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Tutorial4

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

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CSC5350 Game Theory in Computer Science CHEN Wenhao [email protected] SHB 905 Tutorial 4

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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)))

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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

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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 )

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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 with perfect information A strategy of player i A function that assigns an action to each history after which it is player i ’’ s turn to move

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Example 1 Player 1 has 3 strategies assign X assign Y assign Z 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 to the empty history
Example 1 Player 2

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Tutorial4 - CSC5350 Game Theory in Computer Science...

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