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 few years video game conception has had a considerable impression on desktop technological know-how, in particular 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 state-of-the-art. simple chapters on algorithmic equipment for equilibria, mechanism layout and combinatorial auctions are by means of chapters on incentives and pricing, expense sharing, details markets and cryptography and safety. scholars, researchers and practitioners alike have to study extra approximately those interesting theoretical advancements and their frequent functional software.

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The algorithm continues in this manner, iteratively finding prices of sets of goods as they go tight. It terminates when all goods have been assigned prices. 16 The value x ∗ is feasible for the problem restricted to goods in A − S ∗ and buyers in B − (S ∗ ). Furthermore, in the subgraph of G induced on A − S ∗ and B − (S ∗ ), all vertices have nonzero degree. proof In the max-flow computed in N for x = x ∗ , the flow going through nodes in S ∗ completely uses up the capacity of edges from (S ∗ ) to t.

The buyer has a value v for the good, so his payoff is v − p if he buys, and 0 if he does not. Here we are considering a full information game in which seller S is aware of the buyer’s value v, and hence we expect that the seller offers price p just under v, and the buyer buys. ) This game allows the first player to lead, and collect (almost) all the profit. This game is known as the ultimatum game when two players S and B need to divide up v amount of money. The game allows the first player S to make an “ultimatum” (in the form of a price in our context) on how to divide up the money.

Consider now the symmetric game consisting of this matrix: 0 A C= and let (x, y) be a symmetric equilibrium of this game (by x we denote BT 0 the first m components of the vector, where m is the number of rows of A, and by y the rest). It is easy to see that, for (x, y) to be a best response to itself, y must be a best response to x, and x must be a best response to y. Hence, x and y constitute a Nash equilibrium of the original game, completing the proof. Incidentally, it is not known how hard it is to find any Nash equilibrium in a symmetric game (it could be easier than Nash), or to find a nonsymmetric equilibrium in a symmetric game (it could be easier or harder than Nash).

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