Introduction to Management Science (10th Edition)

Competitive game situations can be subdivided into several categories. One classification is based on the number of competitive decision makers , called players , involved in the game. A game situation consisting of two players is referred to as a two-person game . When there are more than two players, the game situation is known as an n-person game .

A two-person game encompasses two players .

Games are also classified according to their outcomes in terms of each player's gains and losses. If the sum of the players' gains and losses equals zero, the game is referred to as a zero-sum game . In a two-person game, one player's gains represent another's losses. For example, if one player wins $100, then the other player loses $100; the two values sum to zero (i.e., +$100 and -$100). Alternatively, if the sum of the players' gains and losses does not equal zero, the game is known as a non-zero -sum game .

In a zero-sum game, one player's gains represent another's exact losses .

The two-person, zero-sum game is the one most frequently used to demonstrate the principles of game theory because it is the simplest mathematically. Thus, we will confine our discussion of game theory to this form of game situation. The complexity of the n-person game situation not only prohibits us from demonstrating it but also restricts its application in real-world situations.

The Two-Person, Zero-Sum Game

Examples of competitive situations that can be organized into two-person, zero-sum games include (1) a union negotiating a new contract with management; (2) two armies participating in a war game; (3) two politicians in conflict over a proposed legislative bill, one attempting to secure its passage and the other attempting to defeat it; (4) a retail firm trying to increase its market share with a new product and a competitor attempting to minimize the firm's gains; and (5) a contractor negotiating with a government agent for a contract on a project.

The following example will demonstrate a two-person, zero-sum game. A professional athlete, Biff Rhino, and his agent, Jim Fence, are renegotiating Biff's contract with the general manager of the Texas Buffaloes, Harry Sligo. The various outcomes of this game situation can be organized into a payoff table similar to the payoff tables used for decision analysis. The payoff table for this example is shown in Table E.1.

Table E.1. Payoff Table for Two-Person, Zero-Sum Game

The payoff table for a two-person game is organized so that the player who is trying to maximize the outcome of the game is on the left and the player who is trying to minimize the outcome is on the top. In Table E.1 the athlete and agent want to maximize the athlete's contract, and the general manager hopes to minimize the athlete's contract. In a sense, the athlete is an offensive player in the game, and the general manager is a defensive player. In game theory, it is assumed that the payoff table is known to both the offensive player and the defensive playeran assumption that is often unrealistic in real-world situations and thus restricts the actual application of this technique.

In a game situation, it is assumed that the payoff table is known to all players .

A strategy is a plan of action that a player follows . Each player in a game has two or more strategies, only one of which is selected for each playing of a game. In Table E.1 the athlete and his agent have two strategies available, 1 and 2, and the general manager has three strategies, A, B, and C. The values in the table are the payoffs or outcomes associated with each player's strategies.

A strategy is a plan of action that a player follows .

For our example, the athlete's strategies involve different types of contracts and the threat of a holdout and/or of becoming a free agent. The general manager's strategies are alternative contract proposals that vary with regard to such items as length of contract, residual payments, no-cut/no-trade clauses, and off- season promotional work. The outcomes are in terms of dollar value. If the athlete selects strategy 2 and the general manager selects strategy C, the outcome is a $20,000 gain for the athlete and a $20,000 loss for the general manager. This outcome results in a zero sum for the game (i.e., +$20,000 20,000 = 0). The amount $20,000 is known as the value of the game .

The value of the game is the offensive player's gain and the defensive player's loss in a zero-sum game .

The purpose of the game for each player is to select the strategy that will result in the best possible payoff or outcome, regardless of what the opponent does. The best strategy for each player is known as the optimal strategy . Next, we will discuss methods for determining strategies.

The best strategy for each player is his or her optimal strategy .