Download Algorithmic Game Theory by Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V. Vazirani PDF

By Noam Nisan, Tim Roughgarden, Eva Tardos, Vijay V. Vazirani

Within the previous few years online game thought has had a considerable effect on machine technological know-how, specifically on web- and e-commerce-related concerns. greater than forty of the head researchers during this box have written chapters that pass from the principles 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, fee sharing, info markets and cryptography and safeguard. scholars, researchers and practitioners alike have to study extra approximately those interesting theoretical advancements and their frequent functional software.

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Extra resources for Algorithmic Game Theory

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Each buyer i has access to only a subset, say Si ⊆ A of the goods. She is indifferent between goods in Si , but is interested in maximizing the total amount of goods obtained. An example of such a situation is when identical goods are sold in different markets and each buyer has access to only a subset of the markets; such a model is studied in Chapter 7. Without loss of generality we may assume that mi = 0, aj = 0, for each buyer i, Si = ∅, and for each good j , there is a buyer i such that j ∈ Si .

On the other hand, B buys from the cheaper seller. To fully specify the game, we have to set a rule for breaking ties. Let us say that if both sellers have the same price, B buys from seller 1. For simplicity, we assume no production costs, so the income of a seller is the sum of the prices at which they sold goods. Now, one strategy for each seller is to set a price of pi = 1, and guarantee an income of 1 from the buyer who does not have a choice. Alternatively, they can also try to compete for buyer B.

Show that for any fixed > 0 and any 2-player game with all nonnegative payoffs, there is an -approximate Nash equilibrium such that both players play the following simple kind of mixed strategy. For each player j , the strategy selects a subset Sˆ j of at most O(log n) of player j ’s pure strategies, and makes player j select one of the strategies in Sˆ j uniformly at random. , may contain the same pure strategy more than once such a strategy is more likely to be selected by the random choice).

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