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Necessary cookies are absolutely essential for the website to function properly. 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.
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 it would thus be optimal to have all games expressed under a non-cooperative framework, 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.
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 games are a special case of constant-sum games in which choices by players can neither increase nor decrease the available resources.
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 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 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 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 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.
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.
Most games studied in game theory are imperfect-information games. Many card games are games of imperfect information, such as poker and bridge.
Games of incomplete information can be reduced, however, to games of imperfect information by introducing " moves by nature ".
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.
There is no unified theory addressing combinatorial elements in games. There are, however, mathematical tools that can solve particular problems and answer general questions.
Games of perfect information have been studied in combinatorial game theory , which has developed novel representations, e. These methods address games with higher combinatorial complexity than those usually considered in traditional or "economic" game theory.
A related field of study, drawing from computational complexity theory , is game complexity , which is concerned with estimating the computational difficulty of finding optimal strategies.
Research in artificial intelligence has addressed both perfect and imperfect information games that have very complex combinatorial structures like chess, go, or backgammon for which no provable optimal strategies have been found.
The practical solutions involve computational heuristics, like alpha—beta pruning or use of artificial neural networks trained by reinforcement learning , which make games more tractable in computing practice.
Games, as studied by economists and real-world game players, are generally finished in finitely many 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 the best way to play such a game, but whether one player has a winning strategy.
The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
Much of game theory is concerned with finite, discrete games that have a finite number of players, moves, events, outcomes, etc. Many concepts can be extended, however.
Continuous games allow players to choose a strategy from a continuous strategy set. For instance, Cournot competition is typically modeled with players' strategies being any non-negative quantities, including fractional quantities.
Differential games such as the continuous pursuit and evasion game are continuous games where the evolution of the players' state variables is governed by differential equations.
The problem of finding an optimal strategy in a differential game is closely related to the optimal control theory.
In particular, there are two types of strategies: the open-loop strategies are found using the Pontryagin maximum principle while the closed-loop strategies are found using Bellman's Dynamic Programming method.
A particular case of differential games are the games with a random time horizon. Therefore, the players maximize the mathematical expectation of the cost function.
It was shown that the modified optimization problem can be reformulated as a discounted differential game over an infinite time interval.
Evolutionary game theory studies players who adjust their strategies over time according to rules that are not necessarily rational or farsighted.
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