Nick Gravin

Simultaneous auctions are (almost) efficient

Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. We study the efficiency of Bayes-Nash equilibrium (BNE) outcomes of simultaneous first- and second-price auctions when bidders have complement-free (a.k.a. subadditive) valuations. While it is known that the social welfare of every pure Nash equilibrium (NE) constitutes a constant fraction of the optimal social welfare, a pure NE rarely exists, and moreover, the full information assumption is often unrealistic. Therefore, quantifying the welfare loss in Bayes-Nash equilibria is of particular interest. Previous work established a logarithmic bound on the ratio between the social welfare of a BNE and the expected optimal social welfare in both first-price auctions (Hassidim et al., 2011) and second-price auctions (Bhawalkar and Roughgarden, 2011), leaving a large gap between a constant and a logarithmic ratio. We introduce a new proof technique and use it to resolve both of these gaps in a unified way. Specifically, we show that the expected social welfare of any BNE is at least 1/2 of the optimal social welfare in the case of first-price auctions, and at least 1/4 in the case of second-price auctions.

Efficient Computation of Approximate Pure Nash Equilibria in Congestion Games

Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general {\sf PLS}-complete. We present a surprisingly simple polynomial-time algorithm that computes

Translational tilings by a polytope, with multiplicity

O(1)

Combinatorial walrasian equilibrium

-approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes

Dynamics of profit-sharing games

(2+\epsilon)

Truthful Generalized Assignments via Stable Matching

-approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and

Procrastination with Variable Present Bias

1/\epsilon

Combinatorial Walrasian Equilibrium

. It also applies to games with polynomial latency functions with constant maximum degree

A note on k -shortest paths problem

d

Dynamics of Profit-Sharing Games

; there, the approximation guarantee is