Team selection with asymmetric agents

Journal of Economic Behavior & Organization Vol. 38 (1999) 421±452

Team selection with asymmetric agents Katerina Sherstyuk* Department of Economics, University of Melbourne, Parkville, Vic. 3052, Australia Received 24 September 1997; accepted 19 February 1998

Abstract In shallow markets where there are mutual gains from cooperation among agents, collusive behavior may occur even if it does not constitute a Nash equilibrium. Yet, such behavior is rarely sustainable. Bolle (1994) reports the results of one-period team selection experiments in which subjects often did not follow the Nash equilibrium behavior but engaged in tacit collusion. We test the robustness of Bolle's findings by introducing asymmetry into subjects' characteristics and repeating the experiment for a number of periods. We find that collusion is not sustainable and the outcomes converge to levels close to the Nash equilibrium. However, the agent's actions stayed slightly above the Nash equilibrium level in all experiments. # 1999 Elsevier Science B.V. All rights reserved. JEL classi®cation: C92; C7; L13 Keywords: Team selection experiments; Competition; Tacit collusion; Fairness

1. Introduction This paper addresses the issue of sustainability of tacit collusion in shallow markets where cooperation among agents in mutually beneficial but does not constitute a Nash equilibrium. Following Bolle (1994, 1995), we consider, in an experimental setting, team selection (TS) games. In these games, each potential team is characterized by its output parameter; potential team members (agents) submit their individual wage demands (asks) to the principal, and the principal selects a team which gives him (or her) the highest profit ± defined as the output of the team net of asks submitted by the team members ± and then pays all the selected agents according to their asks. If the Nash equilibrium payoffs of selected team members are low, the agents may gain from collusion by sticking to a level of asks much higher than the Nash equilibrium level. Some agents then may * Tel.: +61-3-9344-5316; fax: +61-3-9344-6899; e-mail: [email protected] 0167-2681/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 6 8 1 ( 9 9 ) 0 0 0 1 9 - 0

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have to trade off the certainty of being selected under the Nash equilibrium for the higher expected payoffs but probabilistic selection outcome under collusion. The situation is quite similar to many other imperfectly competitive market situations, such as the first price multi-unit auctions with a small number of bidders. An important difference is that in TS games the productivity characteristics are team-specific, whereas in auctions, the values or costs are agent-specific. While the emphasis of experimental research in auctions is on incomplete information games, where each agent knows their value of the item, but not the other bidders' values (e.g. Cox et al., 1984, 1988; Isaac and Walker, 1985; Smith and Walker, 1993), both Bolle (1994) and this study consider the complete information experiments, where all agents are fully informed about the characteristics of all teams. Given that payoff certainly has been earlier shown to facilitate collusion (Plott, 1989; Holt, 1995), we might expect collusion to be more pronounced in the complete information team selection experiments. Another difference between the TS games and auctions is that the former allow to discriminate among the teams according to criteria other than their profitability, such as efficiency, when there is more than one profitmaximizing team. Such discriminatory procedures are uncommon in the auction experiments, where ties are either not allowed (as in open auction markets), or broken on the basis of time preference (as in sealed bid auctions). The possibility to choose among different tie-breaking rules in TS experiments allows creation of an extra dimension of asymmetry among agents, giving an extra control parameter to experimenter. Bolle (1994) conducted a set of TS experiments with two symmetric agents and a principal, in which each subject made one decision in each role (the total of three decisions); under the chosen parameter values, both agents gained positive payoffs under the Nash equilibrium outcomes. Bolle reports that the agents did not always follow the Nash equilibrium prediction in his experiments and exhibited tacit collusion; he therefore argues that fairness and cooperation play an important role in the agents' behavior. In this study we test the robustness of Bolle's findings to repeatedness of the experiment, payoff asymmetry and presence of marginal agents (those whose payoffs are zero in Nash equilibrium). We consider multi-period three-agent experiments with a computerized principal, in which agents have asymmetric contributions to team productivities. Two treatments are considered, which differ in the `tie-breaking' rule only in case when there is more than one profit-maximizing team. We use two subject pools in the experiments, Caltech students (USA) and Melbourne university students (Australia). Previous studies that test the hypothesis of competitiveness of subjects against cooperative or collusive behavioral theories (which cover the fields of industrial organization (e.g. Plott, 1989; Holt, 1995), auctions (Kagel, 1995), and collective actions and public goods experiments (Ledyard, 1995)), indicate that in small groups collusive behavior, even if it does not constitute a Nash equilibrium, often occurs if it is mutually beneficial to the subjects. However, collusion is generally sensitive to such factors as repeatedness of the game, asymmetry of payoffs, players' information about the other players' characteristics and size of the group (e.g. Fouraker and Siegel, 1963; Isaac and Walker, 1985, 1988; Isaac et al., 1985; Holt, 1995).1 Cooperation is more likely to occur 1 The above-mentioned survey articles by Plott (1989), Holt (1995), Kagel (1995) and Ledyard (1995) provide comprehensive coverage of the topic.

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in small groups of symmetric agents, when each agent is informed about the others' payoffs. In this respect, Bolle's result on the presence of collusion in team selection games with two symmetric perfectly informed agents is not surprising. Our design incorporates some features, such as complete information, full information feedback between periods, small number of agents, and high expected gains from collusion for all agents, that facilitate collusion. Its other features, such as one shot nature of the game, repetition and payoff asymmetry, were found elsewhere to decrease cooperation. On the theoretical side, the theory of contestable markets (Baumol et al., 1980) predicts that, due to the presence of a marginal agent, the threat of competition from this agent will drive the asks down to the Nash equilibrium level. Our findings support the competitive predictions rather strongly. Although in the initial periods the agents' asks were significantly above the Nash equilibrium level in all experiments, they kept decreasing from period to period until they reached levels close to the Nash equilibrium. We find very little evidence of fairness along the convergence path to the Nash equilibrium; not only were the marginal agents' payoffs lower than the others' in case the agents were selected, but the marginal agents were selected less often. Yet, interestingly, most of the experiments never fully converged to the Nash equilibrium; the marginal agents were never completely excluded from the market, although in some cases the other agents were worse off under such outcomes than in the Nash equilibrium. We also observed interesting differences in the outcomes of the experiments between the experimental treatments and subject pools. Section 2 briefly discusses the theory behind the TS games. Experimental design is discussed in Section 3. Section 4 summarizes the results. The effects of experience on experimental outcomes is considered in Section 5. In Section 6, we take a closer look at individual behavior. Section 7 concludes the discussion. 2. Theoretical predictions Here we briefly reproduce Bolle's findings on the properties of Nash equilibria of the TS games and then offer characterization of their cooperative solutions. 2.1. The model The following complete information n-person game is considered. There is a set N ˆ {1,. . .,n} of agents, from which a team T  N is to be selected. Each potential team is characterized by its productivity F(T)2R, with F(;) ˆ 0. A team T is called efficient if it has the highest productivity among all possible teams; an agent is called `efficient' if he belongs to every efficient team. In the game, each agent submits his wage demand (ask) vi2R‡ to the principal, and a team is chosen to maximize the principal's profit, defined as: X vi : …T† ˆ F…T† ÿ iT

If there is more than one profit-maximizing team, then two variants of the TS game are considered. In the B-game (after Bolle, 1994), only the most efficient among profit-

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maximizing teams can be selected. In the G-game (`generalized'), all profit-maximizing teams are selected with equal probability. The agents are then paid according to their asks if they are in the selected team, and nothing otherwise. We assume that each agent maximizes his expected payoff,2 which depends on his ask vi and the probability of being selected pi: Ui ˆ pi vi

(1)

2.2. Nash equilibria of TS games Bolle makes the following equilibrium selection assumption which states that agents who are not selected with certainty submit zero asks: Assumption 1 (Equilibrium selection; Bolle, 1995): If, in a Nash equilibrium of a team selection game, for some i2N pi ˆ 0, then vi ˆ 0. The assumption is motivated ``. . .by a strong competitive attitude of the players. If they are not selected, they reduce their requirements as far as possible'' (Bolle, 1995, p.135). However, asking zero is a weakly dominated strategy, especially if players account for the possibility of noisy play of their opponents. We will discuss the validity of this assumption in the review of experimental results. The following observations characterize the existence and properties of pure strategy Nash equilibria of the TS games.3 Proposition 1. Under Assumption 1, 1. (Bolle, 1995) In a pure strategy Nash equilibrium, if it exists, (i) a team can be selected if and only if it is efficient; (ii) only efficient agents may submit positive asks; all inefficient agents submit zero asks. 2. (Bolle, 1995) A pure strategy Nash equilibrium exists for the B-game but not the G-game. 3. For both B and G-games, for any  > 0, there exists an -equilibrium4 of this TS game in which the set of profit-maximizing teams equals the set of efficient teams, and each agent's ask and the principal's profit are arbitrarily close to the corresponding pure strategy Nash equilibrium outcome of the B-game. Example 1. Let n ˆ 3, and suppose F(1,2) ˆ 1100, F(1,3) ˆ F(2,3) ˆ 1000, F(T) ˆ 0 otherwise. Then the Nash equilibrium outcome is vN ˆ (100,100,0), with the principal's profit N ˆ 900. This assumes that, other things being equal, the efficient team {1,2} is selected. If any profit-maximizing team can be selected, then the corresponding equilibrium outcomes of the type described above are given by ve ˆ (100ÿ1,100ÿ2,0), with  ˆ 900 ‡ 1 ‡ 2. Again, the team {1,2} is selected. 2

We therefore assume that agents are risk-neutral, which is done for simplicity of the analysis. The assumption does not affect the analysis of the pure strategy Nash equilibria of the TS games. 3 The proofs of Propositions 1±3 are given in Appendix A. 4 Formally, for any given  > 0, an -equilibrium of a TS game is a set of asks v ˆ (v1,. . .,vn) such that for any i2N, Ui …vi jvÿi †  Ui …~vi jvÿi † ÿ  for all ~vi .

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In a Nash equilibrium, efficient agents decrease their asks to the point where no team can provide a principal with a higher level of profit then an efficient team (it can also be shown that all efficient teams provide the same level of profit in equilibrium). If any profit-maximizing team can be selected by the principal, then in equilibrium efficient teams must provide strictly greater profits than all other teams. This creates an existence problem for the pure strategy Nash equilibria of the TS games, just as it does in the first price auction with complete information, where the bidder with the highest value needs to bid just above the second highest value in order to win, but his Nash equilibrium bid is undefined. To avoid the problem (which vanishes if only discrete increments of, say, $0.01 are allowed), we use the notion of -equilibrium, in which efficient agents decrease their Nash equilibrium asks, if they were positive, by arbitrarily small amounts to guarantee that they are selected with certainty. Thus, TS games and their outcomes may differ depending on fine details of the selection rule. However, if the agents follow the competitive -equilibrium behavior described above, the sets of outcomes of any TS game are essentially the same as the sets of pure strategy Nash equilibria of corresponding B-games. 2.3. Collusive outcomes Let us call an outcome collusive relative to a given Nash equilibrium5 outcome if every agent who can affect the outcome of the game is strictly better off under this outcome than in the given Nash equilibrium. If, in addition, the principal's profit is zero, then the outcome is fully collusive. For a given TS game, we can identify the outcomes of this game with the corresponding vector of strategies vc that induces it; then a vector of strategies is collusive if for all i2N, Ui …vc † > Ui …vN †: In the presence of inefficient (marginal) agents, collusive outcomes may occur when efficient agents trade off their certain but low payoffs under Nash equilibria for higher, but riskier, expected payoffs under collusion. We note the following properties of collusive outcomes. Proposition 2. (i) The sets of collusive outcomes of G-games and B-games may be different. (ii) If each agent i2N `trembles' and submits an ask vi ‡ i, where vi is the intended ask and is are i.i.d. random variables with E(i) ˆ 0, then the sets of collusive outcomes of the B- and G-games are identical. (iii) The teams that are selectable under collusive outcomes of TS games are not necessarily efficient. (iv) The principal's profits are strictly lower under a collusive outcome than under the corresponding Nash equilibrium. Consideration of trembles represents the idea that agents' behavior may be noisy; if so, ties among profit-maximizing teams occur with probability zero and therefore B- and G-games become equivalent. Observe that such random trembles are quite different from 5 For simplicity of exposition, where it does not cause confusion, we will refer to the -Nash equilibria as Nash equilibria.

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the above-mentioned -equilibrium strategies, which involve undercutting the opponents. Besides trembles, collusive outcomes may involve agents using `mixed strategies,' as the example below demonstrates. Example 2: Collusive outcomes. As before, let n ˆ 3, F(1,2) ˆ 1100, F(1,3) ˆ F(2,3) ˆ 1000. The Nash equilibrium outcome is vN ˆ (100,100,0) and the principal's profit is N ˆ 900. The following set of outcomes is collusive under the Ggame, but not the B-game: v ˆ (x,x,xÿ100) with 150 < x  550; the selection probabilities and expected payoffs of agents are p(v|G) ˆ (2/3,2/3,2/3) and U1(v/G) ˆ U 2 (v|G) ˆ 2x/3 > 100, U 3 (v|G) ˆ 2/3(x ÿ 100) > 0; the principal's profit is c ˆ 1000 ÿ 2x < 900. The corresponding outcome with trembles is v ˆ (x ‡ 1, x ‡ 2, x ÿ 100 ‡ 3), E(i) ˆ 0, which is collusive under both B- and G-games. Another example of collusive solutions involves agents using mixed strategies. Suppose that the efficient agents submit asks v1 ˆ v2 ˆ 140, whereas the inefficient agent submits v3 ˆ 140 with probability 1/2, and v3 ˆ 30 with probability 1/2. The solution is collusive under both B- and G-games, with the efficient team being selected half of the times. The corresponding selection probabilities and expected payoffs Q for agents are p(v) ˆ (3/4, 3/4, 1/2), U ˆ (105,105,15); the principal's expected profit is c ˆ 835. 2.4. Summary and discussion The competitive -Nash and the collusive behavioral hypotheses result in different theoretical predictions regarding the outcomes of the TS games. If agents collude, the sets of collusive outcomes differ between the B-games and the G-games; if they behave competitively, the two games produce essentially identical outcomes. In experimental settings, the agents' behavior is likely to be noisy, and therefore the practical difference between the rules of the games in the two treatments may be small. Yet, if the differences in the tie-breaking rules affect the subjects' perceptions about the competitive nature of the game, ± which may include both perceptions of the rules of the game itself and of how competitively their opponents are likely to play it, ± we may expect to see the differences in the subjects' behavior between the two treatments even if ties do not occur very often. 3. Experimental design 3.1. The benchmark Bolle (1994) conducted a set of three-person one-round TS experiments with a principal (being a subject) and two symmetric agents. Each subject made one decision in each role. The productivity parameter values were given by F(1,2) ˆ 100, F(1) ˆ F(2)2{0,40,50,80,100}. Bolle reports that subjects, when making decisions in the roles of agents, did not follow the Nash equilibrium behavior. For different (symmetric) values of agents' productivity parameters, the most common asks were of around 35 each, suggesting a split pattern (35,35,30) of total output of 100 among two agents and the principal. Bolle argues that this result supports the hypothesis that fairness,

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Table 1 Experimental parameter values, the Nash equilibrium predictions and examples of collusive outcomes Solution

NEq (B) e-NEq Coll. (G) Coll. B&G

Team T F(T) P

i2T vi

(T) P

i2T vi

(T) P

i2T vi

(T) P

i2T vi

(T)

{1} ÿ1

{2} ÿ1

{3} ÿ1

100 Ð 100ÿe Ð 550 Ð 700 Ð

100 0 Ð Ð 100ÿe 0 Ð Ð 550 450 Ð Ð 700 300 Ð Ð

{1,2} 1100

{1,3} 1000

{2,3} 1000

200 900 200 ÿ 2 900 ‡ 2 1100 0 1400 Ð

100 900 100ÿ 900‡ 1000 0 1000 0

100 900 100 ÿ  900 ‡  1000 0 1000 0

T selected in B-game

G-game

{1,2}

any

{1,2}

{1,2}

{1,2}

any

{1,3} or{2,3}

in the sense of equal split, and collusion considerations often play an important role in agents' behavior. 3.2. The design We consider the following three-agent TS game with asymmetric productivity values for teams. The team productivities for T  N ˆ {1,2,3} are given by F(1) ˆ F(2) ˆ F(3) ˆ ÿ1, F(1,2), ˆ 1100, F(1,3) ˆ F(2,3) ˆ 1000, F(1,2,3) ˆ 0. Thus, only two-agent teams are selectable, and there are two efficient (Agents 1 and 2) and one inefficient (Agent 3) agents. Asymmetry in teams' productivities and presence of an inefficient (marginal) agent allows us to test the robustness of collusive tendencies in agents' behavior found by Bolle (1994) in symmetric settings. It also allows to distinguish between the B- and G-games, and therefore to consider the effects of differences in selection rules on agents' behavior. Table 1 presents competitive Nash and -Nash equilibria and examples of fully collusive outcomes of the game.6 Note that, in contrast to Bolle's design, (i) There is a unique Nash equilibrium of the game, with Agent 3 getting a zero payoff; moreover, irrespective of others' actions, each efficient agent can guarantee himself a certain payoff of 100 by following the Nash equilibrium strategy; (ii) The sets of collusive outcomes of the B- and G-game differ. Specifically, the G-game provides more opportunities for the agents to collude using pure strategies: Proposition 3. For the given parameter values, 1. If only strategies are considered, then the set of collusive solutions of the G-game includes the set of collusive solutions of the corresponding B-game as a proper subset: Bc  Gc ; 2. Any collusive outcome that results in the principal's expected profit above 800 involves agents using mixed strategies. 6 A more detailed description of the sets of collusive solutions of the games is given in the proof of Proposition 3 in Appendix A.

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Table 2 Features of experimental design as compared to Bolle (1994)

Multiperiod experiments Is principal a subject? Number of agents in a group Symmetric productivities Varying productivity values Unique Nash equilibrium Varying selection rule Varying subjects' roles

Bolle

Our design

No Yes 2 Yes Yes Varies No Yes

Yes No 3 No No Yes Yes No

3. In any symmetric (with respect to the Agents 1 and 2) solution such that the efficient team {1,2} is selected with probability p, the lower bound on the efficient agents' expected ask consistent with collusive behavior is equal to 100/(p ‡ 0.5(1 ÿ p)). Due to asymmetries between efficient and inefficient agents, there is no obvious fair collusive outcome that may serve as a focal point if agents collude. In the G-game, collusive solutions of the form v ˆ (x,x,x ÿ 100),150  x  550, such as v ˆ (550,550,450), have some features of being fair, in the sense that each agent is selected with equal probability 2/3. Trembles can insure that the outcomes are collusive under the B-game as well. Other possibilities for collusion in both games are quite rich, especially if the mixed strategies are considered (see Example 2 in Section 2). Proposition 3(2) indicates that in a certain sense, the region of outcomes with the expected profit below 800 offers more possibilities for collusion than when profits increase above 800. Proposition 3(3) identifies the lower bound on expected collusive asks for efficient agents as a function of their selection probability. We will later consider the selection frequencies and the mean payoffs for inefficient and efficient agents as an indicator of how important collusion and fairness considerations are to the agents. Further features of our experimental design as compared to Bolle's are summarized in Table 2. 3.3. Experimental procedures A total of 14 computerized experiments were conducted, seven B-games and seven Ggames. Two subject pools were used: Caltech students (USA), and Melbourne university students (Australia); most students were undergraduates. All subjects had no previous experience with the game. Four B-games and three G-games were conducted at Caltech, and three B-games and four G-games at Melbourne university. The team selection process was computerized. In each treatment, the subjects were fully informed about the productivity values of all teams and the details of the selection rules.7 Each experiment involved 12 or nine subjects, and included between 28 and 31 repetitions (periods), preceded by two practice periods. The number of periods was 7

The experimental instructions are enclosed in Appendix B.

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unknown to the subjects, but the last period was announced. At the beginning of each period, the subjects were divided by the computer into three-subject groups; subjects' opponents in groups changed from period to period, but their types (efficient or inefficient) stayed unchanged to eliminate possible repeated game reciprocity effects. Each period was organized as a sealed bid, independent for each group.8 The subjects submitted their asks (constrained to be non-negative) to the computer, and at the end of the period, were informed about the asks submitted, the corresponding profits of teams, the team selected and the payoffs of subjects in their group. At the end of the experimental session, the subjects were paid in accordance with their accumulated payoffs in the experiment using the exchange rate $0.005 per franc (U.S. and Australian dollars, respectively, were used). The participation fee of $8 (the value of which was private information) was added for the payoff of inefficient agents to compensate for the potential differences in earnings. 4. Experimental results The aggregate findings on the presence of competitive tendencies, sustainability of collusion, fairness, and on the differences between treatments and subject pools in TS experiments are summarized in Results 1±4 below. The experimental data presented in this section are pooled by subject pools and selection rules and are structured into group data, which include descriptive statistics regarding level of profits and efficiency of the teams, and individual data on subjects' asks. We have also divided the data into three time intervals: Time 1 ± Periods 1±10; Time 2 ± Periods 11±20; Time 3 ± all the following periods. According to the proximity to the Nash equilibrium prediction, the observations are divided into two regions: the region of outcomes consistent with a wide range of collusive behavior where the profits of selected teams are below 800, (the `collusive' region); and the outcomes close to the Nash equilibrium, with profits above 800.9 We first consider the evidence on the presence of competitive or collusive tendencies in the subjects' behavior. Result 1. In all TS experiments, most of the time, the subjects followed neither fully collusive nor competitive Nash equilibrium behavior. Yet, competitive tendencies in the subjects' behavior were present and rather persistent: the individual asks decreased from period to period, and the principal's profits from the selected teams increased. On the contrary, collusive tendencies, if they were present, were not sustainable.

8 A curious observation is that an English auction cannot be used to test one shot team selection games, since the English auction which allows ties among teams induces a quite different sequential game among the agents. Moreover, a large number of collusive solutions can be supported as Nash equilibria in an English auction version of this game. In a pilot experiment conducted at Caltech as an English auction which allowed ties, the subjects did use such strategies to sustain collusion (see also Sherstyuk, 1999). 9 We will also report more detailed statistics regarding treatments, experiments or subjects, when they are of special interest. The full data for each experiment and each individual are available from the author upon request.

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Fig. 1. Average per period profits of selected teams in B- and G-experiments. `all' ± all data; `cit' ± Caltech data; `mel' ± Melbourne data; `nash' ± the Nash equilibrium profit.

Support: Figs. 1 and 2; Tables 3±5. In all experiments and all time intervals, both median and average per period asks were above the Nash equilibrium levels but below the fully collusive levels. The profits of selected teams were within the `collusive' range (below 800 francs) in 60 percent of group observations, but were in the region close to the Nash equilibrium in the other 40 percent of group observations (see Table 6). The percentage of efficient teams selected was around 50 percent in all time intervals, which is below the fully competitive level of 100 percent, but above the `fair' collusive level of 33 percent. Table 3 Experimental results, summary statistics, all data Mean (SD)

Period 1

Time 1

Time 2

Time 3

Total

Nash Eq prediction

Average asks, efficient agents

344.7 (561.6) 315.6 (585.6) 582.0 (238.3) ± ±

292.1 (548.7) 252.8 (644.6) 653.2 (161.0) 47.8 (8.9)

195.8 (77.02) 150.9 (485.1) 733.2 (105.6) 50.2 (8.0)

195.2 (295.3) 125.4 (402.8) 785.7 (102.3) 48.0 (12.0)

224.8 (356.0) 172.5 (515.6) 726.4 (136.3) 48.5 (5.3)

100

Average asks, inefficient agents Average profits % efficient teams selected

0 900 100

Asks are per person averages of all submitted asks, in francs. Profits are the averages for selected teams, in francs. Percentage of efficient teams selected is the average across experiments, in %. Standard deviations are given below the means.

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Fig. 2. Median per period asks in B- and G-games. Above: efficient agents; below: inefficient agents. `all' ± all data; `cit' ± Caltech data; `mel' ± Melbourne data; `nash' ± the Nash equilibrium asks; `coop' ± lowest asks consistent with collusive behavior in pure strategies.

Yet, the dynamics of the data, ± the changes in individual asks and profits of the teams over periods, ± strongly argue for the presence of competitive tendencies in the subjects' behavior in both B- and G-experiments. In fact, in their dynamics, the experiments were similar to a wide range of market experiments (Plott, 1989), with a possible trend of convergence towards the competitive Nash equilibrium. Although the process was slow and noisy, the trend is apparent from Figs. 1 and 2. Table 4 presents the results of

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Table 4 Ordinary least squares regressions of profits of selected teams and asks of selected agents on the time trend (period) and dummy variables corresponding to different subject pools (cit ˆ 1 for Caltech), selection rules (srule ˆ 1 for G-games), agents' roles (role ˆ 1 for efficient agents) and their products. Dependent variable

Profit

Independent variable

Estimated coefficient

t-Statistic

Estimated coefficient

t-Statistic

one cit srule citrule role citrole period periodrule citperiod citperiodrule periodrole citperiodrole

644.74c 36.54b 50.70c ÿ287.37b Ð Ð 5.19c ÿ1.99b 1.79a 6.04c Ð Ð

58.70 2.44 3.43 ÿ13.54 Ð Ð 8.93 ÿ2.54 2.20 5.19 Ð Ð

161.23c ÿ27.68c ÿ16.34c 134.23c 44.46c 22.75b ÿ3.47c 0.71a ÿ1.02a ÿ2.70c 1.49c ÿ0.15

24.00 ÿ2.99 ÿ2.77 15.87 6.55 2.38 ÿ9.82 2.27 ÿ1.96 ÿ5.82 4.19 ÿ0.29

No. of observations Corrected R2

Individual ask

1668

0.406

3328 0.465

a

Significant at 5 percent level. Significant at 2 percent level. Significant at 1 percent level.

b c

Table 5 Last period selection and directions of individual ask changes, percent Selected last period % of asks increased % of asks unchanged % of asks decreased

All data

Caltech

Melbourne

Yes

No

Yes

No

Yes

No

43.5 30.1 26.4

10.6 11.9 77.6

37.1 30.3 32.7

7.8 11.1 81.0

49.5 30.0 20.5

13.1 12.5 74.4

Table 6 Average per period individual ask changes in the two regions of outcomes, francs Profit < 800

All data Efficient agents Inefficient agents Caltech, B-games Caltech, G-games Melbourne, B-games Melbourne, G-games

Profit  800

Total

Ask change

% of obs.

Ask change

% of obs.

Ask change

% of obs.

ÿ20.63 ÿ13.49 ÿ34.91 ÿ30.90 ÿ17.42 ÿ27.26 ÿ10.51

60.0 60.0 60.0 38.3 89.1 68.5 51.4

3.95 1.01 9.85 0.34 ÿ0.53 15.96 2.76

40.0 40.0 40.0 61.7 10.9 31.5 48.6

ÿ10.80 ÿ7.68 ÿ16.99 ÿ11.64 ÿ15.57 ÿ13.67 ÿ4.05

100 100 100 100 100 100 100

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regressions of profits of selected teams and individual asks of selected agents10 on the time trend and a set of dummy variables corresponding to different subject pools, selection rules, and agent-types.11 The results indicate that the increasing trend in profits and the decreasing trend in asks are statistically significant at the 1 percent level (according to the t-test) for both treatments and subject pools. Directions of individual ask changes (Table 5) are also largely consistent with the competitive behavioral hypothesis. Table 5 shows that 77.6 percent of the time the subjects decreased their asks, that is, acted in the direction of the Cournot best response to the last period asks following their non-selection. Moreover 26.4 percent of the times the subjects competed even more ``aggressively,'' decreasing their asks after being selected in the previous period. On the issue of fairness, we note the following. Results 2. Overall, the evidence for the presence of fairness in the subjects' behavior is limited: not only were the inefficient agents' payoffs lower than the payoffs of efficient agents in the case of selection, but they were selected less often. Support: As noted above, the percentage of efficient teams selected was far above the `fair' level of 33 percent (Table 3). On the individual level, inefficient agents were selected about 50 percent of the times, as compared to about 75 percent for the efficient agents. These selection frequencies were about the same for both treatments, both subject pools and both regions of outcomes (with profits above and below 800 francs). In Caltech experiments, inefficient agents had a slightly higher selection frequency in the G-games (55.4%) than in the B-games (50.5%); in Melbourne experiments, the inefficient agents were selected slightly more often (50.1% of the times) in the `collusive' region of outcomes than in the region of outcomes close to the Nash equilibrium (42.4%). Thus, the evidence for competitive tendencies is rather strong. However, there is an interesting feature that distinguishes these experiments from the perfectly competitive (or contestable) experimental markets. We observe that the agents' behavior in the region of outcomes close to the Nash equilibrium was often at odds with the competitive prediction. Results 3. Contrary to the competitive prediction, none of the experiments fully converged to the Nash equilibrium. In the region of outcomes close to the Nash equilibrium (Profit > 800), the average per period individual ask changes were positive. Although in Time 3 of the experiments about 40 percent of efficient agents were worse off than under the Nash equilibrium in terms of their average payoffs, the inefficient agents were never fully excluded from the market. Support: Tables 3 and 6 and Fig. 3. Even if the agents' actions follow a multi-period disequilibrium adjustment process, the competitive prediction implies that inefficient agents' asks should decrease as long as they are positive, and then stabilize at zero; the 10 The data on the selected teams and agents were used in the regression analysis to reduce the noise. Similar tests were performed using profits of all teams and asks of all agents, producing the same qualitative results. 11 Similar regressions were carried out allowing for each period effects to be different. In all cases, the restriction of a linear time trend was accepted according to the F-test.

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Fig. 3. Distribution of average per period per person payoffs in Time 3, Caltech and Melbourne experiments.

inefficient agents should then be excluded from the market, and the efficient agents be selected with certainty. Yet, Table 6 shows that in the range of outcomes with (Profit > 800), the average per period ask change for inefficient agents was positive (9.85 francs),12 while the selection frequencies for inefficient teams in late periods of the experiments did not change much (Table 3). Fig. 3 indicates that, in terms of the average per period payoffs, the outcomes were beneficial to inefficient agents, but not to all efficient agents. To take a closer look at these tendencies, consider representative dynamics of individual asks and profits of selected teams, presented in Fig. 4. The figure demonstrates that, contrary to the competitive prediction, in the regions of outcomes close to the Nash equilibrium, the inefficient agents either stabilized at some low but positive level of asks, or, in several cases, even increased their asks. The former tendency can be explained by 12 In fact, in this range of outcomes, the inefficient agents increased their asks, following their non-selection, in 21.2 percent of cases, as compared to 9.7 percent of cases in the range of outcomes with Profit  800.

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Fig. 4. Representative dynamics of inefficient agents' asks (above) and profits of selected teams (below). Left: Caltech data; right: Melbourne data. Above & ± individual asks; 4 ± Nash equilibrium asks. Below: 4 ± output of selected team; & ± profit of selected team; * ± Nash equilibrium level of profit.

the presence of a positive reservation payoff for the subjects. The latter may only be perceived as noise within the framework of a one-shot game. However, it is possible that some inefficient agents, facing low payoffs, tried to induce cooperation in subsequent periods of the experiment by sending cooperative signals. Such `cooperative' attempts were weak and unstable; the profits of selected teams did not decrease (Fig. 1) Nevertheless, these `non-competitive' actions were beneficial for the inefficient agents as compared to the Nash equilibrium. The efficient agents evidently tried to fine-tune their actions to the actions of the inefficient agents, either in an attempt to support cooperation, or rationally responding to weakened competition and following the match-and-undercut strategy. Remarkably, the efficient agents were not uniformly better off under these outcomes than under the Nash equilibrium; in Time 3, 42.6 and 38.9 percent of efficient subjects for Caltech and Melbourne, correspondingly, had average per period payoffs below the Nash Equilibrium level of 100. (The corresponding numbers for Time 1 are

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11.1 and 18.5 percent, and for Time 2 are 25.9 and 18.5 percent for Caltech and Melbourne, respectively; since the number of efficient agents `losing out' increased in time, it cannot be attributed simply to the subjects not understanding the game.) Per experiment average payoffs for efficient agents in Time 3 were below the Nash equilibrium level in three out of seven Caltech experiments and four out of seven Melbourne experiments. We thus observe, on the part of the efficient agents, some evidence of behavior inconsistent with payoff maximization. It may be explained by either bounded rationality, or, possibly, fairness considerations which became important at the margin. The next group of results regards the differences in experimental outcomes between the treatments, subject pools, and behavior of efficient and inefficient agents. Results 4. 1. In Caltech experiments, the outcomes of B-games were significantly different from the G-games: they were closer to the Nash equilibrium prediction than the G-games. In Melbourne experiments, the differences between the outcomes of B- and Gexperiments were less significant. 2. Caltech subjects were, overall, more competitive than Melbourne subjects: their average per period asks decreased faster. 3. In both B- and G-experiments, inefficient agents were, overall, more competitive than efficient agents: their average per period asks decreased faster. Support. Figs. 1 and 2 and Tables 4 and 6. In Caltech experiments, the average profits of selected teams were 789.4 francs in B-games, as compared to 615.6 francs in G-games. Consequently, only 38.3 percent of the outcomes of B-experiments at Caltech were in the `collusive' region (Profit < 800) for B-games, as compared to 89.1 percent for G-games (Table 6). In Melbourne experiments, the initial levels of profit were lower in B-games than in G-games, but per period rates of profit increase were higher; overall, the difference between the average profits was small (730.4 francs in B-games as compared to 747.8 francs in G-games). Table 4 indicates that all the differences between the treatments, subject pools and subject roles reported in Result 4 are statistically significant at the 5 percent level according to the t-test. Two questions are of immediate interest. First, what can explain the subject pool differences? We observed that Caltech subjects were more competitive than Melbourne subjects in terms of overall rates of ask decreases; moreover, the behavior of Melbourne subjects was similar across treatments, while Caltech subjects behaved quite differently in B- and G-experiments. Second, can the differences between the treatments in Caltech subjects' behavior be attributed to differences in the tie-breaking rules? We do not have an immediate answer to the first question. One possible explanation for the differences between Caltech and Melbourne University subject pools is the previous exposure to economic experiments. Some Caltech subjects who took part in TS experiments had previously participated in other economic experiments, while most Melbourne subjects had no prior experiences with economic experiments. It is quite possible that a number of Caltech subjects, due to their prior experiences with market-type experiments, paid more attention to the details of tie-breaking rules in

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Table 7 Frequencies of ties between profit maximizing teams, percent %

B-games G-games

Caltech experiments

Melbourne experiments

Profit < 800

Profit  800

Total

Profit < 800

Profit  800

Total

4.6 11.4

4.2 6.4

4.4 10.8

9.2 11.3

13.3 6.1

11.2 9.6

experimental instructions, and also had a better idea of how to compete in similar environments.13 The effect of experience on the behavior of Melbourne subjects will be discussed in the next section. On the second issue of whether the differences in the outcomes between the treatments in Caltech experiments were due to differences in the tie-breaking rules, the evidence is mixed: 1. On the one hand, the differences are largely explained by higher first period asks rather than by a slower rate of ask decrease (Table 4); in fact, per period asks decreased faster in the G-games than the B-games! Thus, even if the subjects had more cooperative predispositions in the early periods of the G-games as compared to the Bgames, these cooperative tendencies were not sustainable. 2. On the other hand, the data on the frequencies of ties occurring between profitmaximizing teams (Table 7) are quite revealing. In Caltech experiments, ties occurred more than twice as often in the G-games (10.8% of times) then in the B-games (4.4% of times). Ties were especially frequent in Experiment 2 (B-game) at Caltech (18.5% of times), which involved, among other subjects, several graduate students, and was the most collusive of all experiments by a number of indicators; for example, the agents' asks decreased, on average, by only 5.89 francs per period (as compared to the average of 20.63 francs for the `collusive' region of outcomes). Consequently, the outcomes stayed within the `collusive' region in all periods of this experiment; in Time 3, the mean profits of selected teams were as low as 660.3 francs, as compared to the overall average of 785.7 francs. We thus have a weak indication that fine details of selection rules may make a difference for the outcomes. Yet, we do not have direct evidence, with the exception of one experiment, that G-games induced relatively more cooperative behavior in terms of slower rates of ask decrease. The evidence is not supported by the Melbourne data. In the latter subject pool, ties occurred, overall, more often than in Caltech experiments, once again indicating a relatively lower level of competitiveness of Melbourne subjects. 13 Alternative explanations of a more competitive behavior of Caltech subjects as compared to Melbourne subjects may include a stronger game-theoretic background in the former subject pool (subjects' records indicate that a small number of Caltech subjects were able to calculate the Nash equilibrium outcomes of the given TS game early in the experiment; there is no such evidence for Melbourne subjects), or even differences in national characters. Unfortunately, we do not have the data that would allow us to test the above hypotheses: The subjects were not asked about the number of economic experiments they have previously participated in, or about the level of game theory courses they have taken; neither did they go through a screening test for competitive predispositions.

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5. Effect of experience Although we observed strong competitive tendencies in the subjects' behavior, none of the TS experiments fully converged to the Nash equilibrium prediction (Result 3). Further, on average, Melbourne experiments converged to the Nash equilibrium slower than Caltech experiments (Result 4). To test whether these phenomena were simply due to subjects' inexperience with TS games, and whether experience would result in a faster dynamics and in complete convergence to the Nash equilibrium, we conducted two additional experiments at Melbourne university with the subjects who had previously participated in one of the earlier TS experiments. The first experiment (Experiment 1E) was a B-game which involved 12 subjects and lasted for 42 periods; the second (Experiment 2E) was a G-game with nine subjects which lasted for 52 periods. Fig. 5 shows the representative dynamics from each of the two experiments, and Table 8 displays average asks, selection rates and payoffs of subjects in the last periods of

Fig. 5. Representative dynamics of experiments with experienced subjects. Above: asks of a representative inefficient agent; below: mean profit of selected teams. & ± actual data; 4 ± the Nash equilibrium prediction.

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Table 8 Average per period per person asks, selection rates and payoffs in experiments with experienced subjects, starting from Period 30 Mean (SD)

Efficient agents Inefficient agents

Experiment 1E Ask (francs) 111.2 (22.4) 26.2 (24.5)

Experiment 2E Sel.rate (%)

Payoff (francs)

Ask (francs)

Sel.rate (%)

Payoff (francs)

0.79 (0.13) 0.41 (0.22)

83.2 (45.6) 6.8 (9.9)

172.0 (102.1) 156.0 (185.1)

0.74 (0.18) 0.51 (0.16)

109.4 (87.9) 29.2 (56.0)

Standard deviations are listed below the means.

the experiments (Period 30 and onwards). As it is evident from Fig. 5, the first 30 periods of each experiment were clearly characterized by decreasing trends in individual asks and increasing trends in profits of selected teams. In fact, regression analysis analogous to the one reported in Table 4 can be employed to show that per period profits increased in the first 30 periods of these experiments significantly faster than in earlier Melbourne experiments, but still slower that in Caltech experiments.14 However, the two experiments exhibited quite different dynamics in later periods. Experiment 1E converged to the levels very close to the Nash equilibrium. The inefficient agents' asks in the last 10 periods were positive but negligible.15 Yet, remarkably, the efficient agents' asks also stayed just above the Nash equilibrium level, resulting in uncertain selection outcomes and, on average, lower payoffs (83.2 francs) then in the Nash equilibrium (100 francs). Experiment 2E was completely different. The dynamics of the experiment changed, after about 30 periods, from a steady convergence to the Nash equilibrium to a noisy oscillation significantly above the Nash equilibrium level of asks. In fact, in the late periods of the experiment, many subjects behaved in a manner not inconsistent with collusive behavior: the average payoff of efficient agents in this time interval was 109.4 francs, or slightly above the Nash equilibrium payoff. Thus, we observed that experience may lead to faster convergence rates at the initial stages of the experiment, but may also help to sustain noisy collusive outcomes later in the experiment. We did not observe that experience led to complete convergence to the Nash equilibrium and to exclusion of inefficient agents from the market, even if the realized outcomes yielded lower payoffs for efficient agents. 6. Individual behavior Due to instability of collusion, competitive tendencies may prevail in an experiment even if many agents have collusive predispositions. In this section, we consider the 14 The descriptive statistics, regression results, and the full data on the experiments are available from the author upon request. 15 Given the exchange rates, the asks were of the magnitude of 5±10 cents. Positive asks may be attributed to the subjects' reservation costs of being selected, as discussed earlier.

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competitive and cooperative tendencies in the subjects' behavior on an individual level. We identify factors that were important in determining individual asks, and then use simple econometric analysis to distinguish the individuals who actively induced competition (defined as a decreasing trend in asks) from those who merely followed the trend induced by others. 6.1. Individual behavioral rules As it is evident from the earlier data analysis, the agents' asks followed a disequilibrium adjustment process. Under such a process, each individual's optimal action depends, in general, on his or her expectations about other agents' asks. To specify the agents' expectations about their opponents' actions in each period, we assume that agents take the asks observed in the previous period as a signal about the current level of asks.16 We further assume that each subject uses a simple linear ask adjustment rule of the form: vit ˆ 0i ‡ 1i  vitÿ1 ‡ 2i  fi …vtÿ1 †

(2)

where vit is individual i's ask in period t, vitÿ1 is his own ask in the previous period, and fi(vtÿ1) is a function of the observed outcomes of the previous period. For each agent, fi(vtÿ1) could be either (1) the Cournot±Nash best response ask,17 resulting in a best response (BR) type rule; or (2) the `median' ask, that is, the highest observed winning ask adjusted for role differences,18 resulting in a median ask (MA) type rule; or (3) a zero± one dummy variable indicating the agent's selection outcome, resulting in a selection response (SR) type rule. For each of 162 subjects who participated in the experiments, we used the least squares estimations to evaluate the coefficients on alternative rules19 and further selected one of the three rules as a model of individual behavior using the Davidson and McKinnon J-test

16 The similar approach is employed in the Nash±Cournot best response model which has been proposed and used before to analyze subjects' behavior in the context of public good experiments (Ledyard, 1978; Chen and Plott, 1996). We performed a pooled least squares regression of individual period-to-period ask changes on the observed history variables for up to three periods back. The results support the validity of the one period lag history dependence hypothesis. 17 The exact formulas for calculating the best response ask are given in Appendix C. 18 To adjust the asks to symmetric levels, we may either decrease the efficient agents' asks by 100 for inefficient agents or increase the inefficient agent's ask by 100 for efficient agents. 19 The presence of the non-negativity constraint on subject's asks, vi  0, introduces a potential censorship problem into the data, and could bias the least squares estimations for individuals with many observations close to zero. This is especially a concern for inefficient agents whose Nash equilibrium asks are equal to zero. However, Tobit analysis cannot be applied since the data contain no observations at zero. More importantly, the rationality argument implies that the constraint vi  0 is not binding and therefore cannot affect the subjects' behavior. Maximum likelihood estimations with models which allowed the distribution of a random variable to depend of on the values of the regressors produced very similar results to OLS estimations even for subjects with many observations close to zero. We therefore adopt the least squares estimation method. Further, since the regression models contain lagged dependent variables, we performed the h-test for the presence of autocorrelation as described in Green (1990, p.454). According to the results, in most cases the hypothesis of no autocorrelation was sustained.

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Table 9 Frequencies of individual behavioral rules, percent %

BR ask (BR)

Med. ask (MA)

Selection (SR)

Unclassified

Total

All data Efficient agents Inefficient agents

25 21 32

40 41 37

25 28 20

10 10 11

100 100 100

Caltech, all data Efficient agents Inefficient agents

15 13 19

49 52 44

22 22 29

14 13 8

100 100 100

Melbourne, all data Efficient agents Inefficient agents

33 30 41

30 30 30

26 25 26

11 15 3

100 100 100

(Green, 1990, p.231) for comparison of non-nested linear models20 as well as F-statistics for the significance of regressions.21 Table 9 displays the results, indicating how often each of the three rules ± BR, MA and SR ± was used by the individuals. Overall, all three rules were used in significant proportions. In Caltech experiments, the median ask rules were used more often than other rules (49% of the subjects), whereas in Melbourne experiments, there were slightly more subjects using the best response type of rules. 6.2. Competitiveness of rules We next consider the competitiveness of individual rules by analyzing the dynamics of asks that a given rule induces in a population of identical agents who submit equal adjusted asks in every period. If a rule induces a statistically significant decreasing (increasing) trend of asks, we classify the corresponding behavior as competitive (cooperative); if the trend is insignificant, we classify the behavior as neutral. The assumption of identical agents' asks allows to capture the idea of a given rule actively inducing a given trend in asks, as opposed to following the trend induced by others. We also test whether, according to a given rule, the asks were converging to the Nash equilibrium asks. A detailed description of statistical procedures used to analyze the competitiveness of the rules is given in Appendix C. Here we turn to the results. Result 5. The share of actively competitive subjects was significant in all experiments. Furthermore, the differences in experimental outcomes between the treatments and subject pools can be traced back to the differences in the shares of competitive subjects among all subjects.

20

The Davidson and McKinnon test as well as a number of other estimation techniques discussed below rely on asymptotic properties of the estimates. Since the number of observations for each individual is relatively low (about 30), the results of such procedures may not be very accurate. Still, they will allow us to highlight some interesting qualitative features in individual behavior. 21 We considered the individual behavior unclassified if none of the three regression models had a value of F-statistic for the significance of the regression above the 5 percent significance level.

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Table 10 Classification of individual behavior by competitive categories, percent %

Agent-type

Neutral

Competitive All

All data

Cooperative

Unclassified

Converging to Nash? yes

no

Efficient Inefficient

42 44

47 41

43 19

4 22

1 4

10 11

Efficient Inefficient Efficient Inefficient

37 20 46 67

56 53 46 25

53 33 42 8

3 20 4 17

0 0 4 0

7 27 4 8

Efficient Inefficient G-games Efficient Inefficient Experiment 1E (B) Experiment 2E (G)

50 33 37 60 33 33

50 59 36 26 58.5 67

46 17 33 13 50 33.5

4 42 3 13 8.5 33.5

0 8 0 7 8.5 0

0 0 27 7 0 0

Caltech B-games G-games Melbourne B-games

Support: Table 10. The share of the competitive subjects was, on average, over 40 percent among both efficient and inefficient agents, and varied from 25 to 67 percent by the agent's type, treatment and subject pool. The share of competitive subjects in Caltech experiments was, overall, higher than in Melbourne experiments, especially if we consider the agents whose rules induced convergence to the Nash equilibrium. This explains why Caltech experiments produced more competitive outcomes than Melbourne experiments. Further, for Caltech subjects, the share of competitive agents was higher in the B-experiments than in the G-experiments, resulting in more competitive outcomes in the B-games. Next, interestingly, the results reflect the changes in the inefficient agents' behavior in the regions close to the Nash equilibrium discussed in Sections 4 and 5. The share of competitive inefficient agents whose rules decreased over some range of asks was high in all cases. Yet, for a relatively high number of inefficient agents (19% for Caltech and 26% for Melbourne) their rules did not prescribe the convergence of asks all the way down to the Nash equilibrium, but induced a `switch' from competitive to non-competitive behavior. In most cases, the switch occurred close to the Nash equilibrium level of asks (in fact, the stationary points for such inefficient agents often were not significantly above 50±100 francs), thus explaining the change in the dynamics of many experiments in the region of outcomes close to the Nash equilibrium. Note that practically none of the competitive efficient agents exhibited such a switch in their behavior. We conclude: Results 6. The changes in the dynamics of experiments in the region of outcomes close to the Nash equilibrium are explained by a high number of competitive inefficient agents who changed their behavior in this region.

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The above conclusions are reinforced by considering the behavior of experienced subjects (Experiments 1E and 2E). Both experiments had a share of actively competitive subjects above 50 percent, which was at least as high as in earlier experiments. The difference between Experiments 1E and 2E is also transparent on an individual level. While individual rules of most subjects in Experiment 1E prescribed convergence to the Nash equilibrium, in Experiment 2E all inefficient agents exhibited a switch in their behavior, thus distorting the convergence process. Finally, let us turn to the issue of collusion. As Table 10 shows, the number of neutral agents who were not actively inducing competition was high in all experiments. However, we have detected practically no cooperative-types who would actively sustain collusive tendencies. Cooperation efforts were mostly passive (neutral behavior) or late in the experiment (`switching' behavior); the asks decreased close to the Nash equilibrium level before the agents ± especially, inefficient agents ± exhibited some cooperation attempts. Neutral behavior by itself does not contradict collusive theories discussed in Section 2. Yet, the collusive solutions are unstable in two ways: first, each agent has an incentive to deviate unilaterally; second, cooperation cannot be sustained if any one subject deviates. Apparently, under both experimental treatments, the presence of a few competitive subjects resulted in the tendency of asks among all subjects to decrease until they approached the levels close to the Nash equilibrium. 7. Conclusions To summarize, our analysis revealed substantial amount of competition in subjects' behavior in both B- and G-experiments. Although the subjects did not strictly follow the Nash equilibrium behavior, they decreased their asks rather consistently until the outcomes reached the range close to the Nash equilibrium. The frequencies of selection of efficient teams indicate that fairness considerations were not decisive in determining subjects' behavior: the efficient agents not only received higher payoffs from selection, but were selected more often than the inefficient agents. Thus our results do not support Bolle's (1994) findings on the presence, ± or, at least, persistence, ± of collusive tendencies in the subjects' behavior. We have demonstrated that cooperation is non-robust to such factors as asymmetries of the agents' roles, small increases in the size of the teams and repeatedness of trials. We found weak evidence, based on the results of Caltech experiments, that more `symmetric' tie-breaking rules of the G-game induced less competitive behavior than the B-game. More importantly, however, the data indicate that the cooperative tendencies were not sustainable under either treatment. The TS experiments clearly demonstrated that the behavior of inefficient (marginal) agents was crucial in determining the outcomes of each experiment. In this respect the TS games exhibited many properties in common with traditional competitive markets, but, due to a small number of agents, their outcomes were more sensitive to the behavior of individuals. The asks decreased as long as the inefficient agents' payoffs were high enough in the case of selection. In the regions close to the Nash equilibrium outcomes, where inefficient agents' payoffs, even in the case of selection, were low, the dynamics of

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the experiments depended on whether the inefficient agents tried to break the convergence pattern by increasing or stabilizing their asks. Given that submitting a zero ask was a weakly dominated strategy, the inefficient agents kept their asks above zero in all cases, and the experiments never quite converged to the Nash equilibrium. Remarkably, that was so even when the outcomes stabilized at the levels of asks inconsistent with collusive behavior, and efficient agents were, on average, worse-off than under the Nash equilibrium outcomes. It is left unresolved whether this was due to the efficient agents' bounded rationality or because the efficient agents considered it unfair to keep inefficient agents excluded; the phenomenon did not vanish with experience. Thus it is quite possible that, after all, fairness played some role in the agents' behavior. While collusive tendencies were not sustainable, the competitive Nash equilibrium prediction also was not fully supported by the experimental results. Acknowledgements I would like to thank David Grether, John Ledyard, Charles Plott and David Porter for their help. I also benefited from discussions with many faculty members at the Division of Social Sciences at Caltech and the Department of Economics at Melbourne University. Financial support by Melbourne University (research grants 3-31608 and 3-31615) is gratefully acknowledged. My special thanks are to Wes Boudville for developing the experimental software. All errors are my own. Appendix A Proofs of the statements A.1. Proof of Proposition 1(3) Observe that in any pure strategy Nash equilibrium of a TS game, the sum of the asks submitted by the agents of any efficient team T* is necessarily higher than the sum of the ~ ~ If F(T*) > F(T) asks submitted profit-maximizing team T: Pby the agents of any P inefficientP P * ~ and F(T ) ÿ iT  vi ˆ F…T† ÿ iT~ vi then iT  vi > iT~ vi . By Proposition 1(1), in equilibrium only efficient agents may submit positive asks. Let k* denote the number of efficient agents who submit positive asks in equilibrium; then, by definition, every efficient team contains exactly k* agents who submit positive asks. We also know that all efficient teams are profit-maximizing in equilibrium. Hence, from the above, if an inefficient team is profit-maximizing, then it contains strictly fewer agents who submit positive asks than any efficient team: ~k < k . Take a pure strategy Nash equilibrium of a B-game; denote by v* ˆ (v1 ; :::; i ; :::n ) the equilibrium asks and let k* be the number of agents who submit positive asks. For any  > 0 consider the outcome of a corresponding TS game in which the agents use the following vector of strategies v: for every i2N, vi ˆ vi* ÿ i, where i ˆ 0 if vi* ˆ 0, and 0 < i < minfvi ; =k g if i > 0. Given such v, a team is profit-maximizing if and only if

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it is efficient. Indeed, for any efficient team T* and other team T~ that is profit-maximizing under the Nash equilibrium of the B-game, we obtain, by construction: X X i  i ; iT 

iT~

and, therefore, ~ v†; …T  ; v†  …T; with strict inequality in both expressions above if and only if T~ is inefficient. It is then straightforward to show that v is an -equilibrium strategy profile. A.2. Proof of Proposition 2 (i), (iii): Demonstrated in Example 2 in the text. (ii): If agents `tremble' as described in the proposition, then profits of any two teams are equal with probability zero; hence any team that is profit-maximizing in expectation has an equal chance of being selected in both B- and G-games. (iv): Follows from the definition which states that the agents' collusive asks are strictly higher than in the corresponding Nash equilibrium. A.3. Proof of Proposition 3 Consider the sets of collusive solutions of B- or G-games for the given parameter values. Since in actual experiments, efficient agents alternated between Roles 1 and 2, we restrict our attention to collusive outcomes that are symmetric with respect to the efficient agents' payoffs. Consider the solutions that do not involve agents using mixed strategies first. Under the risk-neutrality assumption, any collusive solution v should, by definition, satisfy:22 p1 …v†v1 > 100; p2 …v†v2 > 100; p3 …v†v3 > 0: Collusive solutions of the B-game. The only type of such solutions possible is where (only) Teams {1,3} and {2,3} are profit-maximizing and are selected with equal probability 1/2 each. The set of collusive solutions is Bc  f…v1 ; v2 ; v3 †jv1 ˆ v2 > 200; v3 > 0; v1 ‡ v3  1000; v1 ÿ v3 > 100g: Collusive solutions of the G-game. Two types of symmetric collusive solutions in pure strategies are possible: 1. The set of solutions where Teams {1,3} and {2,3} are selected with probability 1/2 each; this is the same as the set Bc of the B-game; 2. The set of solutions where Teams {1,2}, {1,3} and {2,3} are selected with probability 1/3 each. It is defined by: G2  f…v1 ; v2 ; v3 †jv1 ˆ v2 > 150; v1 ˆ v3 ‡ 100; v1 ‡ v3  1000g: 22 If the agents are risk-averse, than the lowest expected payoff which they consider equivalent to their Nash equilibrium payoff is higher than in the risk-neutral case, and therefore the set of collusive solutions will shrink.

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The Statements (1) and (2) of Proposition 3 then follow. (Note, however, that if mixed strategies are considered, then the set of collusive solutions expands considerably and (1) any level of expected profit strictly below 900 may be supported by collusive behavior; (2) there are solutions that are collusive under the G-game but not the B-game and vice versa. The B-game favors efficient agents in terms of selection probabilities and therefore allows lower asks to be consistent with collusive behavior; conversely, the G-game allows more collusive opportunities for inefficient agents.) Proposition 3(3) follows from the definition of collusive outcomes and the risk-neutrality assumption, and applies to symmetric collusive solutions in both pure and mixed strategies. Appendix B Experimental instructions B.1. Instructions This is an experiment in decision making. The instructions are simple and if you follow them carefully, you may earn a considerable amount of money that will be paid to you IN CASH at the end of the experiment. During the experiment all units of account will be in francs. Upon concluding the experiment the amount of francs you earned will be converted into dollars at a conversion rate of ... dollars per franc. Your earnings plus a lump sum amount of ... dollars will be paid to you in private. Do not communicate with the other participants except according to the specific rules of the experiment. If you have a question, feel free to raise your hand. An experiment monitor will come over to where you are sitting and answer your question in private. This experiment will last several PERIODs. At the beginning of each period, you will be assigned to a UNIT with two other participants. Each person within a unit will be given a role denoted by an A, B or C. You will not be told which of the other participants are in your unit. WHAT HAPPENS IN YOUR UNIT HAS NO EFFECT ON THE PARTICIPANTS THAT ARE NOT IN YOUR UNIT AND VICE VERSA. B.2. Market organization and payoffs In each period of this session you are going to participate in a GROUP SELECTION process. During the period, you will submit to the market your ASK, which is the amount of francs you want to be paid in case you are in the selected group. Only non-negative asks between 0 and 10,000 francs are allowed. During a period, you can submit only ONE ASK. You will not be informed about the asks submitted by the other participants in your unit until the end of the period. The period ends when all participants submit their asks. Once the period is over, a mechanism, which will be specified below, will select a group of participants on the basis of their asks.

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In each period, your EARNINGs are equal to your ask if a group containing yourself is selected; you earn zero otherwise. A group containing yourself cannot be selected if you do not submit an ask during the period. Example 1. Suppose in period 1 you are in role A, you submit an ask of 15, and in your unit the group {B,C} is selected. Then you earn zero in period 1. Suppose in period 2 you are in role B, your ask is 15, and the group {A,B} in your unit is selected. Then you earn 15 in period 2. Your earnings table for this case is given below. EARNINGS TABLE (in francs) Period # 1 2 Total

Your role A B

Your ask 15 15

Group selected {B,C} {A,B}

Your earnings 0 15 15

At the end of each period you are required to enter your ask, the group selected and your earnings for the period in the enclosed record sheet. B.3. The group selection mechanism Each group of participants in a unit is characterized by a VALUE which can be realized only if the group is selected by the mechanism. Table 1 (enclosed) contains the values for every possible group of participants in a unit. These values are the same for every unit and stay unchanged in every period of the experiment. The table below provides a hypothetical example of group values for a unit.

Group value

Group {A} 0

{B} 0

{C} 0

{A,B} 22

{A,C} 30

{B,C} 13

{A,B,C} ÿ5

For example, group {B} has a value of 0, group {A,C} has a value of 30, and group {A,B,C} has a value of ÿ 5. The key to the group selection mechanism is the RESIDUAL of a group. Given the asks of the participants in a unit, the residual of a group is as follows: RESIDUAL of a group ˆ VALUE of the group ÿ ASKS submitted by the group members Suppose, for example, that participant's A, B and C asks are 5, 7 and 15, respectively. Then, if the value of group {A,B} is 22, its residual equals 22 ÿ (5 ‡ 7) ˆ 10; if the value of group {A,B,C} is ÿ 5, its residual equals ÿ 5 ÿ (5 ‡ 7 ‡ 15) ˆ ÿ32. At the end of each period, in each unit THE MECHANISM WILL SELECT A GROUP WITH THE HIGHEST NON-NEGATIVE RESIDUAL. No more than one group in a unit can be selected. If there are several groups with the highest non-negative residual, the mechanism will select among them the group with the highest value. If there are several groups with the highest non-negative residual and the highest value, a fair lottery will

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determine which of these groups is selected. If all the groups have negative residuals, no one will be selected.23 After the group selection is made, your computer terminal will display the asks submitted by the participants in your unit, the residuals of the groups and the group selected. You will calculate your earnings for the period as described above. You will be given a residual accounting sheet to help you record the participants' asks and the groups' residuals for every period. This will continue for a fixed number of periods. Your unit and role assignment will change from period to period except for the following. The participants who are assigned the role C in the first period will stay in this role for the whole experiment. The participants who are assigned the roles A or B in the first period will never be assigned the role C in later periods. At the end of the experiment, you will be asked to calculate your total earnings, which is the sum of your earnings over periods. Exercise. Suppose the group values are as given in the table above (page 2 of the instructions), and the following asks are submitted by the participants in a unit: Participant role A B C

Ask submitted 15 2 10

Calculate the group residuals corresponding to these asks and enter them in the practice accounting sheet below. GROUP RESIDUAL ACCOUNTING SHEET

Group value (V) Asks (a) Residual (R) ˆ (V) ÿ (a)

Group {A} 0

{B} 0

{C} 0

{A,B} 22

{A,C} 30

{B,C} 13

{A,B,C} ÿ5

Which group will be selected by the group selection mechanism? Selected group: .... Enter the corresponding earnings of the participants into the table below: Participant role # A B C

Earning

23 The paragraph above was used in the instructions for the B-experiments. In the G-experiments, it was substituted by the following:

At the end of each period, in each unit THE MECHANISM WILL SELECT A GROUP WITH THE HIGHEST NON-NEGATIVE RESIDUAL. No more than one group in a unit can be selected. If there are several groups with the highest non-negative residual, a fair lottery will determine which of these groups is selected. If all the groups have negative residuals, no one will be selected.

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ARE THERE ANY QUESTIONS? Appendix C Behavioral rules C.1. Best response function Given the specific parameters of the game, each agents' asks in the group, depending on the role, is  maxfvj ; 100 ‡ v3 g vi BR ˆ maxf1100 ÿ vj ; 1000 ÿ v3 ; 0g  maxfv1 ÿ 100; v2 ÿ 100; 0g v3 BR ˆ maxf1000 ÿ v1 ; 1000 ÿ v2 ; 0g

agent's best response to the other if 1000 ÿ vj ÿ v3  0; otherwise; if 1100 ÿ v2 ÿ v3  0; otherwise;

where i, j2{1,2}, and j 6ˆ i. C.2. Analysis of individual rules Under the assumption that all agents use a given rule and submit equal adjusted asks every period, we address the following issues: C.2.1. Trends in asks First, we consider what kind of dynamics each rule induces. Given that the rules are linear, they may be of one of three types: (1) rules that are monotonic (decreasing, increasing or stationary) for all vi > 0; or (2) convergence-type rules that are stationary at some vi ˆ vi* > 0, decreasing for all vi > vi*, and increasing for all vi < vi*; (3) divergencetype rules that are stationary at some viˆvi**, increasing for all vi > vi** > 0, and decreasing for all vi < vi**; we refer to vi* and vi** as Type 1 and Type 2 stationary point, respectively. We compute increasing and decreasing regions of asks and stationary points for each individual rule as follows. Under the assumption that in any period all agents submit identical asks, agent i's own ask coincides with both the best response and the median ask. Hence, a best response or a median ask-type rule decreases in the region Vd  R‡ if for all vi2Vd …1 ÿ 1 ÿ 2 †vi > 0 ;

(3)

it increases in the region V in  R‡ if for all vi2V in the above inequality is reversed. It is stationary at a point vi2R‡ which solves vi ˆ 0 =…1 ÿ 1 ÿ 2 †:

(4)

If 1 ‡ 2 < 1, then such vi is a Type 1 stationary point (an asymptote); if 1 ‡ 2 > 1, then such vi is a Type 2 stationary point (a divergence point).

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For the SR-type rules, the regions and stationary points can be computed by substituting the selection shock dummy variable S ˆ f(vt ÿ 1) in the Rule 2 with a constant  equal to an individual selection rate. (Observe that an agent using a such rule switches between two models, vit ˆ …0 ‡ 2 † ‡ 1  vitÿ1 ; and vit ˆ 0 ‡ 1  vitÿ1 ; depending on the previous period selection outcome. Therefore, decreasing and increasing regions will oscillate between the respective regions of the above two rules. Since, on average, 2  S ˆ 2r, substituting the selection shock S in the SR rule by a constant r will produce fair estimates of the decreasing and increasing regions of asks.) Thus, a selection response rule decreases in the region V d  R‡ if for all vi2V d …1 ÿ 1 †vi > 0 ‡ 2  r; in

likewise, it increases in the region V  R‡ if for all vi2V reversed. It is stationary at vi such that vi ˆ …0 ‡ 2  r†=…1 ÿ 1 †;

(5) in

the above inequality is (6)

Such vi is a Type 1 stationary point if 1 < 1, and a Type 2 stationary point if the inequality is reversed. C.2.2. Convergence to the Nash equilibrium We next evaluate confidence intervals on the estimates of stationary points and test the hypothesis whether, according to a given rule, the asks were converging to the Nash equilibrium asks. (To evaluate standard errors on the estimates of stationary points, we use Taylor series approximations of the stationary points as functions of the estimated coefficients of the linear regression models, as described in Green (1990, pp.228±230); we then calculate the values of the test statistic to test whether the estimated stationary points were significantly different from the Nash equilibrium asks and the maximal asks submitted by the agent. The technique relies on asymptotic properties of the least squares.) Formally, we test the hypothesis H0 : vi*  viNash for convergence-type rules, and H0 : vi** > viNash for divergence-type rules. C.2.3. Significance in the trends in asks We next test whether the trends of asks induced by each rule are statistically significant. This allows the individuals who actively induced competition to be distinguished from those who, at most, followed the trend induced by others. Specifically, under identical previous period asks assumption, we test the stationarity of the rule hypothesis H0 : vit ˆ vitÿ1, which takes the form of ~ 0 : …0i ˆ 0; 1i ‡ 2i ˆ 1† H

(7)

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for the best response and the median asks rules, or ~ 0 : …1i ˆ 0† or …0i ‡ ri  2i ˆ 0; 1i ˆ 1†; H

(8)

where ri is the individual selection rate, for the selection response rules. (If we relax the assumption of identical previous period asks, the stationarity of the rule hypothesis has an alternative interesting interpretation. If all agents submit current asks in the range between the minimal and maximal observed previous period adjusted asks (e.g. if every agent's next period ask equals a convex combination of the asks observed in the previous period), then, under the stationarity hypothesis, the asks submitted by the agents in any period will stay within the range of the initial period asks. Such rules do not induce significant changes in the level of asks by themselves, but follow the change if it is induced externally.) We refer to a rule as stationary if the null hypothesis is sustained at a 5 percent significance level (according to an F-test), and non-stationary otherwise. C.2.4. Classification For non-stationary rules, the corresponding behavior is classified as competitive if the lower boundary of the decreasing region of asks for the rule is significantly below the maximal ask submitted by the agent. We classify behavior as cooperative if the lower boundary of the increasing region of asks for the corresponding rule is significantly below the maximal ask submitted by the agent. All behavior with stationary rules is called neutral. Thus, competitive behavior induces a decreasing trend in asks in a relevant range, whereas cooperative behavior induces an increasing trend. Neutral behavior does not initiate any trends, but may follow the trend induced by others. References Baumol, W., Paznar, J., Willig, R., 1980. Contestable Markets and the Theory of Industry Structure, Harcourt, Brace, Jovanovich, San Diego, CA. Bolle, F., 1994. Team selection ± An experimental investigation. Journal of Economic Psychology 15, 511±536. Bolle, F., 1995. Team selection: Factor pricing with discrete and inhomogeneous factors. Mathematical Social Science 29, 131±150. Chen, Y., Plott, C., 1996. The Groves±Ledyard mechanism: An experimental study of institutional design. Journal of Public Economics 59, 335±364. Cox, J.C., Smith, V., Walker, J., 1984. Theory and behavior of multiple unit discriminative auctions. Journal of Finance 39, 983±1010. Cox, J.C., Smith, V., Walker, J., 1988. Theory and individual behavior of first-price auctions. Journal of Risk and Uncertainty 1, 61±99. Fouraker, L.E., Siegel, S., 1963. Bargaining Behavior, McGraw-Hill, New York. Green, W., 1990. Econometric Analysis, Chaps. 7, 15, Macmillan, New York. Holt, C.A., 1995. Industrial organization: A survey of laboratory research, In: Kagel, J., Roth, R., (Eds.), Handbook of Experimental Economics, Princeton University Press, Princeton, NJ, pp. 349±443. Isaac, M.R., Walker, J., 1985. Information and conspiracy in sealed bid auctions. Journal of Economic Behavior and Organization 6, 139±159. Isaac, M.R., Walker, J., 1988. Group size effects in public goods provision: The voluntary contributions mechanism. Quarterly Journal of Economics 103, 179±199. Isaac, M.R., McCue, K., Plott, C., 1985. Public goods provision in an experimental environment. Journal of Public Economics 26, 51±74.

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Ledyard, J., 1978. Allocation processes ± Alternatives to the Cournot hypothesis, Paper presented at the summer meetings of the Econometric Society, Boulder, CO. Ledyard, J., 1995. Public goods: A survey of experimental research, In: Kagel, J., Roth, R., (Eds.), Handbook of Experimental Economics, Princeton University Press, Princeton, NJ, pp. 111±194. Kagel, J., 1995. Auctions: A Survey of experimental research, In: Kagel, J., Roth, R., (Eds.), Handbook of Experimental Economics, Princeton University Press, Princeton, NJ, pp. 501±585. Plott, C., 1989. An updated review of industrial organization: Applications of experimental methods, In: Schmalensee, R., Willig, R., (Eds.), Handbook of Industrial Organization, vol. II, Chap. 19, Elsevier, Amsterdam, The Netherlands . Sherstyuk, K., 1999. Collusion without conspiracy: An experimental study of one-sided auctions. Experimental Economics, in press. Smith, V.L., Walker, J., 1993. Rewards experience and decision costs in first-price auctions. Economic Inquiry 31, 237±245.