Multilateral Bargaining November 20, 2012
Multilateral Bargaining A group of individuals (members of parliament, firms, ...) must reach an agreement. to take some joint action on division of resulting gains from cooperation Examples? 2 / 28
Multilateral Bargaining A group of individuals (members of parliament, firms, ...) must reach an agreement. to take some joint action on division of resulting gains from cooperation Examples? In bilateral bargaining, usually assume that mutual agreement is necessary. In larger groups, different decision rules possible unanimity rules (e.g. in negotiations between independent firms) majority rule (e.g. in political context, within organizations) 2 / 28
Bargaining as a pie-splitting problem (n=3) 3 / 28
Options and preferences This is the two-dimensional simplex Players like points close to their corners 4 / 28
Q: Are there any ‘stable’ points? Pick a point. Is it stable? under majority rule? under unanimity rule? 5 / 28
Q: Are there any ‘stable’ points? Pick a point. Is it stable? under majority rule? under unanimity rule? Observe: All outcomes are stable under unanimity rule. No outcome is stable under majority rule. 5 / 28
Structure induced equilibrium Majority and unanimity rules, per se, do not predict an outcome. Stability of outcomes under majority rule is a puzzle. In reality, not true that every option can be considered against all others at no cost. Formal and informal institutions constrain the process of proposing, voting, etc. Noncooperative approach “In contrast to this instutition-less setting, the theory presented here reflects the sequential nature of proposal making (...) and voting, and models it as a noncooperative (...) game.” Baron and Ferejohn (1989) Where will this lead... All outcomes can be part of an equilibrium under majority rule! Equilibrium ‘refinements’ produce more specific predictions 6 / 28
Model n players (odd), pie of size 1 In each ‘round’, random player ‘recognized’ to propose. Player i is recognized with probability 1 / n A proposal is an allocation x = ( x 1 , ..., x n ) such that � x i ≤ 1. If at least n − 1 players vote ‘yes’, the proposal is passed. 2 If fail to agree, new round with new random proposer Game continues until agreement is reached. If game ends in period t with allocation x , player i ’s utility is u i ( x , t ) = δ t x i where δ is a (common) discount factor. 7 / 28
Questions we want to answer Properties of equilibrium allocations Majoritarian? (Dividing benefits between members of a minimum winning coalition) Universal? (Dividing benefits among all members of the decision making body) How are benefits distributed within the coalition? How long does it take for agreement to be reached? 8 / 28
Questions we want to answer Properties of equilibrium allocations Majoritarian? (Dividing benefits between members of a minimum winning coalition) Universal? (Dividing benefits among all members of the decision making body) How are benefits distributed within the coalition? How long does it take for agreement to be reached? Some intution... When voting, players must consider what they are being allocated under a proposal and compare it to what they can expect to get if the game continues. Importance of beliefs concerning others’ behavior time preference (patience) Only a majority of players must agree. The proposer will probably want to ‘buy’ the cheapest majority he can. 8 / 28
Histories and strategies At any time t , players know the history of the game up to that point who made which proposals at what time how each player voted on those proposals A strategy for player i specifies an action (proposal or vote) to take after every possible history of the game up to every possible time t . Equilibrium concept Players cannot precommit to making certain proposals or voting in certain ways. At each point in time, equilibrium must be self-enforcing : must be in each player’s interest to follow equilibrium strategy. Subgame Perfect (Nash) Equilibrium (SPNE): Induces a Nash Equilibrium within every subgame 9 / 28
Simplified example: 2 Period game If no agreement after period 2, all players get zero Backward induction What will happen if round 2 is reached? 10 / 28
Simplified example: 2 Period game If no agreement after period 2, all players get zero Backward induction What will happen if round 2 is reached? People vote ‘yes’ an anything that gives positive payoff. Proposer offers tiny ǫ > 0 to bare majority Proposal passes, proposer gets (essentially) everything 10 / 28
Simplified example: 2 Period game If no agreement after period 2, all players get zero Backward induction What will happen if round 2 is reached? People vote ‘yes’ an anything that gives positive payoff. Proposer offers tiny ǫ > 0 to bare majority Proposal passes, proposer gets (essentially) everything What do players expect if round 1 ends without agreement? 10 / 28
Simplified example: 2 Period game If no agreement after period 2, all players get zero Backward induction What will happen if round 2 is reached? People vote ‘yes’ an anything that gives positive payoff. Proposer offers tiny ǫ > 0 to bare majority Proposal passes, proposer gets (essentially) everything What do players expect if round 1 ends without agreement? Each has a chance of 1 / n to be proposer Expected payoff = 1 / n Continuation value = δ/ n 10 / 28
Simplified example: 2 Period game If no agreement after period 2, all players get zero Backward induction What will happen if round 2 is reached? People vote ‘yes’ an anything that gives positive payoff. Proposer offers tiny ǫ > 0 to bare majority Proposal passes, proposer gets (essentially) everything What do players expect if round 1 ends without agreement? Each has a chance of 1 / n to be proposer Expected payoff = 1 / n Continuation value = δ/ n What will happen in period 1? People vote ‘yes’ on anything that gives them at least δ/ n Proposer offers δ/ n to bare majority Proposal passes 10 / 28
Proposition 1: Features of SPNE (2-period game) Minimum winning coalition n − 1 non-proposers get δ/ n 2 Proposer gets 1 − n − 1 2 ( δ/ n ) For n = 3, this is 1 − δ/ 3 ≥ 2 / 3 For large n , converges to 1 − δ/ 2 ≥ 1 / 2 Proposer always gets at least half of the surplus! Agreement is immediate Results from majority rule, not impatience. There are MANY such equilibria Proposer could randomly choose his coalition Or he could include specific people In latter case, the ‘value of the game’ will differ between individuals 11 / 28
Infinite horizon - multiple equilibria Backward induction argument does not apply. Proposition 2: Any distribution can be supported in equilibrium if n ≥ 5 and δ large enough. Intuition Since there is always a future, can devise elaborate punishments Choose any allocation x that you want to implement Tell the players... Everyone is to propose x if recognized Everyone is to vote for x if proposed If anyone proposes y � = x , it is to be rejected and that person is to be excluded from subsequent proposals. If anyone deviates from the previous item, proceed accordingly... Note: this involves complicated, history-dependent strategies. 12 / 28
History-dependent strategies History-dependent strategies may be difficult to implement Players may not ‘trust’ that others will use such strategies Perhaps more realistic to assume that actions do not depend on history? Stationarity A stationary strategy is one where the player’s action (proposal / vote) only depends on the current state of the game (proposal being considered), not past behavior. A stationary equilibrium is one in which all players are using stationary strategies Complex punishment strategies are not stationary 13 / 28
Stationary subgame perfect equilibrium Same general properties as in 2-period game. Proof All subgames have same (undiscounted) value v i for Mr. i. Mr. i votes ‘yes’ on proposals such that x i ≥ δ v i Proposer will make a proposal that passes for sure. (No point to waiting.) Therefore v i = 1 n ( what proposer gets ) + ( 1 − 1 n )( what responder gets (average) ) Since the proposer will distribute the entire surplus, this value must be v i = 1 / n Thus the proposer allocates δ/ n to a bare majority (as in 2-period game). 14 / 28
Effects of decision rules If k of n players must agree Proposer offers δ/ n to k − 1 others Keep 1 − k − 1 δ n What happens for k = n ? (assume δ = 1) What about k = 1? In all cases, no delay. 15 / 28
Key predictions (Stationary Equilibrium) Proposers build minimum winning coalitions Distribution within coalition favors the proposer More inclusive decision rules produce more equal payoffs Discounting (impatience) increases inequality of payoffs Agreement is reached immediately under all rules 16 / 28
Experiments on Baron-Ferejohn game McKelvey (1991), Frechette et al. (APSR 2003, JPubE 2005, ECTA 2005, Games 2005) Majority rule only, test effects of amendment rules Diermeier and Morton (2005) Introduce a veto player Miller and Vanberg (2011) Compare majority and unanimity rule 17 / 28
Experimental Design Group size: 3 2 Treatments: Majority vs. Unanimity 15 periods, one paid Strategy method (sort of) Pie 20 GBP 4 GBP Show-up fee zTree Conducted at CESS in Oxford 18 / 28
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