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Download Algorithmic Game Theory by Nisan N. (Ed), Vazirani V. (Ed), Roughgarden T. (Ed) PDF

By Nisan N. (Ed), Vazirani V. (Ed), Roughgarden T. (Ed)

Within the previous few years video game conception has had a considerable impression on machine technology, specially on net- and e-commerce-related matters. greater than forty of the head researchers during this box have written chapters that pass from the rules to the cutting-edge. easy chapters on algorithmic tools for equilibria, mechanism layout and combinatorial auctions are by way of chapters on incentives and pricing, price sharing, details markets and cryptography and safeguard. scholars, researchers and practitioners alike have to study extra approximately those interesting theoretical advancements and their frequent useful software.

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1 Complexity of Finding Equilibria The complexity of finding Nash and correlated equilibria will be discussed in detail in Chapters 2 and 3. Here we give a short overview. We then discuss two-player zero-sum games in more detail and show that for such games a Nash equilibrium can be found efficiently using linear programming. It turns out that even general two-player games have a character different from that of games with three or more players. For example, two-player games where payoffs are rational numbers always admit a solution with rational probabilities, and this is not true for games with three or more players.

We will discuss these ideas in Chapter 15. Here we briefly review a very different proposal for what is a “fair” way to share cost, the Shapley value. One advantage of the Shapley value is that it always exists. However, it may not be in the core, even for games that have nonempty core. 14 (Shapley Value) Shapley value is based on evaluating the marginal cost of each player. If we order the player set N as 1, . . , n and use the notation that Ni = {1, . . , i} then the marginal cost of player i is c(Ni ) − c(Ni−1 ).

Changing the strategy si to some other strategy si ∈ Si the player can change his utility to ui (si , s−i ), assuming that all other players stick to their strategies in s−i . We say that a change from strategy si to si is an improving response for player i if ui (si , s−i ) > ui (s) and best response if si maximizes the players’ utility maxsi ∈Si ui (si , s−i ). Playing a game by repeatedly allowing some player to make an improving or a best response move is perhaps the most natural game play.

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