Traditionally fault tolerance and security have divided processes into “good guys ” and “bad guys”. Work on fault tolerance has focused on assuring that certain goals are met, as long as the number of “bad guys ” is bounded (e.g., less than one third or one half of the total number of players). The viewpoint in game theory has been quite different. There are no good guys or bad guys, only rational players who will make moves in their own self-interest. Making this precise requires assigning payoffs (or utilities) to outcomes. There are various solution concepts in game theory—predictions regarding the outcome of a game with rational players. They all essentially involve players making best responses to their beliefs, but differ in what players are assumed to know about what the other players are doing. Perhaps the best-known and most widely-used solution concept is Nash equilibrium (NE). A profile σ of strategies—that is, a collection of strategies consisting of one strategy σi for each player i—is a Nash equilibrium if no player can improve his payoff by changing his strategy unilaterally, even assuming that he knows the strategies of all the other players. In the notation traditionally used in game theory, σ is a Nash equilibrium if, for all i and all strategies τi for player i, ui(σ−i, τi) ≤ ui(σ): player i does not gain any utility by switching to τi if all the remaining players continue to play their component of σ. (See a standard game theory text, such as , for an introduction to solution concepts, and more examples and intuition.
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