Endogenous Coalition Formation in Contests Santiago Sánchez-Pagés Review of Economic Design 2007.

Endogenous Coalition Formation in Contests

Santiago Sánchez-Pagés

Review of Economic Design

2007

Motivation

• Rivalry– Interests of opposing groups do not

coincide

• Conflict– Winners gain exclusive rights at the

expense of the losers

Reasons for Coalition Formation

• Face fewer rivals

• Higher chance of success due to pooling resources

Conflicts of Interest

• Division of prize

• Free-riding

Previous Literature

• Olson (1965)

• Hart and Kurtz (1983)

• Bloch (1996)

• Baik and Lee (1997,2001) and Baik and Shogren (1995)

• Garfinkel (2004) and Bloch et al. (2006)

Previous Literature

• Olson (1965)– The Logic of Collective Action

• Group-size Paradox– Small groups are more often effective than large

groups

Group-Size Paradox

• The perceived effect of an individual defection decreases as group size increases, leading to greater free-riding

• Individual prizes decrease as group size increases, which is the author’s concept of rivalry within a coalition

Previous Literature

• Hart and Kurtz (1983)– Simultaneous games of exclusive

membership• б-game

– Remaining coalition members remain in coalition if an individual player withdraws

• y-game – Coalition breaks apart if one member withdraws

Previous Literature

• Bloch (1996)

– Sequential game of coalition formation

– Players’ reactions to defection are determined endogenously

Previous Literature

• These three games:

– б-game– y-game– Bloch’s sequential game

• are returned to in subsequent sections of the article.

Previous Literature

• Baik articles

– Three stage model• Players form coalitions• Choose sharing rule for coalition• Coalitions compete

Baik vs. Sanchez-Pages

• Baik uses open membership and sharing rule depends on individual investment.

• SP uses exclusive membership and does not model sharing rule.

Previous Literature

• Garfinkel (2004a,b)

– Members of the winning coalition may engage in a new contest depending on the strength of intra-group rivalry

Previous Literature

• Garfinkel (2004a,b)

– Symmetric and nearly symmetric coalition structures are stable, but not the grand coalition when rivalry is strong

The Model

• Stage 1: Agents form groups

• Stage 2: Coalitions contest prize

• Stage 3: Prize distributed among group members (not modeled)

Agents

• Set N of n players in K≤n coalitions

• Ex-ante identical

• Same strategy set

Coalition Structure

• C ={C1,C2,…,CK}

• |Ck| is the cardinality of C

• Ascending ordering: |Ck| ≤ |Ck+1|

• If |C1| = |CK| then the coalition structure is symmetric

Resource Pooling

• ri denotes the resources expended by agent i

• Rk=∑iЄCk ri

• R(C) = (R1,R2,…,RK)

Contest Success Function

• Tullock CSF

Contest Success Function

• Tullock CSF

Typo

Payoff Function

• All members of the winning coalition

receive пk

Payoff Function

• In Baik пk is modeled explicitly as a

sharing rule.

Payoff Function

• Does the individual payoff function пk

have an effect on the coalition structure?

Conditions on Individual Payoff

Conditions on Individual Payoff

• Anonymity

– Assumption of ex-ante identical players means that individual prizes are independent of the exact identity of the group members

Conditions on Individual Payoff

• Rivalry

– Individual payoff is strictly decreasing in the size of the group.

The Contest Stage

• Active Coalitions

The Contest Stage

• Proof of Lemma 1

The Contest Stage

• F.O.C for individual member of active coalition

• Determining total equilibrium expenditure

The Contest Stage

• Substituting the equilibrium total expenditure into the F.O.C. yields the optimal individual expenditure

The Contest Stage

• Agent i participates only if the last term is positive.

• Therefore:

• Is the requirement for i to expend positive effort

The Contest Stage

• If C contains 2 or more singletons then all non-singleton coalitions will be inactive

Unique Nash Equilibrium

Large Coalitions

• Individual members will spend less than members of smaller coalitions

• Free-riding intensifies

• Value of prize to individual decreases

Equilibrium Payoff

• Termed a valuation

• Depends only on size of individual’s coalition and on size of other coalitions

Positive Externalities

• If the valuation to a specific non-changing coalition increases due to two coalitions merging then there are positive externalities

Positive Externalities

• No active coalition will become inactive after the merge provided C’ remains active

Positive Externalities

• Some previously inactive coalitions may become active due to the merge

• An active coalition will not merge if the new coalition will be inactive

Proposition 3

Exclusive Membership

• Agents announce a possible coalition simultaneously

• Coalitions form according to two rules

The γ-game

• The coalition forms only if all members announce the same coalition

• If one potential member deviates then no coalition forms

The σ-game

• The coalition is composed of all members who announced the same coalition

• If any potential member deviates then the coalition still forms

Stand-alone Stability

• A coalition is stand-alone stable if no individual can improve by becoming a singleton

Unique NE of the σ-game

• In any coalition structure of the σ-game the members of the largest group (including the grand coalition) have an incentive to defect and form a singleton.

Intuition behind NE of σ-game

• By becoming a singleton:

– Obtains maximum prize if victor

– Faces larger and less aggressive opponents

Individual payoff in the γ-game

• ρ≥1

• Measure of intra-group rivalry

• ρ=1 no conflict of interest

• ρ≥2 intense conflict of interest

NE in the γ-game

Characteristics of the NE in the γ-game

• No group will be inactive– If it is its members will form singletons

• When intra-group rivalry is intense– No coalition structure other than singletons

will be supported

Sequential Coalition Formation

• Bloch’s Game (1996)

– First player announces │C1│ which forms– Player │C1│+1 proposes │C2│– Continues until player set is exhausted

Sequential Coalition Formation

• Players will not propose a coalition larger than the smallest in existence

SPE of Bloch’s Game

(13)

Effect of Rivalry

• Low rivalry

– An asymmetric two-sided contest

• First player forms singleton

• Remaining players form a grand coalition

Effect of Rivalry

• High rivalry

– Two possibilities

• All singletons

• Grand coalition

Example

Conclusion

• Simultaneous Coalition Formation

• Larger groups tend to become inactive

• Coalition formation has positive spillovers for non-members

Conclusion

• Sequential Coalition Formation

• Low Rivalry– Two-sided contest

• Intermediate Rivalry– Grand coalition likely

• High Rivalry– Singletons only

Modeling Individual Payoff

• In this model intra-group rivalry may cause another contest

• Individual expenditure in this second contest is denoted si

• Need a sharing rule

Garfinkel and Skaperdas (2006)

A sharing rule to determine individual payoff

μ is the degree of cooperation within the group

Garfinkel and Skaperdas (2006)

Payoff in symmetric NE

Garfinkel and Skaperdas (2006)

• When u=1, there is no conflict

• If prize is divisible it is shared equally

• If indivisible, awarded by lottery

Garfinkel and Skaperdas (2006)

• When u=1, there is no conflict

• This is the function that the Bloch et al. (2006) article examined

• The grand coalition is the most efficient structure when rivalry does not exist

Garfinkel and Skaperdas (2006)

• When u=0, there is complete conflict

• Prize is awarded through contest

Sharing Rule

• Why would a coalition form and then have an additional contest to determine a winner?

• An explicit sharing rule can save the expenditure si

Sharing Rule

• What happens if the individual payoff is determined by contribution to the coalitional effort?

Sharing Rule

• What happens if the individual payoff is determined by contribution to the coalitional effort?

• Then пi = (ri/Rk)*V

Individual Payoff

• What happens if the individual payoff is determined by contribution to the coalitional effort?

• Uki(Ck,R(C)) = Pk* пk - rk

– Becomes:

• (Rk/R)*(rk/Rk)*V-rk

Individual Payoff

• What happens if the individual payoff is determined by contribution to the coalitional effort?

• Uki(Ck,R(C)) = Pk* пk - rk

– Becomes:

• (Rk/R)*(rk/Rk)*V-rk = (rk/R)*V - rk

Individual Payoff

• (rk/R)*V - rk

• When the contribution to the aggregate coalitional effort is the rule which determines individual payoff it appears that any player will be indifferent between joining a coalition of any size and remaining a singleton

Further Research

• What are the effects of other rules determining individual payoff?

• Can Garfinkel and Skaperdas model be interpreted in different ways?

The End