By Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V. Vazirani
Within the previous couple of years video game concept has had a considerable impression on laptop technology, in particular on net- and e-commerce-related concerns. greater than forty of the head researchers during this box have written chapters that cross from the principles to the state-of-the-art. uncomplicated chapters on algorithmic equipment for equilibria, mechanism layout and combinatorial auctions are by way of chapters on incentives and pricing, rate sharing, details markets and cryptography and protection. scholars, researchers and practitioners alike have to examine extra approximately those interesting theoretical advancements and their common functional software.
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Recall that a mixed strategy profile is a Nash equilibrium if the mixed strategy of each player is a best response to the mixed strategies of the rest; that is, it attains the maximum possibly utility among all possible mixed strategies of this player. The following observation is useful here (recall that the support of a mixed strategy is the set of all pure strategies that have nonzero probability in it). 1 A mixed strategy is a best response if and only if all pure strategies in its support are best responses.
Give a polynomial time algorithm to find a correlated equilibrium for such a game. Note that the input to this problem consists of the 2n numbers above. As usual, polynomial means polynomial in this input length. You may use the fact that linear programming is solvable in polynomial time. 6 Consider a 2-person game in matrix form. Assume that both players have n pure strategies. In a Nash equilibrium a player may be required to play a mixed strategy that gives nonzero probability to all (or almost all) of his pure strategies.
It is not hard to construct a symmetric game whose strategies are all literals (variables and their negations) and whose Nash equilibria are all truth assignments. In other words, if we choose, for each of the n variables, either the variable itself or its negation, and play it with probability n1 , then we get a symmetric Nash equilibrium, and all Nash equilibria of the game are of this sort. It is also easy to add to this game a new pure Nash equilibrium (d, d), with lower utility, where d (for “default”) is a new strategy.