Bargaining and Coalition Formation Dr James Tremewan (james.tremewan@univie.ac.at) 1 Noncooperative Models of Multilateral Bargaining 1 This set of slides is adapted from slides by Christoph Vanberg
The Baron Ferejon model Multilateral Bargaining • A group of individuals (members of parliament, firms, countries ...) must reach an agreement. • to take some joint action. • on division of resulting gains from cooperation. • 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, international agreements) • majority rule (e.g. in political context, within some organizations) • Big difference with bilateral bargaining: need to consider possible coalitions... introduces massive complexity. 2/40
The Baron Ferejon model Bargaining as a pie-splitting problem (n=3) 3/40
The Baron Ferejon model Options and preferences • This is the two-dimensional simplex • Players like points close to their corners 4/40
The Baron Ferejon model 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. 5/40
The Baron Ferejon model Structure induced equilibrium • Majority and unanimity rules, per se, do not predict an outcome. • Stability of outcomes under majority rule is a puzzle. • 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 6/40
The Baron Ferejon model 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 (or probability of re-election). 7/40
The Baron Ferejon model 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 (“minimum winning coalition”). 8/40
The Baron Ferejon model 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 9/40
The Baron Ferejon model 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 10/40
The Baron Ferejon model 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 11/40
The Baron Ferejon model 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. 12/40
The Baron Ferejon model 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 13/40
The Baron Ferejon model 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). 14/40
The Baron Ferejon model 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. 15/40
The Baron Ferejon model Application: Government Formation in Parliamentary Systems • In a multiparty parliamentary system, parties must often form coalitions to form a government. • Often many different coalitions would constitute a majority... which one will form? • We can use the Baron-Ferejon model: • Each party is asked to attempt to form a government with probability p i (may depend on size of party). • Recognised party offers share of ministries, v j , to other parties. • If the first party fails to form a government, another party is chosen randomly... • δ is some function of the probability new elections are called and the expected performance of the parties in those elections. • If p i = 1 / 3, solution as in earlier slides. • Baron and Ferejon (1989) give example with three parties, δ = 0 . 8, p 1 = 0 . 45, p 2 = 0 . 35, and p 3 = 0 . 2. They find the probabilities ρ i of a party being in government are ρ 1 = 0 . 46, ρ 2 = 0 . 64, and ρ 3 = 0 . 9; the probability of the two larger parties forming a government is 0.1. (see paper for details). 16/40
Experiments Miller and Vanberg Experiment: Miller and Vanberg (2011) • Tests various hypotheses generated by the three player BF model, under both majority and unanimity rules. • Hypotheses: • 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 17/40
Experiments Miller and Vanberg 18/40
Experiments Miller and Vanberg 19/40
Experiments Miller and Vanberg 20/40
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