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Nash Equilibrium: Games
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In a Nash equilibrium, players' rationality is mutual knowledge. However, both intuition and experimental evidence suggest that players do not know for sure the rationality of opponents. This paper proposes a new equilibrium concept, cautious equilibrium, that generalizes Nash equilibrium in terms of preferences in two person strategic games. In a cautious equilibrium, players do not necessarily know the rationality of opponents, but they view rationality as infinitely more likely than irrationality. For suitable models of preference, cautious equilibrium predicts that a player might take a "cautious" strategy that is not a best response in any Nash equilibrium.
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The Nash equilibrium concept has ... shed light on animal behavior. In the 1970s, biologists John Maynard Smith, of the University of Sussex in England, and the late George Price, then of the Galton Laboratories in London, used game theory to study ritual courtship battles within species (9). They modeled these conflicts by a game in which each player can choose to be either a hawk, who escalates the conflict, or a dove, who backs down if his opponent starts playing rough. The researchers found that in equilibrium, the population will consist of a mix of hawks and doves; the percentage of each type depends on how capable of hurting or killing each other the particular species is. Maynard Smith and Price also showed that if type is inherited from one generation to the next, the population should converge to its equilibrium distribution of doves and hawks.
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The Nash equilibrium is a concept in game theory originated by John Nash, who was awarded The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, effectively the Nobel Prize in economics, for his work in the area. It serves to define a kind of "optimum" strategy for games where no such optimum was previously defined. A basic definition is this: If there is a set of strategies for a game with the property that no player can benefit by changing his strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute a Nash equilibrium.
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In game theory, a Manipulated Nash equilibrium or MAPNASH is a refinement of subgame perfect equilibrium used in dynamic games of imperfect information. Informally, a strategy set is a MAPNASH of a game if it would be a subgame perfect equilibrium of the game if the game had perfect information. MAPNASH were first suggested by Amershi, Sadanand, and Sadanand (1988) and has been discussed in several papers since.
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The short definition provided in Non-Cooperative Game states that the Nash equilibrium is the state of such a game where players cannot change their profit by changing their strategy. John Nash published his work on Non-Cooperative Games in 1951. He expands the theory on zero-sum games provided by Von Neumann and Morgenstern who published their work in "Theory of Games and Economic Behavior". According to Nash, this book analyzes cooperative games, where players are allowed to form coalitions and make decisions with knowledge of other player's strategies (Nash, 1951). Nash's expansion focuses on the Non-Cooperative Game in which players make their decision without knowledge of other player's strategies.
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The most commonly used solution concept in game theory is that of (bayesian)Nash Equilibrium (NE).However, except under fairly restrictive assumptions whose empirical validity often is questionable, many games cannot be solved analytically for NE solutions. As an alternative to NE Armantier, Florens and Richard introduce the concept of Constrained Strategic Equilibrium (hereafter CSE). Essentially, they propose to restrict attention to appropriate subsets of strategies, typically indexed by an auxiliary parameter vector, and to search for an equilibrium solution within such subsets. The authors show that CSE offer a major computational advantage, and they provide a powerful algorithm based upon Monte Carlo simulations to determine the CSE numerically. The concept of CSE appeared to be relevant under two scenarios: the first one is directly related to the general notion of 'bounded rationality' and more specifically to the concept of Rules of Thumb ; in the second scenario, one would use the computational advantage of the CSE with the intent to approximate an analytically untractable NE solution. The objective of the present essay is to establish conditions under which a sequence of CSE approximates a NE, in the context of games of incomplete information.
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