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A Nested Contest: Tullock Meets the All-pay Auction

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  • J. Atsu Amegashie

Abstract

I present a two-player nested contest which is a convex combination of two widely studied contests: the Tullock (lottery) contest and the all-pay auction. A Nash equilibrium exists for all parameters of the nested contest. If and only if the contest is sufficiently asymmetric, then there is an equilibrium in pure strategies. In this equilibrium, individual and aggregate efforts are lower relative to the efforts in a Tullock contest. This leads to the surprising result that if aggregate efforts in the all-pay auction are higher than the aggregate efforts in the Tullock contest, then aggregate efforts in the nested contest may not lie between aggregate efforts in the all-pay auction and aggregate efforts in the Tullock contest. When the contest is symmetric or asymmetric, I find a mixed-strategy equilibrium and describe some properties of the equilibrium distribution function; I also find the equilibrium payoffs and expected bids. When the weight on the all-pay auction component of this nested contest lies in an intermediate range, then there exist multiple non-payoff-equivalent equilibria such that there is an all-pay auction equilibrium as defined in Alcade and Dahm (2010) and another equilibrium which is not an all-pay auction equilibrium; these equilibria cannot be ranked using the Pareto criterion. If the goal of a contest-designer is to reduce aggregate effort (i.e., wasteful rent-seeking efforts), then this nested contest may be better than both the Tullock contest and the all-pay auction.

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Paper provided by CESifo Group Munich in its series CESifo Working Paper Series with number 3976.

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Date of creation: 2012
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Handle: RePEc:ces:ceswps:_3976

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Keywords: all-pay auction; discontinuous games; mixed strategy; pure strategy; Tullock contest;

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References

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  1. Baye, M. & Kovenock, D. & Vries, C. de, 1990. "The All-Pay Auction with Complete Information," Discussion Paper, Tilburg University, Center for Economic Research 1990-51, Tilburg University, Center for Economic Research.
  2. Luis Corchón & Matthias Dahm, 2010. "Foundations for contest success functions," Economic Theory, Springer, Springer, vol. 43(1), pages 81-98, April.
  3. Baye, Michael R & Kovenock, Dan & de Vries, Casper G, 1993. "Rigging the Lobbying Process: An Application of the All-Pay Auction," American Economic Review, American Economic Association, American Economic Association, vol. 83(1), pages 289-94, March.
  4. Baye, M.R. & Kovenock, D., 1993. "The Solution to the Tullock Rent-Seeking Game When R > 2: Mixed Strategy Equilibria and Mean Dissipation Rates," Papers, Tilburg - Center for Economic Research 9368, Tilburg - Center for Economic Research.
  5. Leonid Polishchuk & Alexander Tonis, 2013. "Endogenous contest success functions: a mechanism design approach," Economic Theory, Springer, Springer, vol. 52(1), pages 271-297, January.
  6. José Alcalde & Matthias Dahm, 2007. "All-Pay Auction Equilibria In Contests," Working Papers. Serie AD, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie) 2007-27, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
  7. Ron Siegel, 2009. "All-Pay Contests," Econometrica, Econometric Society, Econometric Society, vol. 77(1), pages 71-92, 01.
  8. Clark, D.J. & Riis, C., 1996. "On the Win Probability in Rent-Seeking Gemes," Memorandum, Oslo University, Department of Economics 05/1996, Oslo University, Department of Economics.
  9. Franke, Jörg & Kanzow, Christian & Leininger, Wolfgang & Schwartz, Alexandra, 2013. "Lottery versus All-Pay Auction Contests: A Revenue Dominance Theorem," Annual Conference 2013 (Duesseldorf): Competition Policy and Regulation in a Global Economic Order 79998, Verein für Socialpolitik / German Economic Association.
  10. Chowdhury, Subhasish & Sheremeta, Roman, 2011. "Multiple Equilibria in Tullock Contests," MPRA Paper 52104, University Library of Munich, Germany.
  11. Hao Jia & Stergios Skaperdas & Samarth Vaidya, 2012. "Contest Functions: Theoretical Foundations and Issues in Estimation," Working Papers 111214, University of California-Irvine, Department of Economics.
  12. Alcalde, José & Dahm, Matthias, 2010. "Rent seeking and rent dissipation: A neutrality result," Journal of Public Economics, Elsevier, Elsevier, vol. 94(1-2), pages 1-7, February.
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  14. Arye L. Hillman & John G. Riley, 1989. "Politically Contestable Rents And Transfers," Economics and Politics, Wiley Blackwell, Wiley Blackwell, vol. 1(1), pages 17-39, 03.
  15. Gil Epstein & Shmuel Nitzan, 2006. "The Politics of Randomness," Social Choice and Welfare, Springer, Springer, vol. 27(2), pages 423-433, October.
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  17. Che, Yeon-Koo & Gale, Ian, 2000. "Difference-Form Contests and the Robustness of All-Pay Auctions," Games and Economic Behavior, Elsevier, Elsevier, vol. 30(1), pages 22-43, January.
  18. Nti, Kofi O, 1999. " Rent-Seeking with Asymmetric Valuations," Public Choice, Springer, Springer, vol. 98(3-4), pages 415-30, March.
  19. José Alcalde & Matthias Dahm, 2007. "Tullock And Hirshleifer: A Meeting Of The Minds," Working Papers. Serie AD, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie) 2007-13, Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie).
  20. J. Atsu Amegashie, 2009. "American Idol: should it be a singing contest or a popularity contest?," Journal of Cultural Economics, Springer, Springer, vol. 33(4), pages 265-277, November.
  21. Gil Epstein & Shmuel Nitzan, 2006. "Reduced prizes and increased effort in contests," Social Choice and Welfare, Springer, Springer, vol. 26(3), pages 447-453, June.
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Cited by:
  1. Christian Ewerhart, 2014. "Mixed equilibria in Tullock contests," ECON - Working Papers, Department of Economics - University of Zurich 143, Department of Economics - University of Zurich.

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