSS varies from 0.0 (no OSS users at all) to 1.0 (monopoly of OSS). Interval is set by 0.1. 4
In real situation there are more than one thousand people in the population. But our pilot simulation results show that the large number doesn’t have an effect on the pattern of the results. For the simulation convenience we reduce the number of agents. 5 The US labor statistics shows that S/W engineer ratio in the total employment is 2.2%. ICT and related industry’s employment rates in the OECD countries are 5.54% and 4.24% respectively. We think that 3% is a little bit higher than real situation, but it makes the PS favorable situation. 6 In some case, public spotlight goes to the one person in the OSS project, like Linus Torvalds. Usually, however, OSS development starts with a small group of people
92
year Apr. 2010 Feb. 2010 Jun. 2009 Jun. 2008 Jun. 2007 Jun. 2006 Jun. 2005 Nov. 2004 Jan. 2003 average (%) β1
members total 2674879 2600806 2329966 1913729 1614261 1338375 1091958 943765 544669
want to learn 7949 7802 7299 6560 5502 4534 3518 3159 1577 17.82% 0.8
competent 24632 24343 23197 21532 19643 17098 14440 13876 7823 61.96% 1.0
wizard 6393 6348 6115 5751 5389 4818 4225 4222 2398 16.98% 1.2
wrote the book 701 689 653 630 598 558 492 516 276 1.90% 1.4
wrote it 479 471 460 448 437 420 333 362 191 1.34% 1.8
Table 2.2: Skill distribution based on Sourceforge.net
The space is created as a grid of which size is related to the population density. In our model, the agent’s density is 81.6% in the 35 × 35 grid torso space. Agents look for and compares with the other agents. Every one-month passes, comparison occurs in 10% of the whole agents. With the 3 cases of software skill distribution, 2 cases of mutation rate, 11 cases of initial market share, we have 66 cases of simulation, and we ran the simulation for 240 months7 .
Arithmatical analysis of the simulation N ∈ N is the total number of the agents. T is the current period (we set it as a month), M is the number of days of one month. η ∈ (0, 1) is the proportion of workers hired by PS firm in the total agents and φ ∈ [0, 1] is the proportion of the OSS users. W is working time in 24-hours, F is the maximum leisure time, and D is the time to use software. ω is the wage per hour. In this model, N = 1, 000, M = 30, W = F = 8, and D = 16. Also, for simplicity, we assume the software development skill level is fixed, βi = β. If N is large enough, an agent i’s decision has little affect to φ. To maximize their happiness, an agent i will compare PS and OSS and allocate their OSS 7
In 1989, the first official open source license, the GNU General Public License was written.
93
develop time dti . The individual agent i’s maximization problem in the period T can be set as below: SOL ∈ {PS, OSS}
max HSOL ,
SOL,dti
(2.7)
The happiness functions of PS and OSS users are equation 2.8 and 2.9 respectively. HP S = DM log[(T − 1)ηN W β] + M α(D − dti ) log HOSS
N T −1 ∑ ∑
= DM log
t
Wω −
βdtjt + M α(D − dti ) log
j
(1+µ)ηW 1−φ
F − dti [
Wω F − dti
(2.8)
] (2.9)
Because the happiness is always decreasing function of dti when individual income is sufficiently large, dti goes to zero if there are no possibility of change their mind to develop OSS (no mutation in our terms). Therefore, the performance of OSS will become constant after some finite period T˜. After period T˜, the gap between PS and OSS can be written as below: [
HP S − HOSS
[
B = DM log(T − 1)A + α log 1 − 1−φ
]] (2.10)
ηN W β A = ∑˜ ∑ T −1 N t j βdtjt B=
2.3
(1 + µ)η ω
The result of the simulation
The figure 2.1 shows the final results after 20-years interaction between different skilled agents. From the top to the bottom, sequentially those reflect 94
the initial market share (from 100% to 0%). At first, it can be said that the OSS users have decreased in every cases. Distribution of software development skill and initial market share doesn’t affect on the decline of OSS users. Even we arbitrarily change the of development skill distribution, the result remains unchanged.
Figure 2.1: The movement of heterogeneous skilled OSS users without mutation (by initial market share controlled).
OSS users continues to decline regardless of initial market shares and there is no stable equilibrium between OSS and PS. However, this declining pattern is changed after we introduce the mutation rate: a certain probability of voluntary participation in OSS development, 0.0001 per using OSS. The figure 2.2 shows the number of OSS users becomes stable with mutation rate and heterogenous skill (β1). If we controlled the skill level as β0 and β2, also they reach the stable equilibrium: those results can be found in the appendix. To confirm the effect of mutation rate, we changed mutation rate as 0.00001 per using OSS. The figure 2.3 shows that there is no stable equilibrium. It says
95
Figure 2.2: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled).
that the certain rate of mutation rate has an effect to make stable equilibrium between OSS and PS. The figure 2.4 and 2.5 show how the different skill levels affect on the results while initial OSS market share was 70% and 30%. EO, SO, and PO respectively means β1, β0, andβ2. With other results which are in the appendix, it is difficult to say that there is effect of different skill levels.
2.4
Concluding remarks
Many people have thought that the voluntary software development would fail in the market competition. In his “Open Letter to Hobbysts”, Gates (1976) famously said that “Who can afford to do professional work for nothing? What hobbyst can put three man-years into programming, finding all bugs, documenting his product, and distribute for free?” However, OSS developments continue at least last 30 years, and some OSS 96
Figure 2.3: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-5 (by initial market share controlled).
Figure 2.4: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 70% (by skill level controlled).
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Figure 2.5: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 and OSS initial market share 30% (by skill level controlled).
such as Firefox, Linux, and Apache have widely adopted. We conjecture that the different programming skill and the sudden change of OSS users devotion to development will affect the sustainability of OSS development. By the simulation results, it is difficult to say that the differences in software development skill have the critical effect on the sustainable OSS development. Otherwise, the result shows that if the OSS project has the high initial market share or the high mutation rate, it would survive in the competition with the traditional proprietary software. However, it is difficult to say that there is no effect of skill difference at all. On the one hand, our model has two logical and technical problems. First, we didn’t full simulated on the several types of skill distributions. We just simulated on the one distribution of skills and two values on skills. Second, the PS firm in the model is not a strategic behavior. We didn’t fully design the model. On the other hand, if our partial simulation results have enough explanations,
98
we should examine the true nature of voluntary provision of public goods and the role of successful governance structure in OSS community.
99
2.A
Appendix: The comparison of simulation results
Figure 2.6: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled and β1).
Figure 2.7: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled and β0).
100
Figure 2.8: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 (by initial market share controlled and β2).
Figure 2.9: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 90% (by skill level controlled).
101
Figure 2.10: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 80% (by skill level controlled).
Figure 2.11: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 70% (by skill level controlled).
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Figure 2.12: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 60% (by skill level controlled).
Figure 2.13: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 50% (by skill level controlled).
103
Figure 2.14: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 40% (by skill level controlled).
Figure 2.15: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 30% (by skill level controlled).
104
Figure 2.16: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 20% (by skill level controlled).
Figure 2.17: The movement of heterogeneous skilled OSS users with mutation rate 1.0E-4 OSS initial market share 10% (by skill level controlled).
105
2.B
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2.C
Korean Abstract
오픈 소스 소프트웨어 (OSS) 에는 명시적인 금전적 유인이 존재하지 않는다. 이러한 OSS 프로젝트에는 무임승차 문제가 필연적으로 발생하지만 그러한 무 임승차자를 처벌할 수 있는 방법이 없다. 이러한 상황에도 불구하고 어떻게 OSS 프로젝트들은 상용 소프트웨어 (PS) 와의 경쟁에서 지속적으로 개발되고 살아남을 수 있는가? 이 문제를 풀기 위해 많은 연구자들이 가설을 제시해왔다. 본 논문에서는 이 문제를 OSS와 PS간의 경쟁이라는 상황과 집단 선택의 관 점에 초점을 맞추었다. 이러한 분석을 위해 본 논문에서는 간단한 행위자 기반 모형을 만들고 시뮬레이션을 수행했다. 이 모델에서 행위자들은 OSS를 무료로 쓰거나, 자발적으로 개발에 참여하거나, 내생적으로 결정된 가격을 지불하고 상용 소프트웨어를 사용할 수 있다. 이 모델에서 행위자는 소프트웨어나 여가를 통해 발생하는 측정 가능한 행복을 극대화한다. 이 모델에서 행위자는 개발 기술의 분포를 이질적으로 설정하였다. 그 결과 OSS 생존여부에 영향을 미치는 주요 변수는 OSS의 초기 시장 점유율좌 OSS 사용자 중 개발에 참여하는 확률 이었으며, 이 두 값이 일정 수준 이상일 때 OSS는 지속할 수 있었다. 반면 개발 기술의 이질성은 OSS의 생존과 관련하여 별다른 영향이 관찰되지 않았다. 핵심어 : 오픈소스 소프트웨어, 이질성, 행위자 기반 모형 JEL Classification Numbers: L17, L86
108
Chapter 3
민간자본 고속도로의 최적 요금 검토 : PoA를 통한 접근 Abstract 천안–논산과 대구–부산 구간은 출발지와 목적지가 같지만, 도로공사 와 민간투자회사로 고속도로의 운영주체와 경로가 이원화 되어 있으며, 이용 요금도 서로 다르다. 두 경로의 특성이 다를 경우, 개별 차량의 비용 극소화 선택은 사회적 최적 선택과 유리될 수 있다. 하지만, 각 경로에 도로 사용 요금을 차등 부과할 수 있다면, 두 요금의 차이가 유인이 되어 사회적 최적 통행 배분을 유도할 수 있다. 본 논문은 도로 요금 산정에 일반적으로 쓰이는 단기 비용 함수 (VDF) 에 기반하여 네트워크 효율성 지표인 “자유선택의 비용 (PoA: Price of Anarchy)”을 사용하였다. PoA 는 총정체비용을 기준으로 계산되었으며, 이를 토대로, 도로공사 요금 대 비 최적 민간투자 고속도로 이용요금을 산출하였다. PoA의 시뮬레이션에 따르면, 천안–논산 구간은 현행 가격보다 높이고, 대구–부산 구간은 현행 가격보다 낮출 경우, 통행 배분을 개선할 수 있다. 핵심어 : 자유선택의 비용 (Price of Anarchy), 민간투자 고속도로, 고 속도로 이용요금, 총정체비용 JEL Classification Numbers: R41, R48
109
3.1
서론
현재 한국의 고속도로 체계는 운영 주체가 한국도로공사와 민간투자회사로 이원화되어 있으며, 요금 체계도 분리되어 있다. 대표적인 이원화 구간인 의 경우, 2010년 12월 현재 민간투자회사 고속도로 (이하 민자 고속도로) 의 도로 사용료는 한국도로공사의 고속도로 (이하 도공 고속도로) 의 도로 사용료보다 단위 거리당 48–56% 더 높다. 이러한 민자 고속도로의 상대적으로 높은 가격은 논란의 대상이 되어 왔다. 하지만, 민자 고속도로는 도공 고속도로보다 26%의 거리 단축효과가 있기 때문에 유류비와 이동시간을 줄일 수 있다. 운전자는 통행요금, 유류비, 이동 시간 등을 고려한 통행비용에 따라 자신 의 경로를 선택할 것이다. 이때 경로마다 다른 고속도로 사용료는 도로 이용 유인에 영향을 줄 것이다. 민자 고속도로를 사용하는 운전자의 경우 목적지에 도착하기까지의 운행거리가 짧기 때문에 시간과 유류비용을 절감할 수 있지만 상대적으로 높은 도로사용료를 지불한다. 반대로 도공 고속도로를 사용하는 운전자는 상대적으로 낮은 도로사용료를 지불하는 대신 더 많은 시간과 유류 비용을 부담해야 한다. 만약 현행 민자 고속도로의 통행 비용이 상대적으로 너무 낮다면 민자 고속 도로의 정체를 유발할 것이다. 반대로 민자 고속도로의 통행 비용이 상대적으로 너무 높다면 도공 고속도로의 정체를 유발하는 비효율성을 낳을 것이다. 한편 으로, 현행 고속도로 통행요금 체계가 민자 고속도로와 도공 고속도로 사이의 가장 최적의 배분을 달성토록 할 수 있다면, 그러한 요금 체계는 상대적으로 효율적이라고 할 수 있을 것이다. 이원화되어 있는 현행 고속도로 통행요금의 효율성을 검토하기 위해 본 연 구는 도로 자유 선택의 비용 (Price of Anarchy, 이하 PoA) 이라는 개념을 사용 하고자 한다. PoA는 개인의 전략적 선택으로 결정된 교통량으로 인해 발생하는 사회적 총비용과 동일 구간의 도로에 대해 사회적 최적으로 배분된 교통량으로 인해 발생하는 사회적 총비용의 비율이다. PoA가 1과 같다면 개인의 전략적 선택에 따른 교통량 배분과 사회적 최적 배분이 같음을 의미하고, 1보다 크다면 개인의 전략적 선택에 의한 통행 배분이 사회적인 최적 배분으로부터 유리됨을 110
의미한다. PoA를 이용하는 경우, 차량 흐름의 증가에 따른 누적적인 효율성 변화를 파악하는 데 유리하다. 한편, 본 논문의 분석은 건설 및 유지 비용 등의 공급측 변수를 고려하지 않았지만, 주어진 가격 체계의 효율성을 재검토하는 데, 정책적 유의미성이 있다고 할 수 있다. 분석 대상은 천안–논산, 대구–부산의 두 구간이며, 민자 고속도로가 건설 되어 있지만 대체 고속도로가 존재하지 않는 부산–울산 등은 제외했다. 분석 시기는 2008년으로 한정했다. 2009년, 천안–논산 민자 고속도로와 연결되는 대전–당진 고속도로가 완공되어 운전자의 경로 선택 및 통행량이 영향받을 수 있겠지만, 당진구간의 동서횡단 구간이기 때문에 남북종단 고속도로인 천안– 논산 구간에 미치는 영향은 제한적인 것으로 판단된다. 통행 비용은 2008년을 기준으로 하였다. 이를 토대로 민자 고속도로요금의 변화에 따른 교통량의 배분 상황을 시뮬레이션하며 PoA의 변화를 추적했다. 본고의 구성은 다음과 같다. 3.3절에서는 PoA의 개념에 대해 설명할 것이다. 3.3절에서는 교통량 지체함수 (VDF: Volume Delay Function) 를 통한 통행시간 비용을 기반으로 단일한 차종, 통행목적, 유류 소비량을 가정했을 때의 이론적 PoA를 산출한다. 3.4절에서는 단일성 가정을 완화하여 차종, 통행 목적, 속도 별 연비를 다양화할 때의 PoA를 시뮬레이션을 통하여 도출한다. 나아가 민자 고속도로의 요금 수준을 변화시키면서, PoA 변화를 살펴보고 가장 바람직한 민자도로요금을 도출한다.
3.2
이원화된 고속도로 요금 체계와 이용 현황 분석
천안–논산 고속도로의 경우 구조는 그림 3.1과 같다. 천안–논산 구간의 도로공 사 구간은 대전을 경유하여야 하여 총 거리가 110.65km이고, 민자 고속도로는 87.66km로, 민자 고속도로가 약 23km 단축 노선이다. 2008년 10월 현재1 , 천 안–논산 구간을 승용차로 이용시 도로공사 구간에서는 5,800원, 민자 고속도로 1 2008년 10월과 2009년 12월에 민자 고속도로의 통행요금 조정이 있었다. 2009년에는 2008 년과 비교하여 차종별 각 100원 정도의 요금 인상이 있었다. 본고에서는 2008년 10월의 요금을 기준으로 한다.
111
Figure 3.1: 천안–논산 고속도로 망의 구조 구분 도공
민자
경로 천안-남이JC 남이JC-회덕JC 회덕JC-논산 천안-풍세 풍세-남논산
편도차선 3차선 4차선 2차선 2차선 2차선
법정 최고속도 100km 100km 100km 110km 110km
거리 39.65km 22.60km 48.40km 8.12km 79.54km
총 거리 110.65km
87.66km
Table 3.1: 천안–논산 구간 고속도로 상세 현황
구간에서는 8,300원을 지불하여야 하며, 가격차는 2,500원이다. 동 구간에 대한 km당 요금은 도공 52.39원2 , 민자 104.35원으로 민자가 약 1.99배이다. 승용차 (1종) 를 기준 2008년 9월 현재 통행료와 유류비용만을 반영하여 기초 통행비 용을 비교해보면 도로공사 구간이 18,906원, 민자 고속도로 구간이 18,983원이 지출된다.3 양쪽 도로 모두 정체가 없다고 가정한다면 명목 비용은 거의 같지만 2
도로공사 구간의 km당 편도 2차선 기준요금은 40.5원이며, 편도 3∼4차선의 경우 20%의 할증, 편도 1차선의 경우 50% 할인이 적용된다. 3 유류비용은 유류비와 연비를 토대로 계산되었다.
112
구분 도공 민자 (도공)
구간 천안–논산 천안-남논산 천안-풍세 풍세-남논산 출처 : 국토해양부
총거리 110.7km 87.66km 8.12km 79.54km
1종 5,800원 8,700원 400원 8,300원
2종 5,900원 8,800원 400원 8,400원
3종 6,100원 9,300원 500원 8,800원
4종 7,900원 12,400원 600원 11,800원
5종 9,200원 14,600원 700원 13,900원
Table 3.2: 천안–논산 구간 고속도로 요금 (2008년 10월 현재) 구분 도공 민자
경로 천안-대전-논산 천안-남논산
거리 110.65km 87.66km
통행료 5,800원 8,600원
유류비용 13,106원 10,383원
총 통행비용 18,906원 18,983원
Table 3.3: 정체 비용을 제외한 천안–논산 구간 고속도로 이용 비용 (승용차 기준)
민자 고속도로가 단축노선이기 때문에 도로공사 고속도로 보다 짧은 시간에 목적지에 도착할 수 있으므로 더 비용을 절감하는 선택이 될 것이다. 하지만, 차량대수가 심각한 정체를 유발할 정도로 늘어나거나, 유인체계의 문제로 적 절한 배분이 이루어지지 않는 경우, 민자 고속도로를 이용하는 것은 더 안좋은 선택이 될 수도 있다. 그렇다면, 천안–논산의 실제 통행량은 얼마나 될까? 본고에서 고려하고자 하는 민자/도공 고속도로의 선택에 직면한 차량의 실제 수를 정확히 알기는 어렵다. 민자 고속도로의 경우 최종 요금소에서 요금 정산을 하므로 통행 차 량의 수가 정확히 파악되지만, 도공 고속도로는 구간을 통과한 운전자의 수를 직접 파악할 수 있는 방법이 없기 때문이다. 다만, 도공 경로를 통과하는 차량은 다른 구간들과 연결되어 있기 때문에 민자 고속도로보다 더 많을 것으로 생각할 수 있다. 본 고에서는 민자 고속도로 이용 차량의 200%를 도공 고속도로 이용 차량의 최대값으로 보았다. 한편, 사용 가능한 민자 고속도로 통행량은 년간 총 이용 차량 수만 제시 되어 있어, 분석 기준이 되는 시간당 통행량을 정확히 확인할 수 없었다. 통상 주간에 승용차, 야간에 화물차의 고속도로 주행이 많으며, 주간이라 하더라도
113
출퇴근 시간을 전후로 통행량이 집중되는 경향을 보이는 것으로 알려져 있다. 이런 점을 감안하여, 민자 고속도로의 시간당 이용자는 년간 총 이용자를 1일 6시간 기준으로 나누어 추정하였다. 천안–논산 구간의 2008년 민자 고속도로 이용 차량은 상행 약 462만대, 하행 약 506만대였다. 차량 통행이 평상시 보다 많을 것으로 보이는 설과 추석에는 각각 상행 약 6만 3천대, 하행 약 8만 2천대가, 상행 약 6만 6천대, 하행 약 6 만 5천대가 이용하였다. 표 3.4는 천안–논산 구간 민자 고속도로의 이용현황을 정리한 것이다. 따라서, 천안–논산 구간에서 경로를 선택해야 하는 차량의 수는 시간 당 최소 2,000대를 넘을 것이며, 최대 30,000대는 넘지 않는다고 볼 수 있다. 구분 상행
요금소 풍세
하행
남논산
년도 2007년 2008년 2007년 2008년
시간당 (년) 2,150대 2,110대 2,330대 2,320대
시간당 (설) 11,490대 10,530대 10,410대 13,800대
시간당 (추석) 9,620대 11,090대 9,950대 10,840대
Table 3.4: 천안–논산 구간 민자고속도로 이용현황 (차량수/시간)
대구–부산 고속도로의 구조는 그림 3.2와 같다. 대구–부산 구간 민자 고 속도로 역시 기존 구간에 대한 단축 노선으로 개발되었으나 천안–논산 구간 고속도로와 비교했을 때 다른 특징이 있다. 대구–부산 구간 민자 고속도로의 경우 김해부산 통게이트 (TG) 에 도착한 차량이 부산 TG로 접근하기 위해서는 도공 관리하의 중앙고속도로 일부 구간인 대동–양산 고속도로를 이용해야 하기 때문이다. 즉, 대구–부산 구간의 두 구간 중 민자 고속도로가 속해 있는 경로에는 민자 고속도로 뿐만 아니라 도공 구간이 포함되어 있다. 대구–부산 구간의 도공 구간은 언양을 경유하여야 하여 총 거리가 124.2km이고 민자 구간을 이용하는 경우 총 102.45km로, 민자 구간이 약 22km 단축 노선이다. 2008년 10월 현재 대구–부산 구간을 승용차로 이용시 도로공사 구간에서 는 6,500원, 민자 고속도로 구간에서는 10,600원을 지불하여야 하며, 가격차는 4,100원이다. 동 구간의 km당 요금은 도로공사 52.33원, 민자 115.79원으로 114
Figure 3.2: 대구–부산 고속도로 망의 구조 구분 도공 민자 (도공) (도공)
경로 대구 (동대구)-언-양 언양–부산 대구 (동대구)–김해부산TG 김해부산TG–양산 JC 양산 JC–부산
편도차선 2/3/4차선 2/3/4차선 2차선 2차선 3차선
법정 최고속도 100km 100km 110km 100km 100km
거리 82.14km 42.06km 79.45km 10.8km 12.2km
총 거리 124.2km 102.45km
Table 3.5: 부산–대구 구간 고속도로 상세 현황
민자가 약 2.21배 높다. 승용차를 기준으로 2008년 10월의 통행료와 유류비용 구분 도공 민자
구간 대구 (동대구)–부산 대구 (동대구)–부산 대구 (동대구)– (민자) 김해부산TG (도공) 김해부산TG–부산 출처 : 국토해양부
총거리 124.2km 102.45km
1종 6,500원 10,600원
2종 6,600원 10,800원
3종 6,800원 11,200원
4종 8,800원 15,200원
5종 10,300원 17,900원
79.45km
9,200원
9,400원
9,800원
13,100원
15,500원
23km
1,400원
1,400원
1,400원
2,100원
2,400원
Table 3.6: 부산–대구 구간 고속도로 요금 (2008년 10월 현재)
만을 반영하여 기초적인 통행비용을 산출해보면, 도로공사 구간이 21,211원, 민자 고속도로 구간이 22,127원이 소요된다. 대구–부산 구간의 경우, 천안–논 115
구분 도공 민자
경로 대구 (동대구)-부산
거리 124.2km 102.45km
통행료 6,500원 10,600원
유류비용 14,711원 11,827원
총 통행비용 21,211원 22,127원
Table 3.7: 정체 비용을 제외한 대구–부산 구간 고속도로 이용 비용 (승용차 기준)
산 구간과 달리 민자 고속도로의 통행비용이 상대적으로 높음을 알 수 있다. 하지만, 이 역시 시간가치가 반영되지 않은 단순 비교이다. 따라서, 정체 발생 정도를 고려해야 정확한 비교가 가능하다. 대구–부산 구간 역시 민자 구간의 통행량만을 정확히 알 수 있다. 대구– 부산 구간의 민자 고속도로 이용량은 2008년 기준, 상행 약 636만대, 하행 약 746만대였다. 설에는 상행 약 7만 8천대, 하행 약 9만 1천대가 이용하였으며, 추석에는 상행 약 8만 1천대, 하행 약 8만 7천대가 이용하였다. 표 3.8은 시간 당 대구–부산 구간 민자 고속도로의 이용현황을 정리한 것이다. 천안–논산 구간의 방식과 마찬가지로 도공 구간의 통행량을 추정해 보면, 대구–부산 구간에서 경로를 선택해야 하는 차량의 수는 시간 당 최소 2,900대를 넘을 것이며, 최대 40,000대를 넘지 않을 것으로 볼 수 있다. 구분 상행
TG 대구
하행
김해부산
년도 2007년 2008년 2007년 2008년
시간당 (년) 2,960대 2,910대 3,470대 3,410대
시간당 (설) 12,350대 13,020대 14,270대 15,210대
시간당 (추석) 13,690대 13,650대 13,940대 14,600대
Table 3.8: 대구–부산간 민자고속도로 이용현황 이후 절에서는 이원화된 요금 체계의 효율성을 검토하기 위해, 효율성 기준 으로 PoA를 소개하고, VDF 함수를 이용한 통행량 시뮬레이션을 통해 천안– 논산과 대구–부산 구간의 PoA를 계산하여, 요금체계에 대한 정책적 함의를 도출하고자 한다.
116
3.3
이원화된 고속도로에서의 PoA 산출
PoA의 개념 PoA는 죄수의 딜레마에서처럼 비협조게임의 균형이 목적함수를 극대화하지 않는다는 점에서 비효율적일 수 있다는 널리 알려진 사실에 착안한다. 그 개 념은 “협조 비율 (Coordination Ratio)”로 먼저 소개되었으나, 현재에는 “자유 선택의 비용 (Price of Anarchy)”이라는 표현으로 사용되고 있다 (Koutsoupiasa and Papadimitriou, 2009). PoA는 혼잡게임 (congestion game) 에서의 자원배분 효율성을 나타낸다 (Moulin, 2008, p. 379). 혼잡게임에서 참가자들은 어떤 출발 지점에서 목표 지점으로 자원을 이동시킬 때 그 경로의 비용은 경로에 부과된 자원의 양에 따 라 증가하고, 각 참가자들은 자원 이동에 들어가는 비용을 최소화 하고자 한다. 이때 각 행위자들의 비용을 최소화하는 경로의 집합은 내쉬 균형이 되지만, 비 협조 혼잡게임에서 개별 내쉬 균형의 집합은 효율적이지 않은 것이 일반적이다. 즉, 개별 비용을 극소화하는 경로 선택은 사회적 총 비용을 극소화하는 경로 선택과 다를 수 있다. PoA는 이러한 사회적 총비용을 극소화하는 경로 선택시 발생하는 총비용과 개별 비용을 극소화하는 경로 선택시 발생하는 총 비용의 비율로 정의된다. P oA ≡
개별선택 총비용 사회적 최적 비용
개별 선택 총 비용은 개별 통행자들이 본인의 개별 비용을 최소화하는 경로 를 선택했을 때 발생하는 비용의 총합을 의미하며, 사회적 최적 비용은 개별 총 비용을 최소화하는 통행량이 분배된 상황에서 발생하는 비용의 총합을 의미한 다. 따라서 PoA는 1보다 작을 수 없으며, 클 수록 비효율적임을 의미한다. 경제학적으로 개별 경로 선택의 비효율성과 이의 해결을 위한 방법은 이미 Pigou (1920)가 논의를 시작한 바 있으며, Knight (1924)를 거쳐 최적 교통량 배분을 위해 도로 이용에 대한 세금 부과 등의 인위적 개입에 대해서는 많은
117
연구가 진행4 되었다. PoA는 컴퓨터와 전자 네트워크 등의 연구에서 활발히 이용 (Roughgarden, 2005)되고 있지만, 교통 문제에 직접 적용한 논문은 많지 않다. Youn et al. (2008)은 뉴욕, 런던, 보스턴의 도로 네트워크를 대상으로 교통 시뮬레이션을 구성한 뒤 PoA를 산출한 결과 개별 최적 선택의 결과로 발생하는 비용이 사회 적 최적 비용과 상당한 (약 30% 수준) 차이를 보이고 있으며, 복잡한 네트워크 상에서는 특정 도로를 폐쇄하는 것이 오히려 더 나은 결과를 가져올 수도 있음5 을 보이고 있다. 하지만 Youn et al. (2008)에서의 연구 대상은 복잡한 시내 도로 네트워크이며 정책적 함의가 도로망의 설계에 있다고 할 수 있어, 본 논문의 목적인 단순 네트워크에서의 최적 요금 검토와는 분석 목표의 차이가 있다.
개별 최적 총비용 민자 고속도로 경로를 1이라고 하고, 도공 고속도로 경로를 2라고 하면, 민자 고속도로를 주행하는 시간당 차량 흐름의 크기를 v1 라고 하고, 도공 고속도로를 주행하는 차량 흐름의 크기를 v2 라고 할 수 있다. 이때, 균형은 각 도로 구간 에 대한 한계 비용이 같은 지점에서 형성되게 된다. 만일 한쪽의 한계비용이 더 크다면 새로 진입하게 될 차량들은 한계비용이 더 적은 쪽을 택할 것이기 때문이다.6 이러한 과정은 두 도로의 한계비용 Ci 가 같아질 때까지 계속된다. 여기에서 i 는 민자 혹은 도공 구간을 나타내고, j 는 차량을, k 는 각 구간을 이루는 도로를 나타내는 하첨자이다. 따라서 두 구간의 한계비용을 일치시키는 v1 , v2 를 각각 v1N E , v2N E 라고 할 수 있다. 여기에서 상첨자 N E 는 내쉬균형을 의미한다. C1 (v1N E ) = C2 (v2N E ) 4
(3.1)
도로요금 연구에 대한 개관은 Morrison (1986)에 잘 정리되어 있다. 도로망을 제거함으로써 오히려 더 개선이 될 수 있는 경우를 Braess 역설이라고 한다. Braess et al. (2005)를 참조하라. 6 그러한 점에서 이 모델은 도로 사정에 대한 정보가 완전하게 알려져 있음을 전제한다. 5
118
본 논문에서는 VDF 함수를 이용하여 통행 시간을 추정하며, 여기에 연비와 도로 이용 요금을 포함하여 도로i 이용의 총 비용함수 Ci 를 정의했다.
Ci =
∑
[ V oTj
j
[ ( ∑ dk k
vkf
( 1 + αk
vi CAPk × lk
)βk )
]] + gj dk + Fjk
(3.2)
차종 및 운전자 특성 j 에 따라 시간가치 V oTj 가 정의되며, 구간 i 를 구성 하는 경로 k 에 대해 거리 dk , 자유속도 vkf , 도로의 설계 용량 CAPk , 차선 수 lk , 킬로미터당 유류 소모량의 화폐가격 gj , 도로 이용 요금 Fjk , 계수 αk ,
βk
를 정하게 된다. 단순화를 위해 모든 차량이 1종 승용차이며, 연비도 일정하다고 가정한다. 이 경우 j 는 구분할 필요가 없어지고, gj (vi , k) 역시 상수가 된다. 이때 C1 , C2 는 아래로 볼록 (convex) 하며, 경계해가 존재하는 경우를 제외하면 유일한 해를 가진다. 또한 위 식 3.1은 아래의 식 3.3과 같이 주어진 총 차량 흐름 V ≡ v1 +v2 에 대해서 나타낼 수 있다. C1 (v1N E ) = C2 (V − v1N E )
(3.3)
사회적 최적 총비용 사회적 효율성의 기준은 지불된 총 비용이 아니라 “총 정체비용 (시간비용+ 유류비용)” 으로 본다. 이 정의를 일반화하여 어떤 경로 i 가 도로 k 로 구성되어 있을 경우 차량 흐름 j 에 의해 유발된 정체비용은 식 3.4와 같다. 이는 기존 비용 계산식에서 도로 요금을 제외한 것이다. [ ( ] ( )βk ) ∑ dk v i C˜ij = V oTj + g j dk 1 + αk CAPk × lk vkf k
(3.4)
개별 운전자의 경우 도로 이용 요금을 포함한 총 이용비용을 고려하여 경로 선택을 하므로 사회적 최적 비용을 개별 운전자 도로 이용 비용의 총합으로 생각할 수도 있다. 그러나 이 경우 지나치게 높은 요금 수준에서 PoA가 상당히 119
넓은 범위에서 1이 된다. 모든 차량들이 상대적으로 요금이 낮은 도로를 가는 것이 개별적으로나 사회적으로 최적이 되기 때문이다. 그러한 높은 요금 수준이 공급측면에서 최적일 때에는 문제가 없을 것이나 그렇지 않은 경우엔 효율성지 표로서의 기능을 상실하게 된다. 본 연구의 목표는 현재의 가격이 공급측면에서 합리적인 가격이라고 가정하에 이 가격 근방에서 교통 흐름을 개선시킬 수 있는 가격구조를 도출하는 것이다. 따라서 가격 조절을 통해 도달하고자 하는 사회적 최적 통행 배분의 기준으로 총 정체비용을 정의한 것이다. 하지만 위와 같이 효율성을 정체 비용에 기반하는 것으로 보는 경우에도 개별 차량 흐름은 개별 정체비용이 아닌 개별 지출비용 (시간비용+유류사용비 용+도로이용비용) 을 최소화하는 선택을 한다. 즉, 개별 운전자의 경로 선택은 달라지지 않는다. 달라지는 것은 운전자의 개별 선택을 사회적으로 평가하는 기준이다. 사회적 최적 수준은 사회적 총비용을 극소화하는 v1SO , v2SO 조합을 찾는 문 제로 정리할 수 있다. 상첨자 SO 는 사회적 최적 (Social Optimum) 을 의미한다. 이를 주어진 총 차량 흐름 V 에 대해서 정식화하면 아래와 같다. min v1 C˜1 + v2 C˜2
v1 ,v2
s.t. v1 + v2 = V
(3.5)
위 극소화 문제를 V, v1 에 대해서 풀면 아래와 같은 결과를 얻을 수 있다. 동질적 가정하에서 2계조건은 언제나 아래로 볼록 (convex) 한 VDF의 특성상 언제나 양이 됨을 쉽게 확인할 수 있다. 따라서 1계 조건을 충족하는 {viSO } 는 모두 비용을 극소화하며, 그러한 점이 존재할 경우 반드시 유일하다. 단, 존재 하지 않는 경우에는 경계해 (boundary solution) 를 가지며, 그때의 경우 모든 차량 흐름이 한 차선을 선택한다. 즉, v1 , v2 중 한 해가 0이다. C˜1 (v1SO ) + v1SO C˜1′ (v1SO ) = C˜2 (V − v1SO ) + (V − v1SO )C˜2′ (V − v1SO ) PoA는 식 3.7과 같이 정의된다. 이 값은 정의상 최소값이 1이다. ∑ NE ˜ NE v Ci (vi ) ≥1 P oA = ∑i iSO ˜ SO i vi Ci (vi ) 120
(3.6)
(3.7)
동질적 차량 구성에 따른 이론적 PoA산출 차종은 승용차, 재차 인원 1명, 통행 목적은 업무 통행, km당 유류 비용 (g) 은 실제 주행속도와 무관하게 60km/h 기준을 가정하였다. 통제 변수는 민자 고속 도로 요금을 사용했다. 앞에서 정의된 비용함수에 따라 고속도로 이용 비용을 계산하가 위해서는 몇 가지 파라미터가 정의되어야 한다.
Ci = V oT
[ ( ∑ dk k
vkf
( 1 + αk
vi CAPk × lk
)βk )
] + g · dk + F k
(3.8)
운전자 특성에 따른 시간가치 V oT 는 한국개발연구원 (2008)에 제시된 방법 을 토대로, 산업별고용구조조사 (2008) 의 원자료와 기업경영분석 (2008) 을 사용 하여 승용차운전자, 버스 운전자, 트럭 운전자의 시간가치를 구하였다. 결과는 표 3.9과 같다. 구분 월급여 월근로시간 시간당 임금 오버헤드 비율 업무통행 시간가치 비업무통행 시간가치
승용차 운전자 2,864,563원 185.62시 15,432원 26.84% 19,574원 6,401원
버스 운전자 1,795,000원 303시 5,927원 29.36% 7,667원 1,250원
트럭 운전자 2,191,000원 291시 7,537원 24.60% 9,391원 1,880원
Table 3.9: 업무통행 시간가치 (2008년 기준)
VDF함수를 구성하는 고속도로 구간을 구성하는 도로경로 k 의 자유속도, 차로 설계 용량, VDF 계수는 (한국개발연구원, 2008, p. 152)의 추정값을 사용 하였다. 킬로미터당 유류 소모비용을 계산하기 위해서는 연비와 유류 가격이 필요 하다. 연비는 한국개발연구원 (2008)에서 제시된 차종별 및 속도별 연비를 사 용하였으며, 유류 가격은 통계청의 제품별 주유소 가격 조사 자료를 활용했다. 고속도로 이용 요금은 아래와 같이 계산하였다. 승용차 기준 천안–논산 구 간의 요금 체계는 표 3.12와 같다. 121
lk 2 >3
vkf 117 119
αk 0.645 0.601
βk 2.047 2.378
CAPk 2,200 2,200
Table 3.10: VDF 함수의 도로 차선 수 (lk ) 에 따른 계수값 구분 무연보통휘발유 경유 단위 : 원/l
평균 1,694.88 1,614.72
최소 1,922.59 1,919.23
최대 1,328.50 1,303.12
Table 3.11: 2008년 제품별 주유소 가격 구간 천안–논산
구분 도공 민자
기본료 862원 0원
km당 요금 40.5원 104.35원
거리 110.65km 79.54km
비고 천안–풍세 (8.12km:400원)
Table 3.12: 천안–논산 고속도로 요금체계 (승용차 기준)
따라서, 두 도로의 요금 차이를 x% 의 비율로 조정한다면 1종 차량을 기준 으로 한 민자 고속도로의 요금은 다음과 같이 쓸 수 있다.
x/100(400 + 104.35 × 79.54) + (1 − x/100){862 + 40.5(79.54 + 8.12)} 승용차 (1종) 기준, 대구–부산 (양산) 구간의 요금 체계는 표 3.13과 같다. 구간 대구–양산
구분 도공 민자
기본료 862원 0원
km당 요금 40.5원 (수정?) 112원
거리 104.1km 79.45km
비고 도공 구간 10.8km 포함
Table 3.13: 대구–부산 (양산) 고속도로 요금체계 (승용차 기준) 따라서, 두 도로의 요금 차이를 x% 의 비율로 조정한 1종 차량의 대구–부산 민자 고속도로의 요금은 다음과 같이 쓸 수 있다. 122
x/100(1400 + 115.8 × 79.45) + (1 − x/100){862 + 40.5(79.45 + 10.8)} 통행요금의 차이를 0%7 부터 300% 까지 변화시키며 PoA 변화를 관찰했다. 그림 3.3은 천안–논산 구간의 PoA 변화를 보여준다. 이 그림에서 나타나는 바와 같이, 가격 정책에 따라 PoA의 최대치를 최소화할 수 있는 수준이 존재하며, 이는 160%로 확인된다. 즉, 현재 가격에서 PoA의 최대치를 최소화할 수 있는 가격 정책은 현재 민자 고속도로 가격의 요율을 8,700원에서 11,280원으로 인 상하는 것이다. CNbusiness
1.09 1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1 30000 25000 20000
250 15000
200 160
10000 100
5000 PCU
50 0
0 Relative Private Price(%)
Figure 3.3: 천안 논산 고속도로의 차량흐름, 가격정책별 PoA 분포 이러한 식으로 시간가치와 고속도로 상태에 따라 PoA를 최소화하는 가격 정책과 그 때의 PoA, 총 차량 흐름 V 를 정리하면 표 3.14와 같다. 이와 같은 단순화된 가정 하에서 천안–논산 고속도로의 가격은 다소 낮고, 대구–부산 고 속도로의 가격은 다소 높다는 결론이 나온다. 7
0%는 민자 구간의 단위 거리당 통행료가 도공 구간의 통행료와 동일함을, 100%는 현행 통행료 수준과 동일한 수준을 의미한다.
123
고속도로 CN DB
업무여부
최적 가격정책 (민자)
현행 PoA
최저 PoA
업무 비업무 업무 비업무
160% 160% 80% 60%
1.026 1.028 1.008 1.024
1.002 1.000 1.001 1.000
최저 PoA일 때의 총통 행량 600 500 400 400
Table 3.14: 동질성 가정하에서 수학적으로 도출된 민자도로 최적 가격정책
3.4
이질성 가정하에서 PoA 산출 : 시뮬레이션
본 절에서는 3.3절의 방법을 확장하여, 차종, 속도별 연비, 재차인원의 업무통행 여부 등의 이질성을 반영한 시뮬레이션을 실시하였다. 앞에서와 마찬가지로 요금체계의 변화를 통한 PoA의 변화를 통해, 현행 요금체계의 적절성을 재검 토하였다.
기본 통행량 산정 시뮬레이션에 앞서 고려해야 할 점 중 하나는, 모든 구간에 분석대상을 출도착지 를 기반으로 하지 않는 차량 흐름이 존재한다는 것이다. 천안–논산 고속도로의 예를 들면, 천안–논산 구간의 도로공사 구간을 이용하는 차량 중에는 천안과 논산을 경유해야만 하는 차량만 있는 것이 아니다. 예를 들어, 그림 3.1을 보면, 대전을 출발하여 회덕 분기점에서 도공 고속도로에 진입하여 논산 분기점으로 나가는 차량은 실제 도로의 용량을 사용하지만 반드시 도공 구간만을 사용해야 하기 때문에 민자 구간과 도공 구간 사이의 선택 문제에 직면하지 않는다. 마찬 가지로 민자 구간만을 사용해야 하는 차량들도 존재한다. 하지만, 이러한 차량들 역시 도로의 용량을 사용하여 정체를 유발하므로, 해당 구간을 이용하는 차량 수를 정해주어야 한다. 민자 고속도로의 경우, 민자 고속도로 구간을 도착지로 하는 차량이 파악되므로, 이 차량 수를 해당 구간에 상시 존재하는 차량으로 보았다. 도공 고속도로의 경우, 분석 대상이 되는 구간 직전의 교통량과 분석 대상 구간의 교통량의 차이와 구간내 출도착 차량의 합으 로 추정하였다. 예를 들어, 천안JC–남이JC의 교통량은 직전 구간인 천안-천안 JC의 교통량과 천안JC–논산JC 교통량의 합에서 천안JC–남이JC 교통량을 뺀 것으로 보았다. 124
구간별 교통량은 도로공사의 구간별 교통량 2008년 자료를 사용하였다. 이는 년간 자료로서 시간 당 교통량으로 전환하기 위해, 앞에서와 마찬가지로 1일 6 시간으로 환산하여 계산하였다. 각 구간에 대한 교통량은 표 3.15와 같다. 천안–논산
분류 도공
대구–부산
민자 도공 민자
구간 천안–남이 JCT 남이 JCT–회덕 JCT 회덕 JCT–논산 천안–논산 동대구 JCT–언양 JCT 언양 JCT–양산 JCT 동대구 JCT–김해부산 김해부산–대동 JCT 대동 JCT–양산 JCT
추정 상시 교통량 (대/시간) 1,392 3,385 271 575 625 236 1,257 0 3,952
Table 3.15: 구간별 추정 상시 교통량 표 3.16과 같이 차종, 재차 인원, 통행 목적, 속도별 연비를 다양화하여 시뮬 레이션을 하였다. 천안–논산, 대구–부산 도로망과 같이 단 두 개의 경로만 존재 하는 단순한 경우에도 차량의 이질성 추가만으로도 계산 시간이 크게 증가하여 차량 흐름을 차량 분포에 따라 임의 배정하고 단위 흐름의 크기를 늘릴 수 밖에 없었다.8 다만, 이 경우에는 최적 통행 흐름을 찾아내는 방법이 알려져 있으므로, 더 빠른 연산장치를 다수 동원한다면 더 높은 차량 흐름에 대해서 더 자세한 분석이 가능하다. 또한 각각의 사회적 최적 흐름은 각 차량 흐름에 대해 독립적으로 계산하므로 관찰하고자 하는 차량 흐름의 범위와 요금대의 범위를 좁히면 좀 더 정확한 최적 요금율을 산출할 수 있다.9 8
2경로 n pcu에서 사회적 최적 흐름 배분을 산출하기 위해 고려해야 하는 경우의 수는 아래의 식과 같다. ( ) n ∑ i = O(n · n!) n i 9
차량흐름은 PCU(Passenger Car Unit) 로 정의되며, 시간별 차량 흐름과 차량 종류별 평균 재차인원의 곱으로 표현된다.
125
총 차량 흐름은 40,000대가 되도록 하였다. 임의 차량 배분으로 인해 발생 하는 특수한 결과를 배제하기 위해서 차량 흐름에 이질성이 존재하는 경우 10 회 반복하여 산출된 평균치를 사용하였다. 번호 1 2 3 4 5
개요 Homo Ot OtBt CtOt CtOtBt
속도별 유류비용 감안여부
차량 이질성
상업통행 이질성
F T T T T
F F F T T
F F T F T
최대차량 흐름
400 400 100 50 40
흐름 단위
100 100 400 800 1000
반복회수
1 1 10 10 10
Table 3.16: 시뮬레이션 개관 (F는 False, T는 True를 의미)
차량 이질성과 PoA형태의 변화 앞서 3.3절에서 산출한 뾰족한 산 모양의 이론적 PoA는 시뮬레이션상에서 더 이상 명료하게 나타나지 않게 된다. 그 이유는 두 가지로 정리할 수 있다. 첫째, 유류 사용량이 속도에 대해 U자형으로 나타나게 되면서 더이상 비용함수가 아 래로 볼록 (convex) 하지 않게 된다. 둘째, 이질성이 증가함에 따라 더이상 높은 차량 흐름의 경우에 PoA가 1로 수렴하지 않게 된다. 그림 3.4에서 확인할 수 있는 사실은, 유류비용이 주행 속도에 대해서 U 자형이 될 경우 형태가 약간 달라진다는 것이다. 후반부에 매끈하게 1로 수렴 하는 시뮬레이션 1(CN1, DB1) 과 달리 U자형 유류비용을 감안한 시뮬레이션 2에서는 1로 수렴하지만 높은 pcu에서 수렴하던 POA가 1을 벗어낫다가 다시 1에 수렴하는 경우들이 관찰된다.(CN2, DB2) 차량의 이질성은 통행목적과 차종에 따라 총 18가지로 분류된다.10 이 두 종류의 이질성 유무에 따라 이질성이 높아지는 순서대로 표 3.16에 정리하였다. 10
차량은 승용차 (1종), 소형버스 (2종), 중형버스 (3종), 대형버스 (4종), 트럭 (크기에 따라 각각 1-5종) 의 총 9종으로 분류되며, 통행목적이 업무/비업무에 따라 다시 2종으로 나뉜다. 이러한 분류방식은 (한국개발연구원, 2008, p.178)를 따랐다.
126
CN1
DB1
1.08
1.08
1.06
1.06
1.04
1.04
1.02
1.02
1
1 100 5600 11100 16600 22100 27600 33100 38600
P0 P300
P50
P100
P150
100 5600 11100 16600 22100 27600 33100 38600
P200
P250
P0 P300
P50
P100
CN2 1.2
1.2
1.15
1.15
1.1
1.1
1.05
1.05
1
P50
P100
P150
P250
P0 P300
P50
P100
1.2
1.2
1.15
1.1
1.1
1.05
1.05
1
P200
P250
1 400
6000 11600 17200 22800 28400 34000 39600
P50
P100
P150
P200
400
P250
P0 P300
6000 11600 17200 22800 28400 34000 39600
P50
P100
CN4
P150
P200
P250
DB4
1.2
1.2
1.15
1.15
1.1
1.1
1.05
1.05
1
1 800
6400 12000 17600 23200 28800 34400 40000
P50
P100
P150
P200
800
P250
P0 P300
6400 12000 17600 23200 28800 34400 40000
P50
P100
CN5 1.3
1.3
1.225
1.15
1.15
1.075
1.075
1 1000
1 1000
7000 13000 19000 25000 31000 37000
P50
P100
P150
P150
P200
P250
DB5
1.225
P0 P300
P150 DB3
1.15
P0 P300
P250
100 5600 11100 16600 22100 27600 33100 38600
P200
CN3
P0 P300
P200
1 100 5600 11100 16600 22100 27600 33100 38600
P0 P300
P150 DB2
P200
P250
P0 P300
7000 13000 19000 25000 31000 37000
P50
P100
P150
P200
P250
Figure 3.4: 이질성 정도에 따른 시뮬레이션 결과 개관 (가로축은 pcu, 세로축은 PoA, P0–P300은 각각 3.A소절의 방식으로 설정한 민자 대비 도공 고속도로의 통행요금 비율, CN은 천안–논산, DB는 대구–부산, 뒤의 1–5는 시뮬레이션 1–5 를 의미하며, 각 설정은 표 3.16에 상술)
127
PoA 곡선의 특성 그림 3.4에서 관찰할 수 있는 PoA 곡선의 일반적 특징은 아래와 같다. 특징1) 약 1–10,000pcu 사이에 PoA가 높아졌다 낮아지는 지점 (초기 정점) 이 존재한다. 특징2) 이질성이 존재할 경우 PoA는 1보다 높은 값에 수렴하며, 이질성이 높을 수록 높은 pcu에서 수렴한다. 특징3) 약 1–10,000pcu 사이에 상대적 민자 고속도로의 통행요금에 따른 PoA 의 편차가 가장 심하다. 특징4) pcu가 높아질수록 가격에 따른 PoA의 편차가 사라진다. 특징5) 초기 정점의 높이와 그 정점에 해당하는 pcu는 양의 관계인 경향이 있 다. 이 특징들은 천안–논산 고속도로가 더 강하게 관찰되는 반면, 대구–부산 고속도로에서는 상대적으로 약하게 관찰된다. 그 이유는 대구–부산 고속도로 의 도로망 구조가 더 복잡하기 때문이다. 또한, 위에 열거한 특징들 중 두번째 특징은 CN5, DB5에서 성립하지 않는 것처럼 보인다. 하지만 그림 3.5에서와 같이 극단적으로 높은 pcu상에서는 수렴하는 추세를 관찰할 수 있다.
Figure 3.5: CN5,DB5를 30×4000 pcu 으로 1회 시뮬레이션했을 때의 PoA 그 래프
128
이질적 정도와 관계 없이, 매우 높은 pcu 수준에서는, 유일한 통제변수인 민 자 고속도로의 통행요금과 관계없이 PoA 곡선들이 유사한 수준에 도달한다. 그 이유는 차량 흐름이 클 수록 시간비용이 다른 비용요소들을 압도하기 때문이다 (특성4). 따라서 차량 플로우가 상대적으로 높은 상황에서는 통제 변수를 통한 조정이 의미가 없다. 구간별 PoA 분석 위 현황은 민자 구간에 국한된 것이므로 도공 구간의 더 높은 도로 수용능력을 감안하면 정체되지 않을 때 대략 4,000–7,000 pcu, 정체될 때 25,000–40,000pcu 정도가 될 것이다. 하지만 앞서 살펴보았듯, 정체될 때에는 어떤 가격대를 설정 하더라도 그러한 설정으로 인한 PoA의 편차는 크지 않다.(특징4) 따라서 후반의 수렴부분은 배제하고 통행 요금별 PoA 편차가 큰 0–15,000pcu 구간에 집중할 필요가 있다. 이 pcu구간의 천안–논산 PoA는 아래 그림 3.6과 같이 크게 세 부류로 나눌 수 있다. 1000 PoA 1.15 1.1 1.05 5 0 5000 10000 15000 pcu P0 P50 P100 P150 P200 P250 P300 CN5-50x300 .05
1
1.05
PoA
1.1
1.15
CN5-50x300
0
5000
pcu P0 P100 P200 P300
10000
15000
P50 P150 P250
Figure 3.6: 천안–논산간 고속도로의 0–15,000pcu 구간에서의 PoA (10회 계산 후 평균값)
129
첫째, 작은 점선으로 표현된 0%, 즉 민자구간의 도로요금이 도로공사의 요 금율로 계산된 경우, 다른 모든 경우들에 비해 초반의 pcu는 낮은 수준이지만 약 5,500–12000pcu 사이에서 매우 높다.11 둘째, 굵은 점선으로 표현된 50%, 100%(현재의 가격배분방식), 150%에 해당되는 경우로 3,000–5,500pcu 사이 에서 대체로 높은 편이며, 초기 극점의 높이는 P0의 경우보다 낮지만 11,500– 12,500pcu 사이에서 P0의 PoA보다 다시 높아진다. 셋째, 실선으로 표시된 200% 이상에 해당되는 경우로, 초반에는 1에 가까운 pcu를 유지하다가 후반에도 매 우 낮은 PoA를 유지하는 경우다. 그림 3.7은 같은 그래프의 평균 pcu 근방인 500–3,000pcu 구간에 국한하여 확대한 것이다. 1.005 PoA 1.025 1.02 1.015 1.01 1.005 500 1000 1500 2000 2500 3000 pcu P0 P50 P100 P150 P200 P250 P300 CN5-50x300: 500~3000pcu
1
PoA 1.005 1.01 1.015 1.02 1.025
CN5-50x300: 500~3000pcu
500
1000
1500
pcu
2000
P0 P100 P200 P300
2500
3000
P50 P150 P250
Figure 3.7: 천안–논산 구간 고속도로의 500–3,000pcu 구간에서의 PoA (10회 계산 후 평균값) 여기에서는 크게 두 경우로 나뉜다. 첫째, 상대적으로 1에서 먼 PoA로, 150%, 200% 가격이 이에 해당된다. 둘째, 1에 매우 가까운 PoA로, 나머지 전 부가 이에 해당된다. 11
90% 유의수준을 감안해도 최고점에서는 더 높다.
130
위 결과를 종합하면 천안–논산 구간의 경우 250% 이상으로 도로요금을 확대하는 것이 효율적이라고 생각할 수 있다. 그림3.8을 보면, 낮은 pcu에서는 150%와 200%로 조정된 요금체계를 제외한 다른 모든 요금체계가 상대적으로 효율적임에 반해 3,000pcu 이상에서는 200% 이상의 높은 가격 차가 유지되는 요금체계가 더 효율적이기 때문이다. 대구–부산 구간의 0–15,000pcu에서는 150%–250%요금이 모든 pcu 사이에 서 효율적인 것으로 나타난다. 300%의 경우에는 7,000pcu까지는 150%–200% 요금과 유사한 패턴을 보이다 그 이후에 상대적으로 PoA가 상승하여 150% 이하인 경우의 PoA와 비슷해지다 후반에는 다시 150%–200%요금일 때의 PoA 와 비슷해진다.(그림3.8) 1000 PoA 1.15 1.1 1.05 5 0 5000 10000 15000 pcu P0 P50 P100 P150 P200 P250 P300 DB5-50x300x10 .05
1
1.05
PoA
1.1
1.15
DB5-50x300x10
0
5000
pcu P0 P100 P200 P300
10000
15000
P50 P150 P250
Figure 3.8: 대구–부산 구간 고속도로의 0–15,000pcu 구간에서의 PoA (10회 계산후 평균값) 이러한 성격은 차량 흐름의 이질성이 없을 경우에도, 이론적으로 고찰하는 경우에도 명백하게 나타난다. 3.3절에서 분석한 계산 결과를 보면 다소 차이는 있지만 PoA가 1을 유지하다 1에서 괴리되고, 다시 1로 돌아오는 패턴이 공통적 으로 나타나는데, 표 3.14에 제시했던 가장 낮은 최대치를 나타내는 PoA는 가장 131
낮은 pcu에서 1로부터 유리된다. 또한, 그림 3.9에서 관찰할 수 있는 점은 그러 한 특성을 나타내는 도로요금대 (천안–논산 업무통행 기준 160%) 를 중심으로 매우 낮은 가격과 매우 높은 가격이 나타내는 PoA 패턴이 유사하다는 것이다. 긴 점선으로 나타난 0%, 300%요금수준의 패턴과 짧은 점선으로 나타낸 100%, 200% 요금수준의 패턴이 비슷함을 확인할 수 있다. 이러한 패턴은 이질성이 높아짐에 따라 높은 pcu에서는 사라지게 되지만, 낮은 pcu에서는 지속적으로 관찰된다. PoA 1.08 1.06 1.04 1.02 1 0 10000 20000 30000 pcu P0 P100 P160 P200 P300 CN_business .02 0000
1
1.02
PoA 1.04
1.06
1.08
CN_business
0
10000
pcu P0 P160 P300
20000
30000
P100 P200
Figure 3.9: 이론적으로 계산한 천안–논산 구간의 PoA (그림 3.3 재구성)
정체비용을 기준으로 PoA를 재계산할 경우 0–5,000pcu 구간에서 처음부 터 1보다 높은 PoA로 시작하는 요금율이 존재함을 확인할 수 있다. 하지만 5,000pcu 이상에서는 기존 총비용 기준 PoA(그림 3.4) 와 유사한 패턴을 보인 다. CN5, DB5의 0–15,000pcu 범위를 나타낸 것이 그림 3.11과 그림 3.12이다. 천안– 논산의 경우, 수정된 PoA 곡선은 크게 두 부류로 나뉘어 4,000– 5,000pcu 근방에서 역전되는 특성을 보이게 된다. 0%–150% 요금체계에서 PoA
132
CN1
DB1
1.3
1.4
1.225
1.3
1.15
1.2
1.075
1.1
1
1 100 5600 11100 16600 22100 27600 33100 38600
P0 P300
P50
P100
P150
P200
100 5400 107001600021300266003190037200
P250
P0 P300
P50
P100
CN2 1.4
1.6
1.3
1.45
1.2
1.3
1.1
1.15
1
P200
P250
1 100 5400 107001600021300266003190037200
P0 P300
P50
P100
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100 5300 10500 15700 20900 26100 31300 36500
P250
P0 P300
P50
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CN3
P150
P200
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DB3
1.4
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1.3
1.3
1.2
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1
1 400
P0 P300
6000 11600 17200 22800 28400 34000 39600
P50
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P0 P300
6000 11600 17200 22800 28400 34000 39600
P50
P100
CN4
P150
P200
P250
DB4
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1
1 800
P0 P300
6400 12000 17600 23200 28800 34400 40000
P50
P100
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P200
800
P250
P0 P300
6400 12000 17600 23200 28800 34400 40000
P50
P100
CN5 1.3
1.225
1.225
1.15
1.15
1.075
1.075
1 1000
1 1000
7000 13000 19000 25000 31000 37000
P50
P100
P150
P150
P200
P250
DB5
1.3
P0 P300
P150 DB2
P200
P250
P0 P300
7000 13000 19000 25000 31000 37000
P50
P100
P150
P200
P250
Figure 3.10: 수정된 PoA로 시뮬레이션 1–5 재계산한 결과 개관
133
1000 PoA_Alt 1.25 1.2 1.15 1.1 1.05 5 0 5000 10000 15000 pcu P0 P50 P100 P150 P200 P250 P300 CN5_Alt_50x300 .05
1
PoA_Alt 1.05 1.1 1.15 1.2 1.25
CN5_Alt_50x300
0
5000
pcu P0 P100 P200 P300
10000
15000
P50 P150 P250
Figure 3.11: CN5: 수정된 PoA로 재계산 (10회 계산 후 평균값)
는 4,000–5,000pcu 이하의 pcu에서 낮은 특성을, 이상의 pcu에서 높은 특성을 보이며, 200%–300% 요금체계에서 PoA는 낮은 pcu에서 높은 특성을, 높은 pcu 에서 낮은 특성을 보인다. 이는 총 정체 비용을 효율성의 기준으로 볼 때 초반에 우회로를 강제하는 경우가 비효율적이라는 것을 의미한다. 이러한 정체 비용을 최소화하는 기준에서 최적의 정책은 고정 가격일 경우 현행 가격의 150% 정도 수준이고, 통행요금을 때에 따라 수정할 수 있다면 평소에는 150% 정도를 유지 하다 정체시에는 250% 정도로 요금을 올리는 것이 적절한 정책이라는 해석을 할 수 있다. 대구–부산의 경우는 수정된 PoA를 감안할 경우 총비용 기준 PoA와 완전히 다른 결론에 도달하게 된다. 10,000pcu까지의 모든 구간에서 50% 요금체계가 약우월하며 10,000–15,000pcu에서는 200% 요금체계의 PoA가 가장 낮지만 다 른 요금대의 PoA와 큰 차이가 나지 않는다. 따라서 정체 비용을 감안했을 때의 대구–부산 고속도로의 경우는 현행 요금보다 도공 대비 50% 수준으로 낮추었을 때 차량을 더 원활하게 통행시킬 수 있다는 해석을 할 수 있다.
134
1000 PoA_Alt 1.3 1.2 1.1 5 0 5000 10000 15000 pcu P0 P50 P100 P150 P200 P250 P300 DB5_Alt_50x300 .1
1
PoA_Alt 1.1 1.2
1.3
DB5_Alt_50x300
0
5000
10000
pcu P0 P100 P200 P300
15000
P50 P150 P250
Figure 3.12: DB5: 수정된 PoA로 재계산 (10회 계산 후 평균값)
3.5
결론
앞서 분석한 천안–논산 고속도로 구간과 대구–부산 고속도로 구간의 PoA분석 결과 도출된 민자구간의 최적 도로요금을 표 3.17에 정리하였다.12 도로구간
효율성 평가기준
현행 요금율
현행 요금
천안논산 천안논산 대구부산 대구부산
총비용 총정체비용 총비용 총정체비용
100% 100% 100% 100%
8,700 8,700 10,600 10,600
최적 요금율
최적 요금
약 250% 약 150% 약 200% 약 50%
15,200 10,900 16,300 7,800
Table 3.17: 평가기준별 민자 고속도로 최적요금 (1종 기준)
민자 고속도로의 최적 가격 결정이라는 문제에서 기존의 연구방식 (한계 편익 분석) 에 비해 PoA 분석이 가질 수 있는 가장 큰 장점은 현실에 가까운 12
하지만, 이러한 요금은 총 비용이던 총 정체비용이던 사회적으로 바람직한 배분을 유도할수 있는 요금수준을 의미하는 것이다. 민자 고속도로의 요금율은 민자 고속도로 운영회사에 이득이 없는한 쉽게 관철할 수 있는 문제가 아니다. 시장에서의 실현 가능성까지 고려하기 위해서는 공 급측의 편익과 비용도 감안해야 한다. 그렇게 감안한 PoA 역시 정의하고 산출할 수 있을 것이다. 다만 그러할 경우 현재 통제 변수로 두고 있는 민자 고속도로의 통행요금이 모델 내에서 결정될 것이므로 정책적 함의보다는 시장에서 결정되는 가격에 의해 달성되는 도로 배분의 효율성을 평가한다는 의미가 될 것이다.
135
이질적 상황에서는 최적 요금수준이 차량 흐름에 따라 달라지기 때문에 pcu에 대한 효율성 지표인 PoA가 더 현실적인 결론을 도출할 수 있다는 것이다. 특히 도로망의 특성에 따라 이러한 불규칙성은 다양한 양상을 보이므로 복잡도가 증가할 수록 PoA분석의 잇점은 더 중요해진다. 다만, 도로망의 구조가 복잡할 경우, 사회적 최적 통행 흐름를 찾아내는 것은 매우 어려운 문제다. 복잡한 대도시의 도로망을 대상으로 한 Youn et al. (2008)의 경우도, 이 문제로 인해 단일 출도착점을 기준으로 PoA를 산출했다.13 다중 출도착을 고려할 경우 사회적 최적 통행 흐름는 차량이 단일한 경우조차 근사값을 찾기도 어렵다. 여기에 차량의 이질성까지 고려한다면 사회적 최적 균형의 도출 문제는 더욱 어려워진다. 또한, 본 분석에 사용한 평가 함수가 시간가치와 유류소모량, 그리고 도로 사용료라는 수요측 단기 비용만으로 이루어져 있다는 문제도 있다. 본 연구는 단기 분석으로, 현행 도로의 구조를 주어진 것으로 보고 (1) 동일 출도착지를 가려 하는 차량들이 두 경로를 선택할 수 있고 (2) 두 경로에 상대적으로 다른 요금을 부과할 수 있을 때 사회적으로 최적인 통행 배분에 가까운 상태를 유도할 수 있는 요금을 최적요금으로 보고 그러한 요금을 부과할 경우 더 나은 결과를 얻을 수 있음을 보이려 한 것이다. 본 논문의 방법을 장기로 확장하기 위해서는 비용 측면에 중장기에는 도로유지보수비용 (혹은 도로의 감가상각비) 을 반영 하고 장기에는 도로 건설비용도 고려에 넣어야 할 것이다. 이 부분의 연구는 향후 과제로 남긴다. 또한 실제 정체상황에서는 국도의 선택도 고려된다는 점을 생각하면 도로망 구조에 국도 네트워크를 고려하는 것도 가능할 것이다. 또 한가지 언급해야 할 것은 본 분석에서 사용한 상당 부분의 파라미터가 상대적으로 오래된 한국개발연구원 (2008)에 의존하고 있다는 것이다. 동일 연구의 2008년판이 나와 있으며, 추후 이를 반영한 계획이다.
13
또한 이 논문에서 사회적 최적 차량 흐름를 도출하기 위해 사용한 standard convex minimum cost flow algorithm이 고려하는 네트워크의 총비용 합산 방식은 동 논문에서 PoA 발생의 예로 들고 있는 피구 네트워크에서 제기되는 네트워크 총비용 합산 방식과 계산 방식이 다르다고 보인다.
136
3.A
부록 : 통행량의 추정
PoA 분석은 운전자가 경로 선택의 상황에 직면하고 있음을 전제한다. 그런데, 도로공사 구간의 경우 천안–논산 구간은 대전을 중심으로, 대구–부산은 울산을 중심으로 도로공사 구간 내에서만 이동해야 하는 교통량이 존재할 것이다. 또한, 민자 고속도로만 접근이 가능한 지역에 가고자 하는 운전자도 존재할 것이다. 이러한 차량은 도공 또는 민자 고속도로 중 어느 도로를 이용할 것인지 선택의 문제에 직면하지 않고 있다. 따라서, 이 차량은 도공과 민자 고속도로 중 어떤 도로를 사용할 것인지 선택할 필요가 없는, 해당 노선에 항시적으로 존재하는 교통량으로 생각해야 한다. 이러한 교통량을 확인하기 위해, 구간별 평일, 주말 차종별 교통량 자료와 출도착지별 년간 명절, 차종별 교통량 자료를 사용하였다. 분석 대상이 되는 구간 직전의 교통량과 분석 대상 구간의 교통량의 차이로서 추정하였다. 예를 들어, 천안JC–남이JC의 순 교통량은 직전 구간인 천안-천안 JC의 교통량과 천안JC–논산JC 교통량의 합에서 천안JC–남이JC 교통량을 뺀 것으로 생각할 수 있다. 구간별 교통량은 도로공사의 구간별 교통량 자료 2008 년을 사용하였다. 따라서 민자 고속도로의 가격비율을 도로공사의 요율로 책정한 경우를 0% 라고 하고, 현행 민자 고속도로의 요율로 책정한 경우를 100% 라고 하면 요율별 주어진 V 에 대한 {viN E , viSO } 를 구할 수 있다. 천안–논산간 고속도로의 가격 비율은 표 3.18과 같다.
현행 (2009): 요율 100% 도로공사요율로 환산 : 요율 0%
민자 구간 (F1 ) 8,700 4,400
도로공사 구간 (F2 ) 5,800 5,800
Table 3.18: 천안–논산 간 고속도로의 구간별 요금과 동일 요율 적용시 도로요금
137
3.B
Bibliography
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138
3.C
영문초록
Cheonan–Nonsan(CN) and Daegu–Busan(DB) expressway sections have peculiar properties: they have different routes with same origin and destination. One route has long distance, low price, high capacity, and is managed by Korea Express Public Corporation and the other route has short distance, high price, low capacity, and is managed by Private Corporation. In this case, individual route choice to minimize traffic cost can be separated from social optimal traffic distribution. But when the road price can be set to different level, there exist some price level to achieve social optimal traffic distribution. This paper tries to find this optimal price by calculating “Price of Anarchy(PoA)” for these sections. Social cost was defined to total cost caused by congestion. Based on this social cost, the optimal price set is calculated by computer simulation. Given the value of time and volume density function, by raising road price of CN section and lowering that of DB section, total congestion cost can be minimized. Keywords: Price of Anarchy, Private Expressway, Road Pricing, Congestion Cost JEL Classification Numbers: R41, R48
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