Game theory is a mathematical framework developed to address problems with conflicting or cooperating parties who are able to make rational decisions. The theory primarily deals with finding the optimal rational decision in various scenarios.
Game theory is a relatively new discipline. Modern game theory was introduced in the works of John von Neumann in the 1920s. Von Neumann, Oskar Morgenstern, and John Nash were the main contributors to the development of game theory. The theory offers a wide number of applications in different fields, including economics, political science, finance, psychology, and biology, among others…
Classifications in Game Theory
Game theory classifications are related to several settings. The scenarios examined include the following:
1. Cooperative/Non-cooperative - This is probably the most common type of game discussed in game theory. In cooperative game settings, the participating players can form binding agreements between each other, and decisions are made by a coalition (a group of players). The decision made by a coalition leads to the payoff that should be distributed among the players. On the other hand, a non-cooperative game considers situations where players cannot form binding agreements. The non-cooperative game theory analyzes possible strategies and payoffs of individual players to determine a Nash equilibrium.
2. Symmetric/Asymmetric - A symmetric game deals with a game setting in which the payoffs primarily depend on the strategy chosen by each player, not on other players’ choices. In an asymmetric game, the payoffs vary among the players. Thus, even if the players employ the same strategy, their payoff will be different.
3. Zero-sum/Non-zero-sum - In a zero-sum game, the gains/losses of one player are balanced with the losses/gains of other players. In non-zero-sum games, the gains/losses of one player do not result in the losses/gains of other players. In other words, a non-zero-sum game may result in a win-win situation.
4. Simultaneous/Sequential - In a simultaneous game, all the participating players make their decisions simultaneously, or they make their decisions without the knowledge of the decisions of other players. In a sequential game, the players take turns to make decisions or have information about the decisions of other players.
5. Perfect information/Imperfect information - The perfect information game considers the situation when all the players are able to access the same information with which to make their decisions. In contrast, in an imperfect information game, the information that is available to one player is inaccessible to the other players.
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Types of Game Theory
Although there are many types (e.g., symmetric/asymmetric, simultaneous/sequential, et al.) of game theories, cooperative and non-cooperative game theories are the most common. Cooperative game theory deals with how coalitions, or cooperative groups, interact when only the payoffs are known. It is a game between the coalitions of players rather than between individuals, and it questions how groups form and how they allocate the payoff among players.
Non-cooperative game theory deals with how rational economic agents deal with each other to achieve their own goals! The most common non-cooperative game happens to be the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed down. A simplistic example of a real-world non-cooperative game is Rock-Paper-Scissors.
Examples of Game Theory: There are several “games” that game theory analyzes.
Below, we will just briefly describe a few of these.
The Prisoner’s Dilemma
The Prisoner’s Dilemma is the most well-known example of game theory. Consider the example of two criminals arrested for a crime. Prosecutors have no hard evidence to convict them. However, to gain a confession, officials remove the prisoners from their solitary cells and question each one of them in separate chambers. Neither prisoner has the means to communicate with each other. Officials present four deals, often displayed as a 2 x 2 box.
If both confess, they will each receive a five-year prison sentence.
If Prisoner 1 confesses, but Prisoner 2 does not, Prisoner 1 will get three years and Prisoner 2 will get nine years.
If Prisoner 2 confesses, but Prisoner 1 does not, Prisoner 1 will get 10 years, and Prisoner 2 will get two years.
If neither confesses, each will serve two years in prison.
The most favourable strategy is then not to confess. However, neither is aware of the other’s strategy and without certainty that one will not confess, both will likely confess and receive a five-year prison sentence! The Nash equilibrium suggests that in a prisoner’s dilemma, both players will make the move that is best for them individually but worse for them collectively.
The expression that is “tit for tat” has been determined to be the optimal strategy for optimizing a prisoner’s dilemma. Tit for tat was introduced by Anatol Rapoport, who developed a strategy in which each participant in an iterated prisoner’s dilemma follows a course of action consistent with his opponent’s previous turn. For example, if provoked, a player subsequently responds with retaliation; if unprovoked, the player does cooperate.
- The prisoners’ dilemma was played once, by two players. The players were given a payoff matrix; each could make one choice, and the game ended after the first round of choices.
- The real world of oligopoly has as many players as there are firms in the core industry. They play round after round: a firm raises its price; another firm introduces a new product, when the first firm cuts its price, then a third firm introduces a new marketing strategy, and so on. An oligopoly game is a bit like a baseball game with an unlimited number of innings one firm may come out ahead after one round, then another will emerge on top another day!
- The Iterated Prisoner's Dilemma: If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma. The iterated prisoner's dilemma game is fundamental to some theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled by a multi-player, iterated, version of the game.
Dictator Game
This is a simple game in which Player A must decide how to split a cash prize with Player B, who has no input into the Player A’s decision. While this is not a game theory strategy per se, it does provide some interesting insights into people’s behaviour. Experiments reveal about 50% keep all the money to them, 5% split it equally, and the other 45% give the other participant a smaller share.
The dictator game is very closely related to the ultimatum game, in which Player A is given a set amount of money, part of which has to be given to Player B, who can accept or reject the amount given. The catch here is if the second player rejects the amount offered, both A and B get nothing. The dictator and ultimatum games hold important lessons for issues pertaining to causes such as charitable giving and philanthropy.
Volunteer’s Dilemma
In a volunteer’s dilemma, someone has to undertake a chore or a job for the common good. The worst possible outcome is realized if nobody volunteers. For example, consider a company in which accounting fraud is rampant, though top management is unaware of it. Some junior employees in the accounting department are aware of the fraud but hesitate to tell top management because it would result in the employees involved in the fraud being fired and most likely to be prosecuted!
Being labelled as a whistleblower may also have some repercussions and consequences down the line. But if nobody volunteers, the large-scale fraud may result in the company’s eventual bankruptcy and the loss of everyone’s jobs.
The Centipede Game
The centipede game is an extensive-form game in game theory in which two players alternately get a chance to take the larger share of a slowly increasing money stash. It is arranged so that if a player passes the stash to his opponent who then takes the stash, the player receives a smaller amount compared to if he had taken the pot.
The centipede game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion! The game has a pre-defined total number of rounds, which are known to each player well in advance.
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Game Theory Explained
Here Prof. Dixit explains game theory and its impact on situations we encounter every day. "If Nash got a dollar for every time someone wrote or said 'Nash equilibrium,'" Dixit has said, "he would be a rich man."
Game theory studies interactive decision-making, where the outcome for each participant or "player" depends on the actions of all. If you are a player in such a game, when choosing your course of action or "strategy" you must take into account the choices of others. But in thinking about their choices, you must recognize that they are thinking about yours, and in turn trying to take into account your thinking about their thinking, and so on.
It would seem that such thinking about thinking must be so complex and subtle that its successful practice must remain an arcane art. Indeed, some aspects such as figuring out the true motives of rivals and recognizing complex patterns do often resist logical analysis. But many aspects of strategy can be studied and systematized into a science -- game theory.
A Theory is Born
This science is unusual in the breadth of its potential applications. Unlike physics or chemistry, which have a clearly defined and narrow scope, the precepts of game theory are useful in a whole range of activities, from everyday social interactions and sports to business and economics, politics, law, diplomacy and war. Biologists have recognized that the Darwinian struggle for survival involves strategic interactions, and modern evolutionary theory has close links with game theory.
Game theory got its start with the work of John von Neumann in the 1920s, which culminated in his book with Oskar Morgenstern. They studied "zero-sum" games where the interests of two players were strictly opposed. John Nash treated the more general and realistic case of a mixture of common interests and rivalry and any number of players. Other theorists, most notably Reinhard Selten and John Harsanyi who shared the 1994 Nobel Memorial Prize with Nash, studied even more complex games with sequences of moves, and games where one player has more information than others.
The Nash Equilibrium
The theory constructs a notion of "equilibrium," to which the complex chain of thinking about thinking could converge. Then the strategies of all players would be mutually consistent in the sense that each would be choosing his or her best response to the choices of the others. For such a theory to be useful, the equilibrium it posits should exist. Nash used novel mathematical techniques to prove the existence of equilibrium in a very general class of games. This paved the way for applications. Biologists have even used the notion of Nash equilibrium to formulate the idea of evolutionary stability. Here are a few examples to convey some ideas of game theory and the breadth of its scope.
The Prisoner's Dilemma
In Joseph Heller's novel Catch-22, allied victory in World War II is a foregone conclusion, and Yossarian does not want to be among the last ones to die. His commanding officer points out, "But suppose everyone on our side felt that way?" Yossarian replies, "Then I'd certainly be a damned fool to feel any other way, wouldn't I?"
Every general reader has heard of the prisoner's dilemma. The police interrogate two suspects separately, and suggest to each that he or she should fink on the other and turn state's evidence. "If the other does not fink, then you can cut a good deal for yourself by giving evidence against the other; if the other finks and you hold out, the court will treat you especially harshly. Thus no matter what the other does, it is better for you to fink than not to fink -- finking is your uniformly best or 'dominant' strategy." This is the case whether the two are actually guilty, as in some episodes of NYPD Blue, or innocent, as in the film LA Confidential. Of course, when both fink, they both fare worse than they would have if both had held out; but that outcome, though jointly desirable for them, collapses in the face of their separate temptations to fink.
Yossarian's dilemma is just a multi-person version of this. His death is not going to make any significant difference to the prospects of victory, and he is personally better off alive than dead. So avoiding death is his dominant strategy.
John Nash played an important role in interpreting the first experimental study of the prisoner's dilemma, which was conducted at the RAND Corporation in 1950.
Real-World Dilemmas
Once you recognize the general idea, you will see such dilemmas everywhere. Competing stores who undercut each other's prices when both would have done better if both had kept their prices high are victims of the dilemma. (But in this instance, consumers benefit from the lower prices when the sellers fink on each other.) The same concept explains why it is difficult to raise voluntary contributions, or to get people to volunteer enough time, for worthwhile public causes.
How might such dilemmas be resolved? If the relationship of the players is repeated over a long time horizon, then the prospect of future cooperation may keep them from finking; this is the well-known tit-for-tat strategy. A "large" player who suffers disproportionately more from complete finking may act cooperatively even when the small fry are finking. Thus Saudi Arabia acts as a swing producer in OPEC, cutting its output to keep prices high when others produce more; and the United States bears a disproportionate share of the costs of its military alliances. Finally, if the group as a whole will do better in its external relations if it enjoys internal cooperation, then the process of biological or social selection may generate instincts or social norms that support cooperation and punish cheating. The innate sense of fairness and justice that is observed among human subjects in many laboratory experiments on game theory may have such an origin.
Mixing Moves
In football, when an offense faces a third down with a yard to go, a run up the middle is the usual or "percentage" play. But an occasional long pass in such a situation is important to keep the defense honest. Similarly, a penalty kicker in soccer who kicks exclusively to the goalie's right, or a server in tennis who goes exclusively to the receiver's forehand, will fare poorly because the opponent will anticipate and counter the action. In such situations it is essential to mix one's moves randomly, so that on any one occasion the action is unpredictable.
Mixing is most important in games where the players' interests are strictly opposed, and this happens most frequently in sports. Indeed, recent empirical studies of serving in tennis grand slam finals, and penalty kicks in European soccer leagues, have found the behavior consistent with the theory.
Commitments
Greater freedom of action seems obviously desirable. But in games of bargaining that need not be true, because freedom to act can simply become freedom to concede to the other's demands. Committing yourself to a firm final offer leaves the other party the last chance to avoid a mutually disastrous breakdown, and this can get you a better deal. But a mere verbal declaration of firmness may not be credible. Devising actions to make one's commitments credible is one of the finer arts in the realm of strategic games. Members of a labor union send their leaders into wage bargaining with firm instructions or mandates that tie their hands, thereby making it credible that they will not accept a lower offer. The executive branch of the U.S. government engaged in international negotiations on trade or related matters can credibly take a firm stance by pointing out that the Congress would not ratify anything less. And a child is more likely to get the sweet or toy it wants if it is crying too loudly to hear your reasoned explanations of why it should not have it.
Thomas Schelling pioneered the study of credible commitments, and other more complex "strategic moves" like threats and promises. This has found many applications in diplomacy and war, which, as military strategist Karl von Clausewitz told us long ago, are two sides of the same strategic coin.
Information and Incentives
Suppose you have just graduated with a major in computer science, and have an idea for a totally new "killer app" that will integrate PCs, cell phones, and TV sets to create a new medium. The profit potential is immense. You go to venture capitalists for finance to develop and market your idea. How do they know that the potential is as high as you claim it to be? The idea is too new for them to judge it independently. You have no track record, and might be a complete charlatan who will use the money to live high for a few years and then disappear. One way for them to test your own belief in your idea is to see how much of your own money you are willing to risk in the project. Anyone can talk a good game; if you are willing to put enough of your money where your mouth is, that is a credible signal of your own true valuation of your idea.
This is a game where the players have different information; you know the true potential of your idea much better than does your prospective financier. In such games, actions that reveal or conceal information play crucial roles. The field of "information economics" has clarified many previously puzzling features of corporate governance and industrial organization, and has proved equally useful in political science, studies of contract and tort law, and even biology. The award of the Nobel Memorial Prize in 2001 to its pioneers, George Akerlof, Michael Spence, and Joseph Stiglitz, testifies to its importance. What has enabled information economics to burgeon in the last twenty years is the parallel development of concepts and techniques in game theory.
Aligning Interests, Avoiding Enrons
A related application in business economics is the design of incentive schemes. Modern corporations are owned by numerous shareholders, who do not personally supervise the operations of the companies. How can they make sure that the workers and managers will make the appropriate efforts to maximize shareholder value? They can hire supervisors to watch over workers, and managers to watch over supervisors. But all such monitoring is imperfect: the time on the job is easily monitored, but the quality of effort is very difficult to observe and judge. And there remains the problem of who will watch over the upper-level management. Hence the importance of compensation schemes that align the interests of the workers and managers with those of the shareholders. Game theory and information economics have given us valuable insights into these issues. Of course we do not have perfect solutions; for example, we are just discovering how top management can manipulate and distort the performance measures to increase their own compensation while hurting shareholders and workers alike. This is a game where shareholders and the government need to find and use better counterstrategies.
From Intuition to Prediction
While reading these examples, you probably thought that many of the lessons of game theory are obvious. If you have had some experience of playing similar games, you have probably intuited good strategies for them. What game theory does is to unify and systematize such intuitions. Then the general principles extend the intuitions across many related situations, and the calculation of good strategies for new games is simplified. It is no bad thing if an idea seems obvious when it is properly formulated and explained; on the contrary, a science or theory that takes simple ideas and brings out their full power and scope is all the more valuable for that.
In conclusion, I offer some suggestions for further reading to those whose appetites are whetted by my sampler of examples.
General interest:
Dixit, Avinash, and Barry Nalebuff. Thinking Strategically: the Competitive Edge in Business, Politics, and Everyday Life. New York: W.W. Norton, 1991.
Schelling, Thomas. The Strategy of Conflict. Revised edition. Cambridge: Harvard University Press, 1980.
Elementary textbook:
Dixit, Avinash, and Susan Skeath. Games of Strategy. New York: W.W. Norton, 1999.
Advanced textbooks:
Fudenberg, Drew, and Jean Tirole. Game Theory. Cambridge, Massachusetts: MIT Press, 1991.
Myerson, Roger . Game Theory: Analysis of Conflict. Cambridge, Massachusetts: Harvard University Press, 1991.
Business applications:
Brandenberger, Adam, and Barry Nalebuff. Co-opetition. New York: Doubleday, 1996.
McMillan, John. Games, Strategies, and Managers. Reprint. New York: Oxford University Press, 1996.
Political science applications:
Brams, Steven. Rational Politics: Decisions, Games, and Strategy. Reprint. Boston: Academic Press, 1989.
Ordeshook, Peter, A Political Theory Primer. New York: Routledge, 1992.
Applications to law:
Baird, Douglas, Gertner, Robert, and Randal Picker. Game Theory and the Law. Cambridge, Massachusetts: Harvard University Press, 1994.
Biology:
Maynard Smith, John. Evolution and the Theory of Games. Cambridge, England: Cambridge University Press, 1982.
Classic books about game theory:
Luce, R. Duncan, and Howard Raiffa. Games and Decisions: Introduction and Critical Survey. New York: Wiley, 1957.
Von Neumann, John, and Oskar Morgenstern. Theory of Games and Economic Behavior. Second edition. Princeton, New Jersey: Princeton University Press, 1947.
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Game theory is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. In the 21st century, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players. The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty.
Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of 2014, with the Nobel Memorial Prize in Economic Sciences going to game theorist Jean Tirole, eleven game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory.
Contents
- 1History
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2Game types
- 2.1Cooperative / non-cooperative
- 2.2Symmetric / asymmetric
- 2.3Zero-sum / non-zero-sum
- 2.4Simultaneous / sequential
- 2.5Perfect information and imperfect information
- 2.6Combinatorial games
- 2.7Infinitely long games
- 2.8Discrete and continuous games
- 2.9Differential games
- 2.10Evolutionary game theory
- 2.11Stochastic outcomes (and relation to other fields)
- 2.12Metagames
- 2.13Pooling games
- 2.14Mean field game theory
- 3Representation of games
- 4General and applied uses
- 5In popular culture
- 6See also
- 7Notes
- 8References and further reading
- 9External links
