## Proof of the Stable Solution Theorem

For a two player, zero sum game, a stable solution is one such that each player always plays the same, unchanging strategy. The Stable Solution Theorem states that there will only be a stable solution when the row maximum of the matrix representing the playoffs for one player equals the column minimum.

If the matrix represents the payoffs for of each combinations of strategies of each player, with and player A's and B's play safe winnings respectively, then

1. For any zero sum game, 2. A zero sum game has a stable solution if and only if Prove 1 first. If both players play safe then wins at least and wins at least so the total winnings are at least but in a zero sum game the total winnings are always zero so Since is the row maximin there is a row in which the smallest element is If the row is the kth row then player will be playing safe by playing strategy Similarly is the column minimax, there is a column minimax, there is a column, say the lth, in which the largest entry is Player will be playing safe by choosing strategy The value of the game to is given by the element in the matrix. since is the smallest number in row and since is the biggest number in column Hence but hence hence and  is then the smallest in row and the biggest in column Player assumes that player will play strategy There is no benefit to in changing strategy since gives the biggest payoff to  also assumes that will play strategy There is no benefit to in changing strategy since gives the biggest payoff to so will always play strategy l. The game is table and is the saddle point.

To prove the 'only if' part, note that if the game is stable there is a saddle point or possibly points that give the game's value. Since this is a zero sum game this must be equal and opposite for each player so  