Editing Minimax (section)

=== Example ===
{| class="wikitable" style="text-align:center; float:right; margin-left:1em"
 |+  align="bottom" style="caption-side: bottom" | Payoff matrix for player A
 ! 
 ! B chooses B1
 ! B chooses B2
 ! B chooses B3
 |-
 ! A chooses A1
 |  +3
 |  −2
 |  +2
 |-
 ! A chooses A2
 |  −1
 |  {{0|+}}0
 |  +4
 |-
 ! A chooses A3
 |  −4
 |  −3
 |  +1
|}
The following example of a zero-sum game, where '''A''' and '''B''' make simultaneous moves, illustrates ''maximin'' solutions. Suppose each player has three choices and consider the [[payoff matrix]] for '''A''' displayed on the table ("Payoff matrix for player&nbsp;A"). Assume the payoff matrix for '''B''' is the same matrix with the signs reversed (i.e., if the choices are A1 and B1 then '''B''' pays&nbsp;3 to '''A'''). Then, the maximin choice for '''A''' is A2 since the worst possible result is then having to pay&nbsp;1, while the simple maximin choice for '''B''' is B2 since the worst possible result is then no payment.  However, this solution is not stable, since if '''B''' believes '''A''' will choose A2 then '''B''' will choose B1 to gain&nbsp;1; then if '''A''' believes '''B''' will choose B1 then '''A''' will choose A1 to gain&nbsp;3; and then '''B''' will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed.

Some choices are ''dominated'' by others and can be eliminated: '''A''' will not choose A3 since either A1 or A2 will produce a better result, no matter what '''B''' chooses; '''B''' will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what '''A''' chooses.

Player '''A''' can avoid having to make an expected payment of more than {{sfrac|1| 3 }} by choosing A1 with probability {{sfrac|1| 6 }} and A2 with probability {{nobr| {{sfrac|5| 6 }}:}} The expected payoff for '''A''' would be {{nobr|  3 × {{sfrac|1| 6 }} − 1 × {{sfrac|5| 6 }} {{=}} {{sfrac|−|1| 3 }}  }} in case '''B''' chose B1 and {{nobr|  −2 × {{sfrac|1|6 }} + 0 × {{sfrac|5| 6 }} {{=}} {{sfrac|−|1| 3 }}  }} in case '''B''' chose B2. Similarly, '''B''' can ensure an expected gain of at least {{sfrac|1| 3 }}, no matter what '''A''' chooses, by using a randomized strategy of choosing B1 with probability {{sfrac|1| 3 }} and B2 with probability {{sfrac|2| 3 }}. These [[mixed strategy|mixed]] minimax strategies cannot be improved and are now stable.