Download Endogenous Coalition Formation in Contests

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