# Game Theory

## Game Theory (Encyclopedia of Science)

Game theory is a branch of mathematics concerned with the analysis of conflict situations. The term conflict situation refers to a condition involving two or more people or groups of people trying to achieve some goal. A simple example of a conflict situation is the game of tic-tac-toe. In this game, two people take turns making Xs or Os in a #-shaped grid. The first person to get three Xs or Os in a straight line wins the game. It is possible, however, that neither person is able to achieve this goal, and the game then ends in a tie or a stand-off.

The variety of conditions described by the term conflict situation is enormous. They range from board and card games such as poker, bridge, chess and checkers; to political contests such as elections; to armed conflicts such as battles and wars.

Mathematicians have long been intrigued by games and other kinds of conflict situations. Is there some mathematical system for winning at bridge? at poker? in a war? One of the earliest attempts to answer this question was the probability theory, developed by French mathematician and physicist (one who studies the science of matter and energy) Blaise Pascal (1623662) and his colleague Pierre de Fermat (1601665). At the request of a gentleman gambler, Pascal and Fermat explored the way to predict the likelihood of drawing certain kinds of hands (a straight, a flush, or three-of-a-kind, for...

(The entire section is 988 words.)

## Game Theory (Encyclopedia of Public Health)

Game theory is a way of reasoning through problems. Although its use can be found throughout history, it was only formally stylized by the economists John von Neumann and Oskar Morganstern in the 1940s. Game theory takes the logic behind complex strategic situations and simplifies them into models that can be used to explain how individuals reach decisions to act in the real world. Game theory models attempt to abstract from personal, interpersonal, and institutional details of problems how individuals or groups may behave given a set of given conditions. This modeling allows a researcher or planner to get at the root of complex human interactions. The major assumption underlying most game theory is that people and groups tend to work toward goals that benefit them. That is, they have ends in mind when they take actions.

The most important application of game theory to public health occurs when the actions of individuals or groups affect the health of others. On some occasions, individual or group strategies for betterment lead to inferior outcomes for the greater population.

Using game theory to model public health problems is not different from using it to model any other type of problem or decision-making scenario. One particularly illustrative game is called the Prisoners' Dilemma, illustrated below. This game is often used to show the need for public resources and services. That is, sometimes individuals who choose certain strategies end up with an inferior outcome because of the incentives they were presented with. In public health, the problem becomes apparent quickly.

In order to place these events into a context in which game theory can be employed, four commonly defined criteria are used:

• Players are the decision makers in the game; a player can be an individual, group, or population that must decide how to use the resources available within given constraints.
• Rules are the constraints; all activity is defined by rules and gives the model an analytical credence to be tested for validity in the real world.
• Strategies are the courses of action open to the players in a game; players may choose their action dependent upon different situations they are presented with.
• Payoffs are the final returns to players, which are usually stated in terms that are objectively understood by each player of the game.

Consider a situation in which two groups of people border a malarial swamp. One group is named Alpha and the other is Beta. The swamp causes both groups to be plagued by malaria and other diseases. The problem could easily be remedied by draining the swampland. However, neither group is willing to act first because no incentives exist to take on the hard labor of draining the swamp alone. The greater utility that would be conveyed to both groups is lost because there is no incentive for either individual group to act.

THE SWAMP: A PRISONERS' DILEMMA

The game called Prisoners' Dilemma can be modeled using game theory. The game matrix shown in Table 1 is an example of a common tool in game theory modeling. The players are named in the

Table 1

 The Swamp: A Prisoner's Dilemma Beta Contribute Not Contribute SOURCE: Courtesy of author. Alpha Contribute Alpha 1 Alpha -1 Beta 1 Beta 2 Not Contribute Alpha 2 Alpha 0 Beta -1 Beta 0

outer boxes, the rule is that the players may not communicate before simultaneously acting, the strategies are to contribute or not contribute, and the payoffs are in the innermost boxes.

Look at the situation as it is presented to the Alpha group. They realize that the outcome depends on the action the Beta group takes. If Beta contributes, it pays Alpha to avoid contributing, for in that instance, Alpha will benefit twice as much as if they worked with Beta to drain the swamp (2 points rather than 1). The reason the payoff for not contributing is greater is that Alpha will receive the benefit of draining the swamp without doing any of the work. However, if Beta does not contribute, Alpha still benefits by not contributing rather than contributing alone (the payoff is 0 instead of ). That is, Alpha will choose not to bear the costs of draining the swamp alone.

The Alpha group reasons that regardless of Beta's action, their own best action is to not help drain the swamp. Because Beta's options are symmetric to Alpha's, they also reason that they benefit most through inaction. As a result, the swamp does not get drained, and both groups end up with an inferior outcome. This game leads to a special equilibrium called a Nash equilibrium, which means both players' strategies will lead them to the same payoff regardless of the strategy chosen by the opposing player.

PUBLIC HEALTH IMPLICATIONS

The implication for public health is that the best strategies for individuals or groups are sometimes not the best strategies for everyone taken as a whole. Public health professionals need to be vigilant to these special circumstances and use interventions to create better incentive systems. For example, Alpha and Beta could each be levied a tax, by some authority over both, to pay for the draining of the swamp. The disincentives for progress would then be circumvented and both groups would benefit.

Game theory has been used to model a number of subjects important to public health, including organ donation, ethics, and the patient-provider relationship. Game theory provides a strong modeling device for public health professionals and illustrates the need of public intervention when the incentives of individuals impede progress for the group.

PETER S. MEYER

NANCY L. ATKINSON

ROBERT S. GOLD

BIBLIOGRAPHY

Hirshleifer, J., and Glazer, A. (1992). Price and Applications. Englewood Cliffs, NJ: Prentice Hall.

Nash, J. (1951). "Non-Cooperative Games." Annals of Mathematics 54:28695.

Nicholson, E. (1998). Microeconomic Theory. Fort Worth, TX: Harcourt Brace.

O'Brien, B. J. (1988). "A Game-Theoretic Approach to Donor Kidney Sharing." Social Science and Medicine 26(11):1109116.

Parkin, M. (1990). Microeconomics. New York: Addison-Wesley.

Schneiderman, K. J.; Jecker, N. S.; Rozance, C.; Klotzko, A. J.; and Friedl, B. (1995). "A Different Kind of Prisoner's Dilemma." Cambridge Quarterly of Healthcare Ethics 4(4):53045.

Von Neumann, J., and Morgenstern, O. (1944). The Theory of Games in Economic Behavior. New York: Wiley.

Wynia, M. K. (1997). "Economic Analyses, the Medical Commons, and Patients' Dilemmas: What Is the Physician's Role?" Journal of Investigative Medicine 45(2):353.

## Game Theory (Encyclopedia of Business)

Game theory is the study of cooperative and noncooperative approaches to games and social situations in which participants must choose between individual benefits and collective benefits. The games or hypothetical situations involve scenarios where participants must make decisions that affect not only the individual participants but also all the other participants as well. Consequently, game theory is also called theory of social situations in some fields. Two of the central questions of game theory are: what games have a best strategy and how do participants identify the best or most rational strategy.

The field is called "game theory" because its focus is often limited to hypothetical situations or models and games where the interaction between participants can be analyzed easily and general reasons for participants' decisions can be determined. Game theory provides the most satisfying and conclusive information and analyzation in simpler games or scenarioshose with fewer decision makers and fewer choices. Because of the inherent complexities of more complicated games with more than two decision makers, game theory becomes more speculative when applied to such games. In these scenarios, decision makers confront forces they cannot control, which makes it difficult to monitor and describe rational behavior.

While game theory has been applied to participants in parlor games and game theory scenarios, it has also been applied to a variety of real-life situationso general human and institutional behaviorhich has led the social sciences to work with game theoretic models. Numerous fields including biology, computer science, economics, politics, psychology, mathematics, philosophy, and sociology use game theory models.

While game theory has antecedents dating back to ancient times and the 18th century, it emerged as a scientific discipline early in the 20th century, the result of efforts to apply quantitative analysis to abstract cognitive dilemmas. Because of poorly developed methodologies and a lack of demonstrably consistent application to more widely useful purposes, it was largely ignored until the 1940s.

During the early years of the Cold War, however, significant breakthroughs were made that legitimized it as a science with testable proofs. As a result, certain aspects of game theory could be applied to war games and global military strategy, economic and pricing strategies, social issues, and even labor bargaining.

Game theory frequently has created difficult marriages of behavioral and mathematical assumptions that yielded such ridiculously mechanical computations as "megadeaths" in military strategy, where certain scenarios for nuclear war could be extrapolated into forecasts of how many millions of people would be killed. In other areas, however, the capabilities of game theory were limited to simple marketing programs for products such as cigarettes, automobiles, and toothpaste. Here, applications of game theory could be tested for validity while certain shortcomings could be readily identified, and adjustments made.

Game theory may also be considered a modeling technique that is used to anticipate and explain the actions of all agents involved in competitive situations and to test and determine the relative optimality of different strategies. Its primary applications are in linear programming, statistical decision making, operations research, and military and economic planning. Recent extensions of game theory include using game theory models to solve social and ethical dilemmas as well as to choose when to implement cooperative and competitive strategies in business.

THE DEVELOPMENT OF GAME THEORY

Although American mathematician John von Neumann (1903-1957) is usually credited with founding the modem discipline of game theory, other 20th century thinkers helped set the stage for von Neumann and subsequent theorists. Early game theory worksncluding von Neumann'socused on winner-takes-all (zero-sum) games with pure competition, such as chess and checkers. For example, one of the first game theory works of the century was a theorem developed for zero-sum games propounded in 1913 by Ernst Zermelo, who argued that games such as chess are strictly determined because all information in the game is immediately revealed and because prescribed rules govern the actions of all players. The solution to chess is so complex, however, that no one can actually determine it. Players win a game of chess not by executing a solution, but executing an imperfect strategy and capitalizing on the opponent's mistakes. In contrast, people can determine the solution to tic-tac-toe and easily obtain stalemates.

Harold Kuhn expanded on Zermelo's theorem by arguing that strategies in plays in a game will be in equilibrium if decisions made by the players are rational. These equilibria may be extended to combinations of plays, and even to the game as a whole, removing the limitation that games be zero-sum and individually rational. Another extension of the theory was made by Rufus Isaacs, who established the concept of the differential game, where perfect information may not exist.

In 1928 von Neumann published his seminal article, "Theory of Parlor Games," in which he discussed bluffing in poker, addressed the economic and military applications of game theory, and developed the "minimax" strategy where decision makers attempt to minimize the maximum amount of losses other decision makers can inflict.

American economist Oskar Morgenstern (1902-1977) recognized the interactive nature of economic activitieshat each person's decisions depend on the decisions of every other person's. In their groundbreaking book, The Theory of Games and Economic Behavior, published in 1944, Morgenstern and von Neumann developed applications of game theory to the problems raised by interactivity.

Moreover, von Neumann and Morgenstern provided a basis for new concepts in game theory, including the idea of cooperative and noncooperative games. A cooperative game is one in which sets of players are bound by agreements to work in mutual, rather than individual, interestven when noncooperative play would be more beneficial.

In the 1950s, John Nash and others expanded the study of noncooperative game theory and created the framework for cooperative game theory. Nash's identification of equilibrium in game theory models contributed significantly to modem game theory. Nash addressed noncooperative strategies, particularly in cooperative games where subversion is a strategic consideration. In 1951 he provided applications of the von Neumann-Morgenstern interdependency theorem, previously used mostly in military strategy, to economic problems.

Central to Nash's approach is the concept of individual rationality, where each player may determine when it is best to abrogate an agreement. Nash said an equilibrium is established in any noncooperative game where agreements are self-enforcinghere the benefits of cheating a partner are outweighed by future retribution. The Nash equilibrium is observed in games where players adopt mixed strategies, or pursue a course of play with two or more different, but related, goals.

Nash equilibria may be said to make noncooperative games cooperative, and games featuring it are therefore somewhat predictable. But they illustrate another facet of game theory, that of Pareto efficiency, named for Italian economist Vilfredo Pareto (1848-1923). Cooperative players will negotiate an alliance that is individually Pareto-optimal; their payoff will be less, but their chances of winning something are better.

THE PRINCIPLES OF GAME THEORY

Most game theory models involve the following five conditions:

1. Each decision maker has two or more choices or sequences of choices ("plays").
2. All possible combinations of decisions or plays result in a clear outcome: win or lose.
3. The scenarios have a well-defined outcome and decision makers receive a "payoff (the value of the outcome to the participants). That is, participants will gain or lose something depending on the outcome.
4. The decision makers know the rules of the game as well as the payoffs of the other decision makers.
5. The decision makers are rational: when faced with two alternatives, players will choose the option that provides the greatest benefits.

While decision makers know the rules and their opponents' options, they do not know their opponents actual decisions in advance. Hence, decision makers must choose options based on assumptions of what their opponents will choose. Some game theory scenarios are also zero-sum games, meaning that one decision maker wins what another loses. Others, however, allow mutual gains and losses. Moreover, these scenarios or games involve several strategies: minimizing the maximum losses another decision maker can cause and making decisions based on probability.

In addition, games and scenarios admit different degrees of information. Perfect information games such as chess and checkers contain no surprises: each player has a finite number of moves and each player sees the opponent's moves and can respond to them immediately. But other games and scenarios, such as the prisoner's dilemma (discussed below), contain surprises and more guesswork.

John Harsanyi postulated that in differential or asymmetric games (even including chess, checkers, etc.), each player is certain about only his own utility function, but must speculate about other players' utility functions and, more importantly, other players' conception of every other players' utility functions. Simply put, each player must know his own payoff probability, guess other players' payoff probabilities, and also guess what other players' are guessing about his own payoff probability. Furthermore, the player must guess what other players are guessing about his guesses about them, forming an infinite regress.

COMPETITIVE GAMES AND STRATEGIES

Unlike games of chance such as roulette or coin flipping, the games and scenarios dealt with by game theory require participants to consider what the other participants are thinking and what they will do. Consequently, probability alone will not help decision makers in game theory scenarios as it does in some games. Game theory explores a variety of games beginning with two-person zero-sum games and extending to non-zero-sum games with an infinite number of participants. Zero-sum games include chess, checkers, poker, and bridge, and they are characterized by: one player winning at the expense of another, all players trying to win, and their finiteness (a player has just a finite number of options each turn). If both players are trying to win, there can be no cooperation in two-player zero-sum games.

A central task of game theory is describing the strategies players adopt to win. The strategy is a specific way to play a game or the group of decisions a player makes in an effort to win the game. Von Neumann's early work identified the best strategy for these kinds of games with his minimax principle. In Prisoner's Dilemma, William Poundstone offered a simple scenario to demonstrate this principle. Suppose two children must divide a piece of cake between them and one of the children gets to cut the cake, while the other gets to choose first. This method of dividing the cake avoids having the children argue over who got the bigger piece in that if the cutter cuts the cake unevenly, the chooser will select the larger piece. The children have two options each. While the cutter wants as much cake as possible, he knows that his best option is to cut the cake as evenly as possibleince if he makes one piece bigger, the chooser will select it. Hence, the cutter cuts the cake as evenly as possible to minimize the maximum amount the chooser can gethat is, to avoid the worst.

COOPERATIVE GAMES AND STRATEGIES

In contrast to zero-sum games, cooperative games and scenarios involve decision makers with mutual interests who may make their decisions in a collaborative mannerr where a cooperative strategy would yield the most benefits for all parties involved. These games or scenarios may include two or more participants.

A precondition to cooperation and noncooperation is that the reward, or "payoff," has transferable utilityhat is, that every player values the reward equally. Likewise, each player must equally value the threats associated with losing. If there are differences in either case, player motivations will differ accordingly, possibly affecting the terms of agreement between players.

In addition, noncooperative games may involve both cooperative and noncooperative strategies. In a game of Monopoly, for instance, where there are six players, each player may purchase and hold certain properties to deny other players a monopoly on the group to which that property belongs, thereby precluding the placement of houses and hotels by opponents. A group of two players, however, might find it advantageous to cooperate by trading properties that the other wants, enabling each to build greater threats to other players.

As the game progresses, the other four playersho are in strategically weaker positionsre likely to be victimized by this strategy and bankrupted. At this point, the purpose of the cooperation has run its course and the two players begin oppositional noncooperative play.

Cooperative games remain cooperative only as long as the conditions of their agreement remain enforceable and effective. They become noncooperative when prospective individual rewards overwhelm the basis for cooperation.

In games with more than two players, groups of players might form coalitions to reap mutual benefits. Parliamentary forms of government provide a good example of how coalitions work. When an election denies a party an absolute majority, it may form coalitions with other parties to achieve a majority. These coalitions are based on agreements to pool interestsn unstable situation, given that parties in a coalition have appreciable conflicts. If a smaller coalition partner feels it has been ill-served by the coalition, it may dissolve its partnership and bring down the government.

Another example of a game theory scenario that requires a cooperative strategy comes from the prisoner's dilemma, which was first articulated in this way by Al Tucker. Tucker used the example of two suspects who participated in crime. Prosecutors have sufficient evidence to prove both are accomplices, but cannot prove which was the leader. They offer each suspect a proposition: either testify against the other or remain silent.

If both prisoners remain silent, they go to jail for one year. If they implicate each other they go to jail for two years. But if prisoner A plea bargains and implicates prisoner B, and prisoner B remains silent, prisoner A gets no jail time and prisoner B gets six years. The reverse applies if B implicates A and A remains silent. The dilemma may be expressed in a matrix (see Figure 1).

The dilemma Tucker identified is whether each prisoner should confess or stay silent. If each wishes at all costs to avoid spending six years in jail, he will be motivated to confess and may spend no time in jail or he may spend as many as two years in jail, depending on whether his accomplice confesses. But if one of these prisoners wishes for the mutually beneficial option, he may risk being silent and hope his partner stays silent as well in order to spend just one year behind bars. The cooperative strategy of both prisoners remaining silent would result in each prisoner spending only a year in jail. Each prisoner, however, cannot be sure that the other will also remain silent. Hence, the cooperative strategyf not adopted by both prisonersill lead to one prisoner being played for a sucker and going to jail for six years, while the other is set free.

The cooperative dilemmas facing players in this scenario become more complex when they are not limited to single "one-shot" games, but are extended to additional rounds. These dynamic games, including repeated games, provide players with a basis of knowledgeamely, the behavior of other playershat may be employed in future situations. This knowledge is defined in the Folk Theorem, which states that breach of cooperative agreements may be dealt with in subsequent scenarios and deceivers will be punished next time around. The Nash Theorem suggests that risk-averse players will be rewarded more often for cooperating than for not cooperating.

Although the prisoner's dilemma may seem rudimentary, it can easily be adapted to complex decision making scenarios. Consider that five airline companies are competing in a three-city market. The matrix thus expands to three sets, or dimensions, of five-by-five matrices, yielding 75 possible outcomes. Add the dimension of time, and the number of decisions is multiplied at every period in question.

In dealing with outcomes indicated by matrices, a player may seek to maximize opportunities to finish ahead of his opponents by adopting a strategy wherein the sum of his minimum payoffs is greater than the sum of his opponents' sum of minimum payoffs. This is called the minimax criteria, and ensures that this player's outcomes will exceed, or dominate, those of his opponents.

While the game theory models have been applied to numerous disciplines, they originally were developed to explain economic phenomena. Hence, much of the previous discussion has direct relevance to business and economics. According to Adam M. Brandenburger and Barry J. Nalebuff, authors of Coopetition, the primary use of game theory for business is to help company decision makers determine when to cooperate and when to compete. They found that cooperation generally is useful for expanding a market and competition for divvying up a market.

In their application of game theory to business, Brandenburger and Nalebuff also point out that each participant in any given game or scenario has a certain amount of power, or "added value," which equals the payoff minus the amount the payoff would be if a specific participant were not involved. The added value represents, for example, the negotiating power of a company and its distributors. The video game producer Nintendo wields far more power than the stores that sell its game cartridges, because the company maintains only a small supply of games. As a result, it may not have enough to ship to all retailers at any given point. Hence, stores have been willing to sell Nintendo games on small profit margins just to be able to offer them at all, because they need Nintendo much more than Nintendo needs them.

Furthermore, the noncooperative models can be used to explain company decisions and actions. For example, a number of companies have and continue to approach business as a win-or-lose situation and hence base their marketing decisions on a noncooperative two-player zero-sum model. For example, competition between companies such as Coke and Pepsi, Ford and General Motors, and McDonald's and Burger King has taken the form of this kind of noncooperative model. Some companies adopting this strategy have had success. General Motors, for instance, used it to gain its commanding market share of the automobile industry. In the 1920s, General Motors launched an attack on Ford by entering the market for low-priced cars then dominated by Ford and offering cars with more features at a slightly higher price. The strategy worked and reversed the roles of the automakers: General Motors came to control about 50 percent of the car market, whereas Ford previously had controlled that market share. Hence, General Motors in effect won the payoff at Ford's expense.

The zero-sum model, however, applies mostly to states of perfect competition and to duopolies and many markets do not resemble simple two-player zero-sum models. As previously discussed, players in these more complex multiparticipant contexts attempt to obtain the best possible outcome, not to beat all other players. For example, alliances in the computer industry have resulted in mutual benefits. Microsoft and Intel, for example, have benefited from cooperating and not trying to win or steal each other's market. Microsoft benefits when Intel produces faster processors and Intel benefits when Microsoft produces advanced software that makes use of its faster processors. In contrast, Apple and IBM-compatible computer companies competed, and the IBM-compatible companies wound up with the payoff at Apple's expense. In addition, while Coke and Pepsi battle fervently for share of the cola market, they cooperated to pressure NutraSweet's maker to cut its price by threatening that both companies would switch to an alternative artificial sweetener.

The cooperative game model is especially applicable to economics because, while some economic policies and practices limit cooperation (e.g., antitrust acts and laws), economic activities rely on cooperation by their nature. For example, many aspects of the economy are not competitive: police, schools, libraries, public transportation, and utilities (although they are becoming more competitive). In addition, the Federal Reserve Board controls the country's currency. Moreover, the relationship between the buyer and the seller is also cooperative for the most part. Both consensually participate in a transaction and both benefit from it.

[John Simley,

updated by Karl Hell]

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