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Finitely Repeated Games: A Generalized Nash Folk Theorem Julio Gonz alez-D az Department of Statistics and Operations Research Faculty of Mathematics Universidade de Santiago de Compostela Introduction Outline Outline Finitely

  1. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion The State of Art The Folk Theorems Nash Subgame Perfect Fudenberg and Maskin (1986) Infinite The “Folk Theorem” (1970s) Abreu et al. (1994) Horizon Wen (1994) Finite Horizon Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  2. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion The State of Art The Folk Theorems Nash Subgame Perfect Fudenberg and Maskin (1986) Infinite The “Folk Theorem” (1970s) Abreu et al. (1994) Horizon Wen (1994) Finite Benoˆ ıt and Krishna (1987) Horizon Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  3. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion The State of Art The Folk Theorems Nash Subgame Perfect Fudenberg and Maskin (1986) Infinite The “Folk Theorem” (1970s) Abreu et al. (1994) Horizon Wen (1994) Benoˆ ıt and Krishna (1985) Finite Benoˆ ıt and Krishna (1987) Smith (1995) Horizon Gossner (1995) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  4. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion The State of Art The Folk Theorems Nash Subgame Perfect Fudenberg and Maskin (1986) Infinite The “Folk Theorem” (1970s) Abreu et al. (1994) Horizon Wen (1994) Benoˆ ıt and Krishna (1985) Finite Benoˆ ıt and Krishna (1987) Smith (1995) Horizon Gossner (1995) Necessary and Sufficient conditions Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  5. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion The State of Art The Folk Theorems Nash Subgame Perfect Fudenberg and Maskin (1986) Infinite The “Folk Theorem” (1970s) Abreu et al. (1994) Horizon Wen (1994) Benoˆ ıt and Krishna (1985) Finite Benoˆ ıt and Krishna (1987) Smith (1995) Horizon Gossner (1995) Necessary and Sufficient conditions Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  6. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  7. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  8. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs For each player i there is a Nash Equilibrium a i of G such that ϕ i ( a i ) > v i Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  9. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs For each player i there is a Nash Equilibrium a i of G such that ϕ i ( a i ) > v i Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  10. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs For each player i there is a Nash Equilibrium a i of G such that ϕ i ( a i ) > v i Result Every payoff in ¯ F can be approximated in equilibrium Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  11. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs For each player i there is a Nash Equilibrium a i of G such that ϕ i ( a i ) > v i Result Every payoff in ¯ F can be approximated in equilibrium For each u ∈ ¯ F and each ε > 0, there are T 0 and δ 0 such that for each T ≥ T 0 and each δ ∈ [ δ 0 , 1], there is a Nash Equilibrium σ of G ( δ, T ) satisfying that � ϕ T δ ( σ ) − u � < ε Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  12. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Assumption for the game G Existence of strictly rational Nash payoffs For each player i there is a Nash Equilibrium a i of G such that ϕ i ( a i ) > v i Result Every payoff in ¯ F can be approximated in equilibrium For each u ∈ ¯ F and each ε > 0, there are T 0 and δ 0 such that for each T ≥ T 0 and each δ ∈ [ δ 0 , 1], there is a Nash Equilibrium σ of G ( δ, T ) satisfying that � ϕ T δ ( σ ) − u � < ε Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  13. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Idea of the proof We want to approximate the payoff u > v in equilibrium Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  14. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Idea of the proof We want to approximate the payoff u > v in equilibrium Assume the existence of a Nash Equilibrium a of G such that for each i ∈ N, ϕ i ( a ) > v i Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  15. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Idea of the proof We want to approximate the payoff u > v in equilibrium Assume the existence of a Nash Equilibrium a of G such that for each i ∈ N, ϕ i ( a ) > v i Equilibrium path u, u, . . . , u, u, ϕ ( a ) , . . . , ϕ ( a ) � �� � � �� � T − L stages L stages Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  16. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion (Benoit & Krishna 1987) Idea of the proof We want to approximate the payoff u > v in equilibrium Assume the existence of a Nash Equilibrium a of G such that for each i ∈ N, ϕ i ( a ) > v i Equilibrium path u, u, . . . , u, u, ϕ ( a ) , . . . , ϕ ( a ) � �� � � �� � T − L stages L stages Deviation of agent i u i , . . . , u i , M i , v i , . . . , v i � �� � � �� � T − L +1 stages L stages Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  17. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion Why Nash Equilibrium? Example (A game for which the Nash folk theorem is needed) L M R T 2,2 9,1 1,0 M 1,9 0,0 0,0 B 0,1 0,0 0,0 Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  18. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion Why Nash Equilibrium? Example (A game for which the Nash folk theorem is needed) L M R T 2,2 9,1 1,0 M 1,9 0,0 0,0 B 0,1 0,0 0,0 subgame perfection + Smith (1995) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  19. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion Why Nash Equilibrium? Example (A game for which the Nash folk theorem is needed) L M R T 2,2 9,1 1,0 M 1,9 0,0 0,0 B 0,1 0,0 0,0 subgame perfection + Smith (1995) − → (2 , 2) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  20. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion Why Nash Equilibrium? Example (A game for which the Nash folk theorem is needed) L M R T 2,2 9,1 1,0 M 1,9 0,0 0,0 B 0,1 0,0 0,0 subgame perfection + Smith (1995) − → (2 , 2) Nash + Benoˆ ıt and Krishna (1987) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  21. Finitely Repeated Games Definitions and Classic Results Our Contribution Finite Horizon Nash Folk Theorem Discussion Why Nash Equilibrium? Example (A game for which the Nash folk theorem is needed) L M R T 2,2 9,1 1,0 M 1,9 0,0 0,0 B 0,1 0,0 0,0 subgame perfection + Smith (1995) − → (2 , 2) Nash + Benoˆ ıt and Krishna (1987) − → (5 , 5) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  22. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Outline Finitely Repeated Games 1 Definitions and Classic Results Finite Horizon Nash Folk Theorem Our Contribution 2 Minmax Bettering Ladders The New Folk Theorem The Generalized Folk Theorem Discussion 3 Unobservable Mixed Actions Conclusions Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  23. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Smith (1995) : Recursively distinct Nash payoffs Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  24. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  25. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Minmax Payoff (0,0,0) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  26. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Minmax Payoff (0,0,0) Nash Equilibrium (T,l,L), Payoff (0,0,3) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  27. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Minmax Payoff (0,0,0) Nash Equilibrium (T,l,L), Payoff (0,0,3) (B-K not met) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  28. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Minmax Payoff (0,0,0) Nash Equilibrium (T,l,L), Payoff (0,0,3) (B-K not met) Player 3 can be threatened Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  29. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  30. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  31. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R The profile α 3 =(T,l,R) is a Nash Equilibrium of the reduced game with Payoff (0,3,-1) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  32. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R The profile α 3 =(T,l,R) is a Nash Equilibrium of the reduced game with Payoff (0,3,-1) Now player 2 can be threatened Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  33. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example r l m r l m 1,-1,-1 T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  34. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example r l m r l m 1,-1,-1 T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R and player 2 to play r Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  35. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example r l m r l m 1,-1,-1 T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R and player 2 to play r The profile α 32 =(T,r,R) is a Nash Equilibrium of the reduced game with Payoff (1,-1,-1) Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  36. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Example r l m r l m 1,-1,-1 T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Player 3 is forced to play R and player 2 to play r The profile α 32 =(T,r,R) is a Nash Equilibrium of the reduced game with Payoff (1,-1,-1) Now player 1 can be threatened Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  37. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  38. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  39. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players game G Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  40. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players game G σ 1 “Nash equilibrium” Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  41. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 game G σ 1 “Nash equilibrium” Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  42. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 game G G ( a N 1 ) σ 1 “Nash equilibrium” Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  43. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 game G G ( a N 1 ) σ 1 σ 2 “Nash equilibrium” Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  44. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . game G G ( a N 1 ) . . . σ 1 σ 2 “Nash equilibrium” . . . Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  45. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 game G G ( a N 1 ) . . . σ 1 σ 2 “Nash equilibrium” . . . Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  46. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 game G G ( a N 1 ) . . . G ( a N h − 1 ) σ 1 σ 2 “Nash equilibrium” . . . Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  47. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 game G G ( a N 1 ) . . . G ( a N h − 1 ) σ 1 σ 2 σ h “Nash equilibrium” . . . Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  48. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) σ 1 σ 2 σ h “Nash equilibrium” . . . Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  49. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  50. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − A minimax-bettering ladder of a game G is a triplet {N , A , Σ } Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  51. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − A minimax-bettering ladder of a game G is a triplet {N , A , Σ } N := {∅ = N 0 � N 1 � · · · � N h } subsets of N Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  52. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − A minimax-bettering ladder of a game G is a triplet {N , A , Σ } N := {∅ = N 0 � N 1 � · · · � N h } subsets of N A := { a N 1 ∈ A N 1 , . . . , a N h − 1 ∈ A N h − 1 } Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  53. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − A minimax-bettering ladder of a game G is a triplet {N , A , Σ } N := {∅ = N 0 � N 1 � · · · � N h } subsets of N A := { a N 1 ∈ A N 1 , . . . , a N h − 1 ∈ A N h − 1 } Σ := { σ 1 , . . . , σ h } Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  54. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Formal Definition ∅ reliable players N 1 . . . N h − 1 N h game G G ( a N 1 ) . . . G ( a N h − 1 ) − − − σ 1 σ 2 σ h “Nash equilibrium” . . . − − − A minimax-bettering ladder of a game G is a triplet {N , A , Σ } N := {∅ = N 0 � N 1 � · · · � N h } subsets of N A := { a N 1 ∈ A N 1 , . . . , a N h − 1 ∈ A N h − 1 } Σ := { σ 1 , . . . , σ h } N h is the top rung of the ladder Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  55. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Some properties Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  56. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Some properties A ladder with top rung N h is maximal if there is no ladder with top rung N h ′ such that N h � N h ′ Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  57. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Some properties A ladder with top rung N h is maximal if there is no ladder with top rung N h ′ such that N h � N h ′ A game G is decomposable as a complete minimax-bettering ladder if it has a minimax-bettering ladder with N as its top rung Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  58. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem Minmax Bettering Ladders Some properties A ladder with top rung N h is maximal if there is no ladder with top rung N h ′ such that N h � N h ′ A game G is decomposable as a complete minimax-bettering ladder if it has a minimax-bettering ladder with N as its top rung Lemma All the maximal ladders of a game G have the same top rung Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  59. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Assumption for the game G Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  60. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Assumption for the game G Existence of a complete minmax bettering ladder Result Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  61. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Assumption for the game G Existence of a complete minmax bettering ladder Result Every payoff in ¯ F can be approximated in equilibrium Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  62. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Assumption for the game G Existence of a complete minmax bettering ladder Result Every payoff in ¯ F can be approximated in equilibrium Remark Unlike Benoˆ ıt and Krishna’s result, this theorem provides a necessary and sufficient condition Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  63. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Assumption for the game G Existence of a complete minmax bettering ladder Result Every payoff in ¯ F can be approximated in equilibrium Remark Unlike Benoˆ ıt and Krishna’s result, this theorem provides a necessary and sufficient condition Why the word generalized ? Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  64. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Example (Idea of the proof) l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  65. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Example (Idea of the proof) l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Nash Equilibrium: α =(T,l,L), payoff (0,0,3). Hence, player 3 is reliable Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  66. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Example (Idea of the proof) l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Nash Equilibrium: α =(T,l,L), payoff (0,0,3). Hence, player 3 is reliable “Nash Equilibrium”: α 3 =(T,l,R), payoff (0,3,-1). Hence, player 2 is reliable Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  67. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Example (Idea of the proof) l m r l m r T 0,0,3 0,-1,0 0,-1,0 0,3,-1 0,-1,-1 1,-1,-1 M -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 B -1,0,0 0,-1,0 0,-1,0 -1,0,-1 -1,-1,-1 0,-1,-1 L R Nash Equilibrium: α =(T,l,L), payoff (0,0,3). Hence, player 3 is reliable “Nash Equilibrium”: α 3 =(T,l,R), payoff (0,3,-1). Hence, player 2 is reliable “Nash Equilibrium”: α 32 =(T,r,R), payoff (1,-1,-1). Hence, player 1 is reliable Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  68. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  69. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  70. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  71. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages The ladder Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  72. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages α (0 , 0 , 3) The ladder Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  73. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages α α 3 (0 , 0 , 3) (0 , 3 , − 1) The ladder Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  74. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages α α 3 (0 , 0 , 3) α 32 (0 , 3 , − 1) (1 , − 1 , − 1) The ladder Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  75. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages α α 3 (0 , 0 , 3) α 32 (0 , 3 , − 1) (1 , − 1 , − 1) The ladder u Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

  76. Finitely Repeated Games Minmax Bettering Ladders Our Contribution The New Folk Theorem Discussion The Generalized Folk Theorem The New Folk Theorem P L i stages (Julio Gonz´ alez-D´ ıaz 2003) Idea of the proof We want to approximate the payoff u > v in equilibrium. Equilibrium Path ϕ ( α 32 ) , . . . , ϕ ( α 32 ) u, u, . . . , u, u, . . . ϕ ( α ) , . . . , ϕ ( α ) � �� � � �� � � �� � T − L 1 stages L 3 stages α α 3 (0 , 0 , 3) α 32 (0 , 3 , − 1) (1 , − 1 , − 1) The ladder u Julio Gonz´ alez-D´ ıaz Finitely Repeated Games: A Generalized Nash Folk Theorem

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