To introduce the notation used in game theory, let us consider a simple game. Suppose there are only two lighting fixture stores, X and Y, in Urbana, Illinois. (This is called a duopoly.) The respective market shares have been stable up until now, but the situation may change. The daughter of the owner of store X has just completed her MBA and has developed two distinct advertising strategies, one using radio spots and the other newspaper ads. Upon hearing this, the owner of store Y also proceeds to prepare radio and newspaper ads.

The $2\times 2$ payoff matrix in Table M4.1 shows what will happen to current market shares if both stores begin advertising. By convention, payoffs are shown only for the first game player—X, in this case. Y’s payoffs will just be the negative of each number. For this game, there are only two strategies being used by each player. If store Y had a third strategy, we would be dealing with a $2\times 3$ payoff matrix.

A positive number in Table M4.1 means that X wins and Y loses. A negative number means that Y wins and X loses. It is obvious from the table that the game favors competitor X, since all values are positive except one. If the game had favored player Y, the values in the table would have been negative. In other words, the game in Table M4.1 is biased against Y. However, since Y must play the game, he or she will play to minimize total losses. To do this, player Y would use the minimax criterion, our next topic.