Game Theory

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.

Two key Assumptions: (a) Both players are rational (b) Both players choose their strategies solely to increase their own welfare.

Payoff Table Player 2 Strategy 1 2 3 1 2 3 1 2 4 1 0 5 0 1 -1 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.

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 0 1 -1 Player 1 For player 1, strategy 3 can be eliminated. ( 1 > 0, 2 > 1, 4 > -1)

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 )

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.

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 1 2 3 Minimum 1 2 3 -3 -2 6 2 0 2 5 -2 -4 -3 -4 Player 1 Maximum: 5 0 6 Minimax value Maximin value

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 1 2 3 Minimum 1 2 3 -3 -2 6 2 0 2 5 -2 -4 -3 -4 Player 1 Maximum: 5 0 6 Saddle point

There is no saddle point. An unstable solution Player 2 Strategy 1 2 3 Minimum 1 2 3 0 -2 2 5 4 -3 2 3 -4 -2 -3 -4 Player 1 Maximum: 5 4 2

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 =

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

Graphical Solution Procedure Player 2 Probability Pure Strategy Probability 1 2 3 1 2 0 -2 2 5 4 -3 Player 1 Expected Payoff

Expected Payoff Expected payoff for player 1 =

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

The optimal mixed strategy for player 1 is So the value of the game is The optimal strategy (1)

When player 1 is playing optimally ( ), this inequality will be an equality, so that (2) Because is a probability distribution,

because would violate (2), Because the ordinate of this line must equal at , and because it must never exceed ,

To solve for and , select two values of (say, 0 and 1), The optimal mixed strategy for player 2 is

Solving by Linear Programming Expected payoff for player 1 = The strategy is optimal if

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,

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.

Example Player 2 Probability Pure Strategy Probability 1 2 3 1 2 0 -2 2 5 4 -3 Player 1

The dual Player 2 Probability Pure Strategy Probability 1 2 3 1 2 0 -2 2 5 4 -3 Player 1

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.