Abstract
We analyse the all-pay auction with incomplete information and variance-averse bidders. We characterise the unique symmetric equilibrium for general distributions of valuations and any number of bidders. Variance aversion is a sufficient assumption to predict that high-valuation bidders increase their bids relative to the risk-neutral case while low types decrease their bid. Considering an asymmetric two-player environment with uniformly distributed valuations, we show that a variance-averse player always bids higher than her risk-neutral opponent with the same valuation. Utilising our analytically derived bidding functions we discuss all-pay auctions with variance-averse bidders from an auction designer’s perspective. We briefly consider possible extensions of our model, including noisy signals, type-dependent attitudes towards risk, and variance-seeking preferences.
Article PDF
Similar content being viewed by others
References
Agranov, M., Bisin, A., Schotter, A.: An experimental study of the impact of competition for Other People’s Money: the portfolio manager market. Exp. Econ. 17(4), 564–585 (2014). https://doi.org/10.1007/s10683-013-9384-6
Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’École Américaine. Econometrica 21(4), 503–46 (1953). https://doi.org/10.2307/1907921
Amann, E., Leininger, W.: Asymmetric all-pay auctions with incomplete information: the two-player case. Games Econ. Behav. 14, 1–18 (1996). https://doi.org/10.1006/game.1996.0040
Åstebro, T.B.: The return to independent invention: Evidence of unrealistic optimism, risk seeking or skewness loving? Econ. J. 113, 226–39 (2003). https://doi.org/10.1111/1468-0297.00089
Baron, D.P.: On the utility theoretic foundations of mean-variance analysis. J. Finance 32, 1683–97 (1977). https://doi.org/10.1111/j.1540-6261.1977.tb03363.x
Barut, Y., Kovenock, D., Noussair, C.: A comparison of multiple-unit all-pay and winner-pay auctions under incomplete information. Int. Econ. Rev. 43, 675–708 (2002). https://doi.org/10.1111/1468-2354.t01-1-00031
Blavatskyy, P.R.: Modifying the mean-variance approach to avoid violations of stochastic dominance. Manag. Sci. 56(11), 2050–2057 (2010). https://doi.org/10.1287/mnsc.1100.1224
Chen, Z.C., Ong, D., Segev, E.: Heterogeneous risk/loss aversion in complete information all-pay auctions. Eur. Econ. Rev. 95, 23–37 (2017). https://doi.org/10.1016/j.euroecorev.2017.03.002
Chipman, J.S.: The ordering of portfolios in terms of mean and variance. Rev. Econ. Stud. 40(2), 167–90 (1973). https://doi.org/10.2307/2296646
Chiu, H.: Consistent mean-variance preferences. Oxf. Econ. Pap. 63, 398–418 (2010). https://doi.org/10.1093/oep/gpq015
Cingottini, I., Menicucci, D.: On the profitability of reducing competition in all-pay auctions with risk averse bidders. Econ. Lett. 91, 260–6 (2006). https://doi.org/10.1016/j.econlet.2005.12.014
Cornes, R., Hartley, R.: Loss aversion in contests. University of Manchester, Economics Discussion Paper EDP-1204 (2012a)
Cornes, R., Hartley, R.: Risk aversion in symmetric and asymmetric contests. Econ. Theor. 51, 247–75 (2012b). https://doi.org/10.1007/s00199-009-0490-9
Coyle, B.T.: Risk aversion and price risk in duality models of production: a linear mean-variance approach. Am. J. Agric. Econ. 74, 849–59 (1992). https://doi.org/10.2307/1243182
Dechenaux, E., Kovenock, D., Sheremeta, R.M.: A survey of experimental research on contests, all-pay auctions and tournaments. Exp. Econ. 51(2), 315–50 (2015). https://doi.org/10.1007/s10683-014-9421-0
Eisenhuth, R., Grunewald, M.: Auctions with loss averse bidders. Int. J. Econ. Theory 16(2), 129–52 (2020). https://doi.org/10.1111/ijet.12189
Esö, P., Futó, G.: Auction design with a risk averse seller. Econ. Lett. 65, 71–74 (1999). https://doi.org/10.1016/S0165-1765(99)00115-9
Esö, P., White, L.: Precautionary bidding in auctions. Econometrica 72, 77–92 (2004). https://doi.org/10.1111/j.1468-0262.2004.00478.x
Fang, K., Kotz, S., Ng, K.: Symmetric Multivariate and Related Distributions. Chapman & Hall, London (1987)
Fibich, G., Gavious, A.: Asymmetric first-price auctions: a perturbation approach. Math. Oper. Res. 28(4), 836–52 (2003). https://doi.org/10.1287/moor.28.4.836.20510
Fibich, G., Gavious, A., Sela, A.: Revenue equivalence in asymmetric auctions. J. Econ. Theory 115, 309–21 (2004). https://doi.org/10.1016/S0022-0531(03)00251-5
Fibich, G., Gavious, A., Sela, A.: All-pay auctions with risk-averse players. Int. J. Game Theory 34, 583–99 (2006). https://doi.org/10.1007/s00182-006-0034-5
Fu, Q., Lu, J.: Micro foundations of multi-prize lottery contests: a perspective of noisy performance ranking. Soc. Choice Welfare 38, 497–517 (2012). https://doi.org/10.1007/s00355-011-0542-5
Gneezy, U., Niederle, M., Rustichini, A.: Performance in competitive environments: gender differences. Q. J. Econ. 118(3), 1049–74 (2003). https://doi.org/10.1162/00335530360698496
Hanoch, H., Levy, M.: A note on indifference curves and uncertainty. Swed. J. Econ. 71(3), 206–8 (1970). https://doi.org/10.2307/3439370
Hillman, A.L., Samet, D.: Dissipation of contestable rents by small numbers of contenders. Public Choice 54, 63–82 (1987). https://doi.org/10.1007/BF00123805
Holt, C.A.: Competitive bidding for contracts under alternative auction procedures. J. Polit. Econ. 88, 433–45 (1980). https://doi.org/10.1086/260878
Hu, A., Matthews, S., Zou, L.: Risk aversion and optimal reserve prices in first and second-price auctions. J. Econ. Theory 145, 1188–202 (2010). https://doi.org/10.1016/j.jet.2011.10.005
Ingersoll, J.E.J.: The Theory of Financial Decision Making. Rowman and Littlefield Publishing, Maryland (1987)
Ireland, N.: Risk aversion and aggression in tournaments. University of Warwick, Working Paper (2004). https://www2.warwick.ac.uk/fac/soc/economics/staff/academic/ireland/risk_aversion_3.pdf
Jindapon, P., Yang, Z.: Risk attitudes and heterogeneity in simultaneous and sequential contests. J. Econ. Behav. Org. 138, 69–84 (2017). https://doi.org/10.1016/j.jebo.2017.04.011
Johnstone, D., Lindley, D.: Mean-variance and expected utility: the Borch paradox. Stat. Sci. 28(2), 223–37 (2013). https://doi.org/10.1214/12-STS408
Kaplan, T., Zamir, S.: Asymmetric first-price auctions with uniform distributions: analytic solutions to the general case. Econ. Theor. 50, 269–302 (2012). https://doi.org/10.1007/s00199-010-0563-9
Kirkegaard, R.: A mechanism design approach to ranking asymmetric auctions. Econometrica 80, 2349–64 (2012). https://doi.org/10.3982/ECTA9859
Konrad, K.A., Schlesinger, H.: Risk aversion in rent-seeking and rent-augmenting games. Econ. J. 107(445), 1671–83 (1997). https://doi.org/10.1111/j.1468-0297.1997.tb00074.x
Lazear, E.P., Rosen, S.: Rank order tournaments as optimal labor contracts. J. Polit. Econ. 89, 841–64 (1981). https://doi.org/10.1086/261010
Leininger, W.: Auction theory from an all-pay view: Buying binary lotteries. CESifo Working Paper Series, 382 (2000). http://www.cesifo-group.de/DocDL/cesifo_wp382.pdf
Levy, H., Markowitz, H.M.: Aproximating expected utility by a function of mean and variance. Am. Econ. Rev. 69(3), 308–17 (1979)
Liu, L.: A new foundation for the mean-variance analysis. Eur. J. Oper. Res. 158(1), 229–42 (2004). https://doi.org/10.1016/S0377-2217(03)00301-1
Lizzeri, A., Persico, N.: Uniqueness and existence of equilibrium in auctions with a reserve price. Games Econ. Behav. 30(1), 83–114 (2000). https://doi.org/10.1006/game.1998.0704
Markowitz, H.M.: Portfolio selection. J. Finance 7(1), 77–91 (1952). https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Markowitz, H.M.: Portfolio Selection: Efficient Diversification of Investments, 2nd edn. Wiley, New York (1959)
Markowitz, H.M.: Mean-variance approximations to expected utility. Eur. J. Oper. Res. 234(2), 346–55 (2014). https://doi.org/10.1016/j.ejor.2012.08.023
Markowitz, H.M., Reid, D.W., Tew, B.V.: The value of a blank check. J. Portf. Manag. 20(4), 82–91 (1994). https://doi.org/10.3905/jpm.1994.409480
Maskin, E., Riley, J.: Optimal auctions with risk averse buyers. Econometrica 52(6), 1473–1518 (1984). https://doi.org/10.2307/1913516
Maskin, E., Riley, J.: Asymmetric auctions. Rev. Econ. Stud. 67, 413–38 (2000). https://doi.org/10.1111/1467-937X.00137
Matthews, S.: Comparing auctions for risk averse buyers: a buyer’s point of view. Econometrica 55, 633–46 (1987). https://doi.org/10.2307/1913603
Mermer, A.G.: Contests with expectation-based loss-averse players. Working Paper, Tilburg University, October 25 (2013). http://hosting01.uc3m.es/semanal3/documents/Contests_with_Expectation_Based_Loss_Averse_paper.pdf
Meyer, J.: Two-moment decision models and expected utility maximization. Am. Econ. Rev. 77, 421–30 (1987)
Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004)
Nakamura, Y.: Mean-variance utility. SSRN Discussion paper, #2615290 (2015). https://doi.org/10.2139/ssrn.2615290
Niederle, M., Vesterlund, L.: Do women shy away from competition? Do men compete too much. Q. J. Econ. 122(3), 1067–101 (2007). https://doi.org/10.1162/qjec.122.3.1067
Noussair, C., Silver, J.: Behavior in all-pay auctions with incomplete information. Games Econ. Behav. 55, 189–206 (2006). https://doi.org/10.1016/j.geb.2005.01.005
Parreiras, S.O., Rubinchik, A.: Contests with three or more heterogeneous agents. Games Econ. Behav. 68(2), 703–715 (2010). https://doi.org/10.1016/j.geb.2009.09.007
Pollatsek, A., Tversky, A.: Theory of risk. J. Math. Psychol. 7, 540–53 (1970). https://doi.org/10.1016/0022-2496(70)90039-8
Post, T., Levy, H.: Does risk seeking drive asset prices? A stochastic dominance analysis of aggregate investor preferences. Rev. Financial Stud. 18(3), 925–53 (2005). https://doi.org/10.1093/rfs/hhi021
Qu, X.: Subjective mean-variance preferences without expected utility. Math. Soc. Sci. 87, 31–39 (2017). https://doi.org/10.1016/j.mathsocsci.2017.02.001
Riley, J.G., Samuelson, W.F.: Optimal auctions. Am. Econ. Rev. 71, 381–92 (1981)
Robson, A.: Contests between players with mean-variance preferences. Griffith Business School Discussion Paper, 2012–07. (2012). https://doi.org/10.2139/ssrn.2143008
Rothschild, M., Stiglitz, J.: Increasing risk: I. a definition. J. Econ. Theory 2, 225–43 (1970). https://doi.org/10.1016/0022-0531(70)90038-4
Saha, A.: Risk preference estimation in the nonlinear mean standard deviation approach. Econ. Inq. 35, 770–82 (1997). https://doi.org/10.1111/j.1465-7295.1997.tb01963.x
Sahm, M.: Risk aversion and prudence in contests. Econ. Bull. 37(2), 1122–32 (2017)
Sargent, T., Heller, S.: Macroeconomic Theory, 2nd edn. Academic Press, New York (1987)
Savage, L.J.: The Foundations of Statistics, 2nd edn. Wiley, New York (1954). https://doi.org/10.1002/nav.3800010316
Siegel, R.: All-pay contests. Econometrica 77, 71–92 (2009). https://doi.org/10.3982/ECTA7537
Simaan, Y.: What is the opportunity cost of mean-variance investment strategies? Manag. Sci. 39, 578–87 (1993). https://doi.org/10.1287/mnsc.39.5.578
Smith, J.L., Levin, D.: Ranking auctions with risk-averse bidders. J. Econ. Theory 68, 549–61 (1996). https://doi.org/10.1006/jeth.1996.0031
Strack, P.: Risk-taking in contests: The impact of fund-manager compensation on investor welfare. SSRN, Working paper (2016). https://ssrn.com/abstract=2739177
Tobin, J.: Liquidity preference as behavior towards risk. Rev. Econ. Stud. 25(2), 65–86 (1958). https://doi.org/10.2307/2296205
Treich, N.: Risk-aversion and prudence in rent-seeking games. Public Choice 145(3–4), 339–49 (2010). https://doi.org/10.1007/s11127-009-9569-x
Tsiang, S.C.: The rationale of the mean-standard deviation analysis, skewness preference, and the demand for money. Am. Econ. Rev. 62, 354–71 (1972)
von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Olivier Bos, Thomas Giebe, Todd Kaplan, Dan Kovenock, Sergio Parreiras, and Ron Siegel for helpful comments and discussions. The paper has benefitted from the comments of multiple anonymous referees and the associate editor. Schweinzer appreciates the generous hospitality of CESifo Munich, Germany, and the University of Adelaide, Australia. This project was supported by funds of the Oesterreichische Nationalbank (Austrian Central Bank, Anniversary Fund, project number: 17663) and through the Austrian Science Fund (FWF), project number P31248.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Klose, B., Schweinzer, P. Auctioning risk: the all-pay auction under mean-variance preferences. Econ Theory 73, 881–916 (2022). https://doi.org/10.1007/s00199-020-01332-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00199-020-01332-7