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Tutorial3

Tutorial3 - CSC5350 Game Theory in Computer Science...

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CSC5350 Game Theory in Computer Science CHEN Wenhao [email protected] SHB 905 Tutorial 3

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Outline Mixed strategies Mixed strategy Nash equilibrium Evolutionary equilibrium or evolutionarily stable solution (ESS) Examples
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
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 H D

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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?
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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Tutorial3 - CSC5350 Game Theory in Computer Science...

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