Aranyak Mehta

AdWords and generalized online matching

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

Playing large games using simple strategies

We prove the existence of ε-Nash equilibrium strategies with support logarithmic in the number of pure strategies. We also show that the payoffs to all players in any (exact) Nash equilibrium can be ε-approximated by the payoffs to the players in some such logarithmic support ε-Nash equilibrium. These strategies are also uniform on a multiset of logarithmic size and therefore this leads to a quasi-polynomial algorithm for computing an ε-Nash equilibrium. To our knowledge this is the first subexponential algorithm for finding an ε-Nash equilibrium. Our results hold for any multiple-player game as long as the number of players is a constant (i.e., it is independent of the number of pure strategies). A similar argument also proves that for a fixed number of players m, the payoffs to all players in any m-tuple of mixed strategies can be ε-approximated by the payoffs in some m-tuple of constant support strategies.We also prove that if the payoff matrices of a two person game have low rank then the game has an exact Nash equilibrium with small support. This implies that if the payoff matrices can be well approximated by low rank matrices, the game has an ε-equilibrium with small support. It also implies that if the payoff matrices have constant rank we can compute an exact Nash equilibrium in polynomial time.

AdWords and generalized on-line matching

How does a search engine company decide what ads to display with each query so as to maximize its revenue? This turns out to be a generalization of the online bipartite matching problem. We introduce the notion of a tradeoff revealing LP and use it to derive two optimal algorithms achieving competitive ratios of 1-1/e for this problem.

Online Stochastic Matching: Beating 1-1/e

We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of

A note on approximate Nash equilibria

1-{1\over e} \simeq 0.632

Progress in approximate nash equilibria

, a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the

Inapproximability results for combinatorial auctions with submodular utility functions

1 - {1\over e}

Online Matching and Ad Allocation

bound.Our main result is a

A note on approximate nash equilibria

0.67

Beyond moulin mechanisms

-approximation online algorithm for stochastic bipartite matching, breaking this