Considering Equilibrium Accessibility in Conflict Resolution over Demolition Disputes Qin Zhongfu; Qiu Qiongjie; Wen Haizhen* Institute of Construction Management Zhejiang University Hangzhou, China E-mail: *
[email protected] Abstract—An innovative methodology named Equilibrium Accessibility Analysis based on modified Incentive Algorithm is presented for resolving two-decision maker demolition disputes. Incentive Algorithm is modified by taking into account the irreversible moves and pathdependent transitions. Thus, reachable is identified at the modeling stage, and the path generation step at analysis stage is modified. The major contribution of this research is the Equilibrium Accessibility Analysis, which considers whether the equilibriums are accessible. The systematic procedure of Equilibrium Accessibility Analysis is put forward, and five equilibrium accessibility definitions under five cases are proposed. Thus, equilibriums can be refined reasonably and effectively. A real-life demolition dispute is used to illustrate how this new methodology can be conveniently and effectively applied in practice. Keywords-demolition dispute; modified Algorithm; Equilibrium Accessibility Analysis
I.
Incentive
INTRODUCTION
Conflicts and disputes always arise between demolition participants for many reasons, including the lack of coordination, unrealistic expectations and disagreements about the compensations. Any one of these factors can lag duration, increase costs and even lead to complicated litigation. Resolving demolition disputes, therefore, is an essential task for demolition decision makers (DMs). Game theory, proposed by Von Neumann and Morgenstern, has become an important tool to study social problems and conflicts. Especially, Metagame Theory [1], Conflict Analysis [2] and the Graph Model for Conflict Resolution (GMCR) [3] have been fully used in practice and have provided DMs with effective decision supports, as in [4]~[6].These formal tools assume that each DM never unilaterally moves to less preferred states. However, DMs in real life sometimes take strategies which are favorable in far-sights but unfavorable in short-sights. Incentive Algorithm is such a tool for conflict resolution in accordance with this phenomenon. However, Incentive Algorithm simply thinks that each DM’s unilateral move is reachable, disregarding the irreversible moves and pathdependent transitions [7], which are often present in the real-world. Furthermore, all these methodologies never consider the accessibility of equilibriums. It is important because no DM is able or willing to leave equilibriums for other states, but maybe no DM is able or willing to come to certain equilibriums from other states, either.
The goal of this research is to present an innovative methodology named Equilibrium Accessibility Analysis based on modified Incentive Algorithm for resolving 2DM demolition disputes. Incentive Algorithm is modified firstly. Irreversible moves and path-dependent transitions will bring changes in the construction of reachable list and path generation. Next, Equilibrium Accessibility Analysis is developed to analyze whether the equilibriums are accessible so as to refine them reasonably and effectively. This represents the major contribution of this research. Modified Incentive Algorithm and Equilibrium Accessibility Analysis are presented in the next two sections respectively, followed by a real-life case study. II.
MODIFIED INCENTIVE ALGORITHM
A. Incentive Algorithm Overview Incentive Algorithm is a 2-DM decision support system for conflict resolution considering that DMs take far-sight strategies. In other words, DMs unilaterally move to less preferred states temporarily for long-term interests. The basic logic and ideas of Incentive Algorithm are as follows. The stability of every feasible state should be analyzed. While analyzing, one state is assumed to be the current state. Then the focal DM and its opponent begin to move unilaterally in turns until reach a terminal state. These moves in turns constitute a path. Write down all paths from the current state and analyze. If the terminal states of all paths are all less preferred by the focal DM, the current state is stable for the focal DM; otherwise, it is unstable. In the latter case, the focal DM has incentive moves (IM) from the current state and those more preferred terminal states are called incentive points. If the current state is stable for all DMs, it is an equilibrium. The systematic procedure for applying Incentive Algorithm follows two stages: modeling and analysis. At the modeling stage, the conflict problem is structured by determining the DMs, the feasible states and each DM’s preference vector. Next, at the analysis stage, the stability of every state is measured by Incentive Algorithm. Generally, Incentive Algorithm itself includes four steps: path generation, path verification, path adjustment and determination of equilibriums. B. Modified Incentive Algorithm Irreversible moves and path-dependent transitions are first proposed in GMCR. An irreversible move occurs
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when a DM can cause a conflict to move from state k to q by a unilateral move, but cannot make the transition back from q to k [7]. Path-dependent transitions take place when there are several ways for two or more DMs to move jointly from one state to another [7]. The differences between Modified Incentive Algorithm, considering irreversible moves and path-dependent transitions, and the initial methodology are as follows. 1) Reachable List At the modeling stage, reachable list should be identified. The reachable list for a given DM and starting state is the set of states in which the particular DM has changed his or her strategy or option selection with respect to his initial strategy, while the strategies of all the other DMs have remained fixed [2]. However, some moves in the initial reachable list will become unreachable while irreversible moves and path-dependent transitions are taken into consideration. Remove these unreachable moves, just like removing the infeasible states while determining the feasible states. 2) Modified Path Generation At the analysis stage, paths including unreachable moves should be deleted while generating paths. This step can be called modified path generation. Therefore, Modified Incentive Algorithm itself includes four steps: modified path generation, path verification, path adjustment and determination of the stable states. III.
EQUILIBRIUM ACCESSIBILITY ANALYSIS
A. Systematic Procedure of Equilibrium Accessibility Analysis Formal methodologies, such as Metagame, Conflict Analysis, GMCR and Incentive Algorithm, are analytical tools used for modeling and analyzing the interactive decision problems. Equilibriums are those states stable for all DMs. In other words, no DM is able or willing to leave equilibriums for other states. However, maybe no DM is able or willing to move to certain equilibriums from other states, either. Therefore, an innovative Equilibrium Accessibility Analysis is developed to analyze this phenomenon. The systematic procedure of Equilibrium Accessibility Analysis follows two stages. 1) Equilibrium Accessibility Analysis for the focal DM Each equilibrium should be verified whether it is accessible for the focal DM. An equilibrium for the focal DM can either be accessible (A) or inaccessible (I). Equilibrium Accessibility Analysis for the focal DM is present in section III part B. 2) Equilibrium Accessibility Analysis for all DMs One equilibrium’s accessibility for a 2-DM conflict can be represented in a matrix, as shown in Table I. If the equilibrium is accessible for at least one DM, the equilibrium is an accessible equilibrium (AE); otherwise, it is an inaccessible equilibrium (IE). TABLE I.
ONE EQUILIBRIUM’S ACCESSIBILITY FOR A 2-DM CONFLICT
for DM j for DM i A I
A
I
AE AE
AE IE
B. Equilibrium Accessibility Analysis for the Focal DM Five equilibrium accessibility definitions for 2-DM conflict under five cases are shown in Table II. It should be noted that the last move of every incentive path is a UI for the DM who moves last. For simplicity, the conventional symbols and their meanings are given first: N is the set of DMs, N={i, j}; so is status quo; S, Sp, Sq, S 'p are the set of feasible states, equilibriums, states that i can move to sp by unilateral improvements (UI), and incentive points that j has incentive moves (IM) from all elements of Sq respectively; sp, sq and s 'p are the elements of Sp, Sq and
S 'p respectively. Case1: For i∈N, sp is A for DM i, if sp=s0. In case 1, the current equilibrium is status quo. Undoubtedly, sp is an accessible equilibrium for DM i. Case2: For i∈N, sp is I for DM i, if sp≠s0 and Sq=Ф. In case 2, DM i cannot move to sp by UI. Hence, sp is an inaccessible equilibrium for DM i. Case3: For i∈N, sp is A for DM i, if sp≠s0, Sq≠Ф and S 'p =Ф. In case 3, though DM i can move from any sq to sp by UI, DM j has no IMs from any sq. That is to say, DM j has no willing to move from any sq. Hence, sp is an accessible equilibrium for DM i. Case4: For i∈N, sp is A for DM i, if sp≠s0, Sq≠Ф, S 'p ≠Ф and ∃s 'p : s p i s 'p . In case 4, DM j has several IMs from Sq. However, at least one incentive point is less preferred compared with sp by i. As a conservative person, DM i prefers to move from sq to sp. Hence, sp is an accessible equilibrium for DM i. Case5: For i∈N, sp is I for DM i, if sp≠s0, Sq≠Ф, S 'p ≠Ф and ∀s 'p : s p i s 'p . In case 5, DM j has several IMs from Sq and every incentive point is more preferred compared with sp by DM i. Undoubtedly, DM i prefers to wait for DM j’s response. Hence, sp is an inaccessible equilibrium for DM i. IV.
DEMOLITION DISPUTE: REAL-LIFE CASE STUDY
A. Introduction A real-life demolition dispute, Chongqing nail house case, is used to illustrate that the Equilibrium Accessibility Analysis based on modified Incentive Algorithm can be conveniently and effectively applied in practice. During the negotiation period, from Sep. 2004 to Apr. 2007, the relocation household and the developer negotiated with each other but failed to reach a consensus. Finally, the dispute ended under the coordination in court. More details about this demolition dispute are provided by Hua [8]. B. Modeling Step 1: Identify the DMs and Their Options Two main DMs were identified: (1) Y: Mr. and Mrs. Yang, the relocation household; (2) ZR: Chongqing Zhirun real estate Co., LTD, the developer. The involved DMs and their options are listed in Table III.
TABLE II.
EQUILIBRIUM ACCESSIBILITY ANALYSIS FOR THE FOCAL DM
Case
Symbolic Description
Case 1
sp=s0
Graphic Description status sp quo sq
Sq=Ф
Case 2
sq S =Ф ' p
Case 3 sp≠s0
∃s 'p : s p
i
s 'p
∀s 'p : s p
i
s 'p
Case 4
sq
S ≠Ф ' p
Case 5
Step 3: Identify Each DM’s Preference vector Based on a given DM’s preference, the feasible states are ranked for that DM from the most preferred state on the left to the least preferred state on the right, as shown in Table V. For example, Y most prefers State 8, for which ZR provides surplus compensation and Y choose to move. Step 4: Identify Reachable Lists The reachable list is displayed in Table V. It should be noted that irreversible moves are considered when constructing the reachable lists. For example, Y can move from State 3 to State 2, but cannot move from State 2 to State 3. This means that once Y reject its option (i.e., Y choose to move), it cannot logically or practically accept it again. Path-dependent transitions for DMs should also be considered. For example, ZR cannot leave state 2 until Y accept its option. As mentioned above, Y cannot accept its option again, so ZR cannot leave state 2.
ZR
i: UI j: IM
Step 2: Determine Feasible States Each DM can either accept (1) or reject (0) any of its options and since there are 5 options in total, the total number of decision states is 25=32. However, three types of states are infeasible and can be safely deleted: (1) ZR chooses more than one “compensation” option simultaneously; (2) ZR does not take any of its options; (3) ZR takes legal action without any “compensation”. Therefore, 12 feasible states remain, as shown in Table IV.
DM Y
i: UI j: no IM
Sq≠Ф
TABLE III.
i: UI
Equilibrium Accessibility for DM i A
sp
I
sp A sp’ sp
A
sp’
I
C. Analysis Step1: Stability Analysis In this step, stability analysis is performed under the framework of modified Incentive Algorithm, using the decision support system developed by Hua [9]. The output includes the states’ stability and incentive points. Two things should be noted: (1) DM’s sight length is set as 2 by considering people’s bounded rationality [10] and for simplicity; and (2) the output should be adjusted by considering irreversible moves and path-depended transitions. The stability analysis is displayed in Table V by no shading. step2: Equilibrium Accessibility Analysis According to five definitions given in section III part B, Equilibrium Accessibility Analysis can be performed, as shown in Table V by shading. Take accessibility analysis of equilibrium 4 for Y for example, as shown in Fig.1. For Y ∈ N={Y, ZR}, sp(=4)≠s0(=3), Sq={5}≠Ф, S 'p ={8,18,20,24}≠Ф, and there exits states 18, 20 and 24 such that 4 Y 8 , 20, 24. In other words, ZR has several IMs from state 5. However, three incentive points 18, 20 and 24 are less preferred compared with 4 by Y. As a conservative person, Y prefers to move from 5 to 4. Hence, 4 is an accessible equilibrium (case 4). D. Discussion After Equilibrium Accessibility Analysis, the number of equilibriums has been refined from 6 to 4, by reducing inaccessible equilibriums 2 and 18.
DMS AND THEIR OPTIONS
Options Refuse to move (refuse). Provide unfair compensation (unfair). Provide fair compensation (fair). Provide surplus compensation (surplus). Take legal action (legal).
5
Y: UI ZR: IM
TABLE IV. DM Y
Options refuse unfair fair ZR surplus legal Decimal Number
0 1 0 0 0 2
1 1 0 0 0 3
0 0 1 0 0 4
FEASIBLE STATES 1 0 1 0 0 5
Feasible States 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 8 9 18 19
0 0 1 0 1 20
1 0 1 0 1 21
0 0 0 1 1 24
1 0 0 1 1 25
4
E4 is A for Y.
8
4 ≺Y 8
18
4
Y
18
20
4
Y
20
24
4
Y
24
Figure 1. Accessibility Analysis of Equilibrium 4 for Y
TABLE V.
STABILITY AND EQUILIBRIUM ACCESSIBILITY ANALYSIS
Y Preference Vector Reachable Lists Incentive Points Stability Equilibrium Accessibility for Y ZR Preference Vector Reachable Lists
8 r A
9 8 u
4 r A
5 4 u
24 r A
25 24 u
20 r A
21 20 u
18 r I
19 18 u
3 (2) r -
2 r I
2 -
4 -
8 -
18 -
20 -
24 -
Incentive Points
-
-
-
-
-
-
5 9 21 25 19 3 8 18 20 24
9 5 21 25 19 3 4 18 20 24
21 5 9 21 19 3 4 8 18 24
25 5 9 21 19 3 4 8 18 20
19 5 9 21 25 3 4 8 20 24
Stability Equilibrium Accessibility for ZR Stability State Equilibrium Accessible Equilibrium
r I
r I
r I
r I
r I
r I
u
u
u
u
u
3 5 9 21 25 19 4 8 18 20 24 u
2 E ×
3 ×
4 E AE
5 ×
8 E AE
9 ×
18 E ×
19 ×
20 E AE
21 ×
24 E AE
Equilibrium 2 means that ZR provides unfair compensation and Y chooses to move. Stability analysis based on Modified Incentive Algorithm implies that both Y and ZR are not able to move away from state 2. Hence, state 2 is an equilibrium. However, Equilibrium Accessibility Analysis implies that both Y and ZR cannot move to 2 from any other states by their own UI. Hence, equilibrium 2 is an inaccessible equilibrium (case 2). Equilibrium 18 means that ZR provides unfair compensation and takes legal action and Y chooses to move. Stability analysis based on Modified Incentive Algorithm implies that both Y and ZR are not able to move away from the state 2. Hence, state 18 is an equilibrium. However, Equilibrium Accessibility Analysis implies: (1) every ZR’s incentive point is more preferred compared with 18 by Y. Hence, equilibrium 18 is inaccessible for Y (case 5); (2) ZR can not move from any other states to 18 by UI. Hence, equilibrium 18 is inaccessible for ZR (case 2). Equilibrium 18 is inaccessible for both of Y and ZR. Therefore, equilibrium 18 is an inaccessible equilibrium. V.
CONCLUTIONS
An innovative methodology named Equilibrium Accessibility Analysis based on modified Incentive Algorithm for resolving demolition disputes is presented for handling negotiation in the presence of two DMs. Incentive Algorithm is modified in consideration of irreversible moves and path-dependent transitions, which bring changes in the construction of reachable list and path generation. In particular, considering that maybe no DM is able or willing to come to certain equilibrium from another state, Equilibrium Accessibility Analysis is developed to analyze whether the equilibriums are accessible. With the help of this new methodology, equilibriums can be refined reasonably and effectively and the analysis results are more useful for the DMs. In the real-life case study, the
25 ×
accessibility of six equilibriums is examined and equilibrium 2 and 18 are found as inaccessible equilibriums, and the validity of the proposed methodology is demonstrated. The Equilibrium Accessibility Analysis in section III is exclusive to 2-DM case. Real-life cases always consist of more than two DMs. Hence, further extension of this analysis tool to the n-DM Equilibrium Accessibility Analysis will be more widely applicable.
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