Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 nonchanging 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 │C │+1 proposes │C │ – Continues until player set is exhausted 1 2 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