Conventions and Coalitions in Repeated Games S. Nageeb Ali Ce Liu (Will) Penn State UCSD May 29, 2018
Cooperative game theory takes a “bird’s eye” view to group decisions. Studies stable “static conventions” immune to group deviations. Theory of repeated games studies dynamic ongoing relationships. Studies stable “dynamic conventions” immune to individual deviations.
Cooperative game theory takes a “bird’s eye” view to group decisions. Studies stable “static conventions” immune to group deviations. Theory of repeated games studies dynamic ongoing relationships. Studies stable “dynamic conventions” immune to individual deviations. Our objective: Develop a theory of repeated games for coalitional behavior.
the enterprise Framework nests both noncooperative and cooperative environments. Results speak both to repeated play of a cooperative game as well as coalitional deviations in the repeated play of a non-cooperative game. Solution-concept combines farsightedness and limited commitment.
two results
two results Perfect monitoring in both NTU + TU games: Folk Theorem . Every payoff in the set of feasible and strictly individually rational payoffs is supported as δ → 1 . ⇒ coalitional deviations do not refine play if players are patient.
two results Perfect monitoring in both NTU + TU games: Folk Theorem . Every payoff in the set of feasible and strictly individually rational payoffs is supported as δ → 1 . ⇒ coalitional deviations do not refine play if players are patient. TU games + secret transfers: Anti-Folk Theorem . Only payoffs in the β -core are supported, and any such payoff is supported as δ → 1 . ⇒ secret transfers + coalitional deviations undo dynamic incentives.
4 Folk Theorem for NTU 1 Introduction 5 Transferable Utility 2 Examples 6 Related Lit + Summary 3 Model
the roommates problem 2 rooms, 4 people Bob Ann Carol
the roommates problem 2 rooms, 4 people Bob Ann Carol David
the cycle Arrows point to first preference.
the cycle Arrows point to first preference. No one is a worse roommate than David.
the cycle Arrows point to first preference. No one is a worse roommate than David. U i ( ij ) = 4 − k if j is i ’s k th favorite partner.
no stable match
no stable match
no stable match
no stable match
Suppose individuals play repeatedly. Coalitions cannot commit to future behavior, but can jointly deviate in the stage-game. Our solution-concept seeks robustness to one-shot coalitional deviations . The key idea: • Coalitions have some short-term commitment (which one models implicitly in the cooperative stage-game). • But coalitions lack long-term commitment.
A stable convention if δ > 1 2 .
A stable convention if δ > 1 2 .
A stable convention if δ > 1 2 .
A stable convention if δ > 1 2 .
A divide-the-dollar example Suppose that { 1, 2, 3 } are dividing a dollar. Any division requires approval of at least two players. Player 1 has veto power, and must be included in any winning coalition.
A divide-the-dollar example Suppose that { 1, 2, 3 } are dividing a dollar. Any division requires approval of at least two players. Player 1 has veto power, and must be included in any winning coalition. Core of stage-game: • Player 1 captures entire dollar. • Any other outcome has a profitable deviation for { 1, 2 } or { 1, 3 } .
repeated divisions: perfect monitoring Suppose the environment is repeated and coalitions cannot commit to long-term deviations. We claim that one can now support every division as δ → 1.
( 0, 1 2 , 1 ) is supportable using a stable convention if δ ⩾ 1 2 . Claim: 2
( 0, 1 2 , 1 ) is supportable using a stable convention if δ ⩾ 1 2 . Claim: 2 Pf: Use a core-reversion convention : ( 0, 1 2 , 1 ) • Choose every period on path of play. 2 • Revert to ( 1, 0, 0 ) if any other allocation is chosen.
( 0, 1 2 , 1 ) is supportable using a stable convention if δ ⩾ 1 2 . Claim: 2 Pf: Use a core-reversion convention : ( 0, 1 2 , 1 ) • Choose every period on path of play. 2 • Revert to ( 1, 0, 0 ) if any other allocation is chosen. Even if 1 offers entire surplus to coalition-partner: ( 1 − δ )( 1 ) + δ ( 0 ) ⩽ 1 2 . Key idea : Veto player can’t recruit anyone to join his coalition.
( 0, 1 2 , 1 ) is supportable using a stable convention if δ ⩾ 1 2 . Claim: 2 Pf: Use a core-reversion convention : ( 0, 1 2 , 1 ) • Choose every period on path of play. 2 • Revert to ( 1, 0, 0 ) if any other allocation is chosen. Even if 1 offers entire surplus to coalition-partner: ( 1 − δ )( 1 ) + δ ( 0 ) ⩽ 1 2 . Key idea : Veto player can’t recruit anyone to join his coalition. Core-reversion supports any payoff vector u where min { u 2 , u 3 } ⩾ 1 − δ .
anti-folk theorem with secret transfers Suppose players can make “secret transfers.” Convention conditions only on identity of the blocking coalition, and chosen allocation, but not eventual payoffs.
anti-folk theorem with secret transfers Suppose players can make “secret transfers.” Convention conditions only on identity of the blocking coalition, and chosen allocation, but not eventual payoffs. Stable convention supports only stage-game core: ( 1, 0, 0 ) . (regardless of δ ).
4 Folk Theorem for NTU 1 Introduction 5 Transferable Utility 2 Examples 6 Related Lit + Summary 3 Model
primitives: NTU stage-game Choice over alternatives: a ∈ A . Generated Payoffs: v : A → R n . Set of Coalitions: C ≡ 2 N \ ∅ . If alternative a is recommended, E C ( a ) is the set of alternatives to which coalition C can deviate, assuming no one else deviates. E C ( · ) is coalition C ’s effectivity function.
example: strategic-form game a is action profile of the stage-game. Effectivity function extends usual notion of individual deviations to coalitional deviations. { } a ′ ∈ A : a ′ E C ( a ) ≡ − C = a − C
example: cooperative game a is a partition of N , and A is the set of all partitions. v i ( a ) is the payoff from partition a . E C ( a ) specifies partitions that coalition C can generate, where “untouched” coalitions continue to follow a . E C ( a ) ⊆ { a ′ : C ′ ∩ C = ∅ and C ′ ∈ a ⇒ C ′ ∈ a ′ }
examples Voting: Convention specifies a default policy. But a coalition can introduce an alternative and block the default if it has more than a majority of players. Matching: Convention specifies who matches with whom. But pairs of players can defect and match differently.
assumptions on effectivity Transitivity: For every C , a ′ ∈ E C ( a ) & a ′′ ∈ E C ( a ′ ) ⇒ a ′′ ∈ E C ( a ) . Reflexivity: For every C and a , a ∈ E C ( a ) . Omnipotence of the Grand Coalition: E N ( a ) = A for every a . (For the β -core result)
repeated interaction For a path p where a t is alternative in period t , player i ’s payoff is ∞ ∑ δ t v i ( a t ) . U i ( p ) ≡ ( 1 − δ ) t = 0 ( ) Outcome at t ≡ chosen alternative, coalition that blocks (if any) . A history is a sequence of outcomes. A convention is a mapping, σ : Histories → A .
repeated interaction For a path p where a t is alternative in period t , player i ’s payoff is ∞ ∑ δ t v i ( a t ) . U i ( p ) ≡ ( 1 − δ ) t = 0 ( ) Outcome at t ≡ chosen alternative, coalition that blocks (if any) . A history is a sequence of outcomes. A convention is a mapping, σ : Histories → A . a ( h | σ ) ≡ σ ( h ) is alternative recommended by σ after history h . p ( h | σ ) is path recommended by σ after history h . (computed recursively from a ( h | σ ) ).
Definition. A convention is stable if for every history h , ∄ coalition C and a ′ ∈ E C ( a ( h | σ )) such that for all i in C , ( 1 − δ ) v i ( a ′ ) + δU i ( p ( h , ( a ′ , C ) | σ )) > U i ( p ( h | σ )) . No coalition wishes to deviate at any history, anticipating no deviations at all other histories.
features of stable conventions Recursivity. Relationship to core of stage-game. One-shot Deviation Principles?
recursivity Can use APS to characterize implementable outcomes for a fixed δ .
relationship to core of stage-game Definition. An alternative a is a core-alternative of stage-game if ∄ coalition C and a ′ ∈ E C ( a ) such that for all i in C . v i ( a ′ ) > v i ( a ) .
relationship to core of stage-game Definition. An alternative a is a core-alternative of stage-game if ∄ coalition C and a ′ ∈ E C ( a ) such that for all i in C . v i ( a ′ ) > v i ( a ) . [SPE of G ∞ : NE of G ]. [Stable convention of G ∞ : Core of G ] ≃
Formally, warding off one-shot coalitional deviations. Because this includes singleton coalitions, this is a refinement of SPE. But the sufficiency of examining only one-shot deviations (the so-called “one-shot deviation principle”) is typically a result, not an assumption. What comes from assuming it?
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