   #copyright

Game theory

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   Game theory is a branch of applied mathematics and economics that
   studies situations where players choose different actions in an attempt
   to maximize their returns. First developed as a tool for understanding
   economic behaviour 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.
   Beginning in the 1970s, game theory has been applied to animal
   behaviour, including species' development by natural selection. 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. Finally, 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.

   Although similar to decision theory, game theory studies decisions that
   are made in an environment where various players interact. In other
   words, game theory studies choice of optimal behaviour when costs and
   benefits of each option are not fixed, but depend upon the choices of
   other individuals.

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. There are two ways of representing games
   that are common in the literature.

   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). Here
   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.

Extensive form

   An extensive form game
   Enlarge
   An extensive form game

   The extensive form can be used to formalize games with some important
   order. Games here are 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. Either a
   dotted line or circle is drawn around two 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).

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. 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.

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). 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.

   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. This need not be perfect
   knowledge about every action of earlier players; it might be very
   little information. 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.

Perfect information and imperfect information

   A game of imperfect information (the dotted line represents ignorance
   on the part of player 2)
   Enlarge
   A game of imperfect information (the dotted line represents ignorance
   on the part of player 2)

   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
   know the strategies and payoffs of the other players but not
   necessarily the actions.

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. 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.

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. 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
   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.

Descriptive

   A three stage Centipede Game
   Enlarge
   A three stage Centipede Game

   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.

   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).

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 behaviour 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. Some scholars believe that this
   demonstrates the failure of game theory as a recommendation for
   behaviour.

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 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.

   Finally, biologists have used the hawk-dove game (also known as
   chicken) to analyze fighting behaviour and territoriality.

Computer science and logic

   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.

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. 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. He shows how the
   political candidates will converge to the ideology preferred by the
   median voter. For more recent examples, see the books by Gene M.
   Grossman and Elhanan Helpman, or David Austen-Smith and Jeffrey S.
   Banks.

   A game-theoretic explanation for the democratic peace is that the
   public and open debate in democracies send clear and reliable
   information regarding the 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.

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. 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).

   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; Sober and Wilson 1999).

Sociology

   There are fewer applications of game theory in sociology than in its
   sister disciplines, economics and political science. A game theoretic
   analysis of interactions among prisoners is conducted by Kaminski
   (2004).

History of game theory

   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
   culminated in the 1944 book The Theory of Games and Economic Behaviour
   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, Reinhard Selten introduced his solution concept of subgame
   perfect equilibria, which further refined the Nash equilibrium (later
   he would introduce trembling hand perfection as well). In 1967, John
   Harsanyi developed the concepts of complete information and Bayesian
   games. He, along with John Nash and Reinhard Selten, won the Bank of
   Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1994.

   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 were introduced and
   analyzed.

   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. Aumann contributed
   more to the equilibrium school, developing an equilibrium coarsening
   correlated equilibrium and developing extensive analysis of the
   assumption of common knowledge.

   Retrieved from " http://en.wikipedia.org/wiki/Game_theory"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
