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Collective intelligence Collective action Selforganized criticality Herd mentality Phase transition Agentbased 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 Selfreplication Spatial evolutionary biology. Rational choice theory Bounded rationality Irrational behaviour. Game theory is the study of mathematical models of strategic interaction among rational decisionmakers. Originally, it addressed zerosum games , in which each participant's gains or losses are exactly balanced by those of the other participants. Today, 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. Modern game theory began with the idea of mixedstrategy equilibria in twoperson zerosum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixedpoint 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 , cowritten 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 decisionmaking under uncertainty. Game theory was developed extensively in the s by many scholars. It was explicitly applied to biology 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. John Maynard Smith was awarded the Crafoord Prize for his application of game theory to biology. Discussions of twoperson games began long before the rise of modern, mathematical game theory. The first known discussion of game theory occurred in a letter believed to be written in by Charles Waldegrave, an active Jacobite and uncle to James Waldegrave , a British diplomat. One theory postulates Francis Waldegrave as the true correspondent, but this has yet to be proven. 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 nonexistence of mixedstrategy equilibria in finite twoperson zerosum games , a conjecture that was proved false by von Neumann. Game theory did not really 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 twoperson zerosum 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 nplayer, nonzerosum not just 2player zerosum noncooperative 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 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "titfortat" program that cooperates on the first step, then, on subsequent steps, does whatever its opponent did on the previous step. The same winner was also often obtained by natural selection; a fact widely taken to explain cooperation phenomena in evolutionary biology and the social sciences. 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 [11] 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 noncooperative if players cannot form alliances or if all agreements need to be selfenforcing 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 noncooperative game theory which focuses on predicting individual players' actions and payoffs and analyzing Nash equilibria. Cooperative game theory provides a highlevel approach as it describes only the structure, strategies, and payoffs of coalitions, whereas noncooperative game theory also looks at how bargaining procedures will affect the distribution of payoffs within each coalition. As noncooperative game theory is more general, cooperative games can be analyzed through the approach of noncooperative 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 noncooperative 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. Zerosum games are a special case of constantsum games in which choices by players can neither increase nor decrease the available resources. In zerosum 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 zerosum 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 nonzerosum games, because the outcome has net results greater or less than zero. Informally, in nonzerosum games, a gain by one player does not necessarily correspond with a loss by another. Constantsum 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 zerosum 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. 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 imperfectinformation 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 realworld 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 nonnegative 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. 1.3 What is Sociological Theory?, time: 5:00
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