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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.
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which an advantage that is won by one of two sides is lost by the other. If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero. Thus, cutting a cake, where taking a more significant piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally. Other examples of zero-sum games in daily life include games like poker, chess, and bridge where one person gains and another person lose, which result in a zero-net benefit for every player. In the markets and financial instruments, futures contracts and options are zero-sum games as well. Nevertheless, the situation like the stock market etc. is not a zero-sum game because investors could gain profit or loss from share price influences by profit forecasts or economic outlooks rather than gain profit from other investors lost.
In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players and no player has anything to gain by changing only their own strategy. The principle of Nash equilibrium dates back to the time of Cournot, who applied it to competing firms choosing outputs.
In the card game of poker, a bluff is a bet or raise made with a hand which is not thought to be the best hand. To bluff is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the bluffer. By extension, the phrase "calling somebody's bluff" is often used outside the context of poker to describe situations where one person demands that another proves a claim, or proves that they are not being deceptive.
In economics and game theory, a participant is considered to have superrationality if they have perfect rationality but assume that all other players are superrational too and that a superrational individual will always come up with the same strategy as any other superrational thinker when facing the same problem. Applying this definition, a superrational player playing against a superrational opponent in a prisoner's dilemma will cooperate while a rationally self-interested player would defect.
The year 1713 in science and technology involved some significant events.
The St. Petersburg paradox or St. Petersburg lottery is a paradox related to probability and decision theory in economics. It is based on a theoretical lottery game that leads to a random variable with infinite expected value but nevertheless seems to be worth only a very small amount to the participants. The St. Petersburg paradox is a situation where a naive decision criterion which takes only the expected value into account predicts a course of action that presumably no actual person would be willing to take. Several resolutions to the paradox have been proposed.
In game theory, the best response is the strategy which produces the most favorable outcome for a player, taking other players' strategies as given. The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other players' strategies.
A coordination game is a type of simultaneous game found in game theory. Players are faced with multiple pure strategy Nash equilibria where they choose the same or corresponding strategies. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. A simple example of this is represented in figure 1
Kuhn poker is an extremely simplified form of poker developed by Harold W. Kuhn as a simple model zero-sum two-player imperfect-information game, amenable to a complete game-theoretic analysis. In Kuhn poker, the deck includes only three playing cards, for example a King, Queen, and Jack. One card is dealt to each player, which may place bets similarly to a standard poker. If both players bet or both players pass, the player with the higher card wins, otherwise, the betting player wins.
In game theory, the centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round, but after an additional switch the potential payoff will be higher. Therefore, although at each round a player has an incentive to take the pot, it would be better for them to wait. Although the traditional centipede game had a limit of 100 rounds, any game with this structure but a different number of rounds is called a centipede game.
In game theory, more descriptively known as "interactive decision theory", a player's strategy is any of the options which he or she chooses in a setting where the outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines.
In game theory, battle of the sexes (BoS) is a two-player coordination game, however the game also involves elements of conflict. The traditional method of BOS was to use male and female players, however some authors refer to the game as Bach or Stravinsky and designate the players simply as Player 1 and Player 2, rather than assigning sex.
In game theory, a Bayesian game is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.
In game theory, rationalizability is a solution concept. The general idea is to provide the weakest constraints on players while still requiring that players are rational and this rationality is common knowledge among the players. It is more permissive than Nash equilibrium. Both require that players respond optimally to some belief about their opponents' actions, but Nash equilibrium requires that these beliefs be correct while rationalizability does not. Rationalizability was first defined, independently, by Bernheim (1984) and Pearce (1984).
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their private observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from their strategy, the distribution from which the signals are drawn is called a correlated equilibrium.
In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. The theorem aims to justify a puzzling aspect of mixed strategy Nash equilibria: that each player is wholly indifferent amongst each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game. When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.
Essay d'analyse sur les jeux de hazard is a book on combinatorics and mathematical probability written by Pierre Remond de Montmort and published in 1708. The work applied ideas from combinatorics and probability to analyse various games of chances popular during the time. This book was mainly influenced by Huygens' treatise De ratiociniis in ludo aleae and the knowledge of the fact that Jakob Bernoulli had written an unfinished work in probability. The work was intended to re-create the yet unpublished work of Jakob Bernoulli called Ars Conjectandi. The work greatly influenced the thinking of Montmort's contemporary, Abraham De Moivre.
