**Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction (with Eva Tardos) **

**Renato Paes Leme**

**Monday October 18 , 2010
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__Abstract:__

Generalized Second Price Auction, also knows as Ad Word auctions, and its variants has been the main mechanism used by search companies to auction positions for sponsored search links. In this paper we study the social welfare of the Nash equilibria of this game. It is known that socially optimal Nash equilibria exists (i.e., that the Price of Stability for this game is 1).

This paper is the first to prove bounds on the price of anarchy. Our main result is to show that under some mild assumptions the price of anarchy is small. For pure Nash equilibria we bound the price of anarchy by 1.618, assuming all bidders are playing un-dominated strategies. For mixed Nash equilibria we prove a bound of 4 under the same assumption. We also extend the result to the Bayesian setting when bidders valuations are also random, and prove a bound of 8 for this case. Our proof exhibits a combinatorial structure of Nash equilibria and use this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest.

link: **http://www.cs.cornell.edu/%7Erenatoppl/papers/paesleme_tardos_focs10_manuscript.pdf **