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 ={C 1,C 2,…,C K } |C k | is the cardinality of C Ascending ordering: |C k | ≤ |C k+1 | If |C 1 | = |C K | then the coalition structure is symmetric

Resource Pooling r i denotes the resources expended by agent i R k =∑ iЄCk r i R(C) = (R 1,R 2,…,R K )

Contest Success Function Tullock CSF

Contest Success Function Tullock CSF Typo

Payoff Function All members of the winning coalition receive п k

Payoff Function In Baik п k is modeled explicitly as a sharing rule.

Payoff Function Does the individual payoff function п k have an effect on the coalition structure?

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 │C 1 │ which forms –Player │C 1 │+1 proposes │C 2 │ –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 s i 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 s i

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 п i = (r i /R k )*V

Individual Payoff What happens if the individual payoff is determined by contribution to the coalitional effort? U k i (C k,R(C)) = P k* п k - r k –Becomes: (R k /R)*(r k /R k )*V-r k

Individual Payoff What happens if the individual payoff is determined by contribution to the coalitional effort? U k i (C k,R(C)) = P k* п k - r k –Becomes: (R k /R)*(r k /R k )*V-r k = (r k /R)*V - r k

Individual Payoff (r k /R)*V - r k 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