Tutorial3

Tutorial3 - CSC5350 Game Theory in Computer Science...

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CSC5350 Game Theory in Computer Science CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905 Tutorial 3
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Outline ± Mixed strategies ± Mixed strategy Nash equilibrium ± Evolutionary equilibrium or evolutionarily stable solution (ESS) ± Examples
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Mixed strategies and Mixed strategy Nash equilibrium ± Mixed strategy ± A probability distribution over the set of actions ± Mixed strategy Nash equilibrium ± Every players are using their best response mixed strategies
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Evolutionary equilibrium ± b* is an ESS if and only if for any b b*, either
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Example 1 ± 2 cats in a population always fight over a rat ± Cats can behave like a dove (D) or like a hawk (H) ½ , ½ 0, 1 1, 0 ½ (1-c), ½ (1-c) D H
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Example 1 ± Q.1 (a) Show a mixed strategy Nash equilibrium if c >1. (b) Show that this equilibrium mixed strategy is the only ESS. ± Q.2 What if c < 1?
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Q.1(a) ½ , ½ 0, 1 1, 0 ½ (1-c), ½ (1-c) Let (b*, b*) be the mixed strategy Nash equilibrium u 1 (e(D), b*) = u 1 (e(H), b*) ½ b*(D) + 0 x b*(H) = 1 x b*(D)+ ½ (1-c) x b*(H) ½ (c-1) b*(H) = ½ b*(D) (c-1) b*(H) = b*(D) --- (1) D D H H b*(H) + b*(D) = 1 --- (2) Substitute (1) into (2), b*(H) + (c-1)b*(H) = 1 b*(H) = 1/c So for c>1, b*(D) = 1-1/c (b*, b*) = ((D(1-1/c), H(1/c)), ((D(1-1/c), H(1/c)))
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Q.1(b)
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This note was uploaded on 04/23/2010 for the course CSC CSC5350 taught by Professor Leunghofung during the Winter '09 term at CUHK.

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Tutorial3 - CSC5350 Game Theory in Computer Science...

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