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In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players' strategies as given. The concept of a best response is central to John Nash's most well-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response to the other player's strategy[1]. 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. ...
In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ...
In Game theory, an outcome is a set of actions or strategies taken by the players, or their payoffs resulting from the actions or strategies taken by all players. ...
John Forbes Nash, Jr. ...
In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...
Best response correspondence
 Fig. 1 - Reaction correspondence for player Y in the Stag Hunt game. Reaction correspondences, also known as best response correspondences, are used in the proof of the existence of mixed strategy Nash equilibria[2][3][4]. Reaction correspondences are not "reaction functions" since functions must only have one value per argument, and many reaction correspondences will be undefined , i.e. a vertical line, for some opponent strategy choice. One constructs a correspondence  , for each player from the set of opponent strategy profiles into the set of the player's strategies. So, for any given set of opponent's strategies σ − i, bi(σ − i) represents player i 's best responses to σ − i. Image File history File links Reaction-correspondence-playery. ...
Image File history File links Reaction-correspondence-playery. ...
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings. ...
A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...
Partial plot of a function f. ...
 Fig. 2 - Reaction correspondence for player XY in the Stag Hunt game. Response correspondences for all 2x2 normal form games can be drawn with a line for each player in a unit square strategy space. Figures 1 to 3 graphs the best response correspondences for the stag hunt game. The dotted line in Fig. 1 shows the optimal probability that player Y plays 'Stag' (in the y-axis), as a function of the probability that player X plays Stag (shown in the x-axis). In Fig.2 the dotted line shows the optimal probability that player X plays 'Stag' (shown in the x-axis), as a function of the probability that player Y plays Stag (shown in the y-axis). Note that Fig.2 plots the independent and response variables in the opposite axes to those normally used, so that it may be superimposed onto the previous graph, to show the Nash equilibria at the points where the two player's best responses agree in Fig.3. Image File history File links Reaction-correspondence-playerx. ...
Image File history File links Reaction-correspondence-playerx. ...
In game theory, normal form is a way of describing a game. ...
A line, or straight line, can be described as an (infinitely) thin, (infinitely) long, perfectly straight curve (the term curve in mathematics includes straight curves). In Euclidean geometry, exactly one line can be found that passes through any two points. ...
The unit square in a Cartesian coordinate system with coordinates (x,y) is defined as the square consisting of the points where both x and y lie in the unit interval from 0 to 1. ...
Space has been an interest for philosophers and scientists for much of human history. ...
In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ...
In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that...
Informally, probable is one of several words applied to uncertain events or knowledge, being closely related in meaning to likely, risky, hazardous, and doubtful. ...
In experimental design, a dependent variable is a variable dependent on another variable (called the independent variable). ...
In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...
 Fig.3 - Reaction correspondence for both players in the Stag Hunt game. Nash equilibria shown with points, where the two player's correspondences agree, ie. cross There are three distinctive reaction correspondence shapes, one for each of the three types of symmetric 2x2 games: coordination games, discoordination games and games with dominated strategies (the trivial fourth case in which payoffs are always equal for both moves is not really a game theoretical problem). Any payoff symmetric 2x2 game will take one of these three forms. Image File history File links Reaction-correspondence-stag-hunt. ...
Image File history File links Reaction-correspondence-stag-hunt. ...
In game theory, 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. ...
Coordination games Games in which players score highest when both players choose the same strategy, such as the Stag hunt and Battle of the sexes are called coordination games. These games have reaction correspondences of the same shape as Fig.3, where there is one Nash equilibrium in the bottom left corner, another in the top right, and a mixing Nash somewhere along the diagonal between the other two. In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ...
The Battle of the Sexes is a two player game used in game theory. ...
In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...
Discoordination games Games such as the Game of chicken and Hawk-dove game in which players score highest when they choose opposite strategies, ie discoordinate, are called discoordination games. They have reaction correspondences (Fig.4) that cross in the opposite direction to coordination games, with three Nash equilibria, one in each of the top left and bottom right corners, where one player chooses one strategy, the other player chooses the opposite strategy. The third Nash equilibrium is a mixed strategy which lies along the diagonal from the bottom left to top right corners. If the players do not know which one of them is which, then the mixed Nash is an evolutionarily stable strategy (ESS), as play is confined to the bottom left to top right diagonal line. Otherwise an uncorrelated asymmetry is said to exist, and the corner Nash equilibria are ESSes. The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ...
The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ...
It has been suggested that Peace war game be merged into this article or section. ...
A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...
In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ...
In game theory an uncorrelated asymmetry is an informational asymmetry in a game which is otherwise symmetrical. ...
 Fig.4 - Reaction correspondence for both players in the Hawk-Dove game. Nash equilibria shown with points, where the two player's correspondences agree, ie. cross Image File history File links Reaction-correspondence-hawk-dove. ...
Image File history File links Reaction-correspondence-hawk-dove. ...
Games with dominated strategies
 Fig. 5 - Reaction correspondence for a game with a dominated strategy. Games with dominated strategies have reaction correspondences which only cross at one point, which will be in either the bottom left, or top right corner in payoff symmetric 2x2 games. For instance, in the single-play Prisoner's dilemma, the Cooperate move is not optimal for any probability of opponent Cooperation. Fig.5 shows the reaction correspondence for such a game, where the dimensions are "Probability play Cooperate", the Nash equilibrium is in the lower left corner where neither player plays Cooperate. If the dimensions were defined as "Probability play Defect", then both players best response curves would be 1 for all opponent strategy probabilities and the reaction correspondences would cross (and form a Nash equilibrium) at the top right corner. Image File history File links Reaction-correspondence-dominated. ...
Image File history File links Reaction-correspondence-dominated. ...
In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ...
Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with...
Other (payoff asymmetric) games A wider range of reaction correspondences shapes is possible in 2x2 games with payoff asymmetries. For each player there are five possible best response shapes, shown in Fig.6. From left to right these are: dominated strategy (always play 2), dominated strategy (always play 1), rising (play strategy 2 if probability that the other player plays 2 is above threshold), falling (play strategy 1 if probability that the other player plays 2 is above threshold), and indifferent (both strategies play equally well under all conditions).
 Fig. 6 - The five possible reaction correspondences for a player in a 2x2 game., The axes are assumed to show the probability that the player plays their strategy 1. From left to right: A) Always play 2, strategy 1 is dominated, B) Always play 1, strategy 2 is dominated, C) Strategy 1 best when opponent plays his strategy 1 and 2 best when opponent plays his 2, D) Strategy 1 best when opponent plays his strategy 2 and 2 best when opponent plays his 1, E) Both strategies play equally well no matter what the opponent plays. While there are only four possible types of payoff symmetric 2x2 games (of which one is trivial), the five different best response curves per player allow for a larger number of payoff asymmetric game types. Many of these are not truly different from each other. The dimensions may be redefined (exchange names of strategies 1 and 2) to produce symmetrical games which are logically identical. Image File history File links Download high-resolution version (851x157, 10 KB) All five possible reaction correspondences for a player in a 2x2 game - image created by Pete. ...
Image File history File links Download high-resolution version (851x157, 10 KB) All five possible reaction correspondences for a player in a 2x2 game - image created by Pete. ...
Matching Pennies One well-known game with payoff asymmetries is the Matching pennies game. In this game one player, the row player - graphed on the y dimension, wins if the players coordinate (both choose heads or both choose tails) while the other player, the column player - shown in the x-axis, wins if the players discoordinate. Player Y's reaction correspondence is that of a coordination game, while that of player X is a discoordination game. The mixing Nash equilibrium is an ESS. Matching Pennies is the name for a simple example game used in game theory. ...
In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ...
 Fig. 7 - Reaction correspondences for players in the matching pennies game. The leftmost mapping is for the coordinating player, the middle shows the mapping for the discoordinating player. The sole Nash equilibrium, an evolutionarily stable strategy is shown in the right hand graph. Image File history File links Reaction-correspondence-matching-pennies. ...
Image File history File links Reaction-correspondence-matching-pennies. ...
Matching Pennies is the name for a simple example game used in game theory. ...
In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ...
Best response dynamics In evolutionary game theory, best response dynamics represents a class of strategy updating rules, where players strategies in the next round are determined by their best responses to some subset of the population. Some examples include: Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ...
- In a large population model, players choose their next action probabilistically based on which strategies are best responses to the population as a whole.
- In a spatial model, players choose (in the next round) the action that is the best response to all of their neighbors (see Elison 1993).
Importantly, in these models players only choose the best response on the next round that would give them the highest payoff on the next round. Players do not consider the effect that choosing a strategy on the next round would have on future play in the game. This constraint results in the dynamical rule often being called myopic best response.
Smoothed best response
 A BR correspondence (black) and smoothed BR functions (colors) Instead of best response correspondences, some models use a smoothed best response functions. These functions are similar to the best response correspondence, except that the function does not "jump" from one pure strategy to another. The difference is illustrated in the diagram to the right, where black represents the best response correspondence and the other colors each represent different smoothed best response functions. In standard best response correspondences, even the slightest benefit to one action will result in the individual playing that action with probability 1. In smoothed best response as the difference between two actions decreases the individual's play approaches 50:50. Image File history File links SmoothBRColor. ...
Image File history File links SmoothBRColor. ...
There are many functions that represent smoothed best repsonse functions. The functions illustrated here are several variations on the following function:
 Where  represents the expectation of actions 1 and 2 and gamma represents a paramenter which determines the degree to which the function deviates from the correspondence.
There are several advantages to using smoothed best response, both theoretical and empirical. First, it corresponds to psychological experiments; when individuals are close to indifferent between two actions they appear to choose more or less at random. Second, the play of individuals is determined in all cases, since the it is a function and not a correspondence. Finally, using smoothed best response with some learning rules (as in Fictitious play) can result in players learning to play mixed strategy Nash equilibria. Partial plot of a function f. ...
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings. ...
In game theory, fictitious play is a learning rule first introduced by G.W. Brown (1951). ...
A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...
In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...
References - ^ Nash, John (1950) Equilibrium points in n-person games Proceedings of the National Academy of the USA 36:48-49.
- ^ Fudenberg, D. & Tirole, J. (1991) Game Theory MIT Press, Cambridge MA. (section 1.3.1)
- ^ Gibbons, R. (1992) A primer in game theory (pp. 33-49) Harvester-Wheatsheaf. (section 1.3.B)
- ^ Osborne, M.J. and Rubenstein, A. (1994) A course in game theory MIT Press, Cambridge MA. (section 2.2)
- Ellison, G. (1993) "Learning, Local Interaction, and Coordination" Econometrica 61: 1047-1071
- Fudenberg, D. and D.K. Levine (1998) The Theory of Learning in Games Cambridge: MIT Press.
- Fudenberg, D. and Tirole, J. (1991) Games Theory Cambridge: MIT Press.
- Gibbons, R. (1992) A primer in game theory (pp. 33-49) Harvester-Wheatsheaf.
- Young, H.P. (2005) Strategic Learning and Its Limits Oxford: Oxford University Press.
John Forbes Nash, Jr. ...
Jean Tirole (born 9 August 1953) is a notable contemporary french economist, author of many works in economics, scientific director of the Industrial Economics Institute in Toulouse. ...
Ariel Rubenstein (born April 13, 1951) is an economist who works in game theory. ...
Jean Tirole (born 9 August 1953) is a notable contemporary french economist, author of many works in economics, scientific director of the Industrial Economics Institute in Toulouse. ...
| | Topics in game theory | | Definitions 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. ...
| Normal form game · Extensive form game · Cooperative game · Information set · Preference In game theory, normal form is a way of describing a game. ...
It has been suggested that Game tree be merged into this article or section. ...
A cooperative game is a game where groups of players (coalitions) may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. ...
In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed so far. ...
Preference (or taste) is a concept, used in the social sciences, particularly economics. ...
| | Equilibrium concepts In economics, economic equilibrium often refers to an equilibrium in a market that clears: this is the case where a market for a product has attained the price where the amount supplied of a certain product equals the quantity demanded. ...
In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ...
| Nash equilibrium · Subgame perfection · Bayes-Nash · Trembling hand · Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...
Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ...
In game theory, a Bayesian game is one in which information about characteristics of the other players (i. ...
The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ...
In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. ...
Sequential equilibrium is a refinement of Nash Equilibrium for extensive form games due to David M. Kreps and Robert Wilson. ...
Quasi-perfect equilibrium is a refinement of Nash Equilibrium for extensive form games due to Eric van Damme. ...
In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ...
| | Strategies In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ...
| Dominant strategies · Mixed strategy · Grim trigger · Tit for Tat In game theory, dominance (also called strategic dominance) occurs when one strategy is better than another strategy for one player, no matter how that players opponents may play. ...
A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ...
Grim Trigger is a trigger strategy in game theory for a repeated game, such as an iterated prisoners dilemma. ...
Tit for Tat is a highly-effective strategy in game theory for the iterated prisoners dilemma. ...
| | Classes of games | Symmetric game · Perfect information · Dynamic game · Repeated game · Signaling game · Cheap talk · Zero-sum game · Mechanism design In game theory, 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. ...
Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ...
In game theory, a sequential game is a game where one player chooses his action before the others chooses theirs. ...
In game theory, a repeated game (or iterated game) is an extensive form game which consists in some number of repetitions of some base game (called a stage game). ...
Signaling games are dynamic games with two players, the sender (S) and the receiver (R). ...
Cheap Talk is a term used in Game Theory for pre-play communication which carries no cost. ...
Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ...
Mechanism design is a sub-field of game theory. ...
| | Games Game theory studies strategic interaction between individuals in situations called games. ...
| Prisoner's dilemma · Chicken · Battle of the sexes · Stag hunt · Ultimatum game · Coordination game · Matching pennies · Minority game · Rock, Paper, Scissors · Pirate game · Dictator game Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with...
The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ...
The Battle of the Sexes is a two player game used in game theory. ...
In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ...
The Ultimatum game is an experimental economics game in which two parties interact anonymously and only once, so reciprocation is not an issue. ...
In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ...
Matching Pennies is the name for a simple example game used in game theory. ...
Minority Game is a game proposed by Yi-Cheng Zhang and Damien Challet from the University of Fribourg. ...
Rock, Paper, Scissors chart Listen to this article · (info) This audio file was created from an article revision dated 2006-07-13, and does not reflect subsequent edits to the article. ...
The Pirate Game is a simple mathematical game. ...
The dictator game is a very simple game in experimental economics, similar to the ultimatum game. ...
| | Theorems | Minimax theorem · Purification theorems · Folk theorem · Revelation principle Minimax is a method in decision theory for minimizing the expected maximum loss. ...
In game theory, the purification theorem was contributed by Nobel laurate John Harsanyi in 1973[1]. 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...
In game theory, folk theorems are a class of theorems which imply that in repeated games, any outcome is a feasible solution concept, if under that outcome the players minimax conditions are satisfied. ...
The revelation principle of economics can be stated as, To any equilibrium of a game of incomplete information, there corresponds an associated revelation mechanism that has an equilibrium where the players truthfully report their types. ...
| | Related topics | Mathematics · Economics · Behavioral economics · Evolutionary game theory · Population genetics · Behavioral ecology · Adaptive dynamics · List of game theorists Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
Face-to-face trading interactions among on the New York Stock Exchange trading floor Economics, as a social science, studies human choice behavior and how it affects the production, distribution, and consumption of scarce resources. ...
Nobel Prize in Economics winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ...
Evolutionary game theory (EGT) is the application of game theory in evolutionary biology. ...
Population genetics is the study of the distribution of and change in allele frequencies under the influence of the four evolutionary forces: natural selection, genetic drift, mutation, and migration. ...
Behavioral ecology is the study of the ecological and evolutionary basis for animal behavior, and the roles of behavior in enabling an animal to adapt to its environment (both intrinsic and extrinsic). ...
Adaptive Dynamics is a set of techniques for studying long-term phenotypical evolution developed during the 1990s. ...
This is a list of notable economists, mathematicians, political scientists, and computer scientists whose work has added substantially to the field of game theory. ...
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