Game Theory

A zero sum game is given by a table that gives the payoff to the row player (player 1) from the column player (player 2). The game table has one row for each of the row player's strategies and one column for each of the column player's strategies.

Consider the two player game given by the following table.

Row/Column	Strategy 1	Strategy 2	Strategy 3	Strategy 4	Strategy 5
Strategy 1
Strategy 2	10
38	-12
57	34
96	75
28	67
-33
If the row player selects strategy (row) 1 and the column player selects strategy (column) 1, the column player pays the row player 10. If the row player selects strategy 1 and the column player selects strategy 2, the column player pays -12 or, in other words, the column player receives 12 from the row player. Row and column must each choose a strategy without knowing what the opponent has selected.

Sometimes the solution is for the players to always select one strategy (termed a pure strategy) and sometimes the solution is for the players to select their strategies randomly (termed a mixed strategy). In either case, this can be determined.

Solution to the example

The solution for this example is displayed next.

Row should play the first strategy 53.25% of the time and the second strategy 46.75% of the time. Column should play the second strategy 59.17% of the time and the fifth strategy 40.83% of the time and never play strategies 1, 3 or 4. If they follow these mixes, the (expected) value of the game is that column will pay row 20.2544. That is, if they played this game a large number of times following their optimal mixes the payoffs would be -12, 67, 57 and -33 and would average 20.2544.

Maximin and minimax

When examining games, we usually begin by finding the maximin and minimax . To find the maximin for the row player, examine each row and find the worst (minimum) outcome. These appear in the column labeled 'row minimum' as -12, and -33 in the following table. Then find the best of these, -12, which is the maximum of the minima or the maximin .

To find the minimax for the column player examine each column and find the worst (maximum since column is paying) payoff. These appear in the row named 'Column Maximum' and are 38, 57, 96, 75 and 67. The minimax is the best (lowest) of these, or 38. The value of the game is between the maximin and minimax as appears in this game, with a value of 20.2544, which is between -12 and 38.

Expected values for row

The table below displays the computations (multiplications) expected value for each of row's strategies. Since row should use both strategies the expected values are the same and match the expected value of the game.

Expected values for column

Similarly, if column plays column 2 or 5, column will achieve the value of the game. However, if column selects column 1, 3 or 4, then he or she will pay more than the value of the game as shown by the expected values in the table below.

Graphs are available if either or both players have at most two strategies.