Supplementary Appendix Commitment and (In)Efficiency: a Bargaining Experiment

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1 Supplementary Appendix Commitment and (In)Efficiency: a Bargaining Experiment Marina Agranov Matt Elliott July 28, 2016 This document contains supporting material for the document Commitment and (In)Efficiency: A Bargaining Experiment, which herein we refer to as the main document. 1 Derivation of Theoretical Predictions 1.1 Symmetric Pairwise Bargained Outcomes This approach extends Nash bargaining to networks. A player s disagreement payoff is the surplus they could obtain by just enticing someone else to match with them. Of course, this depends on the agreements others have reached, and so the solution boils down to finding a fixed point of a large system of equations. Worker i s disagreement payoff when matching to j is given by ( ) u i = max 0, max s ik v k, k F \j and the firms disagreement payoffs are defined analogously. Given these disagreement payoffs, an outcome is an SPB outcome if and only if the match is efficient and the payoffs solve the following system of equations: Division of the Humanities and Social Sciences, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA USA. magranov@hss.caltech.edu. Division of the Humanities and Social Sciences, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA USA and Cambridge University, Faculty of Economics. melliott@hss.caltech.edu. 1

2 u i = u i + 1 ( siµ 2 (i) u i v µ (i)) v j = v j + 1 ( ) sµ 2 (j)j u µ (j) v j for all workers i (1) for all firms j. (2) It is shown in Rochford (1984) that a solution to this system of equations always exists. While in principle there can be multiple solutions in all the games we study it is unique. In the four-player games we study, the system of equations we need to solve to find the fixed point reduces to u A = (s AD v D ) (20 (s AD v D ) 0) u B = (20 0 (s AD u A )) v C = (20 0 (s AD v D )) v D = (s AD u A ) (20 (s AD u A ) 0) for s AD {15, 25, 30}. Solving this system of equations yields the predictions stated. Remark 1. In all the games we study there is a unique symmetric pairwise bargained outcome, and this outcome corresponds to (i) the nucleolus, (ii) the kernel, and (iii) the pre-kernel Core In all pairwise stable/core outcomes, the match that is implemented must maximize the total surplus. As generically there is a unique match with this property, pairwise stability alone pins down who must be matched to whom and there is no scope for inefficiency. While pairwise stability, or equivalently the core, pins down the match, many payoff vectors can typically be supported as core outcomes. In particular, Shapley and Shubik (1972) show that there is a core outcome in which all agents on one side of the market simultaneously receive their minimum possible core payoff, 1 Equivalence of the kernel, pre-kernel, and SPB outcomes holds generally for all assignment games and follows from results in Rochford (1984) and Driessen (1998). The nucleolus is contained in the kernel, and so equivalence of it with the other solution concepts follows the uniqueness of the SPB outcomes for the games we study. 2

3 while all agents on the other side of the market simultaneously receive their maximum possible core payoff. Defining u as the workers minimum core payoffs, letting u be the workers maximum core payoffs, defining v as the firms minimum core payoffs, and letting v be the firms maximum core payoffs, the bargaining outcomes (u, v, µ ) and the bargaining outcomes (u, v, µ ) are in the core. Moreover, as the core is convex, the mid-point of these outcomes, ( 1 2 (u + u ), 1 2 (v + v ), µ ), is also in the core. As u i + v µ (i) = s iµ (i), these outcomes simplify to u i = u i + 1 ( sij u i v 2 j) v j = v j + 1 ( sij u i v 2 j) for all workers i (3) for all firms j (4) This is the same payoff structure that we found in Equations (1) and (2), but with each disagreement payoff replaced by the minimum payoff that player could receive in any core outcome. In the four-player games we study, u B = v C = 0. In Game 15, u A = v D = 0; in Game 25, u A = v D = 5; and in Game 30, u A = v D = 10. For completeness, and despite making only set-valued predictions of the payoffs, we also report the range of core payoffs each player can receive. 2 All the cooperative theories we consider capture the idea that the simple threat of players A and D reaching agreement and leaving B and C unmatched should be enough to induce B and C to reach agreements that do not leave A and D with a profitable deviation. 1.3 Markov Perfect Equilibria In this section we derive the MPE that we test. A more formal and general derivation is provided in Elliott and Nava (2015). We start with Game 25. Let W (δ) be the continuation value of players in the subgames where they are bargaining bilaterally with their efficient partners. By Rubinstein (1982) there is a unique perfect equilibrium in these subgames and lim δ 1 W (δ) = 10. Letting V i be the continuation value of player i when no one has yet been matched, we look for a symmetric solution in which V A = V D := V S is the continuation value for both the strong players and V B = V C := V W is the continuation value for both the weak players. We guess and verify that in Game 25 there is an equilibrium in which the strong players mix between offering to each other and offering to their efficient partners. Letting q be the probability that that the strong players, A and D, offer inefficiently to each other if either is selected as the proposer, we then have the following system of equations. 2 Some papers, such as Cole et al. (2001), look for results that are robust to any selection from the core. 3

4 V S = 1 4 (20 δv W + δ(1 + q)v S + (2 q)δw (δ)) V W = 1 4 (20 δv S + (1 q)δv W + (2 q)δw (δ)) 20 δv W = 25 δv S, The first two equations state the continuation values of the strong and weak players as determined by the possible transitions in states that can occur and the payoffs associated with these transitions. The last equation is an indifference condition that must be satisfied for the strong players to strictly mix who they offer to. Solving this system of equations and taking limits, q V S V W 6.45 = 0.56 Given these continuation values, it is easily verified that no players have a profitable deviation. As players are always offered their continuation values, acceptance is optimal, and as the strong players mix, by construction they are indifferent between offering to each other and offering to their efficient partners. Finally, delaying is unprofitable. In the limit, by deviating and delaying a weak player receives an expected payoff of 6.45 < , while a strong player receives an expected payoff of < = For Game 30, players A and D strictly prefer offering to each other. The system of equations is then V S = 1 4 (30 δv S + 2δV S + δw (δ)) V W = 1 4 (20 δv S + δw (δ)) Solving this system of equations and taking limits, we get V S 40 3 = V W 25 6 =

5 It is again easily verified that this is an equilibrium. For example, were a strong player to deviate and make an offer to a weak player, the lowest acceptable offer they could make would leave the strong player with a payoff of < Efficient Perfect Equilibria To construct an efficient perfect equilibrium, we need to create the right system of rewards and punishments for players to play efficiently. One measure of the complexity of strategies is the extent to which the players prescribed actions vary with the history of play (see, for example, Kalai and Stanford (1988)). In this sense Markovian stratagies are particularly simple, as they depend only on the history through the state in this case the set of active players. In order to incentivize the players to play efficiently, more complicated strategies are necessary to create the right system of rewards and punishments. We start by considering a particularly simple class of efficient perfect equilibria, those where (only) reversion to the Markov perfect equilibrium is used as a punishment. Thus equilibrium play depends only on the state and whether there has been a deviation. It doesn t matter who deviated or when. In Game 25 there is a perfect equilibrium that can be supported by reversion to the MPE. Let the expected MPE payoff of player i be Vi M (δ), and as before let W (δ) be the payoff of any player in the unique perfect equilibrium of the subgame where all other players except their efficient partner has exited. We construct an efficient perfect equilibrium in which, on path, player i offers their efficient partner µ (i) a payoff δvi M and their partner accepts. Any deviation from this play is punished by moving to the Markov perfect equilibrium. Thus, by construction, player i best responds by accepting the offer. Indeed, given that deviations are supported by reversion to the MPE, player i must offer their efficient partner exactly δvµ M (i). Anything less would be rejected by µ (i). If the strategy prescribed i offering anything more than δvµ M (i) to µ (i), then i would have a profitable deviation to offer a little less and, knowing that because i has deviated play will revert to the MPE strategies thereafter, µ (i) would accept. It is easily verified that δvi M < 20 δvµ M (i) for all players and thus all players prefer making the prescribed offers to delaying. The final deviation to check is that the strong players cannot do better by offering to each other. In Game 25 this requires that 25 δvs M 20 δvw M. As a strong player offering to another strong player constitutes a deviation, thereafter the MPE will be played. Hence each strong player will just be willing to accept an offer from the other strong player that leaves them with a payoff of δvs M. As in the MPE the strong players mix between making offers to each other and offering to their efficient partner, the indifference condition implies that 25 δvs M = 20 δv W M. Thus the inequality is satisfied and the strong players do 5

6 not have a profitable deviation. However, this is not the case for Game 30. In Game 30 the strong players will not have a profitable deviation if 30 δvs M 20 δvw M (where these MPE continuation values are for Game 30 and not for Game 25 as before). As in Game 30 the strong players strictly prefer offering to each other than offering efficiently in the MPE such that 30 δvd M > 20 δv C M. Thus in Game 30 Markov reversion does not provide sufficient incentives for the strong players to offer efficiently and there is no efficient MPE with Markov revision for Game 30. In the efficient PE with MPE reversion for Game 25, the limit payoffs of the players are V A = 1 ( 20 δv M 4 C + δva M + δ2w (δ) ) V B = 1 ( 20 δv M 4 D + δvb M + δ2w (δ) ) 8.75 V C = 1 ( 20 δv M 4 A + δvc M + δ2w (δ) ) 8.75 V D = 1 ( 20 δv M 4 B + δvd M + δ2w (δ) ) To further illustrate that there is no such efficient PE for Game 30, and letting Vi M now refer to the expected MPE payoff of player i in game 30 (as opposed to Game 25 above), the limit payoffs of the players would be V A = 1 ( 20 δv M 4 C + δva M + δ2w (δ) ) V B = 1 ( 20 δv M 4 D + δvb M + δ2w (δ) ) 7.71 V C = 1 ( 20 δv M 4 A + δvc M + δ2w (δ) ) 7.71 V D = 1 ( 20 δv M 4 B + δvd M + δ2w (δ) ) But then if selected as the proposer, A can either stick with the prescribed strategy that offers C a payoff δvc M, leaving A with a limit payoff of 15.83, or deviate and offer D a payoff δvd M, which D would accept, leaving A with a limit payoff of To find a limit MPE for Game 30 we need to consider more complicated strategies, in which players are both rewarded for rejecting off-path offers and punished for making off-path offers. The rewards are necessary because we can impose punishment only if the offer is rejected. On path, we look for an equilibrium in which players B and C make efficient acceptable offers that leave them with a payoff of x and accept 6

7 A C B (a) Surpluses D A or B Defect Punish A, B On Path C or D Defect C or D Defect Punish C, D A or B Defect A or B Defect C or D Defect (b) Transitions 20 x A B x A 30 δv D B 20 δv D 50 3 A B x C D 20 x C D δv A C 30 δv A D (c) On path (d) Punish A,B (e) Punish C,D Figure 1: Constructing an efficient perfect equilibrium for Game 30. Panel (b) shows the transitions between states when players deviate from the prescribed play, while panels (c) (e) show how players play in each state. Red arrows indicate whom a player offers to if selected as the proposer; and the numbers next to the arrows indicate the payoffs that the offering players will keep. offers of x from their efficient partners. This is shown in panel (c) of Figure 1. On path, after the first efficient pair of players exit the market the remaining efficient pair bargain bilaterally with each other. In such subgames there is a unique perfect equilibrium (Rubinstein, 1982) and the remaining active players receive payoffs W (δ) that converge to 10. Thus, in any efficient equilibrium the last weak player to reach agreement receives a limit payoff of 10. In order to get these weaker players to accept and make offers that give them a payoff of x < 10 we need to punish them if they deviate. We construct off-path punishments that are credible and create the appropriate incentives for players to remain on path. This is achieved by defining two different punishment states, prescribing play in each of these states and a rule for transitioning between them in a way that creates the appropriate incentives. These transitions are such that they occur only if someone deviates from their prescribed strategy, in which case the person who initiated the deviation is punished by moving to the state that punishes her. Importantly, and unlike with MPE reversion, these transitions also 7

8 reward all the players to whom the punished player is linked. These transitions are illustrated in panel (b) of Figure 1. To show that the punishments are credible, suppose we are in the Punish A, B state. If everyone plays as prescribed we remain in this state and the payoffs of the players are given by the following value functions: V A = 1 (30 δ 4 V D + 2δW (δ) + δ V ) A = 30 δ V D + 2δW (δ) 4 δ V B = 1 (20 δ 4 V D ) + δ V B = 20 δ V D δ 9 V C = 1 ( δ 4 V ) 2W (δ) C + 2W (δ) = 4 δ 62 3 V D = 1 (16 23 ) 4 + 3δ V D = δ By symmetry, the punish C, D state value functions of the players are Ṽ A = δ Ṽ B = 2W (δ) 4 δ 62 3 Ṽ C = 20 δ V D δ Ṽ D = 30 δ V D + 2W (δ) 4 δ Consider now the deviations available to the players in the punish A, B state. First, suppose that A deviates and offers D less than δ V D. By rejecting the offer, D ensures that we remain in the same state and that he will receive, in expectation, δ V D. Alternatively, A may delay, in which case A receives δ V A < 30 δ V D. Finally, A could offer to C. C would accept anything greater than δ V C and reject anything less, because we would remain in the punish A,B state. Thus A must offer C a limit payoff of 6 2, leaving A with This inequality becomes strict, while the others remain satisfied, when the offer D makes to B is increased slightly. 3 The alternative deviations available to B are to offer D less than δ V D, which D would reject, leaving B with 2 1 < 3 1, or to delay, which would also leave B with 2 1 < 3 1. The only We make this incentive constraint tight to find the full range of payoffs these punishment strategies can support. 8

9 deviation available to C is to make an offer to A. As rejecting C s offer will result in a switch of states, A would accept only a limit payoff which is weakly greater than 16 2, leaving C with 3 1 < 6 2. Finally, D could deviate. As a deviation by D would result in a switch of states if rejected, for an off-path offer to be accepted D must offer B at least 6 2 in the limit, or A at least 16 2 in the limit. Both deviations are thus 3 3 unprofitable. Finally, as delay would also result in a switch of states, that alternative is unprofitable for D as well. This covers all the possible deviations from the punish A, B state. By symmetry, there are no profitable deviations from the punish C, D state. For these punishments to be effective, in the on-path state C and B must be required to accept only offers, in the limit, of weakly more than 2 2 in the limit, or to 9 make offers that leave them with at least 2 2 in the limit. Similarly, A and D must 9 be required to accept only offers of weakly more than 11 1 in the limit, or to make 9 offers that leave them with at least 11 1 in the limit. Thus for any x (2 2, 8 8) there exists an efficient perfect equilibrium. This places bounds on the offers that can be supported when all the players are active. In the subgame reached once an efficient pair has exited, the remaining players get limit payoffs of 10. Thus the weak players will have limit expected payoffs in the range (6 1, 9 4 ), while the strong players will 9 9 have limit expected payoffs in the range (10 5, 13 8). 9 9 This construction of an efficient perfect equilibrium for Game 30 also works for Game 25. In that case, in the punish A, B state we would need to make A offer D no more than 11 2, leaving A with 13 1, so that A does not want to deviate and instead 3 3 will offer to C. As A offers D his continuation value, this implies that V D = 11 2, 3 which means that B also offers 11 2 to D and that D offers 8 1 to B. This gives a 3 3 limit payoff to B of V B = 5 5. As before, C s limit payoff is V 9 D = 6 2. Finally, A s 3 limit payoff is Given these strategies and limit payoffs, it can be verified that all 9 the incentive constraints are satisfied. Thus, there is an efficient perfect equilibrium for any x (5 5, 8 8 ). As limit payoffs in the subgame are again 10, weak players have 9 9 limit expected payoffs in the range (7 7, 9 4 ), while strong players have limit payoffs in 9 9 the range (10 5, 12 2) Behavioral MPE In this section we derive equilibria for Game 25 and Game 30 in which a proportion of the players play behaviorally. We refer to these equilibria as the behavioral MPE and their derivations are similar to those of the MPE. An important difference is that there is now private information about players types. We thus look for strategy and belief profile pairs that together constitute a Markov perfect Bayesian equilibrium, 4 4 More formally, we refine the set of perfect Bayesian equilibria by looking for equilibria in which the strategies played are history dependent only through the current Markov state. The set of 9

10 where the Markov state reflects both the set of active players and beliefs about these players types. In our setting there is only limited updating of beliefs on-path. As behavioral weak players, by assumption, always offer 10 while their rational counterparts will not offer 10 in equilibrium, the public observation of a weak player offering 10 informs everyone that the weak player in question is behavioral. Similarly, a weak player rejecting an offer less than 10 that a rational weak player would accept on-path, reveals that player to be behavioral. However, other than these events, all on-path play turns out to be the same regardless of whether the acting player is rational or behavioral and so prior beliefs are retained. We look for an equilibrium in which current beliefs are also preserved off-path. Finally, in any subgame with two players either all players receive limit payoffs of 10 regardless of their types, or else no match is possible and both players receive limit payoffs of 0 for sure. Thus the key Markov states we need to consider are: (i) All players are active and none have been revealed to be behavioral; (ii) All players are active and only B has been revealed to be behavioral; (iii) All players are active and only C has been revealed to be behavioral; (iv) All players are active and both B and C have been revealed to be behavioral. Game 25: To solve for the behavioral MPE of Game 25 we first consider the relevant subgames. First, in the subgames in which only players A and C (B and D) are active, both players receive limit payoffs of 10 regardless of their type. Second, in the subgames in which only B and C are active, both players receive payoff of 0 regardless of their type. We now turn to the subgames associated with the different Markov states enumerated above. If both weak players have taken actions that, applying Bayes rule, reveal them to be behavioral we denote the continuation value of player i by Ŵi(δ). In this case, as both weak players are behavioral, they always make offers of 10. These offers are rejected in equilibrium and the strong players only offer to each other. Thus, in this subgame, strong players receive limit payoffs of Ŵi = 25/2, while the weak players receive limit payoffs of 0. Consider now the subgames in which player C has been revealed to be behavioral, but not player B. In these subgames there is an equilibrium in which C s offers are rejected, B always offers to D, D always offers to B and A mixes, possibly Markov states is determined by first partitioning histories into those with same number of active players as before, and then partitioning these states further by the beliefs held about the active players types. 10

11 degenerately, between delaying and offering to D. Given this play and letting ρ(x, δ) be the probability that A offers to D, equilibrium continuation values are given by the solution to the following system of equations. 4 V B = (20 δ V D ) + (3 ρ)δ V B 4 V D = (1 x)(20 δ V B ) + 2xδŴD(δ) + (3 x)δ V D 4 V A = ρ(25 δ V D ) + (1 ρ)δ V A + 2xδŴA(δ) + 2(1 x)δw (δ) + δ V A where, VB is the continuation value of B conditional on B being rational in this subgame. In addition, if ρ (0, 1) we have the indifference condition (25 δ V D ) = δ V A. Solving this system of equations and letting δ 1, ρ(x) (0, 1) for x < 2/3 and ρ(x) = 1 for x 2/3. Thus, for x < 2/3 and for x 2/3 V B x + 2.5x2 1 x V D x + 2.5x2 1 x V A x + 5x x 3, (1 x) 2 V B x V D x 1 + 3x V A x + 15x x A third new subgame to consider is one in which player B but not C has been revealed to be behavioral. By symmetry, equilibrium limit payoffs in this subgame for x < 2/3 are 11

12 Ṽ C x + 2.5x2 1 x Ṽ A x + 2.5x2 1 x Ṽ D x + 5x x 3, (1 x) 2 and for x 2/3 are Ṽ C x Ṽ A x 1 + 3x Ṽ D x + 15x x Now we have solved for the on-path subgames we can consider equilibrium play in the initial state. In equilibrium in the initial state, as in the MPE, the strong players mix between offering to each other and making efficient offers while the weak players always offer to the strong players. To find an equilibrium we have to pin down beliefs when off-path actions are observed. We do this by assuming that prior beliefs are always retained. 5 Given the offer strategies specified above, equilibrium continuation values are given by the solution to the following system of value equations, 5 There is one instance in which this assumption is a little unnatural, but here the assumption does not matter. In equilibrium, when offering to a weak player, a strong player offers the weak player her discounted continuation value for the current state conditional on her being rational. So on-path such offers are accepted by rational weak players but rejected by behavioral weak players, and Bayes rule pins down beliefs. If an off-path offer is made by a strong player to a weak player and rejected, then we assume prior beliefs about the type of responding player are retained and we remain in the same Markov state. When the off-path offer is less than the continuation value of a rational weak player and also less than 10, or more than 10 and also more than the continuation value of a rational weak player, these beliefs are natural. However, if an offer greater than the continuation value of a weak rational player but less than 10 is rejected, it might be more reasonable to let the weak player be believed to be behavioral for sure. Changing beliefs in this way does not matter insofar as there is an equilibrium with the same on-path play also supported by this belief system. 12

13 4V A = (1 q)[(1 x)(20 δv C ) + xδ V A ] + q[25 δv D ] +(1 q)[(1 x)δw + xδṽa] + qδv A +(1 x)δv A + xδ V A + (1 x)δw + xδṽa 4V D = (1 q)[(1 x)(20 δv B ) + xδṽd] + q[25 δv A ] +(1 q)[(1 x)δw + xδ V D ] + qδv D +(1 x)δv D + xδṽd + (1 x)δw + xδ V D 4V C = (20 δv A ) + (1 q)δv C + (1 q)[(1 x)δw + xδṽc] + (1 x)δw + xδṽc 4V B = (20 δv D ) + (1 q)δv B + (1 q)[(1 x)δw + xδ V B ] + (1 x)δw + xδ V B and the indifference conditions 25 δv D = (1 x)(20 δv C ) + xδ V A, 25 δv A = (1 x)(20 δv B ) + xδṽd, where V C is the continuation value of C conditional on C being rational and V B is the continuation value of B conditional on B being rational. Solving this system yields the limit payoffs for strong players plotted in panel (a) of Figure 5 in the main document. To find the expected ex-ante payoff of a weak player we take the weighted average of the expected payoff of a rational weak player, obtained by solving the above system of equations, and the expected payoff of a behavioral weak player, weighted by the proportion of behavioral players. This gives the limit payoffs for the weak behavioral players also shown in panel (a) of Figure 5 in the main document. Finally, to calculate the probability that the efficient match is reached, we consider all possible realizations of offer sequences and offers that will result in the efficient match, and the sum the probability of all such events. How the probability the efficient match is reached varies with the proportion of behavioral players is shown in panel (a) of Figure 4 in the main document. Game 30: We solve for the behavioral MPE of Game 30 in much the same way as in game 25, although the calculations are a bit simpler. A notable difference is that in the subgame in which B (C) but not C (B) has been revealed to be behavioral, strong players both always offer to each other with probability 1 in equilibrium. Secondly in the initial state in equilibrium, as in the MPE, the strong players offer to each other for sure instead of mixing as in Game 25. Solving the associated systems of equations we get the following limit continuation values for the various subgames: 13

14 Ŵ A = ŴD 15 Ŵ B = ŴC 0 V B = ṼC 5x 1 + 2x 25x + 20 V D = ṼA 1 + 2x V A = ṼD 5x2 + 30x x The continuation values in the initial Markov state are then given by solution to the following system 4V A = (30 δv D ) + δv A + (1 x)δv A + xδ V A + (1 x)δw + xδṽa 4V D = (30 δv A ) + δv D + (1 x)δv D + xδṽd + (1 x)δw + xδ V D 4V C = (20 δv A ) + (1 x)δw + xδṽc 4V B = (20 δv D ) + (1 x)δw + xδ V B. Solving this system yields the expected payoffs of strong players. The ex-ante expected payoff of weak players is again found by taking a weighted average of the expected payoff of a weak player conditional on being rational and the expected payoff of a weak player conditional on being behavioral. Finally, the probability the efficient match is reached is found by considering all the ways in which on path play can yield the efficient match as before. These expected payoff and the probability the efficient match is reached are reported in panel (b) of Figure 5 in the main document and panel (b) of Figure 4 in the main document respectively. 2 Instructions 2.1 Instructions for Exit Treatment Welcome. You are about to participate in an experiment on decision-making and you will be paid for your participation in cash privately at the end of the session. Please turn off all electronic devices, especially phones. During the experiment you are not allowed to open or use any other applications on these laboratory computers, except for the interface of the experiment. Structure of the experiment. The experiment consists of 10 games. Each game consists of several rounds. Before the beginning of each game, you will be 14

15 randomly divided into groups of 4 people and assigned an ID letter (A, B, C or D). Your group assignment and your ID letter will be the same in all rounds of the same game, but will vary from game to game. In other words, at the end of each game, you will be randomly divided into new groups and you will be assigned new ID letters. The game number, the round number and your ID letter will be clearly displayed on the top of the screen. To determine your payment, at the end of the experiment, the computer will select one game from the 10 games played. Each game is equally likely to be chosen for payment. Your earnings will be equal to your earnings in this randomly selected game. In addition you will receive $15 for completing the experiment. All the payoffs on the computer screen are in dollars. What happens in each game. In each game you will engage in anonymous bargaining. A network describes who is connected to whom and how many dollars pairs of players will receive if they are matched at the end of the game. Throughout the game, there will be opportunities for pairs to become matched and reach agreement on how to split the amount of dollars the match will generate. Each person can be matched with at most one other person in her group. We will explain below in details what it means to be matched with another person. The screen has three main parts: the top-left part depicts the diagram of a network, the top-right part keeps track of matches established in your group in each round and the bottom-right part is where you will make your decisions. We will describe now in details each of these parts and the game. Here is an example of a diagram of a network that depicts the connections between people and amounts of dollars available for matches between these participants. A B C D This is a 4-person network with each person identified by the letters A, B, C and D. Different people in the network can have different numbers of connections. For instance, A is connected to C and to D, while B is only connected to D. The number next to the line connecting two people represents the surplus (the number of dollars) that pair would generate by matching with each other. For example, if A matched to C, A and C will receive a combined total of 10 dollars. Each game will consist of several rounds. 15

16 First round. At the beginning of a round, each member of the network can choose one of two actions: 1) propose a match or 2) do nothing. Proposing a match means choosing a person in the network with whom you are connected and proposing how to divide available dollars between the two of you. In other words, a proposal of a match is a suggestion of how many dollars you would receive and how many dollars the other person would receive. To make a proposal, please choose the ID letter of the member you want to propose a split using the drop-down menu. Then underneath that type the number of dollars that you propose to keep to yourself. The remaining dollars will be allocated to the member you chose to propose to if he/she accepts your proposal. Here is how the interface looks like (bottom-right corner of the screen): In the example shown in the network diagram above, there are 10 dollars to be divided between members A and C, while there are 13 dollars available for division between members A and D. Therefore, if, for instance, member A makes a proposal to member D and proposes to keep 2 dollars, this means that if member D accepts this proposal she will receive 11 dollars. Once you finalize your proposal, please click SUBMIT button. At this moment you won t be able to modify your proposal any more in this round. If you do not wish to make a proposal in this round, you can press DO NOTHING button, located to the right of SUBMIT button. After all members of the group made their moves (submitted a proposal or done nothing), the computer will select one participant at random. Each participant is equally likely to be selected. Only the move of the selected member will be implemented. If selected member chose to do nothing, then the current round will be over and the group will move on to the next round of the same game. If the selected member proposed a match then the person to whom a match was proposed will be prompted to respond to the proposal. This person may choose to accept or to reject the proposal. If a proposal is accepted, the match is formed. If a proposal is rejected, then we move onto the next round without anything changing; both the selected member and the 16

17 participant proposed to remain unmatched. If a match is formed, then it will be displayed in subsequent rounds in top-right diagram located on the screen. Here is an example of what a match between A and D would look like: 2 A B A B C D C D 11 This diagram mimics the diagram on the left, indicating the connections between people in this network with thin lines. It also indicates your position in the network with the yellow circle. A thick line connecting two subjects indicates that a match between these two subjects was formed. The numbers outside the circles indicate the number of dollars each subjects receives according to this agreement. For instance in the example above, in the first round A and D formed a match and agreed to split 13 dollars so that A gets 2 and D gets 11 dollars. Second and the following rounds. The second, and all the remaining rounds in this game, look very similar to the first round of the except for one feature. At the beginning of each round, all members of the group are asked to choose either to propose a match or to do nothing (just like in the first round). However, while in the very first round all members of the group are unmatched, in the subsequent rounds some members might be matched based on agreements they have reached in the previous rounds. Since any person can be matched with at most one other person in the group, people that formed matches in the previous rounds have no active choice in the subsequent rounds of the game and will be prompted to choose DO NOTHING button. Other people, those that are not matched yet, can either propose a match to someone with whom they have a potential connection (indicated by a thin line on the top-right or top-left diagrams) or DO NOTHING. History. The right-hand side diagram will keep track of the current status of all members of the network in each round. This right-hand side diagram will also allow people to observe how the matches have evolved over the course of the previous rounds for the current game by clicking arrow buttons below the diagram. Notice there is fast-back button. If pressed, this button will show the very first round of the game. There is also a fast-forward button which if pressed will show the current round of the game. The simple arrows are to go back and forth one round at a time. The number between arrows indicates the round number that the diagram is showing. 17

18 When does game end and your payment in a game. There are two possibilities for how a game may come to an end. The first possibility involves chance. At the end of each round, the computer randomly chooses an integer number between 1 and 100 (inclusive), with each number being equally likely. If the chosen number is below 100, then game proceeds to the next round. However, if the computer chose the number 100, then current game ends. In other words, there is 1% chance that the current round is the last round in this game and 99% chance game is not over and group proceeds to the next round. The second possibility is the one in which players who proposed new matches cannot form matches without players who chose to do nothing. In other words, there are no possible matches between any two players who are both still proposing. If that is the case, then the current game comes to an end. When game comes to an end, each member of the group receives the amount of dollars given by the current agreement (last round agreement). The dollars each person will receive from the current agreement are shown on the right-hand side diagram of the network for the current round. If a person is unmatched when the game ends, that participant receives zero dollars. At the end of each game, you will observe the message that indicates why this game ended. Payment. At the end of the experiment, the computer will randomly choose one of the 10 games that you just played and the number of dollars that you earned in this game will be paid to you together with the participation fee. Each game is equally likely to be selected for payment. Are there any questions? 2.2 Quiz for Exit treatment SCREEN 1 Consider the following game: Question 1: Suppose that in round 1 C makes an offer to A in which C keeps 3 dollars. Suppose this offer is selected. How many dollars each player would get if A accepted the offer and then the game ended? 1. A gets 3, C gets 27, B and D get 0 2. A gets 27, C gets 3, B and D get 0 3. A gets 27, C gets 3, B gets 35 and D get 10 18

19 A B 30 C D 4. A gets 27, C gets 3, B get 0 and D get 10 Correct answer is 2. Question 2: Suppose, instead, that in round 1 D makes an offer to A in which D keeps 7 dollars. Suppose this offer is selected. How many dollars each player would get if A rejected the offer and then the game ended? 1. A gets 3, B gets 15, C gets 20 and D gets 7 2. A gets 3, D gets 7, B and C get 0 3. All players get 0 4. Not enough information is given to work it out Correct answer is 3. Question 3: How do subjects get assigned to groups? 1. Participants are reshuffled into new groups in each game and in each round 2. Participants stay in the same groups in all games and in all rounds 3. Participants are reshuffled into new groups in each game, but stay in the same group throughout the game Correct answer is 3. SCREEN 2 Diagram below shows what has happened in round 1 of the game: Question 4: Which player are you? 19

20 25 A B 30 C D 5 1. A 2. B 3. C 4. D Correct answer is 1. Question 5: Which players are currently matched (reached an agreement)? 1. A and D as well as B and C 2. A and C as well as A and D 3. A and C 4. B and D Correct answer is 3. Question 6: Can players A and C make new proposals in round 2? 1. No because they are already matched and once matched you cannot make new proposals 2. Yes because both A and C have another link available: A can be matched with D and B with C 20

21 Correct answer is 1. Question 7: Suppose that in round 2, B proposed to C and D proposed to A. Will the game continue to the next round? 1. No because B and D have no link with each other and they can t match 2. Yes because B and C have a link with each other and so are A and D Correct answer is 1. 21

22 2.3 Screenshots from Exit 25 treatment 22

23 2.4 Instructions for Stay 25 treatment The instructions for the Stay treatments are the same as the instructions for the Exit treatments except for the description of the second and the following rounds of the game as well as the subjects payments. We present here only these two parts which are different between the Exit and the Stay treatments. Second and the following rounds. The second, and all the remaining rounds in this game, look very similar to the first round of the except for one feature. At the beginning of each round, all members of the group are asked to choose either to propose a match or to do nothing (just like in the first round). However, while in the very first round all members of the group are unmatched, in the subsequent rounds some members might be matched based on agreements they have reached in the previous rounds. If a currently matched person is selected by the computer to be the proposer, and makes an offer to a different player who decides to accept this new offer, then the proposer will incur a separation cost. In this eventuality, the previous match that has been agreed upon will be dissolved and the new match will be formed in its place. If a matched person s proposal is rejected, the separation cost is not paid. If a matched person makes a proposal to another participant, but this proposal is not selected by the computer to be implemented, then the separation cost is also not paid. Finally, if a currently matched person receives a proposal and decides to accept it, that person must pay the separation cost and the previous match this person was involved in will be dissolved. The separation cost will be subtracted from the final number of dollars earned in the current game. If a person is responsible for dissolving more than one match, that person will pay the separation cost for each such dissolved match. In today?s experiment the separation cost is 10 cents in all games. On the top of the screen you will be able to see how many times you have paid the separation cost up until the current round: In the current game, you have paid separation costs times If you have already formed a match in a previous round and wish to keep this match as is, you do not need to re-form it. In other words, if both participants involved in the match reject proposals from other participants if such proposals come along and do not propose new matches to other participants themselves, then the previously formed matches remain intact. All the remaining details of a round are the same as in the first round. When game comes to an end, each member of the group receives the amount of dollars given by the current agreement (last round agreement) minus the total 23

24 separation costs each member has incurred. The dollars each person will receive from the current agreement, not including the separation costs incurred, are shown on the right-hand side diagram of the network for the current round. The number of times each person paid separation cost is indicated on the top of the screen. If a person is unmatched when the game ends, that participant receives zero dollars less the separation costs that participant has incurred. 2.5 Quiz for Stay treatment The Quiz at the end of the instruction period for the Stay treatments consisted of 8 questions. The first 5 questions were the same as in the Exit treatment. The remaining 3 questions were different and were presented on the second screen of questions following the same picture as in the Exit treatment. Question 6: Suppose that in round 2, A and C chose DO NOTHING, B proposed to C and D proposed to A. Will the game continue to the next round? 1. No because B and D cannot form a match with each other 2. Yes because D has a link with A and B has a link with C Correct answer is 1. Question 7: Suppose instead in round 2, A and D chose DO NOTHING, B proposed to C and C proposed to B. Will the game continue to the next round? 1. No because A and D chose DO NOTHING 2. Yes because B and C have a link with each other and can form a match Correct answer is 2. Question 8: Suppose that in round 2, B proposed to C and C proposed to B. B was selected to implement her proposal and C accepted this new proposal. What will happen? 1. Players B and C will form a new match and C will pay the separation cost for breaking the previous match she was involved in 2. Players B and C will form a new match and B will pay the separation cost for breaking the match between A and C 3. The previous match between A and C remain intact, because C cannot form new matches 24

25 4. Player C will be matched to both player A and player B. Correct answer is Screenshot for Stay 25 treatment 3 Additional Analysis 3.1 Evolution of Efficiency Figure 2 depicts the evolution of final match efficiency for each market separately as subjects gain experience with the game. As evident from this figure, starting from the 4th repetition onwards, three treatments separate out and the efficiency of the final matches decreases monotonically from Game 15 to Game 25 to Game

26 Figure 2: Evolution of final match efficiency, by market Exit 15 Exit 25 Exit % 80% 60% 40% 20% 0% repe$$on 3.2 Efficiency and the Network Position of the First Mover In all the theories except MPE an efficient agreement is reached regardless of the identity of the first mover. According to MPE, whether or not the market clears efficiently in Games 25 and 30 depends on which player was randomly selected to make the first move. If the first mover is a strong player (a player with two links), then the market is predicted to end up with an inefficient outcome with positive probability, while if a weak player (a player with one link) moves first it is predicted that an efficient outcome will be reached with certainty. This pattern follows directly from two observations: (a) On the equilibrium path all the players make offers that are accepted, and so there is immediate agreement, and (b) the strong players propose to each other with positive probability in Games 25 and 30. In Game 15, by contrast, efficiency is predicted to always be reached irrespective of the identity of the first mover, which is consistent with our data, as we observe that all the final matches in the last five repetitions of Game 15 are efficient. Table 1 reports the results of a regression analysis in which we study the link between the network position of the first mover and the final match efficiency. For each game, we run two regressions; in both of them, the dependent variable is an indicator for the efficiency of the final matches. In regression (1) the right-hand side variables include a binary variable that takes value 1 if the first mover is a strong player and 0 otherwise. In regression (2) the right-hand side variables include a binary variable that takes value 1 if the first accepted offer was made by a strong player and 0 otherwise. While MPE predicts that on path no offers should ever be rejected, such behavior is present in our data, which is why we also run regression (2). 26

27 Table 1: Effect of network position of the first mover on final match efficiency, experienced games Game 25 Game 30 Regression (1) Regression (2) Regression (1) Regression(2) First mover is strong 0.37 (0.11) 0.49 (0.10) First accepted offer 0.81 (0.07) 0.93 (0.04) by strong player Constant 0.66 (0.07) 0.97 (0.06) 0.53 (0.06) 0.95 (0.04) # of observations # of sessions overall R-sq Notes: Random-effects GLS regressions. Dependent variable is an indicator of an efficient final match. Standard errors are clustered at the session level. ** indicates significance at 5% level. Consistent with the MPE predictions, we observe that in both Game 25 and Game 30 efficiency is significantly lower in games in which the first mover is a strong player rather than a weak player. The same is true if we condition the efficiency of matches on the network position of the player who makes the first accepted offer. In regression (2) the higher estimated coefficient in Game 25 than in Game 30 is consistent with the MPE predictions regarding frequencies of efficient proposals by strong players in these two games: In Game 25 strong players are supposed to mix between proposing efficiently and not, while in Game 30 they should always propose inefficiently. 3.3 Sessions Conducted at UCI versus UCSB Here we compare sessions conducted at the two universities: UC Irvine and UC Santa Barbara. Table 2 lists the locations at which we conducted our experimental sessions, by treatment. We will use Game 25 to compare the two subject pools, since this is the only market structure for which we conducted sessions at both locations for both Exit and Stay treatments (3 sessions at UCSB and 1 session at UCI). In Table 3 we report efficiency and players payoffs in the final outcomes observed in each of these sessions. While there are some small differences in outcomes observed in the first half of the experiment in different sessions, these differences disappear as subjects gain experience with the game. In particular, in the last five repetitions we find no significant differences in the frequency of efficient outcomes in any pair of sessions within Exit or Stay treatments (Wilcoxon Ranksum test: p > 0.10 in all pairwise comparisons). Moreover, we detect no significant differences in average payoffs of either strong or weak players in the last five repetitions for any pair of sessions within Exit or Stay treatments (regression analysis: p > 0.10 in all pairwise comparisons). 27

28 Table 2: Locations of experimental sessions UC Irvine UC Santa Barbara Treatment # of sessions # of subjects # of sessions # of subjects EXIT 15 3 sessions 40 subjects EXIT 25 1 session 20 subjects 3 sessions 48 subjects EXIT 30 3 sessions 68 subjects STAY 15 1 session 12 subjects 2 sessions 28 subjects STAY 25 1 session 16 subjects 3 sessions 44 subjects STAY 30 3 sessions 56 subjects Finally, we note that sessions at UCI and UCSB are also very similar in terms of observed individual strategies, market dynamics, and evolution of the players bargaining power. These results are omitted for brevity and are available from the authors upon request. Table 3: Game 25: Efficiency and players payoffs in final match, by session First five repetitions Last five repetitions efficiency B and C A and D efficiency B and C A and D Exit treatment Session 1 (UCSB) 47% 4.63 (0.84) (0.34) 45% 4.25 (0.74) (0.38) Session 2 (UCSB) 60% 5.13 (0.93) (0.39) 56% 4.77 (0.79) (0.26) Session 3 (UCSB) 50% 4.53 (0.70) (0.35) 45% 3.95 (0.71) (0.17) Session 4 (UCI) 68% 6.39 (0.64) (0.23) 56% 4.92 (0.80) (0.17) Stay treatment Session 1 (UCSB) 44% 3.61 (0.86) (0.54) 88% 6.86 (0.81) (0.93) Session 2 (UCSB) 75% 5.70 (0.95) (0.34) 88% 6.14 (0.55) (0.16) Session 3 (UCSB) 75% 6.22 (0.66) (0.33) 85% 6.35 (0.45) (0.16) Session 4 (UCI) 54% 6.44 (0.88) (0.80) 69% 5.41 (0.76) (0.15) Notes: We focus on the groups that finished playing the game naturally rather than those that were interrupted by random termination. For average payoffs of players, we report robust standard errors in the parentheses, where observations are clustered at the individual level. 3.4 Individual Strategies in Exit and Stay Treatments Frequency of Efficient Proposals Figure 3 depicts the frequency of proposing efficiently in each of our markets by strong players in the last five repetitions when markets were complete. This figure replicates Figure 3 from the main document, except that here we use all proposals of this type and not only the first round proposals. As is evident from both Figure 3 and Figure 3 from the main document, consistent with the MPE prediction there is 28

29 a significant shift in strategies used by the strong players as the value of the diagonal link increases, regardless of whether one looks at the first round proposals only (as in Figure 3 from the main document) or at all proposals of this type (as in Figure 3). Figure 3: Frequency of efficient proposals by strong players when markets were complete, last five repetitions Fraction Exit freq of efficient proposals (per subject) Fraction Exit freq of efficient proposals (per subject) Fraction Exit freq of efficient proposals (per subject) Notes: For each subject, we compute the frequency of proposing efficiently, using data from all rounds in which markets were complete in the last five repetitions of each session, conditional on this player being assigned a strong position. Figure 4 depicts the frequency of proposing efficiently in each of our markets by strong players in the last five repetitions when markets were complete, when we assume that the fraction of behavioral types is 50%. As evident from Figure 4, with this proportion of behavioral players, the observed frequencies with which strong players inefficiently offer to each other continues to be consistent with the theory. Figure 4: Frequency of efficient proposals by strong players in behavioral MPE when markets were complete in the last five repetitions Exit 15 Exit 25 Exit Observed Expected Notes: For each subject, we compute the frequency of proposing efficiently, using data from the first round only in the last five repetitions of each session, conditional on this player being assigned a strong position. The predicted frequencies are generated using 50% of behavioral types. Figure 5 depicts the cumulative distribution functions showing frequencies of proposing efficiently by the strong players when markets were complete in the second half of the experiment. This figure replicates Figure 7 from the main document, except that here we use all rounds in which markets were complete instead of only the 29

30 first round in each game as in Figure 7 from the main document. The conclusions reached in Section 8 remain intact. Except for Game 15, in which the vast majority of strong players always propose efficiently in both Exit and Stay treatments, in the remaining Game 25 and Game 30 strong players propose efficiently with higher frequencies when there is a possibility of renegotiation. Regression analysis confirms these results: p < 0.01 in both Exit 25 vs. Stay 25 an Exit 30 vs. Stay 30, while p > 0.10 in Exit 15 vs. Stay 15. Figure 5: CDFs of frequency of efficient proposals by strong players when markets were complete, last five repetitions Exit 15 Stay Exit 25 Stay Exit 30 Stay Notes: We present the cumulative distribution functions summarizing individual frequencies of proposing efficiently in the second half of the experiment when a subject performed a role of strong player and markets were complete. The horizontal axes indicate the likelihood of proposing efficiently, while the vertical axes indicate the values of the CDFs Ask Amounts Figure 6 depicts box plots of average absolute differences between amounts offered in Game 25 and Game 30 and those predicted by MPE, by offer type (strong player to strong player, weak player to strong player, etc.) and constructed for the first five and the last five repetitions in each session separately. The differences were first computed and averaged for each subject, and then combined with those of other subjects. Each of the figures in Figure 6 corresponds to one of the offers on the equilibrium path, as indicated in main text of the paper in Section 5.4. In these figures, each box shows the interquartile range (between the 25th and 75th percentiles), with the median value indicated by the thick dashed line. The length of whiskers is set at the standard 1.5 times the interquartile range. On average these offers deviate a bit from those predicted by MPE. The deviations are largest when weak players make offers to strong players. Weak players often make lower offers to strong players than predicted that, if accepted, would more equitably distribute the surplus between the strong and weak player. Indeed, as discussed in the main text of the paper, we frequently observe weak players demanding perfectly 30

31 Figure 6: Average absolute deviations of the amounts offered by players from the MPE predictions Exit 25: Strong to Weak Exit 25: Strong to Strong Exit 25: Weak to Strong first 5 repe**ons last 5 repe**ons first 5 repe**ons last 5 repe**ons first 5 repe**ons last 5 repe**ons 1 2 Exit 30: Strong to Strong Exit 30: Weak to Strong first 5 repe**ons 1 2 last 5 repe**ons first 5 repe**ons 1 2 last 5 repe**ons Notes: Averages are computed separately for each subject in the first and last five repetitions of a session, and then combined with those of the other subjects. We focus only on cases in which markets were complete. Each box depicts the interquartile range (between the 25th and 75th percentiles), with the median value indicated by the thick dashed line. The length of whiskers is set at 1.5 times the interquartile range. equitable splits of 10 each in both Game 25 and Game 30. In Game 25, 50% of weak subjects ask for an even split when all players are active some time even in the last five repetitions. In Game 30, the corresponding percentage is 59%. Of the 50% in Game 25 who sometimes propose equal splits, the average frequency with which an equal split is demanded is 60%. In Game 30 it is 50%. So in both Game 25 and Game 30, about 30% of the offers made by weak players demand a completely equitable split of 10 each representing a substantial deviation from the MPE. 6 6 These equal offers are even more frequent in the first five rounds of the game, and we can see from Figure 6 that over time average play gets closer to the MPE. To reach this conclusion we have regressed absolute differences between amounts offered and Markov predictions on the dummy variable that indicates the last five repetitions in each session, while clustering observations by the session. We find that the estimated dummy coefficient is negative and different from 0 at the standard 5% significance level in all offers depicted in Figure 6. 31

32 3.4.3 Acceptance/Rejection Behavior of Responders Finally, we consider which offers subjects accept and which they reject, and how that varies with their network position. Figure 7: Responders behavior by network position, first five repetitions offer accepted offer rejected 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected Exit 15 strong responders Exit 25 strong responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected Exit 30 strong responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected Exit 15 weak responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected Exit 25 weak responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected Exit 30 weak responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more Notes: Offers received by responders are depicted on the horizontal axes. The height of each bar represents the number of observations in each offer range. Figures 7 and 8 depicts the acceptance/rejection behavior of our responders in each of the games in the first and the second halves of the experiment as well as the number of observations we have for each offer type. As the value of the diagonal link increases, on average strong responders accept only higher offers, while weak responders are willing to accept smaller shares. Regression analysis confirms this pattern: Shares accepted by strong responders are significantly higher in Game 25 than in Game 15 (p < 0.01) and also significantly higher in Game 30 than in Game 25 (p < 0.01). Similarly, shares accepted by weak responders are significantly lower 32

33 in Game 25 than in Game 15 (p < 0.01) and also significantly lower in Game 30 than in Game 25 (p < 0.01), although we have only a small number of observations in this last case since strong responders almost exclusively propose to each other in this treatment. Figure 8: Responders behavior by network position, last five repetitions offer accepted offer rejected 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected offer accepted offer rejected Exit 15 strong responders Exit 25 strong responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more Exit 30 strong responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more offer accepted offer rejected offer accepted offer rejected Exit 15 weak responders Exit 25 weak responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more Exit 30 weak responders 0.0 to to to to to to to to to to to to to to to to to to to to to to to to to and more Notes: Offers received by responders are depicted on the horizontal axes. The height of each bar represents the number of observations in the indicated offer range. 33