The BIMATRIX.ltx Model

Two-Person Nonconstant Sum Game

There are many situations where the welfare, utility, or profit of one person depends not nly on his decisions but on the decisions of others. A two-person game is a special case of the previously described situation in which (1) there are exactly two players/decision makers, (2) each must make one decision, (3) in ignorance of the other's decision, and (4) the loss incurred by each is a function of bothe decisions. A two-person constant-sum game (frequently more narrowly called a zero sum game) is the special case of the above where (4a) the losses to both are in the same commodity, and (4b) the total loss is a constant independent of players' decisions. Thus, in a constant-sum game, the sole effect of the decisions is to determine how a "constant-sized pie" is allocated. Ordinary linear programming can be used to solve two-person constant-sum games. When (1), (2), and (3) apply but (4b) does not, we have a two-person nonconstant-sum game. Ordinary linear programming cannot be used to solve these games; however, the algorithm commonly applied to quadratic programs does apply. Sometimes a two-person nonconstant-sum game is also called a bimatrix game.

In this example, consider two firms, each of which is about to introduce an improved version of an already popular consumer product. The versions are very similar, so one firm's profit is very much affected by its own advertising decision as well as the decision of its competitor. The major decision for each firm is presumed to be simply the level of advertising.

This example problem illustrates that we might wish our own choice to be (i) somewhat unpredictable by our competitor and (ii) robust in the sense that, regardless of how unpredictable our competitor is, our expected profit is high. Thus, we are lead to the idea of a random strategy. By making our decision random, i.e., flipping a coin, we tend to satisfy (i). By biasing the coin appropriately, we tend to satisfy (ii).

Keywords:

Game Theory | Probabilities | Uncertainty | Bimatrix Strategy |