A general framework for studying contests

Bastani, Spencer and Giebe, Thomas and Gürtler, Oliver (2019): A general framework for studying contests.

Abstract

We develop a general framework for studying contests, including the well-known models of Tullock (1980) and Lazear & Rosen (1981) as special cases. The contest outcome depends on players' efforts and skills, the latter being subject to symmetric uncertainty. The model is tractable, because a symmetric equilibrium exists under general assumptions regarding production technologies and skill distributions. Using a link between our contest model and expected utility theory, we are able to derive new comparative statics results regarding how the size and composition of contests affect equilibrium effort, showing how standard results can be overturned. We also discuss the robustness of our results to changes in the information structure and the implications of our findings for the optimal design of teams.

• Item Type: MPRA Paper
• Language: English
• Keywords: contest theory, symmetric equilibrium, heterogeneity, risk, decision theory
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games; D - Microeconomics > D7 - Analysis of Collective Decision-Making > D74 - Conflict ; Conflict Resolution ; Alliances ; Revolutions; D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D81 - Criteria for Decision-Making under Risk and Uncertainty; J - Labor and Demographic Economics > J2 - Demand and Supply of Labor > J23 - Labor Demand; M - Business Administration and Business Economics ; Marketing ; Accounting ; Personnel Economics > M5 - Personnel Economics > M51 - Firm Employment Decisions ; Promotions

References

Aguinis, H., & O’Boyle Jr., E. (2014). Star performers in twenty-first century organizations. Personnel Psychology, 67, 313–350. doi:10.1111/peps.12054.

Aoyagi, M. (2010). Information feedback in a dynamic tournament. Games and Economic Behavior, 70, 242–260. doi:10.1016/j.geb.2010.01.013.

Auriol, E., Friebel, G., & Pechlivanos, L. (2002). Career concerns in teams. Journal of Labor Economics, 20, 289–307. doi:10.1086/338240.

Barbieri, S., Malueg, D. A., & Topolyan, I. (2014). The best-shot all-pay (group) auction with complete information. Economic Theory, 57, 603–640. doi:10.1007/s00199-014-0847-6.

Barut, Y., & Kovenock, D. (1998). The symmetric multiple prize all-pay auction with complete information. European Journal of Political Economy, 14, 627–644. doi:10.1016/S0176-2680(98)00029-9.

Baye, M. R., Kovenock, D., & de Vries, C. G. (1996). The all-pay auction with complete information. Economic Theory, 8, 291–305. doi:10.1007/BF01211819.

Bernhardt, D. (1995). Strategic promotion and compensation. The Review of Economic Studies, 62, 315–339. doi:10.2307/2297807.

Bjerk, D. (2008). Glass ceilings or sticky floors? Statistical discrimination in a dynamic model of hiring and promotion. The Economic Journal, 118, 961–982. doi:10.1111/j.1468-0297.2008.02157.x.

Brown, J. (2011). Quitters never win: The (adverse) incentive effects of competing with superstars. Journal of Political Economy, 119, 982–1013. doi:10.1086/663306.

Chen, K.-P. (2003). Sabotage in promotion tournaments. Journal of Law, Economics, and Organization, 19, 119–140. doi:10.1093/jleo/19.1.119.

Chowdhury, S., Esteve-Gonzalez, P., & Mukherjee, A. (2019). Heterogeneity, leveling the playing field, and affirmative action in contests. Working Paper.

Chowdhury, S. M., & Gürtler, O. (2015). Sabotage in contests: A survey. Public Choice, 164, 135–155. doi:10.1007/s11127-015-0264-9.

Chowdhury, S. M., & Kim, S.-H. (2017). “Small, yet beautiful”: Reconsidering the optimal design of multi-winner contests. Games and Economic Behavior, 104, 486 – 493. doi:10.1016/j.geb.2017.05.012.

Clark, D., & Riis, C. (1998a). Competition over more than one prize. American Economic Review, 88, 276–89. URL: https://www.jstor.org/stable/i300820.

Clark, D. J., & Riis, C. (1996). On the win probability in rent-seeking games. Discussion Paper in Economics E4/96, University of Tromsø, Norway.

Clark, D. J., & Riis, C. (1998b). Contest success functions: An extension. Economic Theory, 11, 201–204. doi:10.1007/s001990050184.

Cohen, C., Kaplan, T. R., & Sela, A. (2008). Optimal rewards in contests. The RAND Journal of Economics, 39, 434–451. doi:10.1111/j.0741-6261.2008.00021.x.

Corchón, L., & Dahm, M. (2010). Foundations for contest success functions. Economic Theory, 43, 81–98. doi:10.1007/s00199-008-0425-x.

Cornes, R., & Hartley, R. (2005). Asymmetric contests with general technologies. Economic Theory, 26, 923–946. doi:10.1007/s00199-004-0566-5.

DeVaro, J., & Kauhanen, A. (2016). An “opposing responses” test of classic versus market-based promotion tournaments. Journal of Labor Economics, 34, 747–779. doi:10.1086/684911.

DeVaro, J., & Waldman, M. (2012). The signaling role of promotions: Further theory and empirical evidence. Journal of Labor Economics, 30, 91–147. doi:10.1086/662072.

Dewatripont, M., Jewitt, I., & Tirole, J. (1999). The economics of career concerns, part i: Comparing information structures. The Review of Economic Studies, 66, 183–198. doi:10.1111/1467-937X.00084.

Drugov, M., & Ryvkin, D. (2017). Biased contests for symmetric players. Games and Economic Behavior, 103, 116–144. doi:10.1016/j.geb.2016.02.003.

Fang, D., Noe, T., & Strack, P. (2019). Turning up the heat: The discouraging effect of competition in contests. Journal of Political Economy (forthcoming). doi:10.1086/705670.

Fu, Q., & Lu, J. (2009a). The beauty of “bigness”: On optimal design of multiwinner contests. Games and Economic Behavior, 66, 146 – 161. doi:10.1016/j.geb.2008.05.006.

Fu, Q., & Lu, J. (2009b). The optimal multi-stage contest. Economic Theory, 51, 351–382. doi:10.1007/s00199-009-0463-z.

Fu, Q., & Lu, J. (2012). Micro foundations of multi-prize lottery contests: a perspective of noisy performance ranking. Social Choice and Welfare, 38, 497–517. doi:10.1007/s00355-011-0542-5.

Fu, Q., Lu, J., & Pan, Y. (2015). Team contests with multiple pairwise battles. American Economic Review, 105, 2120–40. doi:10.1257/aer.20121469.

Fu, Q., & Wu, Z. (2019). Contests: Theory and topics. Oxford Research Encyclopedia of Economics and Finance, . doi:10.1093/acrefore/9780190625979.013.440.

Fullerton, R. L., & McAfee, R. P. (1999). Auctioning entry into tournaments. Journal of Political Economy, 107, 573–605. doi:10.1086/250072.

Gershkov, A., Li, J., & Schweinzer, P. (2009). Efficient tournaments within teams. The RAND Journal of Economics, 40, 103–119. doi:10.1111/j.1756-2171.2008.00057.x.

Gershkov, A., Li, J., & Schweinzer, P. (2016). How to share it out: The value of information in teams. Journal of Economic Theory, 162, 261–304. doi:10.1016/j.jet.2015.12.013.

Ghosh, S., & Waldman, M. (2010). Standard promotion practices versus up-orout contracts. The RAND Journal of Economics, 41, 301–325. doi:10.1111/j.1756-2171.2010.00101.x.

Gilpatric, S. M. (2009). Risk taking in contests and the role of carrots and sticks. Economic Inquiry, 47, 266–277. doi:10.1111/j.1465-7295.2008.00155.x.

Green, J. R., & Stokey, N. L. (1983). A comparison of tournaments and contracts. Journal of Political Economy, 91, 349–364. doi:10.1086/261153.

Grund, C., & Sliwka, D. (2005). Envy and compassion in tournaments. Journal of Economics & Management Strategy, 14, 187–207. doi:10.1111/j.1430-9134.2005.00039.x.

Gürtler, M., & Gürtler, O. (2015). The optimality of heterogeneous tournaments. Journal of Labor Economics, 33, 1007–1042. doi:10.1086/681207.

Gürtler, M., & Gürtler, O. (2019). Promotion signaling, discrimination, and positive discrimination policies. The RAND Journal of Economics, 50, 1004–1027. doi:10.1111/1756-2171.12303.

Hillman, A. L., & Riley, J. G. (1989). Politically contestable rents and transfers. Economics & Politics, 1, 17–39. doi:10.1111/j.1468-0343.1989.tb00003.x.

Holmström, B. (1982). Managerial Incentive Problems: A Dynamic Perspective. In Essays in Economics and Management in Honor of Lars Wahlbeck, Helsinki 1982. (pp. 169–182). Reprinted in The Review of Economic Studies (1999), 66(1). doi:10.1111/1467-937X.00083.

Holmström, B., & Ricard I Costa, J. (1986). Managerial incentives and capital management. Quarterly Journal of Economics, 101, 835–860. doi:10.2307/1884180.

Hvide, H. K. (2002). Tournament rewards and risk taking. Journal of Labor Economics, 20, 877–898. doi:10.1086/342041.

Imhof, L., & Kräkel, M. (2016). Ex post unbalanced tournaments. The RAND Journal of Economics, 47, 73–98. doi:10.1111/1756-2171.12119.

Jia, H. (2008). A stochastic derivation of the ratio form of contest success functions. Public Choice, 135, 125–130. doi:10.1007/s11127-007-9242-1.

Kirkegaard, R. (2020). Microfounded contest design. Working Paper.

Konrad, K. A. (2009). Strategy and Dynamics in Contests. Oxford University Press.

Kräkel, M. (2008). Optimal risk taking in an uneven tournament game with risk averse players. Journal of Mathematical Economics, 44, 1219–1231. doi:10.1016/j.jmateco.2008.02.002.

Kräkel, M., & Sliwka, D. (2004). Risk taking in asymmetric tournaments. German Economic Review, 5, 103–116. doi:10.1111/j.1465-6485.2004.00096.x.

Krumer, A., Megidish, R., & Sela, A. (2017). Round-robin tournaments with a dominant player. The Scandinavian Journal of Economics, 119, 1167–1200. doi:10.1111/sjoe.12204.

Lazear, E. P. (1989). Pay equality and industrial politics. Journal of Political Economy, 97, 561–580. doi:10.1086/261616.

Lazear, E. P., & Rosen, S. (1981). Rank-order tournaments as optimum labor contracts. Journal of Political Economy, 89, 841–864. doi:10.1086/261010.

Levy, H. (1992). Stochastic dominance and expected utility: Survey and analysis. Management Science, 38, 555–593. doi:10.1287/mnsc.38.4.555.

Malcomson, J. M. (1984). Work incentives, hierarchy, and internal labor markets. Journal of Political Economy, 92, 486–507. doi:10.1086/261238.

Malcomson, J. M. (1986). Rank-Order Contracts for a Principal with Many Agents. The Review of Economic Studies, 53, 807–817. doi:10.2307/2297720.

Moldovanu, B., & Sela, A. (2001). The optimal allocation of prizes in contests. American Economic Review, 91, 542–558. doi:10.1257/aer.91.3.542.

Moldovanu, B., & Sela, A. (2006). Contest architecture. Journal of Economic Theory, 126, 70–96. doi:10.1016/j.jet.2004.10.004.

Moldovanu, B., Sela, A., & Shi, X. (2007). Contests for status. Journal of Political Economy, 115, 338–363. doi:10.1086/518010.

Morath, F., & Münster, J. (2013). Information acquisition in conflicts. Economic Theory, 54, 99–129. doi:10.1007/s00199-012-0710-6.

Münster, J. (2009). Group contest success functions. Economic Theory, 41, 345–357. doi:10.1007/s00199-009-0479-4.

Nalebuff, B. J., & Stiglitz, J. E. (1983). Prizes and incentives: Towards a general theory of compensation and competition. The Bell Journal of Economics, 14, 21–43. doi:10.2307/3003535.

O’Keeffe, M., Viscusi, W. K., & Zeckhauser, R. J. (1984). Economic contests: Comparative reward schemes. Journal of Labor Economics, 2, 27–56. doi:10.1086/298022.

Olszewski, W., & Siegel, R. (2016). Large contests. Econometrica, 84, 835–854. doi:10.3982/ECTA11696.

Owan, H. (2004). Promotion, turnover, earnings, and firm-sponsored training. Journal of Labor Economics, 22, 955–978. doi:10.1086/423160.

Rinne, H. (2014). The hazard rate: Theory and inference - With supplementary MATLAB Programs. Technical Report Justus-Liebig-University, Giessen, Germany.

Rothschild, M., & Stiglitz, J. E. (1970). Increasing risk I: A definition. Journal of Economic Theory, 2, 225–243. doi:10.1016/0022-0531(70)90038-4.

Ryvkin, D., & Drugov, M. (2020). The shape of luck and competition in winnertake-all tournaments. Economic Theory (accepted).

Schotter, A., & Weigelt, K. (1992). Asymmetric tournaments, equal opportunity laws, and affirmative action: Some experimental results. Quarterly Journal of Economics, 107, 511–539. doi:10.2307/2118480.

Schöttner, A., & Thiele, V. (2010). Promotion tournaments and individual performance pay. Journal of Economics & Management Strategy, 19, 699–731. doi:10.1111/j.1530-9134.2010.00267.x.

Schweinzer, P., & Segev, E. (2012). The optimal prize structure of symmetric Tullock contests. Public Choice, 153, 69–82. doi:10.1007/s11127-011-9774-2.

Sela, A. (2012). Sequential two-prize contests. Economic Theory, 51, 383–395. doi:10.1007/s00199-010-0547-9.

Siegel, R. (2009). All-pay contests. Econometrica, 77, 71–92. doi:10.3982/ECTA7537.

Siegel, R. (2010). Asymmetric contests with conditional investments. The American Economic Review, 100, 2230–2260. doi:10.1257/aer.100.5.2230.

Skaperdas, S. (1996). Contest success functions. Economic Theory, 7, 283–290. doi:10.1007/BF01213906.

Tullock, G. (1980). Efficient rent seeking. In J. Buchanan, R. Tollison, & G. Tullock (Eds.), Towards a Theory of the Rent-Seeking Society (pp. 97–112). College Station, Tx.: Texas A&M University Press.

Vojnovi´c, M. (2015). Contest theory: Incentive mechanisms and ranking methods. Cambridge University Press.

Waldman, M. (1984). Job assignments, signalling, and efficiency. The RAND Journal of Economics, 15, 255–267. doi:10.2307/2555679.

Zábojník, J. (2012). Promotion tournaments in market equilibrium. Economic Theory, 51, 213–240. doi:10.1007/s00199-011-0606-x.

Zábojník, J., & Bernhardt, D. (2001). Corporate Tournaments, Human Capital Acquisition, and the Firm Size—Wage Relation. The Review of Economic Studies, 68, 693–716. doi:10.1111/1467-937X.00186.