"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)
"The information obtained by discovering dominances for one player may be of relevance to the others, insofar as the elimination of classes of mixed strategies as possible components of an equilibrium point is concerned. For the t's whose components are all undominated are all that need be considered and thus eliminating some of the strategies of one player may make possible the elimination of a new class of strategies for another player." (John F Nash, "Two-Person Cooperative Games", 1953)
"That strategic rivalry in a long-term relationship may differ from that of a one-shot game is by now quite a familiar idea. Repeated play allows players to respond to each other’s actions, and so each player must consider the reactions of his opponents in making his decision. The fear of retaliation may thus lead to outcomes that otherwise would not occur. The most dramatic expression of this phenomenon is the celebrated "Folk Theorem." An outcome that Pareto dominates the minimax point is called individually rational. The Folk Theorem asserts that any individually rational outcome can arise as a Nash equilibrium in infinitely repeated games with sufficiently little discounting." (Drew Fudenberg & Eric Maskin, "The folk theorem in repeated games with discounting or with incomplete information", Econometrica: Journal of the Econometric Society, 1986)
"Inducement correspondences provide a particularly easy way to explore the Nash equilibrium. Because payoffs are strictly ordered there will always be a single best response in a given inducement correspondence. The inducement correspondence can also be used to describe a number of other solution concepts, including maxi-min and solutions based on dominance." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"The Nash equilibrium is often rationalized using a story about how people think and how their behaviour is related to their thoughts. Economists generally assume that, from a set of alternatives, a player will actively choose the one he likes best. This is the assumption of economic rationality, one of the core assumptions of standard game theory. Rationality alone will not predict behaviour in a game, but it leads us to single out the member of any inducement correspondence that yields the greatest payoff for the player that is making the choice. The resulting behaviour is sometimes described as 'myopic' because it fails to take into account how other players might respond to a given choice." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)
"More precisely, suppose that Player 1 chooses a strategy S and Player 2 chooses a strategy T . We say that this pair of strategies (S,T ) is a Nash equilibrium if S is a best response to T , and T is a best response to S. This concept is not one that can be derived purely from rationality on the part of the players; instead, it is an equilibrium concept. The idea is that if the players choose strategies that are best responses to each other, then no player has an incentive to deviate to an alternative strategy – the system is in a kind of equilibrium state, with no force pushing it toward a different outcome." (David Easley & Jon Kleinberg, "Networks, Crowds, and Markets: Reasoning about a Highly Connected World", 2010)
"To understand the idea of Nash equilibrium, we should first ask why a pair of strategies that are not best responses to each other would not constitute an equilibrium. The answer is that the players cannot both believe that these strategies would actually be used in the game, since they know that at least one player would have an incentive to deviate to another strategy. So a Nash equilibrium can be thought of as an equilibrium in beliefs. If each player believes that the other player will actually play a strategy that is part of a Nash equilibrium, then she has an incentive to play her part of the Nash equilibrium.
No comments:
Post a Comment