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Price of Competition and Dueling Games

Authors Sina Dehghani, Mohammad Taghi Hajiaghayi, Hamid Mahini, Saeed Seddighin



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Sina Dehghani
Mohammad Taghi Hajiaghayi
Hamid Mahini
Saeed Seddighin

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Sina Dehghani, Mohammad Taghi Hajiaghayi, Hamid Mahini, and Saeed Seddighin. Price of Competition and Dueling Games. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 21:1-21:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ICALP.2016.21

Abstract

We study competition in a general framework introduced by Immorlica, Kalai, Lucier, Moitra, Postlewaite, and Tennenholtz and answer their main open question. Immorlica et al. considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user’s satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an ordering over all possible webpages based on the submitted query. The user tends to choose the ordering which displays her requested webpage in a higher rank. The goal of both players is to maximize the probability that her ordering beats that of her opponent and gets the user's attention. Immorlica et al. show this game directs both players to provide suboptimal search results. However, they leave the following as their main open question: "does competition between algorithms improve or degrade expected performance?" (see the introduction for more quotes) In this paper, we resolve this question for the ranking duel and a more general class of dueling games. More precisely, we study the quality of orderings in a competition between two players. This game is a zero-sum game, and thus any Nash equilibrium of the game can be described by minimax strategies. Let the value of the user for an ordering be a function of the position of her requested item in the corresponding ordering, and the social welfare for an ordering be the expected value of the corresponding ordering for the user. We propose the price of competition which is the ratio of the social welfare for the worst minimax strategy to the social welfare obtained by asocial planner. Finding the price of competition is another approach to obtain structural results of Nash equilibria. We use this criterion for analyzing the quality of orderings in the ranking duel. Although Immorlica et al. show that the competition leads to suboptimal strategies, we prove the quality of minimax results is surprisingly close to that of the optimum solution. In particular, via a novel factor-revealing LP for computing price of anarchy, we prove if the value of the user for an ordering is a linear function of its position, then the price of competition is at least 0.612 and bounded above by 0.833. Moreover we consider the cost minimization version of the problem. We prove, the social cost of the worst minimax strategy is at most 3 times the optimal social cost. Last but not least, we go beyond linear valuation functions and capture the main challenge for bounding the price of competition for any arbitrary valuation function. We present a principle which states that the lower bound for the price of competition for all 0-1 valuation functions is the same as the lower bound for the price of competition for all possible valuation functions. It is worth mentioning that this principle not only works for the ranking duel but also for all dueling games. This principle says, in any dueling game, the most challenging part of bounding the price of competition is finding a lower bound for 0-1 valuation functions. We leverage this principle to show that the price of competition is at least 0.25 for the generalized ranking duel.
Keywords
  • POC
  • POA
  • Dueling games
  • Nash equilibria
  • sponsored search

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References

  1. Gagan Aggarwal, Jon Feldman, S. Muthukrishnan, and Martin Pál. Sponsored search auctions with markovian users. In WINE, pages 621-628. 2008. Google Scholar
  2. AmirMahdi Ahmadinejad, Sina Dehghani, MohammadTaghi Hajiaghayi, Hamid Mahini, Saeed Seddighin, and Sadra Yazdanbod. Forming external behaviors by leveraging internal opinions. In Computer Communications (INFOCOM), 2015 IEEE Conference on, pages 1849-1857. IEEE, 2015. Google Scholar
  3. Mahdi Ahmadinejad, Sina Dehghani, MohammadTaghi Hajiaghayi, Brendan Lucier, Hamid Mahini, and Saeed Seddighin. From duels to battefields: Computing equilibria of blotto and other games. AAAI 2016, 2016. Google Scholar
  4. Susanne Albers, Stefan Eilts, Eyal Even-Dar, Yishay Mansour, and Liam Roditty. On nash equilibria for a network creation game. In SODA, pages 89-98, 2006. Google Scholar
  5. Noga Alon, Erik D. Demaine, Mohammad T. Hajiaghayi, and Tom Leighton. Basic network creation games. SIAM Journal on Discrete Mathematics, 27(2):656-668, 2013. Google Scholar
  6. Nir Andelman, Michal Feldman, and Yishay Mansour. Strong price of anarchy. In SODA, pages 189-198, 2007. Google Scholar
  7. Itai Ashlagi, Piotr Krysta, and Moshe Tennenholtz. Social context games. In Internet and Network Economics, pages 675-683. Springer, 2008. Google Scholar
  8. Claude Aspremont, J. Jaskold Gabszewicz, and J.-F. Thisse. On hotelling’s" stability in competition". Econometrica: Journal of the Econometric Society, pages 1145-1150, 1979. Google Scholar
  9. Susan Athey and Glenn Ellison. Position auctions with consumer search. Technical Report 15253, National Bureau of Economic Research, 2009. Google Scholar
  10. J. Bertrand. Book review of théorie mathématique de la richesse sociale and of recherches sur les principes mathématiques de la théorie des richesses. Journal de Savants, 67:499-508, 1883. Google Scholar
  11. Felix Brandt, Felix Fischer, Paul Harrenstein, and Yoav Shoham. Ranking games. Artificial Intelligence, 173(2):221-239, 2009. Google Scholar
  12. George Christodoulou and Elias Koutsoupias. The price of anarchy of finite congestion games. In STOC, pages 67-73, 2005. Google Scholar
  13. Thomas H. Cormen, Clifford Stein, Ronald L. Rivest, and Charles E. Leiserson. Introduction to Algorithms. McGraw-Hill Higher Education, 2nd edition, 2001. Google Scholar
  14. Erik D Demaine, MohammadTaghi Hajiaghayi, Hamid Mahini, and Morteza Zadimoghaddam. The price of anarchy in network creation games. In PODC, pages 292-298, 2007. Google Scholar
  15. Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz. Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. American Economic Review, 97(1):242-259, 2007. Google Scholar
  16. Alex Fabrikant, Ankur Luthra, Elitza Maneva, Christos H. Papadimitriou, and Scott Shenker. On a network creation game. In PODC, pages 347-351, 2003. Google Scholar
  17. Anindya Ghose and Sha Yang. An empirical analysis of search engine advertising: Sponsored search in electronic markets. Management Science, 55(10):1605-1622, 2009. Google Scholar
  18. Harold Hotelling. Stability in competition. The Economic Journal, 39(153):41-57, 1929. Google Scholar
  19. Nicole Immorlica, Adam Tauman Kalai, Brendan Lucier, Ankur Moitra, Andrew Postlewaite, and Moshe Tennenholtz. Dueling algorithms. In STOC, pages 215-224, 2011. Google Scholar
  20. Nicole Immorlica, Li Erran Li, Vahab S. Mirrokni, and Andreas S. Schulz. Coordination mechanisms for selfish scheduling. Theoretical Computer Science, 410(17):1589-1598, 2009. Google Scholar
  21. David Kempe and Brendan Lucier. User satisfaction in competitive sponsored search. In WWW, pages 699-710, 2014. Google Scholar
  22. David Kempe and Mohammad Mahdian. A cascade model for externalities in sponsored search. In Internet and Network Economics, pages 585-596. 2008. Google Scholar
  23. Elias Koutsoupias and Christos Papadimitriou. Worst-case equilibria. In STACS, pages 404-413, 1999. Google Scholar
  24. David M Kreps. A course in microeconomic theory. Harvester Wheatsheaf New York, 1990. Google Scholar
  25. Andreu Mas-Colell, Michael Dennis Whinston, and Jerry R. Green. Microeconomic theory. Oxford university press New York, 1995. Google Scholar
  26. Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani. Algorithmic Game Theory. Cambridge University Press, 2007. Google Scholar
  27. Tim Roughgarden. Selfish routing and the price of anarchy, volume 174. MIT press Cambridge, 2005. Google Scholar
  28. Rahul Telang, Uday Rajan, and Tridas Mukhopadhyay. The market structure for internet search engines. Journal of Management Information Systems, 21(2):137-160, 2004. Google Scholar
  29. Hal R Varian and Jack Repcheck. Intermediate microeconomics: a modern approach. WW Norton &Company New York, NY, 8th edition, 2010. Google Scholar
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