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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 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 1 | 10 | -10 | 24 | 65 |
| Strategy 2 | 28 | 43 | 96 | 25 |
| Strategy 3 | 16 | 37 | 86 | 22 |
If the row player selects row 1 and the column player selects column 1 then the column player pays the row player 10. If the row player selects row 1 and the column player selects row 2 then the column player pays -10 or in other words the column player receives 10 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 below. Row should play the first strategy 5.17% of the time and the second strategy 94.83% of the time and never play row 3. (Notice that row 2 is better than row 3 regardless of the column chosen by the opponent.) Column should play column 1 68.97% of the time and column 4 31.03% of the time and never play columns 2 or 3. If they follow these mixes then the (expected) value of the game is that column will pay row 27.069. That is, if they played this game a large number of times following their optimal mixes the payoffs would be 10, 65 28 or 25 and would average 27.069
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 -10, 25 and 16 in the table below. Then find the best of these - 25 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 28, 43, 96 and 65. The minimax is the best (lowest) of these or 28. The value of the game is between the maximin and minimax as appears in this game with a value of 27.069 which is between 25 and 28.
We can examine what happens if one player decides to select a pure strategy even though he or she should not. For example, if column plays the optimal mix and row always plays strategy 1 or strategy 2 then the value is 27.069 which is the value of the game as shown below. However, if row selects row 3 then the expected value is 17.8621 which is less than the value of the game.
Similarly, if column plays column 1 or 4 then column will achieve the value of the game. However, if column selects column 2 or 3 then he/she will pay more than the value of the game.
Graphs are available if either or both players have at most two strategies.
| 1. Introduction | 2. Example | 3. Main Menu | 4. Printing | 5. Graphs | 6. Modules |
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