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Game Theory
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Game theory is a mathematical theory that deals with the general features of competitive situations. The final outcome depends primarily upon the combination of strategies selected by the adversaries.
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Two key Assumptions: (a) Both players are rational (b) Both players choose their strategies solely to increase their own welfare.
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Payoff Table Player 2 Strategy 1 2 3 1 2 3 1 2 4 1 0 5 Player 1 Each entry in the payoff table for player 1 represents the utility to player 1 (or the negative utility to player 2) of the outcome resulting from the corresponding strategies used by the two players.
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A strategy is dominated by a second strategy if the second strategy is always at least as good regardless of what the opponent does. A dominated strategy can be eliminated immediately from further consideration. Player 2 Strategy 1 2 3 1 2 3 1 2 4 1 0 5 Player 1 For player 1, strategy 3 can be eliminated. ( 1 > 0, 2 > 1, 4 > -1)
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1 2 3 1 2 1 2 4 1 0 5 For player 2, strategy 3 can be eliminated. ( 1 < 4, 1 < 5 ) 1 2 1 2 1 2 1 0 For player 1, strategy 2 can be eliminated. ( 1 = 1, 2 < 0 )
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1 2 1 1 2 For player 2, strategy 2 can be eliminated. ( 1 < 2 ) Consequently, both players should select their strategy 1. A game that has a value of 0 is said to be a fair game.
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Minimax criterion: To minimize his maximum losses whenever resulting choice of strategy cannot be exploited by the opponent to then improve his position. Player 2 Strategy Minimum 1 2 3 -3 -4 Player 1 Maximum: Minimax value Maximin value
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The value of the game is 0, so this is fair game
Saddle Point: A Saddle point is an entry that is both the maximin and minimax. Player 2 Strategy Minimum 1 2 3 -3 -4 Player 1 Maximum: Saddle point
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There is no saddle point.
An unstable solution Player 2 Strategy Minimum 1 2 3 -2 -3 -4 Player 1 Maximum:
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Mixed Strategies = probability that player 1 will use strategy i ( i = 1,2,…,m), = probability that player 2 will use strategy j ( j = 1,2,…,n), Expected payoff for player 1 =
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Minimax theorem: If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax criterion provides a stable solution with (the value of the game), so that neither player can do better by unilaterally changing her or his strategy. = maximin value = minimax value
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Graphical Solution Procedure
Player 2 Probability Pure Strategy Probability 1 2 Player 1 Expected Payoff
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Expected Payoff Expected payoff for player 1 =
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Player 1 wants to maximize the minimum expected payoff
Player 1 wants to maximize the minimum expected payoff. Player 2 wants to minimize the expected payoff. 6 5 4 3 2 1 -1 -2 -3 -4 Maximin point Expected payoff 1.0
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The optimal mixed strategy for player 1 is
So the value of the game is The optimal strategy (1)
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When player 1 is playing optimally ( ),
this inequality will be an equality, so that (2) Because is a probability distribution,
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because would violate (2),
Because the ordinate of this line must equal at , and because it must never exceed ,
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To solve for and , select two values of
(say, 0 and 1), The optimal mixed strategy for player 2 is
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Solving by Linear Programming
Expected payoff for player 1 = The strategy is optimal if
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For each of the strategies where one and the rest equal 0
For each of the strategies where one and the rest equal 0. Substituting these values into the inequality yields Because the are probabilities,
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The two remaining difficulties are
(1) is unknown (2) the linear programming problem has no objective function. Replacing the unknown constant by the variable and then maximizing , so that automatically will equal at the optimal solution for the LP problem.
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Example Player 2 Probability Pure Strategy Probability 1 2 Player 1
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The dual Player 2 Probability Pure Strategy Probability 1 2 Player 1
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Question 1 Consider the game having the following payoff table. (a) Formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. (b) Use the simplex method to find these optimal mixed strategies.
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