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The Role of Commitment in Repeated Games Julio Gonz alez-D az Ignacio Garc a-Jurado Department of Statistics and Operations Research School of Mathematics Universidade de Santiago de Compostela Introduction Outline Motivation

  1. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  2. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Why do we need VSPE? Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  3. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Why do we need VSPE? In our model, we face very large trees Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  4. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Why do we need VSPE? In our model, we face very large trees There can be subgames with no Nash Equilibrium Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  5. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Why do we need VSPE? In our model, we face very large trees There can be subgames with no Nash Equilibrium Hence, Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  6. VSPE Some Examples Unilateral Commitments Formal Definitions Conclusions Discussion Virtually Subgame Perfect Equilibrium Discussion Subgame Perfect Vs Virtually Subgame Perfect Why do we need VSPE? In our model, we face very large trees There can be subgames with no Nash Equilibrium Hence, We cannot use the classic results for the existence of SPE Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  7. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Outline Virtually Subgame Perfect Equilibrium 1 Some Examples Formal Definitions Discussion Unilateral Commitments 2 Definitions Delegation Models and Unilateral Commitments Results Conclusions 3 Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  8. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions The stage game: Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  9. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions   The stage game: G := ( N, A, ϕ )  Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  10. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ )  Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  11. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i  Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  12. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )  Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  13. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )      The repeated game: G T δ := ( N, S, ϕ δ )    Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  14. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ )    Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  15. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )    Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  16. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  17. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  18. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  19. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that ∅ � A c i ⊆ A i , 1 Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  20. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that ∅ � A c i ⊆ A i , 1 j ∈ N 2 A j − α i : � → A i 2 Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  21. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that ∅ � A c i ⊆ A i , 1 → A i and, for each A c ∈ � j ∈ N 2 A j − j ∈ N 2 A j , α i : � 2 α i ( A c ) ∈ A c i . Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  22. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that ∅ � A c i ⊆ A i , 1 → A i and, for each A c ∈ � j ∈ N 2 A j − j ∈ N 2 A j , α i : � 2 α i ( A c ) ∈ A c i . Commitments are Unilateral Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  23. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Definitions  N := { 1 , . . . , n }  The stage game: G := ( N, A, ϕ ) A := � i ∈ N A i ϕ := ( ϕ 1 , . . . , ϕ n )   N := { 1 , . . . , n }   S := � i ∈ N S i  The repeated game: G T δ := ( N, S, ϕ δ ) ( S i := A H i )   ϕ δ  The UC-extension: U ( G ) := ( N, A U , ϕ U ) A U := � i ∈ N A U i , where A U i is the set of all couples ( A c i , α i ) such that ∅ � A c i ⊆ A i , 1 → A i and, for each A c ∈ � j ∈ N 2 A j − j ∈ N 2 A j , α i : � 2 α i ( A c ) ∈ A c i . Complete Information Commitments are Unilateral Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  24. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  25. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  26. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Players: 2 principals, 2 agents Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  27. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes: Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  28. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes: depend on the payoffs Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  29. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Players: 2 principals, 2 agents Compensation Monotonic Function Schemes: depend on the payoffs Contracts: Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  30. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents Compensation Monotonic Function Schemes: depend on the payoffs Contracts: Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  31. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Compensation Monotonic Function Schemes: depend on the payoffs Contracts: Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  32. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Proportional within A c Compensation Monotonic Function Schemes: depend on the payoffs Contracts: Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  33. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Proportional within A c Compensation Monotonic Function Schemes: depend on the payoffs depend on the strategies Contracts: Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  34. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Proportional within A c Compensation Monotonic Function Schemes: depend on the payoffs depend on the strategies Contracts: Public Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  35. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Proportional within A c Compensation Monotonic Function Schemes: depend on the payoffs depend on the strategies Contracts: Public Public Complete Information!!! Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  36. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Delegation Models and Unilateral Commitments Fershtman et al (1991) Our Model Players: 2 principals, 2 agents 2 principals, 2 agents Proportional within A c Compensation Monotonic Function Schemes: depend on the payoffs depend on the strategies Contracts: Public Public Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  37. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  38. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  39. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  40. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: The Folk Theorems Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  41. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: The Folk Theorems Minmax Payoffs: v i = a − i ∈ A − i max min a i ∈ A i ϕ i ( a i , a − i ) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  42. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: The Folk Theorems Minmax Payoffs: v i = a − i ∈ A − i max min a i ∈ A i ϕ i ( a i , a − i ) Feasible and Individually Rational Payoffs: Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  43. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: The Folk Theorems Minmax Payoffs: v i = a − i ∈ A − i max min a i ∈ A i ϕ i ( a i , a − i ) Feasible and Individually Rational Payoffs: F := co { ϕ ( a ) : a ∈ ϕ ( A ) } Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  44. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results Unilateral Commitments Objectives Results: The Folk Theorems Minmax Payoffs: v i = a − i ∈ A − i max min a i ∈ A i ϕ i ( a i , a − i ) Feasible and Individually Rational Payoffs: F := F ∩ { u ∈ R n : u ≥ v } ¯ F := co { ϕ ( a ) : a ∈ ϕ ( A ) } Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  45. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  46. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  47. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Theorem 1 (Garc´ ıa-Jurado et al., 2000) No assumption is needed for the Nash folk theorem with UC. Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  48. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Theorem 1 (Garc´ ıa-Jurado et al., 2000) No assumption is needed for the Nash folk theorem with UC. Proof Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  49. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Theorem 1 (Garc´ ıa-Jurado et al., 2000) No assumption is needed for the Nash folk theorem with UC. Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  50. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Theorem 1 (Garc´ ıa-Jurado et al., 2000) No assumption is needed for the Nash folk theorem with UC. Moreover, the Nash equilibrium of the repeated game with UC can be chosen such that Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  51. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Nash Folk Theorem (without UC) G must have a Nash equilibrium in which some player gets more than his minmax payoff Theorem 1 (Garc´ ıa-Jurado et al., 2000) No assumption is needed for the Nash folk theorem with UC. Moreover, the Nash equilibrium of the repeated game with UC can be chosen such that the subgame that begins after the commitments has a unique Nash payoff Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  52. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  53. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Subgame Perfect Folk Theorem (without UC) G must have a pair of Nash equilibra in which some player gets different payoffs Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  54. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Subgame Perfect Folk Theorem (without UC) G must have a pair of Nash equilibra in which some player gets different payoffs Proposition 1 The counterpart of Theorem 1 for VSPE does not hold. Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  55. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Subgame Perfect Folk Theorem (without UC) G must have a pair of Nash equilibra in which some player gets different payoffs Proposition 1 The counterpart of Theorem 1 for VSPE does not hold. Proposition 2 a ∈ A be a Nash equilibrium of G . Then, the game U ( G ) has Let ¯ a VSPE with payoff ϕ (¯ a ). Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  56. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Subgame Perfect Folk Theorem (without UC) G must have a pair of Nash equilibra in which some player gets different payoffs Proposition 1 The counterpart of Theorem 1 for VSPE does not hold. Proposition 2 a ∈ A be a Nash equilibrium of G . Then, the game U ( G ) has Let ¯ a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  57. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  58. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  59. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) = ⇒ U ( G ( δ, T )) has a Nash Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  60. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) = ⇒ U ( G ( δ, T )) has a Nash 2 Apply Proposition 2 to U ( G ( δ, T )) Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  61. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) = ⇒ U ( G ( δ, T )) has a Nash 2 Apply Proposition 2 to U ( G ( δ, T )) = ⇒ U ( U ( G ( δ, T ))) has a VSPE Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  62. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) = ⇒ U ( G ( δ, T )) has a Nash 2 Apply Proposition 2 to U ( G ( δ, T )) = ⇒ U ( U ( G ( δ, T ))) has a VSPE Moreover, the VSPE can be chosen such that Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  63. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Finite Horizon Theorem 1 No assumptions for the Nash folk theorem with UC. a ∈ A be a Nash equilibrium of G . Then, the Proposition 2 Let ¯ game U ( G ) has a VSPE with payoff ϕ (¯ a ). Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Proof 1 Apply Theorem 1 to G ( δ, T ) = ⇒ U ( G ( δ, T )) has a Nash 2 Apply Proposition 2 to U ( G ( δ, T )) = ⇒ U ( U ( G ( δ, T ))) has a VSPE Moreover, the VSPE can be chosen such that the subgame that begins after the first stage of commitments has a unique Nash payoff Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  64. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Discussion Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  65. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Discussion Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Remarks Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  66. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Discussion Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Remarks Are two stages of commitments natural?? Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  67. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Discussion Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Remarks Are two stages of commitments natural?? We “allow for” commitments on commitments Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  68. VSPE Definitions Unilateral Commitments Delegation Models and Unilateral Commitments Conclusions Results The Folk Theorems Discussion Theorem 2 No assumption is needed for the VSPE folk theorem when we have two stages of commitments. Remarks Are two stages of commitments natural?? We “allow for” commitments on commitments President − → Manager − → Director Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  69. The State of Art Necessary and Sufficient Conditions for the Folk Theorems 2 stages Without UC 1 stage of UC of UC Nash Theorem None Infinite Horizon (Fudenberg and Maskin, 1986) (Virtual) Perfect Th. Non-Equivalent Utilities Infinite Horizon (Abreu et al., 1994) Nash Theorem Minimax-Bettering Ladder Finite Horizon (Gonz´ alez-D´ ıaz, 2003) (Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

  70. The State of Art Necessary and Sufficient Conditions for the Folk Theorems 2 stages Without UC 1 stage of UC of UC Nash Theorem None None Infinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2) (Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon (Abreu et al., 1994) (Prop. 2) Nash Theorem Minimax-Bettering Ladder Finite Horizon (Gonz´ alez-D´ ıaz, 2003) (Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

  71. The State of Art Necessary and Sufficient Conditions for the Folk Theorems 2 stages Without UC 1 stage of UC of UC Nash Theorem None None Infinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2) (Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon (Abreu et al., 1994) (Prop. 2) Nash Theorem Minimax-Bettering Ladder None Finite Horizon (Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000) (Virtual) Perfect Th. Recursively-distinct Finite Horizon Nash payoffs (Smith, 1995)

  72. The State of Art Necessary and Sufficient Conditions for the Folk Theorems 2 stages Without UC 1 stage of UC of UC Nash Theorem None None Infinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2) (Virtual) Perfect Th. Non-Equivalent Utilities None Infinite Horizon (Abreu et al., 1994) (Prop. 2) Nash Theorem Minimax-Bettering Ladder None Finite Horizon (Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000) (Virtual) Perfect Th. Recursively-distinct Minimax-Bettering Ladder Finite Horizon Nash payoffs (Smith, 1995) (Prop. 2, only sufficient)

  73. The State of Art Necessary and Sufficient Conditions for the Folk Theorems 2 stages Without UC 1 stage of UC of UC Nash Theorem None None None Infinite Horizon (Fudenberg and Maskin, 1986) (Prop. 2) (Prop. 2) (Virtual) Perfect Th. Non-Equivalent Utilities None None Infinite Horizon (Abreu et al., 1994) (Prop. 2) (Prop. 2) Nash Theorem Minimax-Bettering Ladder None None Finite Horizon (Gonz´ alez-D´ ıaz, 2003) (Garc´ ıa-Jurado et al., 2000) (Prop. 2) (Virtual) Perfect Th. Recursively-distinct Minimax-Bettering Ladder None Finite Horizon Nash payoffs (Smith, 1995) (Prop. 2, only sufficient) (Th. 2)

  74. VSPE Unilateral Commitments Conclusions Outline Virtually Subgame Perfect Equilibrium 1 Some Examples Formal Definitions Discussion Unilateral Commitments 2 Definitions Delegation Models and Unilateral Commitments Results Conclusions 3 Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

  75. VSPE Unilateral Commitments Conclusions Conclusions Our contribution Julio Gonz´ alez-D´ ıaz, Ignacio Garc´ ıa-Jurado The Role of Commitment in Repeated Games

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