## Game Theory, Payoff Table or Matrix, Maximin, Minimax

"Game Theory is a mathematical theory that deals with the general features of competitive situations in a formal abstract way. It places particular emphasis on the decision-making process of the adversaries." (O.R., Hiller & Liberman) Game Theory should not be confused with 'GAMING' which is a simulation in which decisions are made by live decision makers. Gaming is used in complex military international, and industrial decisions where models are virtually non existent (Mathematical Foundations for Design, Stark & Nichols) Consider first the so called two person zero-sum games where player 1's gain is player 2's loss, i.e. no units enter or leave the game.

p[i;j] is the payoff to player i for action a[i;], from player 2 for action b[;j]. The set of all possible outcomes is called the payoff table or matrix.

Each player might choose to use conservative minimax criterion to select an action.

p[r;s]  maximin i; maximin strategy to play row i; the maximum pure strategy; the upper bound of the game.

p[t;u]  minimax j; minimax strategy to play col j; the minimax pure strategy; the lower bound of the game.

If p[r;s] = p[t;u] (r=t),(s=u) is called a saddle point and the strategies are called the optimal minimax strategies, and the resultant expected payoff is called the value of the game.

End to date, ams 981004