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

Bargaining and Coalition Formation

Dr James Tremewan (james.tremewan@univie.ac.at) Non-Cooperative Solution Concepts in Coalition Formation

Non-Cooperative Solution Concepts

Nash Equilibrium

• The standard solution concept in non-cooperative game theory is

Nash equilibrium.1 Can also be used in coalition formation games.

• NE consider an outcome (or agreement) stable (or

self-enforcing), if no individual can improve their payoff taking

• ther players’ strategies as given.
• Appropriate in a context with no pre-game communication, but

in a coalition formation context it is natural to think that players can communicate.

• The following solution concepts are for situations where pre-play

communication is possible, but commitment is not (differentiating from cooperative solution concepts). Agreements must still be self-enforcing.

1We will only be looking at static games of full information.

2/8

Non-Cooperative Solution Concepts

Strong Nash Equilibrium

• SNE (Aumann, 1959) consider an outcome self-enforcing if no

coalition can improve the payoff of all its members taking other players’ strategies as given.2

• Clearly all SNE are NE (individuals are coalitions of size one).
• All SNE are Pareto efficient... why?
• Incorporates the idea that a group of players can discuss

strategies and simultaneously deviate, but very strong requirement and often no SNE exists.

• e.g. Prisoners’ Dilemma:
• Unique NE: both players defect.
• Not a SNE as the grand coalition can deviate to both

cooperate, and both players gain.

• Sometimes SNE don’t seem reasonable as they must be robust

to deviations to outcomes which themselves are not stable: would a coalition deviate to an outcome that is not stable?

2See Bernheim et al. (1987) for formal definition.

3/8

Non-Cooperative Solution Concepts

Coalition-Proof Nash Equilibrium

• CPNE assume players are farsighted rather than myopic.
• CPNE (Bernheim et al, 1987) consider an agreement

self-enforcing if no coalition can form a self-enforcing agreement that improves the payoff of all its members taking other players’ strategies as given. However only deviations by sub-coalitions of the originally deviating coalition are considered.

• Bernheim et al justify restricting the future deviations considered

to those of a subset of coalitions by appealing to information problems: players within the deviating coalition will have information (about earlier deviations) that those outside do not, making it hard to form agreements between insiders and

• utsiders.
• Presumably this restriction is for tractability, but could make

sense in some contexts, e.g. coalition gaining independence for South Sudan now breaking down into sub-coalitions to fight over newly won resources.

4/8

Non-Cooperative Solution Concepts

Coalition-Proof Nash Equilibrium

• CPNE is less stringent requirement than SNE: clearly all SNE

are CPNE (and all CPNE are NE).

• Prisoners dilemma: both players defect is a CPNE... the

deviation that prevented it from being a SNE is to an unstable agreement: the grand coalition can deviate to both cooperate but then either member of that coalition can unilaterally deviate further to improve their payoff.

• There exist games with no CPNE.
• Bernheim et al extend the idea to games in extensive form.
• The authors also argue that CPNE can be more intuitive than

Pareto dominance refinement (considering only Pareto efficient NE): example on next slide.

5/8

Non-Cooperative Solution Concepts

Coalition-Proof Nash Equilibrium: 3-Player Game

C1 C2 B1 B2 B1 B2 A1 1,1,-5

• 5,-5,0

A1

• 1,-1,5
• 5,-5,0

A2

• 5,-5,0

0,0,10 A1

• 5,-5,0
• 2,-2,0
• Two NE: (A2, B2, C1) and (A1, B1, C2).
• The first Pareto dominates the second so would be chosen by

Pareto dominance refinement.

• But from (A2, B2, C1) A and B have an incentive to jointly

deviate to (A1, B1, C1), and this deviation is self-enforcing (no further deviation by A or B could improve their payoffs). Thus (A1, B1, C2) is the only CPNE.

• There is no SNE as SNE ⊂ CPNE and (A1, B1, C2) is not Pareto

efficient.

6/8

Non-Cooperative Solution Concepts

The Largest Consistent Set

• Chwe (1994) identifies two issues with CPNE:
• It is not always reasonable to restrict consideration to future

deviations by sub-coalitions.

• Far-sighted coalitions may want deviate to agreements that are

initially worse for them but will be followed by a future deviation that benefits them.

• The LCS is a solution concept that incorporates the above two

points:

• Potential deviations are only considered if they are stable with

respect to future chains of deviations involving any combination

• f coalitions which may consist of any players.
• Coalitions may consider deviating to agreements that make

them worse off as long as they will end up better off after some chain of deviations.

• The LCS is intentionally a ”weak” concept: rather than looking

for stable outcomes, it eliminates outcomes that are definitely not stable, so may not make precise prediction.

7/8

Non-Cooperative Solution Concepts

Overview

• We have seen a range of solution concepts which can be used to

analyse non-cooperative games in contexts where it is reasonable to assume that players may communicate, but are unable to commit so that agreements are only stable if they are in some sense self-enforcing.

• As with cooperative solution concepts we face a trade-off: more

restrictive solution concepts make tighter predictions, but may not aways exist.

• Also, the assumptions underlying a solution concept may be

more or less appropriate in a particular context:

• Players may be genuinely myopic (i.e. SNE prefered to CPNE).
• Private information by deviating players may be important (i.e.

CPNE preferred to LCS).

8/8