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Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour. Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics.
Reaction—diffusion systems Partial differential equations Dissipative structures Percolation Cellular automata Spatial ecology Self-replication. Rational choice theory Bounded rationality. Game theory is the study of mathematical models of strategic interactions among rational agents.
Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants.
In the 21st century, game theory applies to a wide range of behavioral relations ; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game 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 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 s by many scholars. It was explicitly applied to evolution in the s, although similar developments go back at least as far as the s.
Game theory has been widely recognized as an important tool in many fields. Wilson , fifteen game theorists have won the economics Nobel Prize.
John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory. Discussions on the mathematics of games began long before the rise of modern mathematical game theory. Cardano 's work on games of chance in Liber de ludo aleae Book on Games of Chance , which was written around but published posthumously in , formulated some of the field's basic ideas.
In , a letter attributed to Charles Waldegrave analyzed a game called "le Her". He was an active Jacobite and uncle to James Waldegrave , a British diplomat. This paved the way for more general theorems. In , the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem. Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games , a conjecture that was proved false by von Neumann.
Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in Von Neumann's work in game theory culminated in this book. This foundational work contains the method for finding mutually consistent solutions for two-person zero-sum games. Subsequent work focused primarily on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In , the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M. RAND pursued the studies because of possible applications to global nuclear strategy. Nash proved that every finite n-player, non-zero-sum not just two-player zero-sum non-cooperative game has what is now known as a Nash equilibrium in mixed strategies. Game theory experienced a flurry of activity in the s, during which the concepts of the core , the extensive form game , fictitious play , repeated games , and the Shapley value were developed.
The s also saw the first applications of game theory to philosophy and political science. In , 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 the s, game theory was extensively applied in biology , largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy.
In addition, the concepts of correlated equilibrium , trembling hand perfection, and common knowledge [a] were introduced and analyzed. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, introducing equilibrium coarsening and correlated equilibria, and developing an extensive formal analysis of the assumption of common knowledge and of its consequences.
Myerson's contributions include the notion of proper equilibrium , and an important graduate text: Game Theory, Analysis of Conflict. In , Alvin E. Roth and Lloyd S. Shapley were awarded the Nobel Prize in Economics "for the theory of stable allocations and the practice of market design".
In , the Nobel went to game theorist Jean Tirole. A game is cooperative if the players are able to form binding commitments externally enforced e. A game is non-cooperative if players cannot form alliances or if all agreements need to be self-enforcing e.
Cooperative games are often analyzed through the framework of cooperative game theory , which focuses on predicting which coalitions will form, the joint actions that groups take, and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria. The lack of formal negotiation leads to the deterioration of public goods through over-use and under provision that stems from private incentives.
Cooperative game theory provides a high-level approach as it describes only the structure, strategies, and payoffs of coalitions, whereas non-cooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As non-cooperative game theory is more general, cooperative games can be analyzed through the approach of non-cooperative game theory the converse does not hold provided that sufficient assumptions are made to encompass all the possible strategies available to players due to the possibility of external enforcement of cooperation.
While using a single theory may be desirable, in many instances insufficient information is available to accurately model the formal procedures available during the strategic bargaining process, or the resulting model would be too complex to offer a practical tool in the real world. In such cases, cooperative game theory provides a simplified approach that allows analysis of the game at large without having to make any assumption about bargaining powers.
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. That is, if the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. The standard representations of chicken , the prisoner's dilemma , and the stag hunt are all symmetric games.
Some [ who? However, the most common payoffs for each of these games are symmetric. The 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 in this section's graphic is asymmetric despite having identical strategy sets for both players. Zero-sum games more generally, constant-sum games are games in which choices by players can neither increase nor decrease the available resources.
In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, always adds to zero more informally, a player benefits only at the equal 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 famed prisoner's dilemma are non-zero-sum games, because the outcome has 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. Constant-sum games correspond to activities like theft and gambling, but not to the fundamental economic situation in which there are potential gains from trade.
It is possible to transform any constant-sum game into a possibly asymmetric zero-sum game by adding a dummy player often called "the board" whose losses compensate the players' net winnings. Simultaneous games are games where both players move simultaneously, or instead 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 information about every action of earlier players; it might be very little knowledge. For instance, a player may know that an earlier player did not perform one particular action, while they do 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.
Often, normal form is used to represent simultaneous games, while extensive form is used to represent sequential ones. The transformation of extensive to normal form is one way, meaning that multiple extensive form games correspond to the same normal form. Consequently, notions of equilibrium for simultaneous games are insufficient for reasoning about sequential games; see subgame perfection. The Cournot competition model involves players choosing quantity of a homogenous product to produce independently and simultaneously, where marginal cost can be different for each firm and the firm's payoff is profit.
The production costs are public information and the firm aims to find their profit-maximising quantity based on what they believe the other firm will produce and behave like monopolies.
In this game firms want to produce at the monopoly quantity but there is a high incentive to deviate and produce more, which decreases the market-clearing price. The Bertrand competition , assumes homogenous products and a constant marginal cost and players choose the prices.
Therefore, the firms have an incentive to deviate from the equilibrium because a homogenous product with a lower price will gain all of the market share, known as a cost advantage. An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players, at every move in the game, know the moves previously made by all other players.
In reality, this can be applied to firms and consumers having information about price and quality of all the available goods in a market. Many card games are games of imperfect information, such as poker and bridge. One of the assumptions of the Nash equilibrium is that every player has correct beliefs about the actions of the other players.
However, there are many situations in game theory where participants do not fully understand the characteristics of their opponents. Negotiators may be unaware of their opponent's valuation of the object of negotiation, companies may be unaware of their opponent's cost functions, combatants may be unaware of their opponent's strengths, and jurors may be unaware of their colleague's interpretation of the evidence at trial.
In some cases, participants may know the character of their opponent well, but may not know how well their opponent knows his or her own character. Bayesian game means a strategic game with incomplete information. For a strategic game, decision makers are players, and every player has a group of actions. A core part of the imperfect information specification is the set of states. Every state completely describes a collection of characteristics relevant to the player such as their preferences and details about them.
There must be a state for every set of features that some player believes may exist. For example, where Player 1 is unsure whether Player 2 would rather date her or get away from her, while Player 2 understands Player 1's preferences as before. Due to the probability involved, the analysis of this situation requires to understand the player's preference for the draw, even though people are only interested in pure strategic equilibrium.
Games in which the difficulty of finding an optimal strategy stems from the multiplicity of possible moves are called combinatorial games. Examples include chess and Go. Games that involve imperfect information may also have a strong combinatorial character, for instance backgammon.
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Collective intelligence Collective action Self-organized criticality Herd mentality Phase transition Agent-based modelling Synchronization Ant colony optimization Particle swarm optimization Swarm behaviour. Evolutionary computation Genetic algorithms Genetic programming Artificial life Machine learning Evolutionary developmental biology Artificial intelligence Evolutionary robotics. Reaction—diffusion systems Partial differential equations Dissipative structures Percolation Cellular automata Spatial ecology Self-replication. Rational choice theory Bounded rationality.
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