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A mathematical method of decision-making in which a competitive situation is analyzed to determine the optimal course of action for an interested party, often used in political, economic, and military planning. Also called theory of games.
The theory of games of strategy can briefly be characterized as the applicaiton of mathematical analysis to abstract models of conflict situations. The first such models analyzed by the theory were parlor games such as chess, poker, and bridge. Since then, models arising from the behavioral sciences such as economics, sociology, and political science have been analyzed. Game theory is used in or closely connected to other areas such as linear programming, statistical decisions, management science, operations research, and military planning. In certain areas, the language and concepts of the theory are sometimes used even though the corresponding mathematics is not. See also
Games in extensive form
The players of a game are called persons, and such a person may actually consist of one or more people (for instance, in bridge the pairs of partners, east-west and north-south, each make up a player in the game). Chance moves occur when hands are dealt from a shuffled pack, dice are rolled, or pointers are spun. One says that all chance moves are allotted to the chance player—a fiction that is useful in abstracting properties of games.
When specified by a list of rules, a game is said to be in extensive form. For mathematical purposes, it is convenient to have games in normalized form, and for that, the idea of a pure strategy is needed.
A pure strategy for a player (not the chance player) is a complete list of choices of legal moves that he or she will make for every possible situation that can occur during the game. This is a much more complete list of decisions than that commonly called a strategy. The number of pure strategies in a game can be astronomical even for childish games such as tic-tac-toe. Because of the enormous number of pure strategies, the actual applications of game theory even to parlor games have been severely limited by computational difficulties. Simplified versions of the games have been developed for which computations have been completely carried out.
Games in normalized form
After players have chosen pure strategies in a game, they need not physically play the game. Instead they could hand their strategies to a neutral person, or umpire, who could then carry out their instructions and make the moves they would have made. This intuitively obvious idea leads naturally to the normalized form of the game.
Assume for the moment there are no chance moves in the game, that is, that there are n real players but no chance player. Denote by s1, s2, …, sn specific pure strategies for players 1, 2, …, n, respectively. Given these, the game must be played in exactly one way and a unique outcome will result. Let Pi(s1, s2, …, sn) be the monetary outcome to player i for this play of the game.
Before the effect of the chance player can be introduced, the important concept of mathematical expectation must be explained. Suppose that O1, O2, …, Ok are mutually exclusive monetary outcomes of a chance event, and suppose further that they happen with probabilities, p1, p2, …, pk where pi > 0 and p1 + p2 + · · · + pk = 1. Then the mathematical expectation E of the chance event is defined to be the sum E = p1O1 + p2O2 + · · ·+ pkOk. See also
If there are chance moves in the game, a set of pure strategies, one for each player, will not determine a unique outcome of the game but merely a set of possible outcomes. These outcomes will be mutually exclusive and have probabilities depending on the chance moves associated with their occurrence. Hence in this case one can let Pi(s1, s2, …, sn) be the expected payoff to player i for each i = 1, 2, …, n.
Now the normalized form of a game is defined as the list of all expected payoffs to each player for every possible combination of pure strategies. In the case of two-person games it is most convenient to list these in tables called matrices. See also
Classification of games
A game is called zero-sum if, for every possible n-tuple of pure strategies s1, …, sn, the sum of the payoffs to all players is zero, that is, Eq. (1) holds. If this
1. 
sum is not zero for some n-tuple of pure strategies, the game is called nonzero-sum.
A game in extensive form is said to have perfect information if every player knows and remembers each move of each of her or his opponents as they are made. A game is said to have perfect recall if each player knows and remembers everything he or she did (but not necessarily what the opponent did). Bridge does not have perfect recall, because the personality of a player (=team) is divided between two actual persons and, for instance, north does not see what is in south's hand (except when south is dummy) even though they are members of the same team.
Games can then be classified according to the number of players they have, whether or not they are zero-sum, and whether or not they have perfect information or perfect recall.
Nonzero-sum games
By far the most satisfactory part of the theory of games consists of the zero-sum two-person cases, that is, in matrix games. Applications of the theory to such areas as economics, sociology, and political science almost invariably lead to many-person nonzero-sum games. Although no universally accepted theory has been developed to cover these games, many interesting and useful attempts have been made to deal with them.
When more than two persons are involved in a conflict situation, the important feature of the game becomes the coalition structure of the game. A coalition is a group of players who band together and, in effect, act as a new player in the game. There are two extremes to be considered. One is the noncooperative game in which such coalitions are banned by some means. Equilibrium-point solutions, discussed below, provide reasonably satisfactory solutions to such games. The other extreme is that in which all the players join together in a coalition to maximize jointly their total payoff. A game in which coalitions are permitted is called a cooperative game.
In the noncooperative game, each player is solely interested in his or her own payoff. By an equilibrium point in such a game is meant a set of mixed strategies s1, …, sn such that Eq. (2)
2. 
holds for each i = 1, …, n for all strategies si, of player i. What this means is that no player can, by changing strategy and assuming that the other strategies stay fixed, improve the payoff. By a theorem of J. Nash, every game has at least one equilibrium-point solution (commonly there are several).
Simple games
An important class of n-person games for application to political behavior are the so-called simple games. Each coalition in such a game can be either winning, losing, or blocking. For instance, a winning coalition may be a set of voters who can elect their candidate, or a group of lawmakers who can pass their bill. The players not in a given winning coalition form a losing coalition. Finally, a coalition is blocking if neither it nor the players not in it can enforce their wishes.
Continuous games
If, in the normalized form of the game, each player is permitted to have a continuous range of pure strategies and the payoff function is permitted to be a function of the two real variables that range over each player's strategies, the result is a continuous game.
Game-playing machines
One of the first applications of large electronic computers to numerical problems was in solving large matrix games. Several methods have been devised for finding such solutions. One such method is the simplex method. So-called decomposition methods can extend these methods to certain problems having thousands or even millions of variables and constraints. The principal application of the method is to solve linear programming problems, which can be shown to be equivalent to matrix games.
Computers have also been programmed to play board games such as checkers and chess. Strictly speaking they do not use the theory of games at all at present, but instead use some of the game sense of the people who devised the codes. See also Decision theory; Information theory; Stochastic process.
A model of optimality taking into consideration not only benefits less costs, but also the interaction between participants.
The prisoner's dilemma described above is illustrated in the following diagram: 
Investopedia Says:
Game theory attempts to look at the relationships between participants in a particular model and predict their optimal decisions. One frequently cited example of game theory is the prisoner's dilemma.
Suppose there are two brokers accused of fraudulent trading activities: Dave and Henry. Both Dave and Henry are being interrogated separately and do not know what the other is saying. Both brokers want to minimize the amount of time spent in jail and here lies the dilemma. The sentences vary as follows:
1) If Dave pleads not guilty and Henry confesses, Henry will receive the minimum sentence of one year, and Dave will have to stay in jail for the maximum sentence of five years.
2) If nobody makes any implications they will both receive a sentence of two years.
3) If both decide to plead guilty and implicate their partner, they will both receive a sentence of three years.
4) If Henry pleads not guilty and Dave confesses, Dave will receive the minimum sentence of one year, and Henry will have to stay in jail for the maximum five years.
Obviously, pleading guilty is the most attractive should the other plead not guilty since the sentence is only one year. However, if the other party also chooses to plead guilty, both will have to serve three years. On the other hand, if both parties plead not guilty, they'd have to serve two years in jail. Consequently, the risk of pleading not guilty is a five-year sentence, should the other choose to confess.
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Analytical approach to competitive situations where two or more participants pursue conflicting objectives. The theory attempts to offer a solution that resolves the conflict among the participants. In games, the participants are competitors; the success of one is usually at the expense of the other. Each person selects and executes those strategies that he believes will result in "winning the game." Game theory attempts to provide a guideline for a variety of game situations.
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:
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 −1). 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.
(SEE ALSO: Community Health; Community Organization; Ethics of Public Health)
Bibliography
Hirshleifer, J., and Glazer, A. (1992). Price and Applications. Englewood Cliffs, NJ: Prentice Hall.
Nash, J. (1951). "Non-Cooperative Games." Annals of Mathematics 54:286–295.
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):1109–1116.
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):530–545.
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):35–43.
— PETER S. MEYER; NANCY L. ATKINSON; ROBERT S. G<
Within national security analysis, Game theory deals with parties making choices that influence each other's interests, where they all know that they are making such choices. Using mathematics, it analyzes the think/doublethink logic of how each adversary sees the other, sees the other's view of it, and so on. Unlike war gaming, where real players assume roles, it involves only mathematical calculations.
John von Neumann and Oskar Morgenstern laid the foundation of game theory in the 1940s. Its application to military problems has been limited but interesting. One World War II example involved submarine warfare. A submarine is passing through a corridor patrolled by submarine‐hunting planes. The submarine must spend some time traveling on the surface to recharge its batteries. The corridor widens and narrows, and the submarine is easier to detect in the narrower parts, with less sea for the hunters to scan. Where should the submarine surface? Where should the hunters focus their effort? The premise that the wide part is the one logical place is self‐refuting. If it were true, the hunters would deduce that, would head there and leave the narrower part alone, making the narrower part better. Choosing the narrow part likewise leads to a contradiction. Game theory advises a “mixed” strategy—do one or the other unpredictably, using exact probabilities calculated from the ease of detection in each section.
Other applications have addressed the problems of when an interceptor aircraft closing on a bomber should open fire, how to allocate antimissile defenses to targets of varying value, and when to fire intercontinental missiles to avoid Soviet nuclear explosions in the stratosphere.
These problems involved specific wartime encounters. Another area is broad strategy. A prevalent misconception is that game theory set the principles of nuclear strategy. In the 1940s, planners hoped that the new mathematics would do this, but strategic problems proved too complex. It was hard even to specify each side's goals. Game theory has not given exact strategic advice, but it has clarified general principles. In a model of crisis confrontation, for example, one side wants to show the adversary that it values winning very highly, to induce the other side to back down. It uses the tactic of sacrifice‐to‐show‐resolve—make some costly military deployment so the adversary will conclude that only a determined government would pay such a cost to prove its determination. The model precisely illustrates the skeletal structure of strategic concepts such as showing resolve or enhancing credibility. By the 1990s, a sophisticated body of academic work had addressed deterrence, escalation, war alliances, and the verification of arms treaties.
[See also Disciplinary Views of War: Political Science and International Relations; Operations Research; Strategy; War Plans.]
Bibliography
The branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant's choice of action depends critically on the actions of other participants. Game theory has been applied to contexts in war, business, and biology.
See the Introduction, Abbreviations and Pronunciation for further details.
This deals with the question of making rational decisions in the face of uncertain conditions by choosing certain strategies in order to outwit an opponent in a formal game. In geography, this strategy is often used to overcome or outwit the environment, and when the environment is unpredictable man has only highly probabilistic notions based on past experience to work on.
Consider a very simplified case. Farmers in the Middle Belt of Ghana can choose either hill rice or maize as a staple crop. The climate may be wet or dry. A pay-off matrix can be set out, showing the likely yield of the crops:
| wet | dry | |
|---|---|---|
| maize | 61 | 49 |
| hill rice | 30 | 71 |
Branch of mathematics that has been applied to politics since c.1960. A game is any situation in which the outcomes (‘pay-offs’) are the product of the interaction of more than one rational player. The term therefore includes not only games in the ordinary sense, such as chess and football, but an enormously wide range of human interactions. (And it has been applied to animal interactions, by assuming that over time animals become genetically programmed to behave as if they were rational economic men.) Any human interaction from ‘Should I drive on the left or the right side of the road?’ to ‘How should I behave in international negotiations?’ may be treated as a game.
There are many ways of classifying games. The two most useful are between games with perfect information and those without; and between zero-sum and non-zero-sum games. Chess is a game of perfect information. It is fully defined by the rules on what constitutes a legal move and what constitutes winning. In theory a computer could look at all the possible combinations of moves and responses to moves and specify a unique best strategy for both Black and White. When this happens, chess will cease to be an interesting game, but it is a long way off (1992 a human beat off a computer challenge in draughts, which has many times fewer available moves than chess). Bridge is a game of imperfect information, in which players must not only calculate what it is rational for the other side to do, but also calculate the probabilities on which player holds each card they cannot see. Most human encounters are not games of perfect information. A zero-sum game is one in which the aggregate pay-off—the sum of the pay-offs for all the players put together—is the same in all outcomes (for instance if a player in a two-person game is paid £100 for winning, £50 for a tie, and nothing for a defeat). A non-zero-sum game is any other. If the pay-offs were £100 for winning, £60 for a draw, and nothing for a defeat, for instance, the players would have an incentive to agree to draw and to split the extra takings between them. This makes the game non-zero-sum, or one of partial cooperation. The games most often studied in politics, especially Chicken and Prisoners' dilemma, are non-zero-sum.
Though first formalized in the 1940s, game theory has a long prehistory. Elements of game-theoretic reasoning can be seen in the writings of many thinkers, including Plato, Hobbes, Rousseau, and Dodgson.
For more information on game theory, visit Britannica.com.
The mathematical theory of situations in which two or more players have a choice of decisions (strategies); where the outcome depends on all the strategies; and where each player has a set of preferences defined over the outcomes. See also convention, Nash equilibrium, prisoners' dilemma.
A treatment of competitive games in which probability theory is used in relation to the advantages and disadvantages of decisions that have to be made in situations involving conflicting interests.
This restriction was overcome by the work of John F. Nash during the early 1950s. Nash mathematically clarified the distinction between cooperative and noncooperative games. In noncooperative games, unlike cooperative ones, no outside authority assures that players stick to the same predetermined rules, and binding agreements are not feasible. Further, he recognized that in noncooperative games there exist sets of optimal strategies (so-called Nash equilibria) used by the players in a game such that no player can benefit by unilaterally changing his or her strategy if the strategies of the other players remain unchanged. Because noncooperative games are common in the real world, the discovery revolutionized game theory. Nash also recognized that such an equilibrium solution would also be optimal in cooperative games. He suggested approaching the study of cooperative games via their reduction to noncooperative form and proposed a methodology, called the Nash program, for doing so. Nash also introduced the concept of bargaining, in which two or more players collude to produce a situation where failure to collude would make each of them worse off.
The theory of games applies statistical logic to the choice of strategies. It is applicable to many fields, including military problems and economics. The Nobel Memorial Prize in Economic Sciences was awarded to Nash, John Harsanyi, and Reinhard Selten (1994) and to Robert J. Aumann and Thomas C. Schelling (2005) for work in applying game theory to aspects of economics.
Bibliography
See J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior (3d ed. 1953); D. Fudenberg and J. Tirole, Game Theory (1994); M. D. Davis, Game Theory: A Nontechnical Introduction (1997); R. B. Myerson, Game Theory: Analysis of Conflict (1997); J. F. Nash, Jr., Essays on Game Theory (1997); A. Rapoport, Two-Person Game Theory (1999).
A branch of mathematical logic which deals with all of the possible reactions to a particular strategy used mainly in systems analysis.
Refers to taking steps to avoid giving an edge to an unfamiliar opponent by calling or betting in a certain way that makes sense based on the odds of certain situations, the number of outs available etc. With opponents one knows well, one will bluff more or less often based on what is known of their calling habits, however against an unknown players one would use game theory to aide in deciding how to take action.
SoundPoker Says: For example, if there are four bets in the pot and your hand can only win through bluffing, you can likely get away with a one-bet bluff more than one-quarter or the time, meaning you profit by making use of this game theory as opposed to making decisions based on what you know of your opponent.
See Also: Bluff, Card Shark, Edge, Fish
Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modelling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimisation approach developed in neoclassical economics.
The field of game theory came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies.
Game theory has played, and continues to play a large role in the social sciences, and is now also used in many diverse academic fields. Beginning in the 1970s, game theory has been applied to animal behaviour, including evolutionary theory. Many games, especially the prisoner's dilemma, are used to illustrate ideas in political science and ethics. Game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.
In addition to its academic interest, game theory has received attention in popular culture. A Nobel Prize–winning game theorist, John Nash, was the subject of the 1998 biography by Sylvia Nasar and the 2001 film A Beautiful Mind. Game theory was also a theme in the 1983 film WarGames. Several game shows have adopted game theoretic situations, including Friend or Foe? and to some extent Survivor. The character Jack Bristow on the television show Alias is one of the few fictional game theorists in popular culture.[1]
Although some game theoretic analyses appear similar to decision theory, game theory studies decisions made in an environment in which players interact. In other words, game theory studies choice of optimal behavior when costs and benefits of each option depend upon the choices of other individuals.
The games studied by game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. Most cooperative games are presented in the characteristic function form, while the extensive and the normal forms are used to define noncooperative games.
The extensive form can be used to formalize games with some important order. Games here are often presented as trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.
In the game pictured here, there are two players. Player 1 moves first and chooses either F or U. Player 2 sees Player 1's move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2.
The extensive form can also capture simultaneous-move games and games with incomplete information. To represent it, either a dotted line connects different vertices to represent them as being part of the same information set (i.e., the players do not know at which point they are), or a closed line is drawn around them.
| Player 2 chooses Left |
Player 2 chooses Right |
|
| Player 1 chooses Up |
4, 3 | –1, –1 |
| Player 1 chooses Down |
0, 0 | 3, 4 |
| Normal form or payoff matrix of a 2-player, 2-strategy game | ||
The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (see the example to the right). More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions. In the accompanying example there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays Up and that Player 2 plays Left. Then Player 1 gets a payoff of 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.
In cooperative games with transferable utility no individual payoffs are given. Instead, the characteristic function determines the payoff of each coalition. The standard assumption is that the empty coalition obtains a payoff of 0.
The origin of this form is to be found the in the seminal book of von Neumann and Morgenstern who, when studying coalitional normal form games, assumed that when a coalition C forms, it plays against the complementary coalition (N\C) as if they were playing a 2-player game. The equilibrium payoff of C is characteristic. Now there are different models to derive coalitional values from normal form games, but not all games in characteristic function form can be derived from normal form games.
Formally, a characteristic function form game (also known as a TU-game) is given as a pair (N,v), where N denotes a set of players and 
is a characteristic
function.
The characteristic function form has been generalised to games without the assumption of transferable utility.
The characteristic function form ignores the possible externalities of coalition formation. In the partition function form the payoff of a coalition depends not only on its members, but also on the way the rest of the players are partitioned (Thrall & Lucas 1963).
A game is cooperative if the players are able to form binding commitments. For instance the legal system requires them to adhere to their promises. In noncooperative games this is not possible.
Often it is assumed that communication among players is allowed in cooperative games, but not in noncooperative ones. This classification on two binary criteria has been rejected (Harsanyi 1974).
Of the two types of games, noncooperative games are able to model situations to the finest details, producing accurate results. Cooperative games focus on the game at large. Considerable efforts have been made to link the two approaches. The so-called Nash-programme has already established many of the cooperative solutions as noncooperative equilibria.
Hybrid games contain cooperative and non-cooperative elements. For instance, coalitions of players are formed in a cooperative game, but these play in a non-cooperative fashion.
| E | F | |
| E | 1, 2 | 0, 0 |
| F | 0, 0 | 1, 2 |
| An asymmetric game | ||
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of chicken, the prisoner's dilemma, and the stag hunt are all symmetric games. Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the ultimatum game and similarly the dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
| A | B | |
| A | –1, 1 | 3, –3 |
| B | 0, 0 | –2, 2 |
| A zero-sum game | ||
Zero sum games are a special case of constant sum games, in which choices by players can neither increase nor decrease the available resources. In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (more informally, a player benefits only at the expense of others). Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose. Other zero sum games include matching pennies and most classical board games including Go and chess.
Many games studied by game theorists (including the famous prisoner's dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
Constant sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade. It is possible to transform any game into a (possibly asymmetric) zero-sum game by adding an additional dummy player (often called "the board"), whose losses compensate the players' net winnings.
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players' actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while he does not know which of the other available actions the first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones.
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games, although there are some interesting examples of perfect information games, including the ultimatum game and centipede game. Perfect information games include also chess, go, mancala, and arimaa.
Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player knows the strategies and payoffs of the other players but not necessarily the actions.
Games, as studied by economists and real-world game players, are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.
The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proven, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are "win" or "lose"—for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
Most of the objects treated in most branches of game theory are discrete, with a finite number of players, moves, events, outcomes, etc. However, the concepts can be extended into the realm of real numbers. This branch has sometimes been called "differential" games, because they map to a real line, usually time, although the behaviors may be mathematically discontinuous. Much of this is discussed under such subjects as "optimization theory" and extends into many fields of engineering and physics.
These are games the play of which is the development of the rules for another game, the target or subject game. Metagames seek to maximize the utility value of the rule set developed. The theory of metagames is related to mechanism design theory.
Games in one form or another are widely used in many different disciplines.
The application of game theory to political science is focused in the overlapping areas of fair division, political economy, public choice, positive political theory, and social choice theory. In each of these areas, researchers have developed game theoretic models in which the players are often voters, states, interest groups, and politicians.
For early examples of game theory applied to political science, see the work of Anthony Downs. In his book An Economic Theory of Democracy (1957), he applies a Hotelling firm location model to the political process. In the Downsian model, political candidates commit to ideologies on a one-dimensional policy space. The theorist shows how the political candidates will converge to the ideology preferred by the median voter. For more recent examples, see the books by George Tsebelis, Gene M. Grossman and Elhanan Helpman, or David Austen-Smith and Jeffrey S. Banks.
A game-theoretic explanation for democratic peace is that public and open debate in democracies send clear and reliable information regarding their intentions to other states. In contrast, it is difficult to know the intentions of nondemocratic leaders, what effect concessions will have, and if promises will be kept. Thus there will be mistrust and unwillingness to make concessions if at least one of the parties in a dispute is a nondemocracy.[1]
Game theory provides a theoretical description for a variety of observable consequences of changes in governmental policies. For example, in a static world where producers were not themselves decision makers attempting to optimize their own expenditure of resources while assuming risks, response to an increase in tax rates would imply an increase in revenues and vice versa. Game Theory inclusively weights the decision making of all participants and thus explains the contrary results illustrated by the Laffer Curve.
Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, fair division, oligopolies, social network formation, and voting systems. This research usually focuses on particular sets of strategies known as equilibria in games. These "solution concepts" are usually based on what is required by norms of rationality. The most famous of these is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents a best response to the other strategies. So, if all the players are playing the strategies in a Nash equilibrium, they have no unilateral incentive to deviate, since their strategy is the best they can do given what others are doing.
The payoffs of the game are generally taken to represent the utility of individual players. Often in modeling situations the payoffs represent money, which presumably corresponds to an individual's utility. This assumption, however, can be faulty.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type. Naturally one might wonder to what use should this information be put. Economists and business professors suggest two primary uses.
The first use is to inform us about how actual human populations behave. Some scholars believe that by finding the equilibria of games they can predict how actual human populations will behave when confronted with situations analogous to the game being studied. This particular view of game theory has come under recent criticism. First, it is criticized because the assumptions made by game theorists are often violated. Game theorists may assume players always act rationally to maximize their wins (the Homo economicus model), but real humans often act either irrationally, or act rationally to maximize the wins of some larger group of people (altruism). Game theorists respond by comparing their assumptions to those used in physics. Thus while their assumptions do not always hold, they can treat game theory as a reasonable scientific ideal akin to the models used by physicists. However, additional criticism of this use of game theory has been levied because some experiments have demonstrated that individuals do not play equilibrium strategies. For instance, in the centipede game, guess 2/3 of the average game, and the dictator game, people regularly do not play Nash equilibria. There is an ongoing debate regarding the importance of these experiments.[2]
Alternatively, some authors claim that Nash equilibria do not provide predictions for human populations, but rather provide an explanation for why populations that play Nash equilibria remain in that state. However, the question of how populations reach those points remains open.
Some game theorists have turned to evolutionary game theory in order to resolve these worries. These models presume either no rationality or bounded rationality on the part of players. Despite the name, evolutionary game theory does not necessarily presume natural selection in the biological sense. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).
| Cooperate | Defect | |
| Cooperate | 2, 2 | 0, 3 |
| Defect | 3, 0 | 1, 1 |
| The Prisoner's Dilemma | ||
On the other hand, some scholars see game theory not as a predictive tool for the behavior of human beings, but as a suggestion for how people ought to behave. Since a Nash equilibrium of a game constitutes one's best response to the actions of the other players, playing a strategy that is part of a Nash equilibrium seems appropriate. However, this use for game theory has also come under criticism. First, in some cases it is appropriate to play a non-equilibrium strategy if one expects others to play non-equilibrium strategies as well. For an example, see Guess 2/3 of the average.
Second, the Prisoner's Dilemma presents another potential counterexample. In the Prisoner's Dilemma, each player pursuing his own self-interest leads both players to be worse off than had they not pursued their own self-interests.
| Hawk | Dove | |
| Hawk | v−c, v−c | 2v, 0 |
| Dove | 0, 2v | v, v |
| The hawk-dove game | ||
Unlike economics, the payoffs for games in biology are often interpreted as corresponding to fitness. In addition, the focus has been less on equilibria that correspond to a notion of rationality, but rather on ones that would be maintained by evolutionary forces. The most well-known equilibrium in biology is known as the Evolutionary stable strategy or (ESS), and was first introduced by John Maynard Smith (described in his 1982 book). Although its initial motivation did not involve any of the mental requirements of the Nash equilibrium, every ESS is a Nash equilibrium.
In biology, game theory has been used to understand many different phenomena. It was first used to explain the evolution (and stability) of the approximate 1:1 sex ratios. Ronald Fisher (1930) suggested that the 1:1 sex ratios are a result of evolutionary forces acting on individuals who could be seen as trying to maximize their number of grandchildren.
Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Maynard Smith & Harper, 2003). The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals. For example, the Mobbing behavior of many species, in which a large number of prey animals attack a larger predator, seems to be an example of spontaneous emergent organization.
Finally, biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.
Game theory has come to play an increasingly important role in logic and in computer science. Several logical theories have a basis in game semantics. In addition, computer scientists have used games to model interactive computations.
Separately, game theory has played a role in online algorithms. In particular, the k-server problem, which has in the past been referred to as games with moving costs and request-answer games.[3]
| Stag | Hare | |
| Stag | 3, 3 | 0, 2 |
| Hare | 2, 0 | 2, 2 |
| Stag hunt | ||
Game theory has been put to several uses in philosophy. Responding to two papers by W.V.O. Quine (1960, 1967), David Lewis (1969) used game theory to develop a philosophical account of convention. In so doing, he provided the first analysis of common knowledge and employed it in analyzing play in coordination games. In addition, he first suggested that one can understand meaning in terms of signaling games. This later suggestion has been pursued by several philosophers since Lewis (Skyrms 1996, Grim et al. 2004).
In ethics, some authors have attempted to pursue the project, begun by Thomas Hobbes, of deriving morality from self-interest. Since games like the Prisoner's Dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is required by self-interest is an important component of this project. This general strategy is a component of the general social contract view in political philosophy (for examples, see Gauthier 1987 and Kavka 1986).[4]
Other authors have attempted to use evolutionary game theory in order to explain the emergence of human attitudes about morality and corresponding animal behaviors. These authors look at several games including the Prisoner's Dilemma, Stag hunt, and the Nash bargaining game as providing an explanation for the emergence of attitudes about morality (see, e.g., Skyrms 1996, 2004; Sober and Wilson 1999).
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. It was not until the publication of Antoine Augustin Cournot's Researches into the Mathematical Principles of the Theory of Wealth in 1838 that a general game theoretic analysis was pursued. In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.
Although Cournot's analysis is more general than Waldegrave's, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. While the French mathematician Borel did some earlier work on games, Von Neumann can rightfully be credited as the inventor of game theory. Von Neumann was a brilliant mathematician whose work was far-reaching from set theory to his calculations that were key to development of both the Atom and Hydrogen bombs and finally to his work developing computers. Von Neumann's work in game theory culminated in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding optimal solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first discussion of the prisoner's dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, John Nash developed a definition of an "optimum" strategy for multiplayer games where no such optimum was previously defined, known as Nash equilibrium. This equilibrium is sufficiently general, allowing for the analysis of non-cooperative games in addition to cooperative ones.
Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of Game theory to philosophy and political science occurred during this time.
In 1965,
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionary stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[5] were introduced and analysed.
In 2005, game theorists Thomas Schelling and Robert Aumann followed Nash, Selten and Harsanyi as Nobel Laureates. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing an equilibrium coarsening, correlated equilibrium, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
In 2007, Roger Myerson, together with Leonid Hurwicz and Eric Maskin, was awarded of the Nobel Prize in Economics "for having laid the foundations of mechanism design theory." Among his contributions, is also the notion of proper equilibrium, and an important graduate text: Game Theory, Analysis of Conflict, published in 1991.
