Editing Zero-sum game (section)

=== Example ===

{| class="wikitable" style="float:right; margin-left:1em;"
|+ align=bottom |''A zero-sum game (Two person)''
! {{diagonal split header|{{red|Red}}|{{blue|Blue}}}}
! {{blue|''A''}}
! {{blue|''B''}}
! {{blue|''C''}}
|-
! {{red|''1''}}
| {{diagonal split header|{{red|30}}|{{blue|−30}}|white}}
| {{diagonal split header|{{red|−10}}|{{blue|10}}|white}}
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
|-
! {{red|''2''}}
| {{diagonal split header|{{red|−10}}|{{blue|10}}|white}}
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
| {{diagonal split header|{{red|−20}}|{{blue|20}}|white}}
|}

A game's [[payoff matrix]] is a convenient representation. Consider these situations as an example, the two-player zero-sum game pictured at right or above.

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

''Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.''

In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.

[[Émile Borel]] and [[John von Neumann]] had the fundamental insight that [[probability]] provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum [[expected value|expected]] point-loss independent of the opponent's strategy. This leads to a [[linear programming]] problem with the optimal strategies for each player. This [[minimax]] method can compute probably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that Red should choose action 1 with probability {{sfrac|4|7}} and action 2 with probability {{sfrac|3|7}}, and Blue should assign the probabilities 0, {{sfrac|4|7}}, and {{sfrac|3|7}} to the three actions A, B, and C. Red will then win {{sfrac|20|7}} points on average per game.