endogenous coalition formation in contests santiago sánchez-pagés review of economic design 2007
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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
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)
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
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
• 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
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
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
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
Effect of Rivalry
• Low rivalry
– An asymmetric two-sided contest
• First player forms singleton
• Remaining players form a grand coalition
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)
• 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?
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