Game Theory (applied science)

Summary

Game theory is a tool that has come to be used to explain and predict decision-making in a variety of fields. It is used to explain and predict both human and nonhuman behavior as well as the behavior of larger entities such as nation-states. Game theory is often quite complex mathematically, especially in newer fields such as evolutionary game theory.

Definition and Basic Principles

Game theory is a means of modeling individual decision-making when the decisions are interdependent and when the actors are aware of this interdependence. Individual actors (which may be defined as people, firms, and nations) are assumed to maximize their own utility—that is, to act in ways that will provide the greatest benefits to them. Games take a variety of forms such as cooperative or noncooperative. They may be played one time only or repeated either indefinitely or with a finite ending point, and they may involve two or more players. Game theory is used by strategists in a variety of settings.

The underlying mathematical assumptions found in much of game theory are often difficult to understand. However, the explanations drawn from game theory are often intuitive, and people can engage in decision-making based on game theory without understanding the underlying mathematics. The formal approach to decision-making that is part of game theory enables decision-makers to clarify their options and enhance their ability to maximize their utility.

Classical game theory has quite rigid assumptions that govern the decision process. Some of these assumptions are so rigid that critics have argued that game theory has few applications beyond controlled settings.

Later game theorists developed approaches to strategy that have made game theory applicable in many decision-making applications. Advances such as evolutionary game theory take human learning into account, and behavioral game theory factors emotion into decision-making in such a way that it can be modeled and predictions can be made.

Background and History

The mathematical theory of games was developed by the mathematicianJohn von Neumann and the economist Oskar Morgenstern, who published The Theory of Games and Economic Behavior in 1944. Various scholars have contributed to the development of game theory, and in the process, it has become useful for scholars and practitioners in a variety of fields, although economics and finance are the disciplines most commonly associated with game theory.

Some of the seminal work in game theory was done by the mathematician John Nash in the early 1950s, with later scholars building on his work. Nash along with John Harsanyi and Reinhard Selten received a Nobel Prize in Economic Sciences in 1994 for their work in game theory. The political scientist Thomas Schelling received a Nobel Prize in Economic Sciences in 2005 for his work in predicting the outcomes of international conflicts.

How It Works

Game theory is the application of mathematical reasoning to decision-making to provide quantitative estimates of the utilities of game players. Implicit in the decision-making process is the assumption that all actors act in a self-interested fashion so as to maximize their own utilities. Rationality, rather than altruism or cooperation, is a governing principle in much of game theory. Actors are expected to cooperate only when doing so benefits their own self-interest.

General Classification. Theories may be classified in several ways—the number of individuals or entities, perfect or imperfect information, constant or variable sum games, and finite or infinite outcome options. The number of individuals or entities may range from a single individual to a sales team, a multinational organization, or a country. Players in perfect information games have a comprehensive understanding of the game components and players, while imperfect information games involve players who lack information about their opponent. For example, chess players have more game information than poker players. Constant-sum game players have opposing interests, and as one player wins, the other loses. In variable-sum games, all parties may win somehow, making communication an important predictor of the outcome.

Cooperative and Noncooperative Games. The most common types of game theories used in applied sciences are cooperative and noncooperative theories. Cooperative theories are variable-sum games that apply to the formation of groups and how payoffs are allocated among players when only potential payoffs are known. Players can communicate and negotiate to create binding agreements or alliances. In noncooperative games, players do not create binding contracts but may communicate. Games are viewed as noncooperative in that the players act only in their own self-interest and will not cooperate, even when doing so might lead to a superior outcome.

Prisoners' Dilemma. A classic and well-known noncooperative game is The Prisoners' Dilemma. In this game, the police arrest Moe and Joe, two small-time criminals, for burglary. The police have enough information to convict the two men for possessing burglar's tools, a crime that carries a sentence of five years in prison. They want to convict the two men of burglary, which carries a ten-year sentence, but this requires at least one man to confess and testify against the other. It is in each man's best interest to remain silent, as this will result in only a five-year sentence for each. The police separate Moe and Joe. Officers tell Moe that if he confesses, he will receive a reduced three-year sentence, but if Joe confesses first, Moe will receive a harsher twelve-year sentence. At the same time, other officers give Joe the same options. The police rely on the self-interest of each criminal to lead him to confess, which will result in ten-year sentences for both men. If the various payoffs are examined, cooperation (mutual silence), which means two five-year sentences, is the most rewarding overall. However, if the two criminals act rationally, each will assume that the other will confess, and therefore, each will confess. Because each man cannot make sure that the other will cooperate, each will act in a self-interested fashion and end up worse off than if he had cooperated with the other by remaining silent.

As some game theorists such as Robert Axelrod have demonstrated, it is often advantageous for players to cooperate so that both can achieve higher payoffs. In reality, as the prisoners' dilemma demonstrates, players who act rationally often achieve an undesirable outcome.

Sequential Games. Many games follow a sequence in which one actor takes an action and the other reacts. The first actor then reacts, and so the game proceeds. A good way to think of this process is to consider a chess game in which each move is countered by the other player, but each player is trying to think ahead so as to anticipate his or her opponent's future moves. Sequential games are often used to describe the decision process of nations, which may lead to war if wrong reactions occur.

Sequential games can be diagramed using a tree that lists the payoffs at each step, or a computer program can be used to describe the moves. Some sequential games are multiplayer games that can become quite complicated to sort out. Work in game theory suggests that equilibrium points change at each stage of the game, creating what are called sequential equilibria.

Simultaneous Games. In some games, players make their decisions at the same time instead of reacting to the actions of the other players. In this case, they may be trying to anticipate the actions of the other player so as to achieve an advantage. The prisoners' dilemma is an example of a simultaneous game. Although rational players may sometimes cooperate in a sequential game, they will not cooperate in a simultaneous game.

Applications and Products

Game theory is most commonly used in economics, finance, business, and politics, although its applications have spread to biology and other fields.

Economics, Finance, and Business. Game theory was first developed to explain economic decision-making, and it is widely used by economists, financial analysts, and individuals. For example, a firm may want to analyze the impact of various options for responding to the introduction of a new product by a competitor. Its strategists might devise a payoff matrix that encompasses market responses to the competitor's product and to the product that the company introduces to counter its competitor. Alternatively, a firm might prepare a payoff matrix as part of the decision process concerning entry into a new market.

Prisoner B stays silent(cooperates)Prisoner B confesses(defects)Prisoner A stays silent(cooperates)Each serves 1 monthPrisoner A: 1 yearPrisoner B: goes freePrisoner A confesses(defects)Prisoner A: goes freePrisoner B: 1 yearEach serves 3 month

Game theory cannot be used to predict the stock market. However, some game theorists have devised models to explain investor response to such events as an increase in the interest rate by the Federal Reserve Board. At the international level, there are various games that can be used to explain and predict the responses of governments to the imposition of various regulations such as tariffs on imports. Because of the large number of variables involved, this sort of modeling is quite complex and still in its infancy.

Some businesses might be tempted to develop a game theoretic response (perhaps using the prisoners' dilemma) to the actions of workers. For example, a company can develop a game-theoretic response to worker demands for increased wages that will enable the company to maximize its utility in the negotiating process. Even if a game does not play out exactly as modeled, a company gains by clarifying its objectives in the development of a formal payoff matrix. Companies can also develop an approach to hiring new employees or dealing with suppliers that draws on game theory to specify goals and strategies to be adopted.

Politics. Some of the most interesting work in game theory has occurred in explaining developments in international affairs, enabling countries to make better decisions in the future. Game theory is often used to explain the decision process of the administration of President John F. Kennedy during the Cuban Missile Crisis. Other game theoretic explanations have been developed to examine a country's decision to start a war, as in the Arab-Israeli conflicts. Game theorists are able to test their formal game results against what actually occurred in these cases so as to enhance their models' ability to predict.

Other game theorists have devised game theoretic models that explain legislative decision-making. Much of the legislative process can be captured by game theoretic models that take into account the step-by-step process of legislation. In this process, members from one party propose legislation and the other party responds, then the first party often responds, and the responses continue until the legislation is passed or defeated. Most of these models are academic and do not seem to govern legislative decision-making, at least not explicitly.

Biology. Some evolutionary biologists have used game theory to describe the evolutionary pattern of some species. One relationship that is often described in game theoretic terms is the coevolution of predators and prey in a particular area. In this case, biologists do not describe conscious responses but rather situations in which a decline in a prey species affects predators or an increase in predators leads to a decline in prey and a subsequent decline in predators. Biologists use this relationship (called the hawk-dove game by the biologist John Maynard Smith) to show how species will evolve so as to better fit an evolutionary niche, such as the development of coloration that enables prey to better conceal itself from predators.

Project management. Game theory is used in project management to study the decision-making process of various participants, such as project managers, investors, contractors, subcontractors, governments, and customers. Game theory is particularly useful in various situations where several bodies are trying to get the same result (either in competition with each other, or in cooperation with each other), but have independent and rational decision-making abilities. A game theory model comes into play for understanding, for example, the behavior of contractors in relation to subcontractors or subcontractors in relation to one another. Each of these parties has to decide how far to push their own interests against the interests of the other parties without putting the whole project at risk, which everybody has an interest in.

Careers and Course Work

Individuals can use game theory to help them make better decisions. It is also used to develop business or political strategies, predict aspects of financial markets, and describe evolutionary processes in biology. Many people practice game theory, often without knowing it. Formal game theory, driven by extensive mathematical modeling and computer applications, is used in industry and government to help guide decision-making. Only a few mathematicians are full-time game theorists. Most game theorists use game theory to enhance decisions or to describe decision processes.

Although game theory is a field of applied mathematics, many graduate programs in social sciences require students to familiarize themselves with at least the basics. Students planning on advanced work in game theory can benefit from courses in statistics and formal logic, as well as any computer programming and any coursework that enhances their ability to deal with quantitative material. As behavioral game theory develops, gaining knowledge of psychology (and possibly neuroscience) will become important for some applications. Many game theorists, however, are self-taught.

Social Context and Future Prospects

Game theory is an evolving field that can become esoteric and divorced from the realities of practical decision-making. It can also be an intuitive, essentially nonmathematical approach to enhancing decision-making. Formal and informal game theoretic approaches are likely to be used in business and government in the future. However, most observers agree that sophisticated actors who are aware of the principles of game theory are likely to prevail over those who follow an ad hoc approach to decision-making.

Game theory is an important driving force in modern economics, biology, computer science, and philosophy's study of ethics. It offers guidance for voting, information systems, bargaining, and more as society addresses new challenges like artificial intelligence (AI) ethics, pandemics, vaccine accesses, and more. Increasingly, game theory is being applied to advanced technology. Developing machine learning models with artificial intelligence—such as multi-agent AI systems, reinforcement or imitation learning, and adversary training in generative adversarial networks—is guided by mathematical game theory.

Game theory is not a perfect guide to decision-making. At times, it has led to overly simplistic approaches derived from a narrow view of utility. With the introduction of newer conceptual frameworks, game theory has become less rigid and better able to model human decision-making. In the future, sophisticated computer simulations based on game theory will likely be used for corporate and governmental decision-making.

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