Game Theory, Evolutionary Dynamics, and Multi-Agent Learning. Prof. Nicola Gatti
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1 Game Theory, Evolutionary Dynamics, and Multi-Agent Learning Prof. Nicola Gatti
2 Game theory
3 Game theory: basics Normal form Players Actions Outcomes Utilities Strategies Solutions
4 Game theory: basics Normal form Players Actions Outcomes Utilities Strategies Solutions
5 Game theory: basics Rock Paper Scissors Normal form Players Actions Outcomes Utilities Strategies Solutions Rock Paper Scissors
6 Game theory: basics Rock Paper Scissors Normal form Players Rock Tie wins wins Actions Outcomes Utilities Strategies Paper wins Tie wins Solutions Scissors wins wins Tie
7 Game theory: basics Rock Paper Scissors Normal form Players Actions Outcomes Utilities Strategies Solutions Rock 0,0-1,1 1,-1 Paper 1,-1 0,0-1,1 Scissors -1,1 1,-1 0,0
8 Game theory: basics Rock Paper Scissors Normal form Players Rock 0,0-1,1 1,-1 1(Rock) Actions Outcomes Utilities Strategies Paper 1,-1 0,0-1,1 1(Paper) Solutions Scissors -1,1 1,-1 0,0 1(Scissors) 2(Rock) 2(Paper) 2(Scissors)
9 Game theory: basics Rock Paper Scissors Normal form Players Rock 0,0-1,1 1,-1 1/3 Actions Outcomes Utilities Strategies Paper 1,-1 0,0-1,1 1/3 Solutions Scissors -1,1 1,-1 0,0 1/3 1/3 1/3 1/3
10 Nash Equilibrium A strategy profile 1 2 arg max arg max 2 n n 1U 1 1U 2 ( 1, 2 ) 2 o 2 o is a Nash equilibrium if and if:
11 Evolutionary dynamics
12 An evolutionary interpretation 33% 33% 33% Rock Paper Scissors
13 An evolutionary interpretation Infinite population of individuals 33% 33% 33% Rock Paper Scissors
14 Bacterials
15 An evolutionary interpretation 33% 33% 30% 50% 33% 20% Rock Paper Scissors Rock Paper Scissors
16 An evolutionary interpretation Rock Paper Scissors 33% 33% Rock 0,0-1,1 1,-1 Paper 1,-1 0,0-1,1 30% 50% 33% Scissors -1,1 1,-1 0,0 20% Rock Paper Scissors Rock Paper Scissors
17 An evolutionary interpretation Rock Paper Scissors 33% 33% Rock 0,0-1,1 1,-1 30% 50% Paper 1,-1 0,0-1,1 33% Scissors -1,1 1,-1 0,0 20% Rock Paper Scissors Rock Paper Scissors At each round, each individual of a population plays against each individual of the opponent s population
18 An evolutionary interpretation Utility = Utility = 0.1 Rock Paper Scissors 33% 33% Rock 0,0-1,1 1,-1 Paper 1,-1 0,0-1,1 30% 50% 33% Scissors -1,1 1,-1 0,0 20% Rock Paper Scissors Rock Paper Scissors Utility = 0.2
19 An evolutionary interpretation Utility = % 33% Utility = % Rock Paper Scissors Utility = 0.2
20 An evolutionary interpretation Utility = % 33% Utility = % Rock Paper Scissors Utility = 0.2 The average utility is 0
21 An evolutionary interpretation Utility = % 33% Utility = % Rock Paper Scissors Utility = 0.2 The average utility is 0 A positive (utility - average utility) leads to an increase of the population A negative (utility - average utility) leads to a decrease of the population
22 An evolutionary interpretation 17% 37% 46% Rock Paper Scissors New population after the replication
23 Revision protocol Question: how the populations change? 1(a, t) = 1 (a, t) Replicator dynamics e a U 1 2 (t) 1(t)U 1 2 (t) Utility given by playing a with a probability of 1 Average population utility
24 Evolutionary Stable Strategies A strategy is an ESS if it is immune to invasion by mutant strategies, given that the mutants initially occupy a small fraction of population Every ESS is an asymptotically stable fixed point of the replicator dynamics
25 Evolutionary Stable Strategies A strategy is an ESS if it is immune to invasion by mutant strategies, given that the mutants initially occupy a small fraction of population Every ESS is an asymptotically stable fixed point of the replicator dynamics Nash equilibria ESS While a NE always exists, an ESS may not exist
26 Prisoner s dilemma Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
27 Prisoner s dilemma Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
28 Prisoner s dilemma 1 Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1 s cooperate probability 0.75 y x 1 s cooperate probability
29 Prisoner s dilemma 1 Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1 s cooperate probability 0.75 y asymptotically stable x 1 s cooperate probability
30 Stag hunt Stag Hare Stag 4,4 1,3 Hare 3,1 3,3
31 Stag hunt Stag Hare Stag 4,4 1,3 Hare 3,1 3,3
32 Stag hunt Stag Hare Stag 4,4 1,3 Hare 3,1 3,3 s stag probability x 1 s stag probability
33 Stag hunt 1 asymptotically stable Stag Hare 0.75 Stag 4,4 1,3 Hare 3,1 3,3 s stag probability asymptotically stable x 1 s stag probability
34 Matching pennies Head Tail Head 0,1 1,0 Tail 1,0 0,1
35 Matching pennies Head Tail Head 0,1 1,0 Tail 1,0 0,1
36 Matching pennies 1 Head Tail 0.75 Head 0,1 1,0 Tail 1,0 0,1 s head probability y x 1 s head probability
37 Matching pennies 1 Head Tail 0.75 Head 0,1 1,0 Tail 1,0 0,1 s head probability y x 1 s head probability
38 Multi-agent learning
39 Markov decision problem
40 Reinforcement learning
41 Q-learning (1) For every pair state/action: Q(s, a) Q(s, a)+ h i r + max a 0 Q(s, a 0 ) Q(s, a) learning rate reward discount factor
42 Example: normal-form games 1 state 2 actions (Cooperate, Defeat) Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
43 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 1(a) = 1(a) = ( 1.0 a = Cooperate 0.0 ( a = Defeat 0.2 a = Cooperate 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
44 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
45 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
46 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
47 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
48 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
49 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
50 Example: normal-form games Q(a) Q(a)+ r Q(a) =0.2 round s action s Q function t =0 Q(Cooperate) = 0 t = 1 a = Cooperate Q(Cooperate) = 0.6 t = 2 a = Defeat Q(Cooperate) = 0.48 t = 3 a = Defeat Q(Cooperate) = t = 4 a = Defeat Q(Cooperate) = t = 5 a = Defeat Q(Cooperate) = t = 6 a = Cooperate Q(Cooperate) = ( 1.0 a = Cooperate 1(a) = 0.0 a = Defeat ( 0.2 a = Cooperate 1(a) = 0.8 a = Defeat Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
51 Q-learning (2) Softmax (a.k.a. Boltzam exploration) i(a) = exp(q(s, a)/ ) P a 0 exp(q(s, a 0 )/ )
52 Q-learning (2) Softmax (a.k.a. Boltzam exploration) i(a) = exp(q(s, a)/ ) P a exp(q(s, a 0 )/ ) 0 temperature
53 Q-learning (2) Softmax (a.k.a. Boltzam exploration) i(a) = exp(q(s, a)/ ) P a exp(q(s, a 0 )/ ) 0 temperature Every action is played with strictly positive probability The larger the temperature, the smoother the function If the temperature is 0, we would have a best response
54 Example: normal-form games Q(Cooperate) Q(Defeat) 1(Cooperate) 1(Cooperate)
55 Example: normal-form games Q(Cooperate) Q(Defeat) 1(Cooperate) 1(Cooperate)
56 Example: normal-form games Q(Cooperate) Q(Defeat) 1(Cooperate) 1(Cooperate)
57 Example: normal-form games Q(Cooperate) Q(Defeat) 1(Cooperate) 1(Cooperate)
58 Self-play Q-learning dynamics
59 Self-play learning Q-learning algorithm Q-learning algorithm Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1
60 Learning dynamics Assumptions: Time is continuous All the actions can be selected simultaneously
61 Learning dynamics Assumptions: Time is continuous All the actions can be selected simultaneously 1(a, t) = 1(a, t) e a U 1 2 (t) 1(t)U 1 2 (t) 1 (a, t) log( 1 (a)) X 1(a 0 ) log( 1 (a 0 )) a 0
62 Learning dynamics Assumptions: Time is continuous All the actions can be selected simultaneously 1(a, t) = 1(a, t) e a U 1 2 (t) 1(t)U 1 2 (t) 1 (a, t) log( 1 (a)) X 1(a 0 ) log( 1 (a 0 )) a 0 exploitation term exploration term
63 Learning dynamics Assumptions: Time is continuous All the actions can be selected simultaneously 1(a, t) = 1(a, t) e a U 1 2 (t) 1(t)U 1 2 (t) 1 (a, t) log( 1 (a)) X 1(a 0 ) log( 1 (a 0 )) a 0 exploitation term exploration term When the temperature is 0, the Q-learning behaves as the replicator dynamics
64 0 Prisoner s dilemma 1 Cooperate Defeat Cooperate 3,3 0,5 Defeat 5,0 1,1 s cooperate probability y asymptotically stable x 1 s cooperate probability
65 Stag hunt 1 asymptotically stable Stag Hare 0.75 Stag 4,4 1,3 Hare 3,1 3,3 s stag probability y asymptotically stable x 1 s stag probability
66 Matching pennies 1 Head Tail Head 0,1 1,0 Tail 1,0 0,1 s head probability y Player x 1 s head probability
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