# Game Theory, Introduction to

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**DOI:**https://doi.org/10.1007/978-3-642-27737-5_240-3

## Keywords

Differential Game Dynamic Game Stochastic Game Repeated Game Stable MatchGame theory is the study of decision problems which involves several individuals (the decision-makers or players) interacting rationally. The models of game theory are abstract representations of a number of real-life situations and have applications to economics, political sciences, computer sciences, evolutionary biology, social psychology, and law, among others. These applications are also important for the development of the theory, since the questions that emerge may lead to new theoretic results.

This session is an attempt to provide the main features of game theory, covering most of the fundamental theoretic aspects under the cooperative, noncooperative, and “general” or “mixed” approaches.

The cooperative approach focuses on the possible outcomes of the decision-maker’s interaction by abstracting from the actions or decisions that may lead to these outcomes. Specifically, cooperative game theory studies the interactions among coalitions of players. Its main question is as follows: Given the sets of feasible payoffs for each coalition, what payoff will be awarded to each player? One can take a positive or normative approach to answering this question, and different solution concepts in the theory lead toward one or the other.

The first cooperative solution concept is the von Neumann-Morgenstern stable sets, treated in “Cooperative Games (Von Neumann-Morgenstern Stable Sets)” (Jun Wako and Shigeo Muto). However, the two best known solution concepts in cooperative game theory are perhaps the core and the Shapley value which are presented and discussed in “Cooperative Games” (Roberto Serrano).

The noncooperative approach focuses on the actions that the decision-makers can take. Historically, the first contribution to the noncooperative game theory is due to Zermelo (1913), but the idea of a general theory of games was introduced by John von Neumann and Oskar Morgenstern in their famous book in 1944 entitled *Theory of Games and Economic Behavior*. These authors argued that most economic questions should be analyzed as games. They introduced the extensive-form and the strategic-form representations of a game, also known as dynamic and static games, respectively.

Dynamic games stress the sequentiality of the various decisions that agents can make. An essential component of a dynamic game is the description of who moves first, who moves second, etc. Static games, on the other hand, abstract from sequentiality of the possible moves and model interactions as simultaneous decisions. All extensive-form games can be modeled as static games, and all strategic-form games can be modeled as dynamic games. However, some situations may be more conveniently modeled as one or the other kinds of game. Dynamic games are examined in “Dynamic Games with an Application to Climate Change Models” (Prajit K. Dutta). The structure, as well as its principal results, is discussed in detail. This entry ends with an important application, the economics of climate change.

The main ideas and results related to static games, as well as some interesting relationships that connect equilibrium concepts with the idea of rationality, are reviewed in “Static Games” (Oscar Volij). In this entry, the general theorem of existence of strategic equilibria, due to Nash (1950), is presented. This result extends to more general games, the minimax theorem, which was proved in von Neumann (1928) for two-player zero-sum games. In the literature, there are two proofs published by Nash. One of them uses Brouwer’s fixed-point theorem. The other one is a simpler proof, attributed to Gale by Nash that uses Kakutani’s fixed-point theorem. Some version of the proof that uses Brouwer’s fixed-point theorem, by Geanakoplos (2003), is presented in this entry as well as some discussion on correlated equilibrium and Bayesian games.

The correlated equilibrium is a game theoretic solution concept proposed by Aumann (1974, 1987) in order to capture the strategic correlation opportunities that the players face when they take into account the extraneous environment in which they interact. The entry “Correlated Equilibria and Communication in Games” (Françoise Forges) focuses on two possible extensions of the correlated equilibrium to Bayesian games: the strategic-form correlated equilibrium and the communication equilibrium. The general framework of games with incomplete information is treated in “Bayesian Games: Games with Incomplete Information” (Shmuel Zamir) with special reference to “Bayesian games.”

Repeated games deal with situations in which a group of agents engage in a strategic interaction over and over. The entry “Repeated Games with Complete Information” (Olivier Gossner and Tristan Tomala) is devoted to repeated games with complete information. In such games, the data of the strategic interaction is fixed over time and is known by all the players. The entry “Repeated Games with Incomplete Information” (Jérôme Renault) discusses repeated games with incomplete information, a situation where several players repeat the same stage game, the players having different knowledge of the stage game which is repeated.

Repeated games have many equilibria, including the repetition of stage-game Nash equilibria. At the same time, particularly when monitoring is imperfect, certain plausible outcomes are not consistent with equilibrium. Reputation effect is the term used for the impact upon the set of equilibria (typically of a repeated game) of perturbing the game by introducing incomplete information of a particular kind. This issue is treated in “Reputation Effects” (George Mailath).

Games with two players are of particular significance. The first two-person game studied in the literature was the zero-sum two-person game, first analyzed by von Neumann and Morgenstern (1944). In such a game, one player’s gain is the other player’s loss. Chess, checkers, rummy, two-finger Morra, and tic-tac-toe are all examples of zero-sum two-person games. The theory for such games is surveyed in “Zero-Sum Two Person Games” (T.E.S. Raghavan). Recent results on stochastic zero-sum games are presented in “Stochastic Games” (Eilon Solan). Stochastic games are used to model dynamic interactions in which the environment changes in response to the behavior of the players. These games are discussed in Solan’s entry.

Signaling games and inspection games are also two-player games. Signaling games are the subject of “Signaling Games” (Joel Sobel). They are games of incomplete information in which one player is informed and the other is not. Players can use the actions of their opponents to make inferences about hidden information. The earliest work on this subject is Spence’s seminal 1972 work, in which education serves as a signal of ability. Inspection games are covered in “Inspection Games” (Rudolf Avenhaus and Morton J. Canty). These games deal with the problem faced by an *inspector* who is required to control the compliance of an *inspectee* to some legal or otherwise formal undertaking. They started with the analysis of arms control and disarmament problems in the early 1960s and have been applied to auditing, environmental control, material accountancy, etc.

Inspections cause conflict in many real-world situations. In economics, there are services of many kinds, the fulfillment or payment of which has to be verified. One example is the problem of principal-agent relationships discussed in detail in “Principal-Agent Models” (David Perez-Castrillo and Inez Macho-Stadler). The principal-agent models provide the theory of contracts under asymmetric information, concerning relationships between owner and manager, insurer and insured, etc. The principal, e.g., an employer, delegates work or responsibility to the agent, the employee, and chooses a payment schedule that best exploits the agent’s self-interests. The agent, of course, behaves so as to maximize his/her own utility given the fee schedule proposed by the principal. The problem faced by the principal is to devise incentives to motivate the agent to act in the principal’s interest. This generates some type of transaction cost for the principal, which includes the task of investigating and selecting appropriate agents, gaining information to set performance standards, monitoring agents, and bonding payments by the agents and residual losses.

The entry “Differential Games” (Marc Quincampoix) is devoted to differential games with focus on two-player zero-sum and antagonist differential games. These are games in which the state of the players depends on time in a continuous way. The positions of the players are solutions to differential equations. Motivated by military applications in the “Cold War,” these games have a wide range of applications from economics to engineering sciences and recently to biology and behavioral ecology.

The mechanism designed is the subject of “Mechanism Design” (Ron Lavi). It studies the construction of mechanisms that aim to reach a socially desirable outcome in the presence of rational but selfish players, who care only about their own private utility. More specifically, the question is how to design a mechanism such that the equilibrium behavior of the players in the game induced by the mechanism leads to the socially desired goal.

The theory of mechanism design has been contributed to the development of other research areas, for example, auction theory, contract theory, and two-sided matching theory. “For having laid the foundations of mechanism design theory,” the 2007 Nobel Prize in economics was awarded to Leonid Hurwicz, Eric Maskin, and Roger Myerson.

A related theory is the theory of implementation, the subject of “Implementation Theory” (Luis Corchon). It reverses the usual procedure, namely, fix a mechanism and see what the outcomes are. More precisely, it investigates the correspondence between normative goals and mechanisms designed to achieve those goals.

A class of “mixed” games is that of two-sided matching games, which has been analyzed since Gale and Shapley, 1962, under both cooperative and noncooperative game theoretic approaches. The two-sided matching theory is surveyed in “Two-Sided Matching Models” (Marilda Sotomayor and Ömer Özak) by focusing on the differences and similarities between some matching models. In their paper, Gale and Shapley formulated and solved the stable matching problem for the marriage and the college admissions markets. The solution of the college admissions problem was given by a simple deferred-acceptance algorithm which has been adapted and applied in the reorganization of admission processes of many two-sided matching markets.

Another class of problems that have been discussed from the perspective of cooperative and noncooperative game theory is the cost-sharing problems, treated in “Cost Sharing” (Maurice Koster). Applications are numerous ranging from environmental issues like pollution and fishing grounds to sharing multipurpose reservoirs, road systems, communication networks, and the Internet. The worth of a “coalition” of such activities is defined as the hypothetical cost of carrying out the activities in that coalition only.

Market games and clubs are treated in “Market Games and Clubs” (Myrna Wooders) with focus on the equivalence between markets – defined as private goods economies where all participants in the economy have utility functions that are linear in the variable money – and games in characteristic function form.

Learning in games is surveyed in “Learning in Games” (John Nachbar). It covers models in which players are “rational” but not necessarily in equilibrium: players forecast, possibly inaccurately, the future behavior of their opponents and optimize or ε optimize with respect to their forecasts.

Fair division is reviewed in “Fair Division” (Steven Brams). It provides a rigorous analysis of procedures for allocating goods, or deciding who wins on what issues, in a dispute.

The following two entries deal with applications to political sciences. The first one, the entry “Voting” (Alvaro Sandroni, Antonio Penta, Jonathan Pogach, Deniz Selman and Michela Tincani), presents a game theoretic analysis of voting systems, which are procedures to choose a winner among a set of candidates from the individual preferences of the voters or more ambitiously allow to rank all the candidates or a part of them. Such a situation occurs in the field of social choice and welfare, in the field of elections, and in many other fields as games, sports, artificial intelligence, spam detection, web search engines and more generally Internet applications, statistics, and so on. From a practical point of view, it is crucial to be able to announce who is the winner in a “reasonable” time. This raises the question of the complexity of the voting procedures. The second entry “Voting Procedures, Complexity of” (Olivier Hudry) details the complexity results about several voting procedures.

The entry “Evolutionary Game Theory” (William Sandholm) deals with applications to biology. This field, known as evolutionary game theory, started in 1972 with the publication of a series of papers by the mathematical biologist John Maynard Smith. Maynard Smith adapted the methods of traditional game theory, which were created to model the behavior of rational economic agents, to the context of biological natural selection.

Network models have a long history in sociology, natural sciences, and engineering. However, only recently economists have begun to think of political and economic interactions as network phenomena and to model everything as games of network formation. The entry “Networks and Stability” (Frank Page and Myrna Wooders) is devoted to stable networks and the game theoretic underpinnings of stable networks.

The entry “Game Theory and Strategic Complexity” (Kalyan Chatterjee and Hamid Sabourian) deals with some aspect of bounded rationality that has generated important work, namely, the presence of constraints on the capacities of players. Various constraints could be considered, for example, limits on the ability to plan ahead in intertemporal decision-making or on the ability to compute best responses.

This entry discusses cognitive costs to players of using strategies that depend on long histories of past play. This is done mainly in the context of bargaining and markets. It is shown that such complexity considerations often enable us to make sharp predictions. It is also considered the issue in the context of repeated games.