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Unformatted text preview: arXiv:1101.2883v1 [cs.GT] 14 Jan 2011 Dueling algorithms Nicole Immorlica Adam Tauman Kalai Brendan Lucier Ankur Moitra Andrew Postlewaite bardbl Moshe Tennenholtz January 17, 2011 Abstract We revisit classic algorithmic search and optimization problems from the perspective of competition. Rather than a single optimizer minimizing expected cost, we consider a zero- sum game in which an optimization problem is presented to two players, whose only goal is to outperform the opponent . Such games are typically exponentially large zero-sum games, but they often have a rich combinatorial structure. We provide general techniques by which such structure can be leveraged to find minmax-optimal and approximate minmax-optimal strategies. We give examples of ranking, hiring, compression, and binary search duels, among others. We give bounds on how often one can beat the classic optimization algorithms in such duels. Department of Electrical Engineering and Computer Science, Northwestern University Part of this work was performed while the author was at Microsoft Research Microsoft Research New England Department of Computer Science, University of Toronto Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. Supported in part by a Fannie Hurts Fellowship. bardbl Department of Economics, University of Pennsylvania Microsoft R&D Israel and the Technion, Israel 1 Introduction Many natural optimization problems have two-player competitive analogs. For example, con- sider the ranking problem of selecting an order on n items, where the cost of searching for a single item is its rank in the list. Given a fixed probability distribution over desired items, the trivial greedy algorithm, which orders items in decreasing probability, is optimal. Next consider the following natural two-player version of the problem, which models a user choosing between two search engines. The user thinks of a desired web page and a query and executes the query on both search engines. The engine that ranks the desired page higher is cho- sen by the user as the winner. If the greedy algorithm has the ranking of pages 1 , 2 ,..., n , then the ranking 2 , 3 ,..., n , 1 beats the greedy ranking on every item except 1 . We say the greedy algorithm is 1 1 /n beatable because there is a probability distribution over pages for which the greedy algorithm loses 1 1 /n of the time. Thus, in a competitive setting, an optimal search engine can perform poorly against a clever opponent. This ranking duel can be modeled as a symmetric constant-sum game, with n ! strategies, in which the player with the higher ranking of the target page receives a payoff of 1 and the other receives a payoff of 0 (in the case of a tie, say they both receive a payoff of 1/2). As in all symmetric one-sum games, there must be (mixed) strategies that guarantee expected payoff of at least 1/2 against any opponent. Put another way, there must be a (randomized) algorithmat least 1/2 against any opponent....
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This note was uploaded on 10/15/2011 for the course CS 1210 taught by Professor M.izzo during the Spring '11 term at Community college of RI.
- Spring '11