Dynamics and Equilibria
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1 Dynamics and Equilibria Sergiu Hart Presidential Address, GAMES 2008 (July 2008) Revised and Expanded (November 2009) Revised (2010, 2011, 2012, 2013) SERGIU HART c 2008 p. 1
2 DYNAMICS AND EQUILIBRIA Sergiu Hart Center for the Study of Rationality Dept of Economics Dept of Mathematics The Hebrew University of Jerusalem SERGIU HART c 2008 p. 2
3 Papers SERGIU HART c 2008 p. 3
4 Papers Hart and Mas-Colell, Econometrica 2000 Hart and Mas-Colell, J Econ Theory 2001 Hart and Mas-Colell, Amer Econ Rev 2003 Hart, Econometrica 2005 Hart and Mas-Colell, Games Econ Behav 2006 Hart and Mansour, Games Econ Behav 2010 Hart, Games Econ Behav 2011 SERGIU HART c 2008 p. 3
5 Papers Hart and Mas-Colell, Econometrica 2000 Hart and Mas-Colell, J Econ Theory 2001 Hart and Mas-Colell, Amer Econ Rev 2003 Hart, Econometrica 2005 Hart and Mas-Colell, Games Econ Behav 2006 Hart and Mansour, Games Econ Behav 2010 Hart, Games Econ Behav SERGIU HART c 2008 p. 3
6 Book c 2008 p. 4 S ERGIU HART
7 Nash Equilibrium SERGIU HART c 2008 p. 5
8 Nash Equilibrium John Nash, Ph.D. Dissertation, Princeton 1950 SERGIU HART c 2008 p. 5
9 Nash Equilibrium EQUILIBRIUM POINT: John Nash, Ph.D. Dissertation, Princeton 1950 SERGIU HART c 2008 p. 5
10 Nash Equilibrium EQUILIBRIUM POINT: "Each player s strategy is optimal against those of the others." John Nash, Ph.D. Dissertation, Princeton 1950 SERGIU HART c 2008 p. 5
11 Dynamics SERGIU HART c 2008 p. 6
12 Dynamics FACT SERGIU HART c 2008 p. 6
13 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 6
14 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "general" SERGIU HART c 2008 p. 6
15 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "general" : in all games SERGIU HART c 2008 p. 6
16 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "general" : in all games rather than: in specific classes of games SERGIU HART c 2008 p. 6
17 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "general" : in all games rather than: in specific classes of games: two-person zero-sum games two-person potential games supermodular games... SERGIU HART c 2008 p. 6
18 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 7
19 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "leading to Nash equilibrium" SERGIU HART c 2008 p. 7
20 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "leading to Nash equilibrium" : at a Nash equilibrium (or close to it) from some time on SERGIU HART c 2008 p. 7
21 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 8
22 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" SERGIU HART c 2008 p. 8
23 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : SERGIU HART c 2008 p. 8
24 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) SERGIU HART c 2008 p. 8
25 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) simple and efficient SERGIU HART c 2008 p. 8
26 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) simple and efficient: computation (performed at each step) SERGIU HART c 2008 p. 8
27 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) simple and efficient: computation (performed at each step) time (how long to reach equilibrium) SERGIU HART c 2008 p. 8
28 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) simple and efficient: computation (performed at each step) time (how long to reach equilibrium) information (of each player) SERGIU HART c 2008 p. 8
29 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural" : adaptive (reacting, improving,...) simple and efficient: computation (performed at each step) time (how long to reach equilibrium) information (of each player) bounded rationality SERGIU HART c 2008 p. 8
30 Dynamics Dynamics that are NOT "natural" : SERGIU HART c 2008 p. 9
31 Dynamics Dynamics that are NOT "natural" : exhaustive search (deterministic or stochastic) SERGIU HART c 2008 p. 9
32 Exhaustive Search SERGIU HART c 2008 p. 10
33 Exhaustive Search SERGIU HART c 2008 p. 10
34 Exhaustive Search SERGIU HART c 2008 p. 10
35 Exhaustive Search SERGIU HART c 2008 p. 10
36 Exhaustive Search SERGIU HART c 2008 p. 10
37 Exhaustive Search SERGIU HART c 2008 p. 10
38 Exhaustive Search SERGIU HART c 2008 p. 10
39 Exhaustive Search SERGIU HART c 2008 p. 10
40 Exhaustive Search SERGIU HART c 2008 p. 10
41 Einstein s Manuscript Albert Einstein, 1912 On the Special Theory of Relativity (manuscript) SERGIU HART c 2008 p. 11
42 Dynamics Dynamics that are NOT "natural" : exhaustive search (deterministic or stochastic) SERGIU HART c 2008 p. 12
43 Dynamics Dynamics that are NOT "natural" : exhaustive search (deterministic or stochastic) using a mediator SERGIU HART c 2008 p. 12
44 Dynamics Dynamics that are NOT "natural" : exhaustive search (deterministic or stochastic) using a mediator broadcasting the private information and then performing joint computation SERGIU HART c 2008 p. 12
45 Dynamics Dynamics that are NOT "natural" : exhaustive search (deterministic or stochastic) using a mediator broadcasting the private information and then performing joint computation fully rational learning (prior beliefs on the strategies of the opponents, Bayesian updating, optimization) SERGIU HART c 2008 p. 12
46 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation (performed at each step) time (how long to reach equilibrium) information (of each player) SERGIU HART c 2008 p. 13
47 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation (performed at each step) time (how long to reach equilibrium) information (of each player) SERGIU HART c 2008 p. 13
48 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation (performed at each step) time (how long to reach equilibrium) information (of each player) SERGIU HART c 2008 p. 13
49 Natural Dynamics: Information SERGIU HART c 2008 p. 14
50 Natural Dynamics: Information Each player knows only his own payoff (utility) function SERGIU HART c 2008 p. 14
51 Natural Dynamics: Information Each player knows only his own payoff (utility) function (does not know the payoff functions of the other players) SERGIU HART c 2008 p. 14
52 Natural Dynamics: Information UNCOUPLED DYNAMICS : Each player knows only his own payoff (utility) function (does not know the payoff functions of the other players) Hart and Mas-Colell, AER 2003 SERGIU HART c 2008 p. 14
53 Natural Dynamics: Information UNCOUPLED DYNAMICS : Each player knows only his own payoff (utility) function (does not know the payoff functions of the other players) (privacy-preserving, decentralized, distributed...) Hart and Mas-Colell, AER 2003 SERGIU HART c 2008 p. 14
54 Games N-person game in strategic (normal) form: Players i = 1, 2,..., N SERGIU HART c 2008 p. 15
55 Games N-person game in strategic (normal) form: Players i = 1, 2,..., N For each player i: Actions a i in A i SERGIU HART c 2008 p. 15
56 Games N-person game in strategic (normal) form: Players i = 1, 2,..., N For each player i: Actions a i in A i For each player i: Payoffs (utilities) u i (a) u i (a 1, a 2,..., a N ) SERGIU HART c 2008 p. 15
57 Dynamics Time t = 1, 2,... SERGIU HART c 2008 p. 16
58 Dynamics Time t = 1, 2,... At period t each player i chooses an action a i t in A i SERGIU HART c 2008 p. 16
59 Dynamics Time t = 1, 2,... At period t each player i chooses an action a i t in A i according to a probability distribution σ i t in (A i ) SERGIU HART c 2008 p. 16
60 Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N SERGIU HART c 2008 p. 17
61 Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N A general dynamic: SERGIU HART c 2008 p. 17
62 Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N A general dynamic: σ i t σi t ( HISTORY ; GAME ) SERGIU HART c 2008 p. 17
63 Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N A general dynamic: σ i t σi t ( HISTORY ; GAME ) σ i t ( HISTORY ; u1,..., u i,..., u N ) SERGIU HART c 2008 p. 17
64 Uncoupled Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N A general dynamic: σ i t σi t ( HISTORY ; GAME ) σ i t ( HISTORY ; u1,..., u i,..., u N ) An UNCOUPLED dynamic: SERGIU HART c 2008 p. 17
65 Uncoupled Dynamics Fix the set of players 1, 2,..., N and their action spaces A 1, A 2,..., A N A general dynamic: σ i t σi t ( HISTORY ; GAME ) σ i t ( HISTORY ; u1,..., u i,..., u N ) An UNCOUPLED dynamic: σ i t σi t ( HISTORY ; ui ) SERGIU HART c 2008 p. 17
66 Uncoupled Dynamics Simplest uncoupled dynamics SERGIU HART c 2008 p. 18
67 Uncoupled Dynamics Simplest uncoupled dynamics: σ i t fi (a t 1 ; u i ) where a t 1 = (a 1 t 1, a2 t 1,..., an t 1 ) A are the actions of all the players in the previous period SERGIU HART c 2008 p. 18
68 Uncoupled Dynamics Simplest uncoupled dynamics: σ i t fi (a t 1 ; u i ) where a t 1 = (a 1 t 1, a2 t 1,..., an t 1 ) A are the actions of all the players in the previous period Only last period matters ( 1-recall ) SERGIU HART c 2008 p. 18
69 Uncoupled Dynamics Simplest uncoupled dynamics: σ i t fi (a t 1 ; u i ) where a t 1 = (a 1 t 1, a2 t 1,..., an t 1 ) A are the actions of all the players in the previous period Only last period matters ( 1-recall ) Time t does not matter ( stationary ) SERGIU HART c 2008 p. 18
70 Impossibility SERGIU HART c 2008 p. 19
71 Impossibility Theorem. There are NO uncoupled dynamics with 1-recall σ i t fi (a t 1 ; u i ) that yield almost sure convergence of play to pure Nash equilibria of the stage game in all games where such equilibria exist. SERGIU HART c 2008 p. 19
72 Impossibility Theorem. There are NO uncoupled dynamics with 1-recall σ i t fi (a t 1 ; u i ) that yield almost sure convergence of play to pure Nash equilibria of the stage game in all games where such equilibria exist. Hart and Mas-Colell, GEB 2006 SERGIU HART c 2008 p. 19
73 Proof Consider the following two-person game, which has a unique pure Nash equilibrium C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 20
74 Proof Consider the following two-person game, which has a unique pure Nash equilibrium (R3,C3) C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 20
75 Proof Consider the following two-person game, which has a unique pure Nash equilibrium (R3,C3) C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 Assume by way of contradiction that we are given an uncoupled, 1-recall, stationary dynamic that yields almost sure convergence to pure Nash equilibria when these exist SERGIU HART c 2008 p. 20
76 Proof Suppose the play at time t 1 is (R1,C1) C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 21
77 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 21
78 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 21
79 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t Proof: Change the payoff function of COLIN so that (R1,C1) is the unique pure Nash eq. C1 C2 C3 R1 1,0 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 21
80 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t Proof: Change the payoff function of COLIN so that (R1,C1) is the unique pure Nash eq. C1 C2 C3 R1 1,1 0,1 1,0 R2 0,1 1,0 1,0 R3 0,1 0,1 1,0 SERGIU HART c 2008 p. 21
81 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t Proof: Change the payoff function of COLIN so that (R1,C1) is the unique pure Nash eq. In the new game, ROWENA must play R1 after (R1,C1) (by 1-recall, stationarity, and a.s. convergence to the pure Nash eq.) SERGIU HART c 2008 p. 21
82 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t Proof: Change the payoff function of COLIN so that (R1,C1) is the unique pure Nash eq. In the new game, ROWENA must play R1 after (R1,C1) (by 1-recall, stationarity, and a.s. convergence to the pure Nash eq.) By uncoupledness, the same holds in the original game SERGIU HART c 2008 p. 21
83 Proof Suppose the play at time t 1 is (R1,C1) ROWENA is best replying at (R1,C1) ROWENA will play R1 also at t SERGIU HART c 2008 p. 21
84 Proof ROWENA is best replying at t 1 ROWENA will play the same action at t SERGIU HART c 2008 p. 21
85 Proof Similarly for COLIN: A player who is best replying cannot switch SERGIU HART c 2008 p. 22
86 Proof Similarly for COLIN: A player who is best replying cannot switch C1 C2 C3 R1 R2 1,0 0,1 1,0 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 22
87 Proof Similarly for COLIN: A player who is best replying cannot switch C1 C2 C3 R1 R2 1,0 0,1 1,0 0,1 1,0 1,0 R3 0,1 0,1 1,1 SERGIU HART c 2008 p. 22
88 Proof Similarly for COLIN: A player who is best replying cannot switch C1 C2 C3 R1 R2 1,0 0,1 1,0 0,1 1,0 1,0 R3 0,1 0,1 1,1 (R3,C3) cannot be reached SERGIU HART c 2008 p. 22
89 Proof Similarly for COLIN: A player who is best replying cannot switch C1 C2 C3 R1 R2 1,0 0,1 1,0 0,1 1,0 1,0 R3 0,1 0,1 1,1 (R3,C3) cannot be reached (unless we start there) SERGIU HART c 2008 p. 22
90 2-Recall SERGIU HART c 2008 p. 23
91 2-Recall: Possibility Theorem. THERE EXIST uncoupled dynamics with 2-RECALL σ i t fi (a t 2, a t 1 ; u i ) that yield almost sure convergence of play to pure Nash equilibria of the stage game in every game where such equilibria exist. SERGIU HART c 2008 p. 23
92 Possibility Define the strategy of each player i as follows: SERGIU HART c 2008 p. 24
93 Possibility Define the strategy of each player i as follows: IF: Everyone played the same in the previous two periods: a t 2 = a t 1 = a; and Player i best replied: a i BR i (a i ; u i ) THEN: At t player i plays a i again: a i t = ai SERGIU HART c 2008 p. 24
94 Possibility Define the strategy of each player i as follows: IF: Everyone played the same in the previous two periods: a t 2 = a t 1 = a; and Player i best replied: a i BR i (a i ; u i ) THEN: At t player i plays a i again: a i t = ai ELSE: At t player i randomizes uniformly over A i SERGIU HART c 2008 p. 24
95 Possibility "Good": SERGIU HART c 2008 p. 25
96 Possibility "Good": simple SERGIU HART c 2008 p. 25
97 Possibility "Good": simple "Bad": SERGIU HART c 2008 p. 25
98 Possibility "Good": simple "Bad": exhaustive search SERGIU HART c 2008 p. 25
99 Possibility "Good": simple "Bad": exhaustive search all players must use it SERGIU HART c 2008 p. 25
100 Possibility "Good": simple "Bad": exhaustive search all players must use it takes a long time SERGIU HART c 2008 p. 25
101 Possibility "Good": simple "Bad": exhaustive search all players must use it takes a long time "Ugly":... SERGIU HART c 2008 p. 25
102 Possibility "Good": simple "Bad": exhaustive search all players must use it takes a long time "Ugly":... SERGIU HART c 2008 p. 25
103 Possibility "Good": simple "Bad": exhaustive search all players must use it takes a long time "Ugly":... SERGIU HART c 2008 p. 25
104 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation time information SERGIU HART c 2008 p. 26
105 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation time information: uncoupledness SERGIU HART c 2008 p. 26
106 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation: finite recall time information: uncoupledness SERGIU HART c 2008 p. 26
107 Dynamics FACT There are no general, natural dynamics leading to Nash equilibrium "natural": adaptive simple and efficient: computation: finite recall time to reach equilibrium? information: uncoupledness SERGIU HART c 2008 p. 26
108 Natural Dynamics: Time HOW LONG TO EQUILIBRIUM? SERGIU HART c 2008 p. 27
109 Natural Dynamics: Time HOW LONG TO EQUILIBRIUM? Estimate the number of time periods it takes until a Nash equilibrium is reached SERGIU HART c 2008 p. 27
110 Natural Dynamics: Time HOW LONG TO EQUILIBRIUM? Estimate the number of time periods it takes until a Nash equilibrium is reached How? SERGIU HART c 2008 p. 27
111 Natural Dynamics: Time HOW LONG TO EQUILIBRIUM? Estimate the number of time periods it takes until a Nash equilibrium is reached How? An uncoupled dynamic A distributed computational procedure SERGIU HART c 2008 p. 27
112 Natural Dynamics: Time HOW LONG TO EQUILIBRIUM? Estimate the number of time periods it takes until a Nash equilibrium is reached How? An uncoupled dynamic A distributed computational procedure COMMUNICATION COMPLEXITY SERGIU HART c 2008 p. 27
113 Communication Complexity SERGIU HART c 2008 p. 28
114 Communication Complexity Distributed computational procedure SERGIU HART c 2008 p. 28
115 Communication Complexity Distributed computational procedure START: Each participant has some private information [INPUTS] SERGIU HART c 2008 p. 28
116 Communication Complexity Distributed computational procedure START: Each participant has some private information COMMUNICATION: Messages are transmitted between the participants [INPUTS] SERGIU HART c 2008 p. 28
117 Communication Complexity Distributed computational procedure START: Each participant has some private information COMMUNICATION: Messages are transmitted between the participants END: All participants reach agreement on the result [INPUTS] SERGIU HART c 2008 p. 28
118 Communication Complexity Distributed computational procedure START: Each participant has some private information COMMUNICATION: Messages are transmitted between the participants END: All participants reach agreement on the result [INPUTS] SERGIU HART c 2008 p. 28
119 Communication Complexity Distributed computational procedure START: Each participant has some private information [INPUTS] COMMUNICATION: Messages are transmitted between the participants END: All participants reach agreement on the result [OUTPUT] SERGIU HART c 2008 p. 28
120 Communication Complexity Distributed computational procedure START: Each participant has some private information [INPUTS] COMMUNICATION: Messages are transmitted between the participants END: All participants reach agreement on the result [OUTPUT] COMMUNICATION COMPLEXITY = the minimal number of rounds needed SERGIU HART c 2008 p. 28
121 Communication Complexity Distributed computational procedure START: Each participant has some private information [INPUTS] COMMUNICATION: Messages are transmitted between the participants END: All participants reach agreement on the result [OUTPUT] COMMUNICATION COMPLEXITY = the minimal number of rounds needed Yao 1979, Kushilevitz and Nisan 1997 SERGIU HART c 2008 p. 28
122 How Long to Equilibrium SERGIU HART c 2008 p. 29
123 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria SERGIU HART c 2008 p. 29
124 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria START: Each player knows his own payoff function [INPUTS] SERGIU HART c 2008 p. 29
125 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria START: Each player knows his own payoff function COMMUNICATION: The actions played are commonly observed [INPUTS] SERGIU HART c 2008 p. 29
126 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria START: Each player knows his own payoff function [INPUTS] COMMUNICATION: The actions played are commonly observed END: All players play a Nash equilibrium [OUTPUT] SERGIU HART c 2008 p. 29
127 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria START: Each player knows his own payoff function [INPUTS] COMMUNICATION: The actions played are commonly observed END: All players play a Nash equilibrium [OUTPUT] COMMUNICATION COMPLEXITY = the minimal number of rounds needed SERGIU HART c 2008 p. 29
128 How Long to Equilibrium Uncoupled dynamic leading to Nash equilibria START: Each player knows his own payoff function [INPUTS] COMMUNICATION: The actions played are commonly observed END: All players play a Nash equilibrium [OUTPUT] COMMUNICATION COMPLEXITY = the minimal number of rounds needed Conitzer and Sandholm 2004 SERGIU HART c 2008 p. 29
129 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if SERGIU HART c 2008 p. 30
130 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES SERGIU HART c 2008 p. 30
131 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES is POLYNOMIAL in the number of players SERGIU HART c 2008 p. 30
132 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES is POLYNOMIAL in the number of players (rather than: exponential) SERGIU HART c 2008 p. 30
133 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES is POLYNOMIAL in the number of players (rather than: exponential) Theorem. There are NO TIME-EFFICIENT uncoupled dynamics that reach a pure Nash equilibrium in all games where such equilibria exist. SERGIU HART c 2008 p. 30
134 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES is POLYNOMIAL in the number of players (rather than: exponential) Theorem. There are NO TIME-EFFICIENT uncoupled dynamics that reach a pure Nash equilibrium in all games where such equilibria exist. Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 30
135 How Long to Equilibrium An uncoupled dynamic leading to Nash equilibria is TIME-EFFICIENT if the TIME IT TAKES is POLYNOMIAL in the number of players (rather than: exponential) Theorem. There are NO TIME-EFFICIENT uncoupled dynamics that reach a pure Nash equilibrium in all games where such equilibria exist. In fact: exponential, like exhaustive search Hart and Mansour, GEB 2010 SERGIU HART c 2008 p. 30
136 How Long to Equilibrium Intuition: SERGIU HART c 2008 p. 31
137 How Long to Equilibrium Intuition: different games have different equilibria SERGIU HART c 2008 p. 31
138 How Long to Equilibrium Intuition: different games have different equilibria the dynamic procedure must distinguish between them SERGIU HART c 2008 p. 31
139 How Long to Equilibrium Intuition: different games have different equilibria the dynamic procedure must distinguish between them no single player can do so by himself SERGIU HART c 2008 p. 31
140 Dynamics and Nash Equilibrium SERGIU HART c 2008 p. 32
141 Dynamics and Nash Equilibrium FACT There are NO general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 32
142 Dynamics and Nash Equilibrium FACT There are NO general, natural dynamics leading to Nash equilibrium RESULT There CANNOT BE general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 32
143 Dynamics and Nash Equilibrium RESULT There CANNOT BE general, natural dynamics leading to Nash equilibrium SERGIU HART c 2008 p. 33
144 Dynamics and Nash Equilibrium RESULT There CANNOT BE general, natural dynamics leading to Nash equilibrium Perhaps we are asking too much? SERGIU HART c 2008 p. 33
145 Dynamics and Nash Equilibrium RESULT There CANNOT BE general, natural dynamics leading to Nash equilibrium Perhaps we are asking too much? For instance, the size of the data (the payoff functions) is exponential rather than polynomial in the number of players SERGIU HART c 2008 p. 33
146 Correlated Equilibrium SERGIU HART c 2008 p. 34
147 Correlated Equilibrium CORRELATED EQUILIBRIUM Aumann, JME 1974 SERGIU HART c 2008 p. 34
148 Correlated Equilibrium CORRELATED EQUILIBRIUM : Nash equilibrium when players receive payoff-irrelevant information before playing the game Aumann, JME 1974 SERGIU HART c 2008 p. 34
149 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game SERGIU HART c 2008 p. 35
150 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: SERGIU HART c 2008 p. 35
151 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: Independent signals SERGIU HART c 2008 p. 35
152 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: Independent signals Nash equilibrium SERGIU HART c 2008 p. 35
153 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: Independent signals Nash equilibrium Public signals ( sunspots ) SERGIU HART c 2008 p. 35
154 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: Independent signals Nash equilibrium Public signals ( sunspots ) convex combinations of Nash equilibria SERGIU HART c 2008 p. 35
155 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when players receive payoff-irrelevant signals before playing the game Examples: Independent signals Nash equilibrium Public signals ( sunspots ) convex combinations of Nash equilibria Butterflies play the Chicken Game ( Speckled Wood Pararge aegeria) SERGIU HART c 2008 p. 35
156 Correlated Equilibria "Chicken" game LEAVE STAY LEAVE 5, 5 3, 6 STAY 6, 3 0, 0 SERGIU HART c 2008 p. 36
157 Correlated Equilibria "Chicken" game LEAVE STAY LEAVE 5, 5 3, 6 STAY 6, 3 0, 0 a Nash equilibrium SERGIU HART c 2008 p. 36
158 Correlated Equilibria "Chicken" game LEAVE STAY LEAVE 5, 5 3, 6 STAY 6, 3 0, 0 another Nash equilibriumi SERGIU HART c 2008 p. 36
159 Correlated Equilibria "Chicken" game LEAVE STAY LEAVE 5, 5 3, 6 STAY 6, 3 0, 0 L 0 1/2 1/2 0 a (publicly) correlated equilibrium SERGIU HART c 2008 p. 36
160 Correlated Equilibria "Chicken" game LEAVE STAY LEAVE 5, 5 3, 6 STAY 6, 3 0, 0 L S L 1/3 1/3 S 1/3 0 another correlated equilibrium after signal L play LEAVE after signal S play STAY SERGIU HART c 2008 p. 36
161 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when the players receive payoff-irrelevant signals before playing the game (Aumann 1974) Examples: Independent signals Nash equilibrium Public signals ( sunspots ) convex combinations of Nash equilibria Butterflies play the Chicken Game ( Speckled Wood Pararge aegeria) SERGIU HART c 2008 p. 37
162 Correlated Equilibrium A Correlated Equilibrium is a Nash equilibrium when the players receive payoff-irrelevant signals before playing the game (Aumann 1974) Examples: Independent signals Nash equilibrium Public signals ( sunspots ) convex combinations of Nash equilibria Butterflies play the Chicken Game ( Speckled Wood Pararge aegeria) Boston Celtics front line SERGIU HART c 2008 p. 37
163 Correlated Equilibrium Signals (public, correlated) are unavoidable SERGIU HART c 2008 p. 38
164 Correlated Equilibrium Signals (public, correlated) are unavoidable Common Knowledge of Rationality Correlated Equilibrium (Aumann 1987) SERGIU HART c 2008 p. 38
165 Correlated Equilibrium Signals (public, correlated) are unavoidable Common Knowledge of Rationality Correlated Equilibrium (Aumann 1987) A joint distribution z is a correlated equilibrium u(j, s i )z(j, s i ) u(k, s i )z(j, s i ) s i s i for all i N and all j, k S i SERGIU HART c 2008 p. 38
166 Dynamics & Correlated Equilibria SERGIU HART c 2008 p. 39
167 Dynamics & Correlated Equilibria RESULT THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA SERGIU HART c 2008 p. 39
168 Dynamics & Correlated Equilibria RESULT THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA Regret Matching Hart and Mas-Colell, Ec ca 2000 SERGIU HART c 2008 p. 39
169 Dynamics & Correlated Equilibria RESULT THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA Regret Matching General regret-based dynamics Hart and Mas-Colell, Ec ca 2000, JET 2001 SERGIU HART c 2008 p. 39
170 Regret Matching SERGIU HART c 2008 p. 40
171 Regret Matching "REGRET": the increase in past payoff, if any, if a different action would have been used SERGIU HART c 2008 p. 40
172 Regret Matching "REGRET": the increase in past payoff, if any, if a different action would have been used "MATCHING": switching to a different action with a probability that is proportional to the regret for that action SERGIU HART c 2008 p. 40
173 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA SERGIU HART c 2008 p. 41
174 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA "general": in all games SERGIU HART c 2008 p. 41
175 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA "general": in all games "natural": SERGIU HART c 2008 p. 41
176 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA "general": in all games "natural": adaptive (also: close to "behavioral") SERGIU HART c 2008 p. 41
177 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA "general": in all games "natural": adaptive (also: close to "behavioral") simple and efficient: computation, time, information SERGIU HART c 2008 p. 41
178 Dynamics & Correlated Equilibria THERE EXIST general, natural dynamics leading to CORRELATED EQUILIBRIA "general": in all games "natural": adaptive (also: close to "behavioral") simple and efficient: computation, time, information "leading to correlated equilibria": statistics of play become close to CORRELATED EQUILIBRIA SERGIU HART c 2008 p. 41
179 Regret Matching and Beyond SERGIU HART c 2008 p. 42
180 Regret Matching and Beyond SERGIU HART c 2008 p. 42
181 Regret Matching and Beyond SERGIU HART c 2008 p. 43
182 Regret Matching and Beyond SERGIU HART c 2008 p. 44
183 Regret Matching and Beyond SERGIU HART c 2008 p. 45
184 Regret Matching and Beyond SERGIU HART c 2008 p. 46
185 Regret Matching and Beyond SERGIU HART c 2008 p. 47
186 Regret Matching and Beyond SERGIU HART c 2008 p. 48
187 Regret Matching and Beyond SERGIU HART c 2008 p. 49
188 Dynamics and Equilibrium SERGIU HART c 2008 p. 50
189 Dynamics and Equilibrium NASH EQUILIBRIUM: a fixed-point of a non-linear map SERGIU HART c 2008 p. 50
190 Dynamics and Equilibrium NASH EQUILIBRIUM: a fixed-point of a non-linear map CORRELATED EQUILIBRIUM: a solution of finitely many linear inequalities SERGIU HART c 2008 p. 50
191 Dynamics and Equilibrium NASH EQUILIBRIUM: a fixed-point of a non-linear map CORRELATED EQUILIBRIUM: a solution of finitely many linear inequalities set-valued fixed-point (curb sets)? SERGIU HART c 2008 p. 50
192 Dynamics and Equilibrium SERGIU HART c 2008 p. 51
193 Dynamics and Equilibrium "LAW OF CONSERVATION OF COORDINATION": SERGIU HART c 2008 p. 51
194 Dynamics and Equilibrium "LAW OF CONSERVATION OF COORDINATION": There must be some COORDINATION SERGIU HART c 2008 p. 51
195 Dynamics and Equilibrium "LAW OF CONSERVATION OF COORDINATION": There must be some COORDINATION either in the EQUILIBRIUM notion, SERGIU HART c 2008 p. 51
196 Dynamics and Equilibrium "LAW OF CONSERVATION OF COORDINATION": There must be some COORDINATION either in the EQUILIBRIUM notion, or in the DYNAMIC SERGIU HART c 2008 p. 51
197 The "Program" SERGIU HART c 2008 p. 52
198 The "Program" A. Demarcate the BORDER between SERGIU HART c 2008 p. 52
199 The "Program" A. Demarcate the BORDER between classes of dynamics where convergence to equilibria CAN be obtained SERGIU HART c 2008 p. 52
200 The "Program" A. Demarcate the BORDER between classes of dynamics where convergence to equilibria CAN be obtained, and classes of dynamics where convergence to equilibria CANNOT be obtained SERGIU HART c 2008 p. 52
201 The "Program" A. Demarcate the BORDER between classes of dynamics where convergence to equilibria CAN be obtained, and classes of dynamics where convergence to equilibria CANNOT be obtained B. Find NATURAL dynamics for the various equilibrium concepts SERGIU HART c 2008 p. 52
202 Dynamics and Equilibrium SERGIU HART c 2008 p. 53
203 Dynamics and Equilibrium Nash Equilibrium SERGIU HART c 2008 p. 53
204 Dynamics and Equilibrium Nash Equilibrium "DYNAMICALLY DIFFICULT" SERGIU HART c 2008 p. 53
205 Dynamics and Equilibrium Nash Equilibrium "DYNAMICALLY DIFFICULT" Correlated Equilibrium SERGIU HART c 2008 p. 53
206 Dynamics and Equilibrium Nash Equilibrium "DYNAMICALLY DIFFICULT" Correlated Equilibrium "DYNAMICALLY EASY" SERGIU HART c 2008 p. 53
207 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
208 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
209 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
210 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
211 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
212 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
213 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
214 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
215 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
216 Dynamics and Equilibrium SERGIU HART c 2008 p. 54
217 My Game Theory SERGIU HART c 2008 p. 55
218 My Game Theory SERGIU HART c 2008 p. 55
219 My Game Theory SERGIU HART c 2008 p. 55
220 My Game Theory INSIGHTS, IDEAS, CONCEPTS SERGIU HART c 2008 p. 55
221 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 55
222 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
223 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
224 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
225 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
226 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
227 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
228 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
229 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
230 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
231 My Game Theory INSIGHTS, IDEAS, CONCEPTS FORMAL MODELS SERGIU HART c 2008 p. 56
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