Bargaining and Coalition Formation Dr James Tremewan - - PowerPoint PPT Presentation

Bargaining and Coalition Formation

Dr James Tremewan (james.tremewan@univie.ac.at)1 Noncooperative Models of Multilateral Bargaining

1This 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
• rganizations)
• Big difference with bilateral bargaining: need to consider

possible coalitions... introduces massive complexity.

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The Baron Ferejon model

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

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The Baron Ferejon model

Options and preferences

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

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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.

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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

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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 = (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 (or probability of re-election).

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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”).

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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

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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

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The Baron Ferejon model

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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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.

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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

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The Baron Ferejon model

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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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 n δ

• What happens for k = n? (assume δ = 1)
• What about k = 1?
• In all cases, no delay.

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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

pi (may depend on size of party).

• Recognised party offers share of ministries, vj, 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 pi = 1/3, solution as in earlier slides.
• Baron and Ferejon (1989) give example with three parties, δ = 0.8,

p1 = 0.45, p2 = 0.35, and p3 = 0.2. They find the probabilities ρi of a party being in government are ρ1 = 0.46, ρ2 = 0.64, and ρ3 = 0.9; the probability of the two larger parties forming a government is 0.1. (see paper for details).

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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

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Experiments Miller and Vanberg 18/40

Experiments Miller and Vanberg 19/40

Experiments Miller and Vanberg 20/40

Experiments Miller and Vanberg

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

• Minimum winning coalitions, i.e. in majority rules, two players take

everything leaving the third with nothing.

• 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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Experiments Miller and Vanberg

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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Experiments Miller and Vanberg 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) 23/40

Experiments Miller and Vanberg

Majority rule (round 1, all periods)

0.1 0.2 Larger share Smaller share 13 12 13 24/40

Experiments Miller and Vanberg

Unanimity rule (round 1, all periods)

0.1 0.2 0.3 0.4 0.5 Larger share Smaller share 13 12 13

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Experiments Miller and Vanberg 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) 26/40

Experiments Miller and Vanberg

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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Experiments Tremewan and Vanberg

Experiment: Tremewan and Vanberg

Main idea

• Majoritarian bargaining with free timing of moves.
• Build on experiments using Baron-Ferejohn game
• Maintain substantive assumptions
• Distributive (divide-a-dollar) decision problem
• Cost of delay (discounting)
• Voting rules (majority, unanimity)
• Relax simplifying technical assumptions (move structure)
• Intervals of real time instead of discete rounds
• All players may propose or agree at any time
• Multiple moves possible each ‘round’

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Experiments Tremewan and Vanberg

Design

• 3 players
• Each ‘round’ = 10 seconds
• Pie shrinks 10% each round
• Treatments (between subjects): Majority

and unanimity rule Details

• Initial pie-size 30 EUR
• 12 periods, one paid
• Random matching
• Matching groups of 9 (some 6)
• Decisions via graphical interface
• 5 EUR show-up fee
• Average payment: 14.11 EUR
• Vienna
• zTree (Fischbacher), ORSEE

(Greiner)

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Experiments Tremewan and Vanberg 30/40

Experiments Tremewan and Vanberg 31/40

Experiments Tremewan and Vanberg

Questions

• Can we replicate previous findings?
• Minimum winning coalitions under majority rule?
• Equal distributions within coalitions?
• More delay under unanimity rule?
• Note that we have no non-cooperative benchmark solution
• We think the BF model predictions substantively apply
• Cooperative concepts may also apply
• Empirical question

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Experiments Tremewan and Vanberg

Final agreements

(Pie size normalized = 12 all rounds)

172 1 15 1 2 32 114 26 8 11 6 7 5 4 2 4 3 2 2 1 1 20 40 60 80 100 120 140 160 180 200

Majority rule (N=228) Unanimity rule (N=192)

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Experiments Tremewan and Vanberg

Final agreements

3 4 7 11 26 114

50 100 150 number of games (10,2,0) (11,1,0) (9,3,0) (8,4,0) (7,5,0) (6,6,0)

MWC agreements (majority rule)

1 1 2 2 2 4 5 6 8 32

10 20 30 number of games (8,3,1) (9,2,1) (5,5,2) (6,5,1) (7,3,2) (7,4,1) (6,4,2) (6,3,3) (5,4,3) (4,4,4)

non-MWC agreements (majority rule)

1 2 15 172

50 100 150 200 number of games (6,3,3) (5,5,2) (5,4,3) (4,4,4)

MWC agreements (unanimity rule)

Minimum Winning Coalitions no yes Majority rule 63 165 27% 73% Unanimity rule 2 190 1% 99%

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Experiments Tremewan and Vanberg

Final agreements - majority rule

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 MWC (M&V) Equal Split (M&V)

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Experiments Tremewan and Vanberg

Final agreements - majority rule

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 MWC Equal Split MWC (M&V) Equal Split (M&V)

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Experiments Tremewan and Vanberg

Final agreements - majority rule

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 MWC Fifty ‐ Fifty MWC (M&V) Fifty ‐ Fifty (M&V)

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Experiments Tremewan and Vanberg

Delay

Rounds before agreement reached

20 40 60 80 5 10 15 5 10 15

Majority rule Unanimity rule

Percent

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Experiments Tremewan and Vanberg

Delay

Average number of rounds by experiment period

0.5 1 1.5 2 2.5 3 3.5 1 2 3 4 5 6 7 8 9 10 11 12 Majority rule Unanimity rule

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Experiments Tremewan and Vanberg

Summary of results (one shot setting) Final agreements

(1) Most agreements are minimum winning coalitions (+ theory, + previous exp.) (2) Most mwc implement equal splits within coalitions (+ previous experiments) (3) More equal splits under majority rule than in previous experiments (no proposer advantage in our setting)

Delay

(4) Some delay under both majority and unanimity rule (- theory, + previous exp.) (5) Significantly more delay under unanimity rule (+ previous experiments)

Conclusion

• Results from previous (more structured) experiments supported.
• Major difference is lack of proposer power in our setting.

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