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Unformatted text preview: Learning—Adjustment with persistent noise 14.126 Game Theory Mihai Manea Muhamet Yildiz Main idea ¡ There will always be small but positive probability of mutation. ¡ Then, some of the strict Nash equilibria will not be “stochastically stable.” 1 General Procedure Stochastic Adjustment 1. Consider a game. A B 2. Specify a state 1. space Θ , e.g., the A 2,2 0,0 # of players playing a strategy . B 0,0 1,1 2. Θ = {AA,AB,BA,BB} 2 Stochastic Adjustment, continued 3. 3. Specify an adjustment AA AB BA BB dynamics, e.g., best response dynamics, 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 AA with a transition matrix AB P, where P = P θ,ξ = Pr( θ at t+1 ξ at t) BA BB φ = a probability distribution, a column vector. Stochastic Adjustment, continued 4. Introduce a small 4. AA AB BA BB noise: Consider P ε , continuous in ε and P ε → P as ε → 0. Make sure that there P ε = exist a unique φ ε * s.t. φ ε * = P ε φ ε *. (1−ε) 2 (1−ε)ε (1−ε)ε ε 2 (1−ε)ε ε 2 (1−ε) 2 (1−ε)ε (1−ε)ε (1−ε) 2 ε 2 (1−ε)ε ε 2 (1−ε)ε (1−ε)ε (1−ε) 2 φ ε * = (1/4,1/4,1/4,1/4) T . 3 Stochastic Adjustment, continued 5. Verify that lim ε→0 φ ε * = φ * exists; compute φ *. (By continuity φ * = P φ *.) 6. Check that φ * is a point mass, i.e., φ *( θ *) = 1 for some θ *. The strategy profile at θ * is called stochastically stable equilibrium . Kandoori, Mailath & Rob 4 Coordination game A B q α ∗ = 1/3 A B (2,2) (0,0) (0,0) (1,1) α ∗ = 1/3 p Adjustment Process ¡ N = population size. ¡ θ t = # of players who play A at t . ¡ u A ( θ t ) = θ t /N u(A,A) + ( N θ t )/N u( A , B ) ¡ θ t +1 = P( θ t ), where P( θ t ) > θ t Ù u A ( θ t ) > u B ( θ t ) & P( θ t ) = θ t Ù u A ( θ t ) = u B ( θ t ). ¡ Example: ⎧ ⎪ N if u A ( θ t ) > u B ( θ t ) P ( θ t ) = BR ( θ t ) = ⎨ θ t if u A ( θ t ) = u B ( θ t ) ⎪ ⎩ 0 if u A ( θ t ) < u B ( θ t ) 5 Noise ¡ Independently, each agent with probability 2ε mutates, and plays either of the strategies with equal probabilities....
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 Spring '05
 MuhammadYildiz
 Game Theory, invariant distribution, replicator dynamics, Stochastic Adjustment

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