## Matrix games

The most general "m by n matrix games" is exactly like an example. X has n possible moves (columns of A). Y chooses from the m rows. The entry aij is the payment when X chooses column J and Y chooses Rows I. A negative entry means a payment to y. This is a zero-sum game. Whatever 1 player loses, The other wins.

X is free to choose any mixed strategy x= (x1,.....,xn). These x1 give The frequencies for the n columns and the add to 1. At every turn x uses a random device to produce strategy I with frequency xi . Y chooses a vector y = (y1,.....,ym), also with yi > 0 and ∑yi =1, which gives The frequencies for selecting rows.

A single play of the game is random. On average, the combination of column J for x and row for y will turn up with probability x i y i. When it does came up, the payoff is aij. The expected payoff to X from this combination is aijxjyi.

The total expected payoff from each play of the same game is ∑∑aijxjyi = yAx:

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