Tutorial4

Tutorial4 - CSC5350 Game Theory in Computer Science CHEN...

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Unformatted text preview: CSC5350 Game Theory in Computer Science CHEN Wenhao whchen@cse.cuhk.edu.hk SHB 905 Tutorial 4 Outline ¡ Extensive game with perfect information ¡ Examples T3 - Q.1(a) ½ (1-c), ½ (1-c) 1, 0 0, 1 ½ , ½ 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))) T3 - Q.1(a) ½ (1-c), ½ (1-c) 1, 0 0, 1 ½ , ½ Let (a*, 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) D D H H (a*, b*) = ((D(1-1/c), H(1/c)), ((D(1-1/c), H(1/c))) u 2 (a*, e(D)) = u 2 (a*, e(H)) ½ a*(D) + 0 x a*(H) = 1 x a*(D)+ ½ (1-c) x a*(H) ½ (c-1) a*(H) = ½ a*(D) (c-1) a*(H) = a*(D) Extensive game with perfect information ¡ Players move one by one ¡ Players know what action other players have made Extensive game with perfect information ¡ Four components ¡ Set of players ¡ Set of terminal histories ¡ the set of all sequences of actions that may occur ¡ Player function ¡ assigns a player to every subhistory ¡ Preferences over the set of terminal histories Example 1 ¡ Extensive game ¡ Set of players: {1, 2} ¡ Set of terminal histories: {(X, Y), (X, Z), (Y, X), (Y, Z), (Z, X), (Z, Y)} ¡ Player function ¡ P( ∅ ) = 1 ¡ P(X) = P(Y) = P(Z) = 2 ¡ Preferences ¡ (Y, Z) = 1 (Z, Y) ≥ 1 (X, Z) = 1 (Z, X) ≥ 1 (X, Y) = 1 (Y, X) ¡ (X, Y) = 2 (Y, X) ≥ 2 (X, Z) = 2 (Z, X) ≥ 2 (Y, Z) = 2 (Z, Y) Example 1 ¡ Express as game tree 1, 3 3, 1 X Y Z X Z 1 2 2, 2 3, 1 X Y 2 1, 3 2, 2 Y Z 2 (Y, Z) = 1 (Z, Y) ≥ 1 (X, Z) = 1 (Z, X) ≥ 1 (X, Y) = 1 (Y, X) (X, Y) = 2 (Y, X) ≥ 2 (X, Z) = 2 (Z, X) ≥ 2 (Y, Z) = 2 (Z, Y) Extensive game...
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Tutorial4 - CSC5350 Game Theory in Computer Science CHEN...

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