The branch of mathematics that analyses a range of problems involving decision-making; also called games theory. Although often illustrated by games of chance, there are important applications to military strategy, economics, ecology, and other applied sciences. Game theory was developed in the 20th-c, principally by French mathematician Emile Borel (18711956) and US mathematician John Von Neumann. Games involving one, two, or more players are distinguished, as in patience, chess, and roulette respectively. Game theory analyses the strategies each player uses to maximize the chance of winning, and attempts to predict outcomes.
First developed as a tool for understanding economic behavior and then by the RAND Corporation to define nuclear strategies, game theory is now used in many diverse academic fields, ranging from biology and psychology to sociology and philosophy. Because of games like the prisoner's dilemma, in which rational self-interest hurts everyone, game theory has been used in political science, ethics and philosophy.In addition to its academic interest, game theory has received attention in popular culture. Several game shows have adopted game theoretic situations, including Friend or Foe? and to some extent Survivor.
Although similar to decision theory, game theory studies decisions that are made in an environment where various players interact.
Representation of games
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.
See also List of games in game theory.
Normal form
|
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 a matrix which shows the players, strategies, and payoffs (see the example to the right).
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.
Extensive form
The extensive form can be used to formalize games with some important order.
In the game pictured here, there are two players.
The extensive form can also capture simultaneous-move games.
Types of games
Symmetric and asymmetric
| 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. Some scholars would consider certain asymmetric games as examples of these games as well.
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.
Zero sum and non-zero sum
| A | B | |
| A | –1, 1 | 3, –3 |
| B | 0, 0 | –2, 2 |
| A zero-sum game | ||
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). 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.
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 and sequential
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.
The difference between simultaneous and sequential games is captured in the different representations discussed above.
Perfect information and imperfect information
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.
Infinitely long games
For obvious reasons, games as studied by economists and real-world game players are generally finished in a finite number of moves.
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.
Uses of game theory
Games in one form or another are widely used in many different academic disciplines.
Economics and business
Economists have long used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, oligopolies, social network formation, and voting systems.
The payoffs of the game are generally taken to represent the utility of individual players.
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation. 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. 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.
Some game theorists have turned to evolutionary game theory in order to resolve these worries. Evolutionary game theory includes both biological as well as cultural evolution and also models of individual learning (for example, fictitious play dynamics).
Kesten Green (2005) obtained empirical evidence that game theory experts were no better than novices at forecasting the decisions that protagonists would make in eight real conflicts that involved interaction between the parties.
Normative analysis
| 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.
Applications in Biology
| 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 biology, game theory has been used to understand many different phenomena.
Additionally, biologists have used evolutionary game theory and the ESS to explain the emergence of animal communication (Maynard Smith & The analysis of signaling games and other communication games has provided some insight into the evolution of communication among animals.
Finally, biologists have used the hawk-dove game (also known as chicken) to analyze fighting behavior and territoriality.
Computer science and logic
Game theory has come to play an increasingly important role in logic and in computer science.
Political science
The application of game theory to political science is focused in the overlapping areas of political economy, public choice, positive political theory, and social choice theory.
For early examples of game theory applied to political science, see the work of Anthony Downs.
Philosophy
| Stag | Hare | |
| Stag | 3, 3 | 0, 2 |
| Hare | 2, 0 | 2, 2 |
| Stag hunt | ||
Game theory has been put to several uses in philosophy.
Finally, 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;
Sociology
There are fewer applications of game theory in sociology than in its sister disciplines, economics and political science.
History of game theory
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713.
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's work culminated in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. 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.
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 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 2005, game theorists Thomas Schelling and Robert Aumann won the Bank of Sweden Prize in Economic Sciences. Schelling worked on dynamic models, early examples of evolutionary game theory.
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