# Zero Sum Game

# Zero Sum Game

A zero-sum game is a term used in connection with game theory and management games. Game theory is a mathematical theory that applies to certain situations in which there are conflicts of interest between two or more individuals or groups. Management games are training or educational activities utilizing game theory models consisting of work situations. A zero-sum game is one type of management game in which all the payoffs for all players total zero; what one player or group gains, the other loses.

## GAME THEORY

To better understand the term zero-sum game, it is beneficial to analyze game theory, as well as management games. Game theory is a significant branch of operations research and is closely related to decision theory and operational gaming. It attempts to answer the question: In a situation of conflict, what choice should the player make?

Game theory deals with abstract models of conflict situations or games of strategy. A game occurs when an individual or teams of people are in competition either against one another or against situations, or both.

**The Game Theory Model.** A game can be represented by the following model:

- There are
*n*players (*n*being a certain number), each of whom is required to make one choice from a specified set of possible choices. - When every player has made a choice, the particular combination of choices they have made determines an outcome that, in some way, affects or interests all players.
- Each player knows what outcome results from each possible combination of choices.
- Each player has an order of preference for the possible outcomes (often each player assigns to each outcome a numerical value, called a payoff, which can be thought of as representing the number of points, or dollars, etc., that he gains or loses from the outcome).
- Each player knows the preferences of the other players (she knows what their payoffs are) and all players are assumed to act so as to gain the most they can from the game.
- But each player makes his choice without knowing what choice the other players are making.

In the game, the competing players are identified as *persons* whether they are individuals, teams, or any other group representing a single set of interests. A *play* of a game is an exercise of the conflict model according to the rules; it consists of one or more *moves* by each player and may involve moves left to chance. The outcome of the game is represented by the payoff, a gain or loss of some utility to each of the players as a result of the positions reached at the end of the game. The *solution* of agameis comprised of the identification from among all the possible alternative courses of action, which ensures the player's expected payoff at a quantity called the *value* of the game.

In a business scenario, for example, the competition between two companies may be structured in game-theory terms. The persons are the companies; the play can be a determined period of time; and the rules are the discipline of the marketplace. Within the rules, management may make a variety of decisions upon which actions may be taken. These are known as the moves.

The firm's master plan is the strategy. In this example, the strategies of the companies would describe the companies' general decisions on such topics as advertising, mergers, and new product lines. The results of the interactions among the strategic choices made by the two firms are manifested by the payoff, which could be chosen to be annual gross sales, net profits, and so on. Only when a situation such as this is structured and quantified is it meaningful to address a solution and value for the game.

The theory is used to calculate the optimum strategy that maximizes the winnings or minimizes the losses of one or more of the players.

**Example of a Game-Theory Game.** Finite games, those in which each player has available a finite number of strategies, may be categorized according to the number of persons, relationships among payoffs, and whether cooperation among the players is allowed. The simplest form is the two-person zero-sum game; zero-sum denoting that the sum of the payoffs to the two players is zero.

A payoff matrix can be arranged to identify the payoffs for each player. The matrix is expressed in terms of the payoff to A, whereas B's payoffs are the negative of A's, thus satisfying the condition that their sum be zero. Positive entries indicate payments by B to A; negative ones, payments by A to B.

The solution can take two forms; the pure strategy case, in which a single strategy will be indicated as optimal; or a mixed strategy case, in which two or more strategies appear along with the relative frequencies with which they must be employed. An example of a two-person zero-sum game given by Derek French and Heather Saward, showing a pure strategy solution, is presented in Exhibit 1.

A's problem is to choose one of his four strategies; while B's is to choose one of his three. For example, the choices of A2 and B2 result in the payment by B to A of three units, while A4 and B3 lead to the payment by A to B of two units. First, consider A's analysis of his problem: A1 is a weak strategy because it nets A less than does the equally available strategy A2, regardless of B's choice. By choosing A4 in an effort to realize the payoff of ten units at A4 and B1 could result in the loss of two units if B selects B3; similar dilemmas exist for the other choices.

Suppose that A takes a conservative point of view and examines the least his choice could produce; a gain of two for A2, a gain of five for A3, and a loss of two for A4. Of these options, A3 and its consequence appear to be the best choice; the five-unit gain represents an assured security level to A since he cannot be driven below this point

by any action taken by B. In essence, A has examined the minimum gain that each row strategy could produce and, striving to maximize his gain, has selected the greatest of these.

This is referred to as A's maximum strategy (R3 in Exhibit 1). At this point, B analyzes the greatest loss he might sustain as a result of his strategy choice; ten units for B1, five units for B2, and eight units for B3. Of these choices, B2 causes the smallest loss on B and establishes his security level by guaranteeing that no action of A's can cause his loss to be above five.

Summarizing, B has identified the maximum loss that each column strategy could produce and, wanting to minimize his loss has selected the least of these; known as B's minimax strategy. The most important feature of this result is the independently arrived-at agreement on the part of the players as to their security levels. This example also possesses a saddle-point, an element that is concurrently the greatest of the row minima and the least of the column maxima. The significance lies in the fact that if either player deviates from this choice; it will result in either decreased gain or increased loss.

The solution is that A always employs A3, B always employs B2, and the value of the game is five. This, of course, is not a fair game since A always wins five units at each play. It can be made fair, however, by requiring A to pay five units to B each time to induce B to play, or by reducing each element of the game matrix by five.

## APPLICATIONS

Game theorists have posited a number of real-world applications for their abstract models. The most notorious use of game theory was utilized by the armed services in the Vietnam War for strategic purposes; however, the theory is noted today for its potential contribution to industrial affairs. Game theory is used to analyze economic policies and international agreements (e.g., whether economic sanctions act as practical incentives or build additional resentment). It is applied in management games, in which managers are grouped into teams representing a manager or the management of one of several competing organizations. The manager must take a sequence of decisions relating to a simulation of a real-life management problem, and is then presented with the results of each decision after it is made.

Since the rise of “reality” television game shows such as *Survivor* (2000–present) and *The Apprentice* (2004–present), and the sudden popularity of televised poker in the mid-2000s, the viewing public has had the opportunity to witness game strategies being used in a real-life context. While these programs are ultimately zero-sum games—in the end, one person wins everything—successful participants have employed various strategies to cooperate with and exploit each other, all in an effort to win.

In all game-theory games, the result of an individual decision is the response, or the next move, of the other competitors. The games are used for several training purposes. They provide experience and they bring rapid feedback on the results of a decision. They also can show cause-and-effect relationships that may be blurred during longer time periods in real-life situations. The end result is to attain more personal involvement, greater attention, and greater retention of new concepts and ideas that have been acquired.

## BIBLIOGRAPHY

Binmore, Ken. *Game Theory: A Very Short Introduction*. Oxford: Oxford University Press, 2007.

Finch, Frank, ed. *The Facts on File Encyclopedia of Management Techniques*. New York, NY: Facts on File, 1985.

French, Derek, and Heather Saward. *Dictionary of Management*. Aldershot, Hants, England: Gower, 1983.

“Game Theory.” *Asia Africa Intelligence Wire*, 29 August 2004.

Goodkey, Kennedy. “Is the Key to Survivor in ‘Non-Cooperative Games’?” *Reality News Online*, 24 December 2002. Available from: http://www.realitynewsonline.com/cgi-bin/ae.pl?mode=4&article=article2575.art&page=1.

Harford, Tim. “All Is Fair in Love, War and Poker.” *BBC News*, 17 August 2006. Avai lable from: http://news.bbc.co.uk/2/hi/business/5260120.stm.

McMillan, John. *Games, Strategies and Managers*. New York, NY: Oxford University Press, 1996.

Poundstone, William. *Prisoners' Dilemma*. New York, NY:Anchor Books, Doubleday, 1993.

# Zero-Sum Game

# Zero-Sum Game

“Zero-sum game” describes a situation in which two “players” with strictly opposed interests each make a decision that results in one player’s winning equaling the opposing player’s loss. Many recreational games, such as chess, poker, and tic-tac-toe, are zero-sum because for one player to win, the opposing player(s) must lose.

The notion of zero-sum games originated in a branch of applied mathematics known as game theory, which has enjoyed extensive application in the social sciences. John von Neumann (1903–1957), a mathematician, is usually credited with creating game theory, and he first explicated the theory of zero-sum games in his seminal work with Oskar Morgenstern, *Theory of Games and Economic Behavior* (1944). Game theory is essentially a study of conflict situations between two or more opponents or players. Each player in the game situation must decide on a course of action, or *strategy*, and the strategy each player chooses affects the outcome for all players in the game. The outcome, or solution, to a zero-sum game specifies how each player should move, and if each player moves accordingly, then the resulting payoff is known as the *value* of the game (Kelly 2003).

The easiest class of games to analyze is *two-person* zero-sum games, and these games typically receive the most scholarly attention among those who study zero-sum games. In *Theory of Games*, von Neumann and Morgenstern focus their attention on *two-person* zero-sum games and show that in this type of game situation there always exists a solution that allows each player to avoid the worst possible outcome. To arrive at this solution, both players base their course of action on what they expect their opponent’s action will probably be. Keeping their opponent’s likely course of action in mind, both players attempt to minimize the opponent’s maximum payoff, thereby maximizing their own minimum payoff. In doing so, the outcome of the game ends up being that both players obtain the best payoff they possibly can, given the nature of the game, and neither is able to do any better. This outcome is known as the *equilibrium* of the game, and this point can be thought of as the outcome in which neither player has any regrets about the course of action chosen. This method of play is known as the *minimax theorem*, and von Neumann and Morgenstern showed that all two-person zero-sum games have a minimax solution.

All zero-sum games can be classified as having either *perfect information* or *imperfect information*. In a game with perfect information, each player in the game is fully aware of all previous moves in the game, meaning that each player knows what actions the opponent has already taken. In tic-tac-toe, for example, after the “X” player’s move, the “O” player knows exactly where the “X” player has placed an “X.” In games of perfect information, there is always at least one optimal or best possible strategy for each player. However, the existence of a best possible strategy does not guarantee that a player will win or even be able to identify that strategy. Using the best possible strategy only guarantees that both players will minimize their losses, regardless of whether they win. But there may also be so many viable strategies to choose from that it becomes impossible to determine what the best strategy is.

When applying the minimax theorem to zero-sum games with perfect information, it is possible to achieve the equilibrium point, or the point that represents the outcome that results from both players using their best possible strategy, also known as the *saddle point*. All zerosum games with perfect information have at least one saddle point, and the saddle points can be determined using the minimax theorem. However, on some occasions the minimax theorem does not necessarily have to be used to determine a game’s saddle points. Occasionally, one player has strategies available that *dominate* the other strategies. A strategy is considered *dominant* if it yields a player a better outcome than any other strategy, despite the actions taken by the opponent. When a strategy is dominated by another, then the dominated strategy is said to be *inadmissible* because, if players are trying to get the best possible outcome, then it cannot make sense to choose a dominated strategy (Kelly 2003).

In games with imperfect information, the players are not fully aware of their opponent’s prior moves. This means that each player must choose an action without knowing what action the opponent has taken or may be taking simultaneously. A simple example of this would be the game rock-paper-scissors. While there may not be one best possible strategy, it is still possible to find a minimax solution to two-person games of imperfect information. This solution can be obtained by using *mixed strategies*. Using a mixed strategy means that a player uses one strategy sometimes, another strategy at other times. The player assigns each strategy a particular probability of being used and chooses a strategy based on these probabilities. When mixed strategies are in equilibrium, meaning that neither player can do better by deviating from these strategies, the strategies are sometimes called *minimax mixed strategies* (Kelly 2003).

Analysis of zero-sum games has been applied to a variety of social science disciplines, but it has probably enjoyed most extensive application in the fields of economics and political science. In political science, for example, most elections can be thought of as zero-sum games given that for one candidate to win, the opposing candidate must lose. Also, when considering the distribution of political resources, some scholars believe that for one group to gain political resources, others must lose resources, thus implying a zero-sum nature to political competition. However, the application of zero-sum games to political and economic phenomena is necessarily limited given that most conflict situations are not zero-sum. In many conflict situations, competitors do not have strictly opposed interests; it is often possible for both players in a game to win, as sometimes is the case with economic competition, or for both players to lose, as can happen with pollution or arms races (McCain 2004). Because of the dearth of real-world zero-sum situations, and thus zero-sum’s limited applicability, most game theoretic applications in the social sciences are not zero-sum.

**SEE ALSO** *Elections; Electoral Systems; First-past-the-post; Game Theory; Information, Asymmetric; Information, Economics of; Mixed Strategy; Politics; Social Science; Strategic Behavior; Strategic Games; Voting Schemes*

## BIBLIOGRAPHY

Dixit, A. K., and B. Nalebuff. 1991. *Thinking Strategically: The Competitive Edge in Business, Politics, and Everyday Life*. New York: Norton.

Kelly, A. 2003. *Decision Making Using Game Theory: An Introduction for Managers*. Cambridge, U.K., and New York: Cambridge University Press.

McCain, R. A. 2004. *Game Theory: A Non-Technical Introduction to the Analysis of Strategy*. Mason, OH: Thomson/SouthWestern.

Von Neumann, J., and O. Morgenstern. 1944. *Theory of Games and Economic Behavior*. Princeton, NJ: Princeton University Press.

*Monique L. Lyle*

# zero-sum

ze·ro-sum • adj. (of a game or situation) in which whatever is gained by one side is lost by the other: *altruism is not a zero-sum game.*

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