Can Bounded Rationality Explain Excess Capacity?∗ Chloé Le Coq†
Jon Thor Sturluson‡
November 8, 2006
Abstract Excess capacity is observed in many markets especially those where a substantial initial investment is required. The theoretical literature often explains this feature by strategic attempts to deter entry or to limit new entrants’ market shares but the empirical evidence for such a rationale is mixed. Moreover, excess capacity has also been observed in experimental studies on capacityconstrained games where there is no entry (and therefore no entry-deterrence motive). This paper explores experimentally another rationale for excess capacity: rather than (in addition to) being a threat to (potential) entrants, excess capacity held by incumbents may constitute a valuable option to reap extra gains from competition with an inexperienced entrant, if he turns out to makes a mistake. In our experimental design we used the level of experience (the number of periods played) as a proxy for the level of rationality and matched subjects with different levels of experience. We find evidence of excess capacity decreasing with opponent’s experience. ∗
This paper is a sustantially revised version of a chapter of Le Coq and Sturluson’s 2003 Stockholm School of Economics Ph.D. thesis. It was before circulated as "Does Opponent’s experience matter?". The authors would like to thank Tore Ellingsen for his insightful comments in the project’s infancy, Urs Fischbacher for allowig us to use the z-Tree software and Hans-Theo Norman for technical help. We thank also seminar participants at the IIOC 2004 (Chicago), EARIE 2003 (Lausanne), SAET 2003 (Rhodos) for helpful comments. We gratefully acknowledge financial support from the Nordic Energy Research Program. † University of California Energy Institute, 2547 Channing Way, Berkeley, CA,
[email protected], ‡ Reykjavik University, Iceland. E-mail:
[email protected].
1
Moreover on average, subjects play as if they significantly underestimates their opponent’s experience, inducing more excess capacity in the market than the theory predicts. JEL classification: C92; L13,L110, 6110. Keywords: Entry, Experiment, Oligopoly, Quantity precommitment, Rationality.
1
Introduction
In several recent experiments on capacity precommitment games, a common result is to observe excess capacity (Davis,1999, Muren, 2000, Anderhub and al., 2004). Until now this has been considered to be a puzzle, as it is not consistent with standard game-theoretical prediction. There is however widespread evidence of significant excess capacities in many real markets (e.g., Masson and Shaanan 1986 and Ghemawat and Caves 1986), especially those where a substantial initial investment is required and incumbents are large, such as the markets for chemicals (Gilbert and Lieberman, 1987) or photographic film (Kadiyali, 1996). Several explanations for excess capacity have been suggested, such as stochastic demand, strategic deterrence, coordination, forebear strategy and so forth. Still, there is no room for such motives in experimental studies on capacity-constrained game where these considerations are absent by design. This paper offers bounded rationality as an alternative explanation for excess capacities. Similarly to Baye and Morgan (2004) that show that little bounded rationality among sellers could explain price dispersion observed on the Internet and in the lab, we argue that little bounded rationality among firms can explain some excess capacity observed on many markets and in the lab. Following the growing literature on bounded rationality in industrial organization (see Ellison, 2005, for a review), the bounded rationality means here that bounded rational players cannot implement their optimal strategies with accuracy. We are in particular interested to see if the difference in bounded rationality among firms could partly explain the excess capacity generally hold by the incumbent firms. To explore this alternative explanation for excess capacities we design a market experiment where subjects simultaneously commit to capacity levels before they compete in prices. The setup is similar to those in the previous experimental studies mentioned above, except that subjects with different levels of experience played 2
against each other. We use the level of experience (the number of periods played) as a proxy for the level of rationality. The implicit assumption is that experienced players can be expected to play more rationally (making fewer mistakes) than inexperienced ones. Given the capacity-constrained experimental game considered, in the second stage, subjects’ strategy consists of a function from chosen capacities to prices. Our main hypothesis is that limited rationality has a crucial effect on firms’ price response. More precisely a bounded rational player has an imperfect price response, i.e., his price choice is less sensitive to a change in capacity, relative to a more rational one. One way to argue for this is to point out the extremes. A perfectly rational (experienced) subject will respond to a deviation in capacity according to the slope of the demand function, while a completely clueless (inexperienced) one will not respond at all. It is then plausible that subjects might expect less than complete price reactions from inexperienced opponents. Hence, a subject is more willing to expand output, because the effect of such an expansion on the market price is smaller when facing an inexperienced subject. In short, imperfect price response, leads to higher capacity than predicted by standard theory based on perfectly rational players and common knowledge. A pattern of this kind was found Lieberman (1987) in the case of chemical product industries. He noticed that incumbents increased their rate of investment when facing new entrants, but reduced it when facing incumbents. The effect was particularly noticeable in the case of concentrated industries. Since entrants are usually less familiar with the market that they enter than the incumbents it is reasonable to expect that (inexperienced) entrants act more erratic than (experienced) incumbents. We find that there is a significant difference in behavior depending on opponents’ experience; moreover we find support for our hypothesis that players facing inexperienced players tend to choose higher capacities than otherwise. Capacities are larger and prices lower, on average, when opponents are inexperienced, compared to when they are experienced. Finally the experimental results suggest that the subject’s own experience matters for the effect of opponent’s experience, suggesting a more complex interaction between players’ own experience and their opponents’ experience than first anticipated. We also develop a model of bounded rationality based on solution concept proposed by McKelvey and Palfrey (1998). Model simulations turn out to be qual3
itatively consistent with the experimental findings, as overcapacity is the general prediction as soon as little bounded rationality among firms is introduced. Interestingly an experienced player always offers excess capacity but the less so, the higher their own level of experience. The remainder of this paper is structured as follows. The next section provides a general description of the experimental design and procedures. Section 3 presents the results of the experiment which support our main hypothesis i.e. that an opponent’s experience level significantly affects behavior. Section 4 presents the equilibrium predictions of the bounded rationality model developed by McKelvey and Palfrey (1998) applied to the capacity-constrained game considered. Section 5 concludes by discussing some important implications of the results and suggestions for further research.12
2
Experimental Design and procedures
2.1
The benchmark model
The game subjects play in this experiment is a standard version of the Kreps and Scheinkman (1983) model, henceforth referred to as KS. Consider a symmetric duopoly model with linear demand. The game consists of two stages. In the first stage, player 1 and player 2 simultaneously choose their capacity levels k1 and k2 ∈ R+ , respectively. The cost of each unit of installed capacity is c. In the second stage, having learned the capacity chosen by their opponent, subjects simultaneously choose prices, p1 and p2 ∈ R+ . Production is costless but cannot exceed capacity. For a given set of prices aggregate capacity can exceed demand, in which case the efficient rationing rule applies. Player i’s payoff is given by:3 1
More precisely, the incumbents initiating competitive actions before entry occurs because of capacity commitment motivation (Dixit’s, 1979), the strategic learning-by-doing (Spence, 1978), the cost-signaling (Milgrom and Roberts, 1982), the long-term contracting environment (Aghion and Bolton, 1987), and the switching costs (Klemperer, 1987). Osborne and Pitchnik (1986) or Allen, Deneckere, Faith, and Kovenock (2000) focused on the role of excess capacity in a price competition in a capacity-constrained game. 2 Some important assumptions made by Kreps and Scheinkman have been criticized for being too restrictive, e.g., concave demand, efficient rationing and perfect information (see for example Davidson and Deneckere (1986) or Reynolds and Wilson (2000). 3 The payoff function was not described to the subjects in algebraic form, but using words. See the complete instructions in Appendix II.
4
⎧ ⎪ if pi < pj , ⎨ pi min (ki , d (pi )) − cki π i (ki , kj , pi , pj ) = pi min (ki , max (d (pi ) − kj , d (pi ) /2)) − cki if pi = pj , ⎪ ⎩ if pi > pj . pi min (ki , max (d (pi ) − kj , 0)) − cki
(1)
where i, j ∈ {1, 2} and i 6= j, and where d (pi ) = α − βpi . In our experimental setup, we choose the parameters α = 120, β = 1 and c = 30. The subgame-perfect equilibrium is unique and equal to the Cournot outcome in terms of capacities, prices and profits:4 ki∗ = 30, p∗i = 60 and π ∗i = 900
for i ∈ {1, 2} .
(2)
In comparison, a competitive market would yield an aggregate output equals to 90, prices equal to the marginal cost, 30, and zero profits. The assumption of rationality is critical for this prediction. As we show in section 4 if we allow for some level of bounded rationality the prediction is biased towards higher capacities and lower prices. The effect of bounded rationality is particularly crucial for the price response. A bounded rational player is likely to be less sensitive to capacity choice, relative to a rational player. For instance, a perfectly rational subject will respond to a deviation in capacity according to the slope of the demand function, while a bounded rational player is more likely to ignore the effects of capacity choice on his payoffs. It is then plausible that subjects might expect less than complete price reactions from bounded rational opponents and consequently find it in their best interest to install more capacity then they would do when facing a perfectly rational opponent. The key design feature of our experiment is that we match subjects with varying level of experience, using experience as a proxy for the level of rationality. By this we can vary players beliefs about their opponents level of rationality, thereby emphasizing the effects of limited individual rationality. This is not to undermine the importance of common knowledge. Indeed, our results support the hypothesis that lack of common knowledge can indeed lead to overcapacity in KS games, when subjects are experienced. 4
Follows straight from Proposition 2 in Kreps and Scheinkman (1983).
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2.2
Experimental procedures
Two groups of subjects (A, B), played the roles of firms in a duopoly market. 60 subjects played the game for 20 periods or two 10 period phases.5 The subjects could practice playing the game for 2 periods before starting each phase. The experimental sessions were run in the following way. Group A started out. In their first phase subjects in group A played against each other. In other words, inexperienced subjects played against other inexperienced subjects. Then group B was introduced to the session. For ten periods subjects in group A played against subjects in group B. That is, experienced subjects played against inexperienced subjects. In their second phase subjects in group B played against each other. In other words, experienced subjects played against other similarly experienced subjects. In the following exposition we refer to different phases using the subscripts 1 and 2 . The label A1 for instance refers to group A in their first phase, when they are inexperienced, while A2 is group A in their second phase when they are experienced. This method of labeling also has a transparent reference to opponents experience, as subjects in group A always play against inexperienced players while subjects in group B always face experienced subjects. Subjects where matched so that they never played against the same opponent more than once (no-repeat matching). All players were carefully and truthfully informed about the matching procedure and the level of their opponents prior experience was particularly emphasized. The experiment was conducted at the Stockholm School of Economics and subjects where all business majors. A full session lasted about an hour and a half, including time spent reading the instructions.6 Subjects were paid according to their total profits earned during a sessions plus a 100kr (11.6$) show-up fee. We used an artificial laboratory currency, “experimental dollars” (e$) where 1kr (0.116 US$) equals 50e$. The participants earned, on average, 302kr (35$), with a minimum of 150kr (17$) and a maximum of 480kr (55$). The experiment was fully computerized. In each period, subjects observed three different screens.7 On the capacity choice screen each subject entered a capacity of 5
One subject had to leave the experiment, but was replaced by an assistant. All observations from that subject are skipped in the following analysis, before and after the substitution. This should have no impact on other players actions, however, since this subject played against other subjects in a different room, and his opponents were not informed about the switch and had no means of learning about it. 6 See Appendix II for full text instructions. 7 We used the z-Tree software package developed by Fischbacher (2002).
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Figure 1: Experimental procedure for sessions A and B his choice in the interval 0 to 90 with a maximum of two decimals. On the price choice screen his previously chosen capacity level and current opponent’s capacity level were displayed. He then entered a price level between 0 and 120 with up to two decimals.8 The result screen then displayed all choices made by him and his opponent and the resulting profits. The capacity choice and price choice screens both featured a profit calculator, where subjects could insert different hypothetical values for their own and their opponent’s capacity and prices and compare the resulting profits.
3
Experimental results
In the following analysis we focus on individual subjects’ choices of both capacity and prices. We find excess capacity to be persistent throughout the experiment and prices to be below the Cournot prediction. Excess capacity is usually understood as production capacity exceeding actual production. That means that some installed capacity is left idle. Here we choose to consider capacity above the Cournot theoretical output level excess capacity. This definition of excess capacity is more appropriate for our analysis, as it enables us to look at subjects’ decisions, rather 8
The feasible capacity and price ranges correspond to all rationalizable strategies (Bernheim, 1984).
7
than only the market outcomes as a whole.9
3.1
Descriptive Statistics
Figure 2 shows the average of capacities and prices for each group of subjects. There is a steady decline in capacities and an increase in prices, on average, as subjects gain experience. There is however considerable excess capacity in both groups. As shown in panel a, in the first five periods, average capacity levels in group B are close to half of the aggregate competitive output level, or 45. Subjects in group A choose even higher levels, or 52 on average. The per-period averages rapidly decline in periods 6 to 10 and level off after that. In the last 5 periods, the average capacity is between 35-37, depending on group. The average capacity is higher in group A than for group B in all but the very last period. Panel c, depicts average capacities when both groups have gained some experience. As before, group B faces relatively more experienced opponents than group A does. The downward trend continues, while the average capacity chosen is on average greater for group A than group B. The distribution of capacity choices are shown in Figures 8 and 9 in Appendix III. They show considerable dispersion in the first periods and slow convergence to a bimodal distribution, with modes at 30 (the Cournot output) and 40. The frequency of capacity levels around 40 is about twice as high for subjects A2 (facing inexperienced opponents) than subjects B2 (facing experienced opponents). These patterns fit with our hypothesis that players facing inexperienced opponents (subject A2 ) tend to choose higher capacities than similarly experienced players facing experienced opponents (subject B2 ). As shown in panels b and d in figure 2, Prices tend to increase, on average, in both groups except in the last few periods of phase 2, when they seem to stabilize. Subjects in group A tend to choose lower prices than subjects in group B. As it turns out, prices are more affected by opponents’ experience than capacity. This is somewhat surprising. It is obvious how variation in capacities translates into variation in prices, but why the effect of opponents’s experience is stronger on prices than capacities is not obvious at all. We need to consider a theoretical prediction 9
It’s particularly problematic to look at subject level data using the first definition. Which subject ends up with idle capacity depends on their dicisions in the price subgame, which can be highly unpredictable when both firms have considerable capacity installed.
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Figure 2: Mean capacity and price, by group, period and phase under assumptions of bounded rationality in more detail before we close this issue. That discussion is postponed, however, until section 4.
3.2
Statistical analysis
To test our hypotheses about differences in group behavior, while allowing for individual effects, we estimate a time-weighted regression model on the whole dataset: 1 t−1 t−1 xit = β i Di + β A DA + β B DB + εit , t t t
(I)
where xit is either observed capacity or price of individual i, Di is a subject dummy variable and DA and DB are group dummies. Moreover, β A , β B are estimates of the converging values (coefficient for the group-dummy variable) in group A and 9
Parameter βA βB β EQ R2 n p∗Courn. p†A vs. B
Model I Phase 1 Capacity 42.99
Table 1: Regression results Model II Phase 2 Phase 1 Price Capacity Price Capacity 41.56 36.56 46.71 42.99
(1.09)
(.99)
(.54)
(.79)
(1.13)
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39.43
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(.96)
(.53)
(.77)
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0.44 590 0.00 0.01
0.28 590 0.00 0.00
0.47 590 0.00 0.08
0.38 590 0.00 0.00
0.44 470 0.01
0.47 470 0.07
Estimation and testing of model (I) with capacities and prices as dependent variables. Standard errors are corrected for heteroscedasticity (White’s method). ∗ The critical level of significance (p-value) for the single-sided test of β i = (30, 60). † The critical level of significance (p-value) for the single-sided test of β A = β B .
B respectively. The index t refers to number of periods of play. The weights to the subject dummy variables ( 1t in front of Di ) are larger for observations in early ) are lower in periods than in later periods. The weights on the group dummies ( t−1 t early periods and higher in later periods. The key feature of the regression model is that it puts relatively more weight on individual behavior in early periods but group behavior in later periods. This allows us to control of individual learning processes while using the full dataset to estimate the group effects. Table 1 lists the main parameter estimates. To begin with it is worth while to consider the main research question in previous KS experiments, namely whether capacities and prices converge to the Cournot outcome. Our result is consistent with the earlier studies. Result 1 Capacities and prices fail to converge to the Cournot values, 30 and 60 respectively. This is the case for inexperienced and experienced subjects. The relevant hypothesis is that the group dummies are equal to the Cournot
10
prediction. The following single-sided F-tests using model (I): Capacities: H0 : β g = 30 H1 : β g > 30 Prices: H0 : β g = 60 H1 : β g < 60 where g ∈ {A, B}, were all rejected at the 1% level of significance. Table 1 reports the p values in each case (see pCourn. ).10 We now turn to our main question, whether subjects’ behavior depends on opponents’ level of experience. In particular, do they choose lower capacities, on average, when playing against experienced subjects relative to when they play against inexperienced subjects. And similarly, do they choose higher prices, on average, when playing against experienced subjects. Result 2 Subjects choose significantly higher capacities and lower prices, on average, when playing against inexperienced subjects than when playing against experienced subjects. To test the hypothesis we compare capacity and price choices of subjects with similar experience facing opponents with varying degree of experience. For a given phase, the opponent of subject A is always less experienced than the opponent of subject B. To analyze if opponent’s lack of experience can cause excess capacity (and underpricing), we test the following hypotheses using model (I): Capacities: H0 : β A = β B H1 : β A > β B Prices: H0 : β A = β B H1 : β A < β B The resulting critical p-values of the F-tests are reported in Table 1 (see pA vs. B ).11 In phase 1, the H0 is rejected for both variables at the 1% level of significance or higher. In the second phase (when subjects are experienced themselves) the hypothesis is rejected for prices at the 1% level and at the 8% level or higher for capacities. 10
Capacities got somewhat closer to the predicted values in a reference session where subjects played in fixed pairs. The outcome in the last few periods was still significantly different from the Cournot outcome. 11 Note that almost identical results were produced using a two-factor fixed-effects panel regression. Such a model has more general individual and time effects but the tests are less intuitive. Nonparametric tests for equal medians are less conclusive, due to the small number of observations in each period, but suggest similar results.
11
This suggests that opponents’ experience affects subjects’ capacity and price choices in the way we predicted. Subjects choose higher capacities and lower prices, on average, when playing against inexperienced subjects than when playing against experienced subjects. The fact that the difference in group behavior is not as significant for experienced subjects as inexperienced subjects, when it comes to capacity choice, is noteworthy. As we show in section 4, a model of bounded rationality can explain why the effect of opponents’ experience is reduced with own experience. We demonstrate that this is not a general result but rather specific to our setup. One reason why subjects might choose larger capacities when facing relatively inexperienced subjects is that inexperienced opponents can be expected not to respond to capacity decisions as strongly as experienced subjects would. Consider, as a benchmark, the subgame-perfect strategy profile, where both subjects choose a capacity of 30 units and a price of 60. If either subject beliefs that a capacity deviation of 40 in the first stage, would not trigger a price response by the opponent in the second stage to accommodate an increase in supply, such a strategy, paired with a slight decrease in own prices, is profitable. Our next result suggests that inexperienced subjects indeed are less likely to change prices to accommodate for previously made capacity choices. Result 3 Experienced subjects choose prices in response to aggregate capacity according to the slope of the demand function (-1). The price choice of experienced subjects are less responsive to capacity. Consider the following regression for each group and phase in turn: 1 t−1 t−1 t−1 + βQ (ki + kj ) + β kj (kj − 30) + εit pit = β i Di + β 0 t t t t
(II)
where the dependent variable is price chosen by subject i in period t. As in equation (I), Di are subject dummies with time dependent weights. β 0 is a common converging constant. In addition we consider two explanatory variables related to the capacity: the aggregate capacity (ki + kj ) and the opponent’s capacity in excess of the Cournot level (kj − 30). These last three variables have time weights emphasizing the observations in later periods. The results are shown in Table 2. The Nash equilibrium prediction is for players to choose prices as if β 0 = 120 and β Q = −1, when a pure strategy equilibrium exists in the subgame. For relatively high capacities the price subgame only has 12
Table 2: Analysis of the relationship between price and capacity Group A B Phase 1 2 1 2 ∗∗ ∗∗ ∗∗ b 83.41 91.77 88.93 109.83∗∗ β0 (3.22) (3.33) (3.13) (3.12) ∗∗ ∗∗ ∗∗ b βQ -0.46 -0.63 -0.52 -0.90∗∗ (0.04) (0.05) (0.04) (0.05) ∗∗ bk β 0.10 0.05 -0.29 0.03 j (0.07) (0.06) (0.05) (0.07) R2 n
0.67 230
0.80 230
0.77 240
0.85 240
bQ when quantities are particequilibria in mixed strategies. That means a lower β ularly high. In all cases, however, aggregate capacity is an explanatory variable. The level of significance is similar for experienced and inexperienced subjects. The bQ parameter is increasing with experience and close to -1 for experienced subjects β in groups B, while lower in group A. Recall that subjects in group A always play against relatively inexperienced subjects while subjects in B play against relatively experienced subjects. The fact that the standard errors of this particular parameter do not change with experience suggests that the result is not merely of consequence of the use of mixed strategies as that would have triggered greater standard errors in the case of inexperienced subjects or high capacity. This result is consistent with previous experimental results (Davis, 1999; Muren, 2000; and Anderhub et al., 2003). Most subjects seem to understand the relationship between capacity and price. Inexperienced subjects adjust their prices to capacities only partially or imperfectly. The adjustment is close to the standard Nash equilibrium prediction for experienced subjects. When an experienced player plays the KS game, one should expect his price choices to be strongly negatively correlated with aggregate capacity choice. An inexperienced player is less likely to fully grasp the effects of predetermined capacities for his optimal price strategy, and so the correlation can be expected to be lower. This conjecture is confirmed by the data. We also see that subjects can and do take advantage of this, as profits are not reduced by increasing capacity moderately above the Cournot level when the opponent is inexperienced. Excess capacity is not generally profitable, however, wen 13
Kernel Fit (Epanechnikov, h= 6.7500) a) Experienced vs. Inexperienced (A2)
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Figure 3: Profits and capacity choice by experienced subjects, scatter plot with kernel fit opponents are experienced. Result 4 Subjects with moderate excess capacity do not earn less than those with no excess capacity, when playing against inexperienced subjects. Excess capacity is not profitable, ex-post, when opponents are experienced. Figure 3 plots actual profits made in the experiment against capacity; profits tend to decrease as capacity is increased. This should be expected based on theoretical predictions, at least for capacities above the Cournot level. On the left panel of Figure 3 we see a scatter plot of capacities and profits for group A2 , experienced players facing inexperienced opponents. Interestingly, profits do not tend to drop when capacity is increased for capacity levels between 30 and 40. A Kernel fit, also shown in the Figure, illustrates this even better. This relation between capacities and profits is quite different in group B2 where experienced subjects face opponents who are also experienced. In that case the relationship between capacities and prices is more strongly negative, especially when comparing the capacity levels 30 and 40. The above results are consistent with our main hypothesis, that opponents’ experience has a negative effect on capacity choice and a positive effect on prices. Furthermore, we have shown that experienced subjects choose prices in accordance with the slope of the demand curve. Inexperienced subjects price responses are 14
imperfect, however. While this is consistent with our story we have reason to belief that both capacity and prices decisions are affected by bounded rationality. To analyze this relation in more detail we apply a version of the Quantal Response Equilibria model to the KS game in the next section.
4
Behavioral predictions
For our analysis of a quantal response equilibrium in the KS game we need to be a bit more formal in the exposition. The interested reader should consult McKelvey and Palfrey (1995 and 1998) for more details. The main conclusions of this section are emphasized in the concluding section.
4.1
A QRE for KS
A quantal response equilibrium can be viewed as an empirical version of Nash equilibrium. Players are not expected to choose best responses with probability one, but rather use probabilistic decision rules. The probability of a particular strategy being chosen is increasing in the expected payoff of that strategy. An equilibrium is found when the decision probabilities are consistent with the expected payoffs that they create. One way to look at his is to say that payoffs are observed with error. Assuming that the error terms are additive and independently distributed according to the extreme value distribution gives rise to the logit-QRE, a rather convenient special case. While standard QRE applies only to normal-form games it can be extended to extensive form games in the agent normal form. In the agent quantal response equilibrium (AQRE) we assume that independent agents make decisions for the players at each information set (McKelvey and Palfrey, 1998). A logit-AQRE for the KS game can be defined as follows. Let b∗ denote a complete strategy profile. π (ki , b∗ ) describes the expected payoff when player i chooses the capacity level ki with probability one while the opponent’s strategy and own price strategy follow b∗ . Similarly, π (pi , b∗ ) is the expected payoff when the strategy profile b∗ is played, except that own price is pi with probability one. The expected payoff functions are calculated from (1) assuming that the beliefs are consistent with b∗ and the associated realizational probabilities12 . 12
The realizational probabilities derive from the selected strategy b. First let ρ (q|b) be the
15
Subject i is assumed to employ a logistic choice function with precision parameter λi ≥ 0 when making his choices, i.e. to choose ki with probability b∗i,ki
∗
eλi π(ki ,b ) = P λi π(ki ,b∗ ) , e
i ∈ {1, 2}
(3)
ki ∈Ak
and to choose pi with probability b∗i,pi |k , conditional on both player’s capacity choices where b∗i,pi |k
∗
eλi π(pi ,b ) = P , 0 ∗ eλi π(pi ,b )
i ∈ {1, 2} .
(4)
p0i ∈Ap
The parameter λi captures the decision maker’s response precision. When λi = 0 the errors completely dominate any information about the payoff function and player i choose all available strategies with equal probability. On the opposite extreme, when λi → ∞ the errors become negligible, in which case player i chooses his best response to b∗ with probability one. In general, players act more rationally the higher their λi parameter is.
4.2
Qualitative prediction
The logit-AQRE is found by solving a system of equations (3) and (4) for all information sets for particular levels of λi .It turns out to be extremely difficult, if not impossible, to find a closed form solution to this problem. The main reason being discontinuities in the payoff function (1). Nevertheless, the problem can be solved numerically given a discreteized action space.13 The following discusprobability of the particular quantity vector q = (q1 , q2 ), given strategy profile b. Then let ρ (p|q, b) be the conditional probability of the vector p when q has occurred and the strategy profile b is selected. Finally ρ (q, p|b) = ρ (p|q, b) ρ (q|b) is the probability of an outcome which involves q and p given strategy b. For a given strategy profile b the expected payoff to player i, and all his agents at the first stage, is X X π i (b) = ρ (qi , pi |b) π i (q, p) . qi ∈Aq pi ∈Ap
where πi (q, p) is defined in (1). The expected payoff at the second stage, when q is common knowledge, is X πi (b|q) = ρ (pi |q, b) πi (q, p) . pi ∈Ap
13
While continuous space techniques exist for normal form games (Anderson et al., 1998) we are unaware of similar tools for extensive form games. The results do not seem to be too sensitive to
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1.5
2.0
Figure 4: Mean capacities and prices in logit-AQRE equilibrium sion is based on simulations where the action space is limited to multiples of five: Ak = Ap = {10, 15, 20, ..., 80}14 The means of capacities and prices generated by logit-AQRE under different assumptions on λi are shown in Figure 4. The graphs are drawn from the perspective of player 1, with his precision level, λ1 , shown on the x-axis. Each curve shows the relationship between player 1’s average capacity and his precision level, for a given precision level for player 2. A change in the opponents’ precision level, λ2 , is represented by a shift in the curve.15 Observe from the left hand graph in Figure 4 that player 1’s average capacity decreases in his own precision, and seems to converge toward the Cournot output level. This is especially true when the opponent’s experience level is high. In other words, when both players observe their payoffs with high precision the logit-AQRE converges to the unique subgame-perfect equilibrium. An increase in λ2 corresponds to a downward shift in the graph to the left implying that player 1’s average capacity is decreasing in λ2 . This effect is important for intermediate λ1 but small for low or high λ1 . the discrete approximation of the action space and we obtain a manageable number of nonlinear equations. 14 The program was solved using the GAMS/PATH mixed complimentarity problem solver. The code can be obtained from the authors upon request. 15 The numerical values of the λ’s are specific to the model and can only be compared in relative terms. The model is not solvable for λ’s much larger than 2.
17
From the right hand graph in we can see that the average price increases with the player’s own precision (λ1 ) but does not converge to the Cournot outcome when the opponent lacks precision (λ2 is low). An increase in λ2 has a positive effect on the expected price of player 1 and the effect is strengthened by an increase in λ1 . In general there is a bias towards more capacity and lower prices when the opponent is bounded rational. The effect of opponent’s experience depends on own experience in a way that is consistent with result 2. Recall that the estimated effect of opponents’ experience was not as large or as significant in a group of experienced subjects compared with less experienced subjects. If we believe that experienced subjects have a λi greater than 0.5 the figure suggests that the effect of opponents’ experience is decreasing in own experience. The patterns observed in Figure 4 are partly due to our choice of action space. The set of rationalizable strategies, that were available to subjects in the experiment and used in the above simulation, is highly asymmetric around the SPE values. If we select an action space such that the expected quantities and prices are equal to the Cournot levels for totally irrational players, λ1 = λ2 = 0, the relationship between mean of capacity and own experience, λ1 , is no longer monotonic. However, for any λ1 > 0 the expected capacity is larger and the price lower than predicted by the Cournot outcome. Further discussion on this issue can be found in Appendix I.
4.3
Quantitative predictions
While the above predictions fit qualitatively well with the experimental results of section 3, the question remains whether the predictions are quantitatively accurate. We can tackle that question by estimating the precision parameters that best fit the experimental data and compare the predicted and actual average capacities and prices. For groups A and B, we estimate the precision parameters (λi ), maximizing the following likelihood function, lnLAQRE =
X
¡ ¢ yit (ki , kj , pi ) × ln bki (λ1 , λ2 ) + bpi |k (λ1 , λ2 ) .
(III)
The summation applies to subjects, a subset of periods and all possible combinations of ki , kj ∈ Ak and pi ∈ Ap . The index variable yit (ki , kj , pi ) takes the value of 1 if subject i selects capacity ki and price pi in round t while his current opponent chooses
18
1.60 1.40
lambda A 95% u.b. 95% l.b.
1.20
lambda B 95% u.b.
1.00
95% l.b.
0.80 0.60 0.40 0.20 0.00
Treatment A1A1
Treatment A2B1
Treatment B2B2
1-2 3-4 5-6 7-8 9-10 1-2 3-4 5-6 7-8 9-10 1-2 periods
3-4 5-6 7-8 9-10
Figure 5: Estimated λ and 95% confidence intervals kj or else the value is zero.16 The b functions are the equilibrium response functions as defined in (3) and (4). Each treatment is broken down into five experience levels with two periods in each. In group A1 ’s first phase and group B2 ’s second phase, where all subjects have the same experience we estimate a single precision parameter, λA or λB respectively, while for A2 and B1 , when the groups play against each other, a separate parameter is estimated for each group. The estimates together with the log-likelihood values (ln LAQRE ) are reported in Table 4 (Appendix III). For comparison the log likelihood of the random model (i.e., when λA = λB = 0) is also displayed.17 The estimates for λA and λB generally increase with the number of periods played (see Figure 5) reflecting the tendency for subjects to choose optimal strategies with more precision as they become more experienced in playing the game. Interestingly λB is higher than λA for any given level of experience. This does not suggest that the speed of learning is somehow affected by opponents experience, since the difference in λ is more or less of the same magnitude for different experience levels. A more 16
Selected quantities and prices are rounded up or down to the nearest point in the discrete action space. 17 A few additional behavioural models where considered in this context, including a model of fictitious play and imitation models. The logit-AQRE gave better predictions of behaviour than the alternative models considered.
19
plausible hypothesis is that inexperienced subjects in group B put more effort into finding their best strategy, compared with similarly experienced A players, from the outset because they know their competition will be challenging. In Table 5 (first two columns), we compare the predictions based on the estimated logit-AQRE model for each experience level, and the actual choice (adjusted for the discrete action space). It is clear that, even though the model provides fairly good qualitative predictions, it systematically underpredicts capacities and overpredicts prices. The predicted standard deviations are close to the actual levels. This suggests that the logit-AQRE model is unable to fully capture the incentives for excess capacity. That does not necessarily undermine opponents’ bounded rationality as a rationale for excess capacity, but rather that a more complete model of behavior is needed to capture actual behavior in more detail.
5
Conclusion
The purpose of this paper is to improve our understanding of why subjects consistently choose capacities above, and prices below, the predicted subgame-perfect equilibrium in experimental Kreps and Scheinkman games. We argue that players’ perceptions of their opponents’ skills, or level of rationality, are important in this context. Using experience in playing the game as a proxy for the level of rationality we find that capacities are relatively high when opponents’ level of experience is relatively low and that prices are relatively low when opponents lack experience. We further explore the experimental results and the behavioral predictions using the quantal response equilibrium framework. Predictions, based on a logit agent-form quantal response equilibrium model, turn out to be qualitatively consistent with the experimental findings. The logit-AQRE model, as specified here, does not, however, give quantitatively accurate predictions. The observed deviations are much larger than predicted by the model, thereby indicating that the current model specification is too restrictive. At least two extensions seem worthy of further research. First, a closer inspection of the distribution of capacity choices, as shown in Figure 8 for group A and Figure 9 for group B, suggest that there may be considerable heterogeneity within each group. The distribution of capacity is bimodal for a reasonably experienced subject pool. In the last two periods of play, almost a third of the subjects in group B
20
chose the Cournot output level 30, while another third chose a capacity level close to 40. Anderhub et al. (2003) find a similar pattern, i.e. a bimodal distribution of capacities. Allowing for different levels of rationality within each group would account for this, and possibly increase the level of the predicted bias. Second, extending the AQRE model to allow for inconsistent response functions is another interesting alternative. Weizsacker’s (2003) extension of the normal form QRE allows for response functions which depend on the perceived opponent’s choice distributions, that need not be consistent with the opponents actual equilibrium strategy. Analysis of experimental data suggest that perceptions are quite frequently biased in the direction of underestimating the rationality of other players. In the logit quantal response framework, this amounts to a downward bias in players’ perception of their opponents’ precision level. If the same bias were to appear in the KS model the predictions of such a model would probably be closer to the actual outcome, as the average quantities should increase, given that beliefs about the opponent’s precision level decrease, as shown in Figure 4.
21
References Anderhub, V., Güth, W., Kamecke, U. and Normann, H.-T., 2003, Capacity choices and price competition in experimental markets, Experimental Economics, v. 6, iss. 1, 27-52. Anderson et al., 1998, Rent-Seeking with Bounded Rationality: An Analysis of the All-Pay Auction, Journal of Political Economy 106: 828-853. Baye, M. R. and John Morgan, 2004, Price Dispersion in the Lab and on the Internet: Theory and Evidence, RAND Journal of Economics 35(3):449-66. Bernheim,B-D., 1984, Rationalizable Strategic Behavior, Recent developments in game theory, Maskin,-Eric-S.(ed). Elgar Reference Collection. International Library of Critical Writings in Economics, vol. 109. Cheltenham, U.K. and Northampton, Mass.: Elgar. Camerer, C.; H. Teck; Chong, K. Models of Thinking, 2003, Learning, and Teaching in Games; American Economic Review, v. 93, iss. 2, pp. 192-95. Davis, D. D.,1999, Advance production and cournot outcomes: An experimental investigation, Journal of Economic Behaviour and Organization, 40(1), 59–79. Davidson, C. and Deneckere, R. , 1986,. Long-run Competition in Capacity, Short-run Competition in Price, and the Cournot Model, RAND Journal of Economics, v. 17(3), 404-15. Dufwenberg, M., Lindqvist, T. and Moore, E., 2003, Bubbles and experience: An experiment on speculation, American Economic Review??? . Fudenberg and Levine, 1998, The Theory of Learning in Games, MIT Press, Cambridge, MA. Fischbacher, U. , 1999,. Z-Tree: Zurich Toolbox for Readymade Economic Experiments. Working Paper No. 21, Institute for Empirical Research in Economics, University of Zurich. Ghemawat,-P. and Caves, R-E, 1986, Capital Commitment and Profitability: An Empirical Investigation, Oxford-Economic-Papers,-N.-S. Suppl. Nov. 1986; 38(0): 94-110 Gilbert,-R-J; and M Lieberman, 1987, Investment and Coordination in Oligopolistic Industries.University of California at Berkeley, Economics Working Papers: 8730. Kreps, D. M. and Scheinkman, J. A.: 1983, Quantity precommitment and bertrand competition yield cournot outcomes, Bell Journal of Economics 14(2), 326–37. 22
Lieberman, M. B., 1987, Postentry Investment and Market Structure in the Chemical Processing Industries, RAND Journal of Economics 18(4):533-49. McKelvey RD and TR Palfrey (1995), Quantal Response Equilibria for Normal Form Games, Games and Economic Behavior 10, 6 - 38. McKelvey, R. D. and Palfrey, T. R.,1998, Quantal response equilibria for extensive form games, Experimental Economics, 1(1), 9–41. Masson,R and Shaanan J. Optimal Oligopoly Pricing and the Threat of Entry: Canadian Evidence, International-Journal-of-Industrial-Organization, September 1987; 5(3): 323-39 Muren, A., 2000, Quantity precommitment in an experimental oligopoly market, Journal of Economic Behavior and Organization, 41(2), 147–157. Pearce, D., 1984, Rationalizable strategic behavior and the problem of perfection, Econometrica, 52, 1029–1050. Rassenti, S., Reynolds, S. S., Smith, V. L. and Szidarovszky, F., 2000, Adaptation and convergence of behavior in repeated experimental cournot games, Journal of Economic Behavior and Organization 41, 117–46. Reynolds, Stanley S.; Wilson, B.J., 2000, Bertrand-Edgeworth Competition, Demand Uncertainty, and Asymmetric Outcomes; By .; Journal of Economic Theory, 92 (1), 122-41. Slonim R.L., 2001, Competing Against Experienced and Inexperienced Players, Mimeo,Department of Economics, Case Western Reserve University. Smith, V. L., Suchanek, G. L. and Williams, A. W., 1988, Bubbles, crashes, and endogenous expectations in experimental spot asset markets, Econometrica 56, 1119–1151. Weizsäcker, G., 2003, Ignoring the rationality of others: Evidence from experimental normal-form games, Games and Economic Behavior 44(1):145-171.
23
Appendix I - Choice of Action Space If players randomize completely, the average capacity is determined by the mean of the action space. In most conceivable configurations of the KS model, the average of all feasible capacity levels is higher than the Cournot output, while the opposite is true for prices. This might affect the prediction of the logit-AQRE model. It is therefore interesting to compare the predictions in section 4 to the case where the action space is symmetric around the Cournot outcomes, Ak = {0, 5, ..., 60} and Ap = {30, 35...90}. In this setup the expected capacity and price chosen by a totally clueless player (with λ = 0) is simply the Cournot outcome. Figure 6 illustrates average capacity and price choices in the logit-AQRE equilibrium profile . With a symmetric action space, average capacity is no longer uniformly decreasing in own precision level (λ1 ). For positive levels of λ2 (opponent’s precision level) it first increases and then decreases. Furthermore, capacity only converges to the Cournot output level when both λ1 and λ2 increase simultaneously. The case of prices is more complicated, as the average price is not monotonic with respect to λ2 either. A symmetric action space would be difficult to impose on our experiment18 and, for this reason, we can expect a further bias towards high capacities in the case of inexperienced subjects, for this reason. It is very hard, if not impossible, to control for how subjects think about the action space. Judging from how subjects started out with average capacities close to 45, the average of rationalizable capacity levels, we believe the specification used to be appropriate. The comparison of these two configurations is still helpful. It suggests that the presence of a bias towards larger capacities and lower prices does not seem to be caused by the choice of action space, although it is an important determinant of the shape of the bias with respect to the level of players’ rationality and their perceptions about the rationality of others. 18
Cournot equilibrium prices and quantities are rarely at the center of what is naturally perceived as feasible prices and quantities. Without any knowledge of what subjects perceive to be a natural action space we specify the action space as those actions belonging to all rationalizable strategies (Bernheim, 1984).
24
Appendix II - Instructions {}: group A, []: for group B and, * *: group C Welcome to this experiment in the economics of decision making, which should take approximately 90 minutes. You will be paid a minimum of SEK 100 for your participation, but you can earn much more if you make good decisions. At the end of the session you will be paid, in private and in cash, an amount that will depend on your decisions. Please read the instructions carefully. If you have any questions please raise your hand, and you will be helped privately. General Rules This experimental session will consist of several periods. In each period you play the role of a firm which produces a good and sells it in a market. {One other firm, represented by a randomly selected participant, sells his product in the same market in each period. For the first 10 periods you will play against other participants sitting in this room. You will never face the same participant more than once. After period 10 the experiment will restart, but now you will play against a different group of participants located elsewhere. Unlike you, these participants have no prior experience of this experiment. As before, you will only play against each of the new participants once.} [One other firm, represented by a randomly selected participant, sells his product in the same market in each period. For the first 10 periods you will play against a different group of participants located elsewhere. Unlike you, these participants have experience of this experiment, they have played the game before. You will never face the same participant more than once. After period 10 the experiment will restart, but now you will play against other participants sitting in this room. As before, you will only play against each of the new participants once.] *One other firm, represented by another participant, sells his product in the same market. You will play against the same participant in all 20 periods. The true identity of your opponent will not be revealed to you, neither will your identity be revealed to him.* By making good decisions you can earn profits in experimental dollars (e-dollars). At the end of the session you will be paid SEK 100 plus the e-dollars you have earned at the exchange rate of 1 SEK for every 50 e-dollars. Simply put, the more experimental dollars you earn the more cash you will receive at the end of the session. 25
In each period you make two separate decisions for your firm. First you decide how much you would like to produce (Q1) and then, after you have observed the production level of your competitor (Q2), you choose your price (P1). Production stage At the beginning of each period you decide how many units of the good to produce (Q1). You make your decision by entering a number in the box on the left hand side of the screen and then press OK. Any positive number between 0 and 90, with up to 2 decimals is acceptable. (Example: 10, 20.6, and 33.33 are valid but -12, 50.123 are not). Please use a dot (.) as the decimal separator. The amount you produce has consequences for your profit in that period, since you have to pay a production cost of 30 e-dollars for each unit; regardless of how much you sell. Note that no inventories can be carried to future periods. Before you enter in your quantity you should think carefully about your choice. You can use the calculator displayed on the right hand side of the computer screen. There you can enter prospective production quantities and prices for your firm and its competing firm, press CALCULATE and observe the results in the Table on the lower right hand side. Table 3 explains the columns. Table 3: Calculator table legend Q1 Your production Q2 Competitor’s production X1 Your sold quantity X2 Competitor’s sold quantity P1 Your price P2 Competitor’s price
26
Price stage When all participants have entered their production levels you will automatically go to the price stage. On the left hand side you can see your own chosen production level as well as your competing firm’s production level. You enter a price of your choice in the box below this information and press OK when you are ready. Any positive number between 0 and 120, with up to 2 decimals is acceptable. (Example: 10, 20.6, and 33.33 are valid but -12, 50.123 are not). Please use a dot (.) as a decimal separator. You may want to do some more calculations before you set your price. You still have the calculator on your right hand side, but this time you can only alter the prices (P1 and P2). The previously chosen quantities (Q1 and Q2) are fixed at this stage. How much you sell is determined by your price (P1) and its relation to your competitor’s price (P2). Consumer demand is calculated by a computer program and follows a simple equation D = 120 − P. where D is demand and P is a price. This means for instance that at the price 0 consumers are willing to purchase 120 units of the product. At the price of 25.5 consumers are willing to purchase 94.5 units. There is no demand for the product at price levels equal to or greater than 120. Figure 7 illustrates demand and unit cost. Consumers strictly prefer buying from the firm offering the lower price. Hence, the firm with the lower price will sell all its production up to the demand level at that price. The firm with the higher price can only sell the product to consumers who are not supplied by the lower pricing firm, and never more than the demand level at its price, or its produced quantity. If both firms choose the same price, demand will be split equally between them up to the quantity limits (the respective production levels). Example 1: Say that Q1 = 35, Q2 = 45, P1 = 45 and P2 = 55. Since demand at the lower price level is 75, you (firm 1) can sell all your produced quantity. Your revenue is 45 × 35 = 1575 and your total cost is 30 × 35 = 1050. Your profit is 1575 − 1050 = 525. Your opponent (firm 2) can only sell 30 units. The demand for his product equals 120 − 55 = 65, by the demand equation. From that we have to subtract what is already supplied by your firm, or 35 units. Hence, he sells 65 − 35 = 30 units at the price of 55. His revenue is 55 × 30 = 1650, his total cost 27
is 45 × 30 = 1350 and he makes a 300 e-dollar profit. Example 2: Say that Q1 = Q2 = 15 and P 1 = P 2 = 55. Demand at this price is greater than the sum of the production levels but you can only sell what you produce. Your profit is 55 × 15 − 30 × 15 = 375. Result display When all participants have entered their prices the result display will appear. You can then see a summary for that period, for yourself and your competitor. Press continue when you have studied the results. Periods To help you familiarize yourself with the computer interface and the calculations, you get to practice for two periods. The result of these periods will not affect your payoff. {Then, you will play for 10 periods, once with each of the participants in your room. Then, after a short break, the experiment restarts, and now you play against inexperienced participants. Again you go through two practice periods (for the others) and then 10 periods, where you can earn money, against each of the inexperienced participants.} [The result of these pratice periods will not affect your payoff. Then you will play for 10 periods, once with each of the experienced participants in the other room. Then, after a short break, the experiment restarts and now you play for 10 periods against participants sitting in your room, who have the same level of experience as you do] *Then you will play for 20 periods for which you can earn money.* Before you leave, we ask you to fill out a short questionnaire about the experiment. We will use the time while you complete it to calculate your earnings. Everything described here is not only valid for you, but also for all other participants in this experiment. Now you should be ready to start the experiment. Please raise your hand if you have any questions. We prefer to answer your questions privately. Good luck!
28
45.0 43.0
61.0
Prices
Capacities 60.0
41.0 59.0
39.0 37.0
58.0
35.0 57.0
33.0
0.00 31.0
56.0
29.0
λ2
27.0
0.00
0.05
0.10
0.25
λ2 55.0
0.50
1.00
λ1
0.25 0.50 2.00
2.00 25.0 0.00
0.10
1.50
54.0 0.00 2.00
0.50
1.00
λ1
1.50
2.00
Figure 6: Mean capacities and prices in logit-AQRE equilibrium - Action space symmetric around Cournot outcome
120
Price / Unit cost
100 80 60 40 20 0 0
20
40
60
80
100
120
Quantity
Figure 7: The demand function (solid line) and the unit cost function (dotted line)
29
Appendix III - More experimental results
30
Table 4: Maximum likelihood estimation of the logit-AQRE model Periods λA λB lnLAQRE lnL
1-2
a) Group A1 -205.9
0.19
-249.1
(0.03)
3-4
0.26
-201.2
-249.1
-187.5
-249.1
-196.9
-249.1
-171.5
-249.1
(0.04)
5-6
0.29 (0.04)
7-8
0.34 (0.04)
9-10
0.54 (0.05)
1-2 3-4 5-6 7-8 9-10
1-2
0.73
b) Group A2 vs.B 1 0.40 -345.2
(0.10)
(0.05)
0.75
0.48
(0.09)
(0.05)
0.86
0.65
(0.10)
(0.05)
0.81
0.69
(0.10)
(0.07)
0.81
0.69
(0.10)
(0.08)
0.86
-509.1
-341.1
-509.1
-310.1
-509.1
-315.2
-509.1
-321.9
-509.1
c) Group B 2 -154.3
-260.0
-157.8
-260.0
-144.8
-260.0
-144.0
-260.0
-133.0
-260.0
(0.11)
3-4
0.82 (0.09)
5-6
1.01 (0.12)
7-8
1.02 (0.11)
9-10
1.141 (0.12)
Standard errors in parantheses, estimated by the BHHH method.
31
Table 5: Actual vs. predicted quantities and prices kA kB pA pB Actual Pred. Actual Pred. Actual Pred. Actual Pred.
1-2 3-4 5-6 7-8 9-10
53.9
38.0
a) Group A1 vs.A1 40.7
(18.5)
(18.2)
(11.5)
(17.5)
50.5
36.1
37.5
50.3
(16.2)
(16.3)
(8.6)
(16.4)
48.9
35.3
37.5
51.2
(16.4)
(15.3)
(10.1)
(15.8)
48.5
44.5
34.5
40.3
52.2
(14.0)
(14.3)
(12.2)
(15.2)
38.4
32.3
47.5
55.8
(13.1)
(10.6)
(13.7)
(12.4)
37.7
31.5
b) Group A2 vs.B 1 45.8 33.3 44.1
56.5
42.9
54.7
(9.2)
(9.0)
(16.6)
(12.2)
(10.0)
(13.2)
37.1
31.4
43.3
32.6
44.7
56.9
44.5
55.6
(8.4)
(8.8)
(15.9)
(11.4)
(10.2)
(11.7)
(11.2)
(12.3)
5-6
37.0
31.1
41.7
31.6
46.1
57.9
46.4
57.3
(8.6)
(8.0)
(16.5)
(9.5)
(10.2)
(10.5)
(10.3)
(10.8)
7-8
37.2
31.2
38.9
31.5
47.0
57.8
46.5
57.4
(7.4)
(8.3)
(14.6)
(9.1)
(8.8)
(10.6)
(8.8)
(10.7)
9-10
36.7
31.2
39.6
31.5
46.6
57.8
46.7
57.4
(8.0)
(8.3)
(12.7)
(9.1)
(8.3)
(10.6)
(8.6)
(10.7)
46.8
58.2
1-2 3-4
1-2 3-4 5-6 7-8 9-10
(12.6)
(9.7)
c) Group B 2 vs. B 2 37.4 31.0 (11.5)
(8.0)
(7.5)
(9.9)
37.1
31.1
47.3
58.1
(11.6)
(8.2)
(7.8)
(10.1)
36.5
30.7
48.2
58.8
(11.8)
(7.3)
(8.0)
(9.2)
35.3
30.7
49.5
58.8
(11.5)
(7.2)
(7.7)
(9.1)
35.2
30.5
51.1
59.1
(11.7)
(6.8)
(8.4)
(8.7)
32
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
Count
20 15 10 5 0
Count
20 15 10 5 0
20
40
60
80
20
40
q
60
80
20
40
q
60
80
20
40
q
60
80
20
40
q
60
80
q
Figure 8: Distribution of quantity choices in group A, each graph indicates a particular experience level (two periods)
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
Count
20 15 10 5 0
Count
20 15 10 5 0
20
40
60
q
80
20
40
60
q
80
20
40
60
q
80
20
40
60
q
80
20
40
60
80
q
Figure 9: Distribution of quantity choices in group B, each graph indicates a particular experience level (two periods)
33
