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

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

Within the previous couple of years online game concept has had a considerable impression on laptop technology, specially on net- and e-commerce-related concerns. greater than forty of the head researchers during this box have written chapters that move from the rules to the cutting-edge. uncomplicated chapters on algorithmic equipment for equilibria, mechanism layout and combinatorial auctions are via chapters on incentives and pricing, price sharing, details markets and cryptography and safety. scholars, researchers and practitioners alike have to examine extra approximately those attention-grabbing theoretical advancements and their frequent sensible software.

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4. 5 Games with No Nash Equilibria Both assumptions in the theorem about the finite set of players and finite strategy sets are important: games with an infinite number of players, or games with a finite number of players who have access to an infinite strategy set may not have Nash equilibria. A simple example of this arises in the following pricing game. 3. Sellers 1 and 2 are selling identical products to buyers A, B, and C. 3. Each buyer wants to buy one unit of the product. Buyers A and C have access to one seller only, namely 1 and 2, respectively.

The basic reason is that every game is guaranteed to have a Nash equilibrium. In contrast, in a typical NP-complete problem such as satisfiability, the sought solution may or may not exist. 3 For, suppose that Nash is NP-complete, and there is a reduction from satisfiability to Nash. This would entail an efficiently computable function f mapping Boolean formulae to games, and such that, for every formula φ, φ is satisfiable if and only if any Nash equilibrium of f (φ) satisfies some easy-to-check property .

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. Games with two players will be discussed in greater detail in Chapter 3. We will discuss the complexity of finding Nash equilibrium in Chapter 2. NPcompleteness, the “standard” way of establishing intractability of individual problems, does not seem to be the right tool for studying the complexity of Nash equilibria.

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