Multilateral Bargaining November 20, 2012 Multilateral Bargaining - - PowerPoint PPT Presentation

Multilateral Bargaining

November 20, 2012

Multilateral Bargaining

A group of individuals (members of parliament, firms, ...) must reach an agreement. to take some joint action

• n division of resulting gains from cooperation

Examples?

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Multilateral Bargaining

A group of individuals (members of parliament, firms, ...) must reach an agreement. to take some joint action

• n 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)

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Bargaining as a pie-splitting problem (n=3)

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Options and preferences

This is the two-dimensional simplex Players like points close to their corners

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Q: Are there any ‘stable’ points? Pick a point. Is it stable?

under majority rule? under unanimity rule?

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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.

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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

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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 = (x1, ..., xn) such that xi ≤ 1. If at least n−1

2

players vote ‘yes’, the proposal is passed. 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 ui(x, t) = δtxi where δ is a (common) discount factor.

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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?

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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.

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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

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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?

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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

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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?

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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

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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

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Proposition 1: Features of SPNE (2-period game)

Minimum winning coalition

n−1 2

non-proposers get δ/n 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

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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.

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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)

• nly 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

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Stationary subgame perfect equilibrium

Same general properties as in 2-period game.

Proof

All subgames have same (undiscounted) value vi for Mr. i.

• Mr. i votes ‘yes’ on proposals such that xi ≥ δvi

Proposer will make a proposal that passes for sure. (No point to waiting.) Therefore vi = 1 n(what proposer gets) + (1 − 1 n)(what responder gets (average)) Since the proposer will distribute the entire surplus, this value must be vi = 1/n Thus the proposer allocates δ/n to a bare majority (as in 2-period game).

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Effects of decision rules

If k of n players must agree Proposer offers δ/n to k − 1 others Keep 1 − k − 1 n δ What happens for k = n? (assume δ = 1) What about k = 1? In all cases, no delay.

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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

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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

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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

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Benchmark (SSPE with n = 3 and δ = .9)

Minimum winning coalitions Proposals Demand Offer majority rule 69 − 70% 30 − 31% unanimity rule 38 − 40% 30 − 31% Offers above (below) equilibrium offers accepted (rejected) No delay

Main hypotheses (contrary to theoretical predictions)

Proposals more often fail under unanimity rule Individuals more often vote no under unanimity rule

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Data Focus on first bargaining round in each period Each subject makes one proposal, votes on two others 15 periods Majority Unanimity sessions 2 2 subjects 24 24 proposals 360 360 votes 720 720

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Proposed own share (Round 1, all periods) Share to non-proposers (Round 1, all periods)

20 40 60 .1 .2 .3 .4 .5 .6 .1 .2 .3 .4 .5 .6 Majority rule Unanimity rule

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Proposed own share (Round 1, all periods) Average own share over time Share to non-proposers (Round 1, all periods)

20 40 60 .1 .2 .3 .4 .5 .6 .1 .2 .3 .4 .5 .6 Majority rule Unanimity rule

Equal splits vs MWC (majority rule) 24 / 28

Majority rule (round 1, all periods)

0.1 0.2 Larger share Smaller share 13 12 13

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Unanimity rule (round 1, all periods)

0.1 0.2 0.3 0.4 0.5 Larger share Smaller share 13 12 13 26 / 28

Fraction of proposals passed (round 1) 27 / 28

Fraction of proposals passed (round 1) Accepted and Rejected Offers (round 1, all periods) 27 / 28

Fraction of proposals passed (round 1) Probability of acceptance (individual level, RE Logit)

• wnshare ≤ 31%
• wnshare > 31%

Unanimity

• 1.136

0.177 (0.492)** (0.298) Proposer’s share

• 4.750
• 1.456

(1.389)*** (0.973) Own share 8.047 5.587 (1.312)*** (1.956)*** Period 0.058 0.021 (0.028)** (0.016) Constant

• 0.632
• .0171

(1.782) (0.739) Observations 438 948 Number of subjects 48 48 *** p < 0.01 ** p < 0.05 * p < 0.1 Accepted and Rejected Offers (round 1, all periods) 27 / 28

Summary Most proposers (eventually) build minimum winning coalitions Proposer’s own share far below benchmark prediction (SSPE) Distributions often close to equal within coalitions More proposals fail under unanimity rule Under unanimity rule, voters more often reject small shares

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