## Abstract

We consider a two-player contest in which one contestant has a headstart advantage, but both can exert further effort. We allow the prize to depend on total performance in the contest and consider the respective cases in which efforts are productive and destructive of prize value. When the contest success function takes a logit form, and marginal cost is increasing in effort, we show that a Nash equilibrium exists and is unique both in productive and destructive endogenous prize contests. In equilibrium, the underdog expends more resources to win the prize, but still his probability of winning remains below that of the favorite. In a productive contest, the underdog behaves more aggressively and wins the prize more often in comparison to a fixed-value contest. Thus, the degree of competitive balance—defined as the level of uncertainty of the outcome—depends upon the (fixed or endogenous) prize nature of the contest.

## Keywords

Endogenous prize contests Productive and destructive effort Competitive balance## Notes

### Acknowledgements

We thank Jürgen Eichberger, Jörg Oechssler, seminar participants at the University of Heidelberg and the audience at the 5th World Congress of the Game Theory Society in Maastricht for their helpful comments and suggestions.

## References

- Amegashie, J. A., & Kutsoati, E. (2005). Rematches in boxing and other sporting events.
*Journal of Sports Economics*,*6*(4), 401–411.CrossRefGoogle Scholar - Baik, K. H. (1994). Effort levels in contests with two asymmetric players.
*Southern Economic Journal*,*61*(2), 367–378.CrossRefGoogle Scholar - Baye, M. R., Kovenock, D., & De Vries, C. G. (2012). Contests with rank-order spillovers.
*Economic Theory*,*51*(2), 315–350.CrossRefGoogle Scholar - Beviá, C., & Corchón, L. C. (2013). Endogenous strength in conflicts.
*International Journal of Industrial Organization*,*31*(3), 297–306.CrossRefGoogle Scholar - Borland, J., & Macdonald, R. (2003). Demand for Sport.
*Oxford Review of Economic Policy*,*19*, 478–502.CrossRefGoogle Scholar - Chang, Y. M., & Luo, Z. (2013). War or settlement: An economic analysis of conflcit with endogenous and increasing destruction.
*Defence and Peace Economics*,*24*(1), 23–46.CrossRefGoogle Scholar - Chang, Y. M., & Luo, Z. (2016). Endogenous destruction in conflict: Theory and extensions.
*Economic Inquiry*,*55*(1), 479–500.CrossRefGoogle Scholar - Chiang, A. C., & Wainwright, K. (2005).
*Fundamental methods of mathematical economics*. New York: McGraw Hill.Google Scholar - Chowdhury, S. M., & Sheremata, R. M. (2011). A generalized Tullock contest.
*Public Choice*,*147*(3–4), 413–420.CrossRefGoogle Scholar - Chung, T.-Y. (1996). Rent-seeking contest when the prize increases with aggregate efforts.
*Public Choice*,*87*(1–2), 55–66.CrossRefGoogle Scholar - Cohen, C., Kaplan, T. R., & Sela, A. (2008). Optimal rewards in contests.
*RAND Journal of Economics*,*39*(2), 434–451.CrossRefGoogle Scholar - Congleton, R. D., Hillman, A. L., & Konrad, K. A. (2008). An overview. In R. D. Congleton, A. L. Hillman, & K. A. Konrad (Eds.),
*40 years of research on rent seeking (vol. 1 and, vol. 2)*(pp. 1–42). Berlin: Springer.Google Scholar - Cornes, R., & Hartley, R. (2003). Risk aversion, heterogeneity and contests.
*Public Choice*,*117*(1–2), 1–25.CrossRefGoogle Scholar - Cornes, R., & Hartley, R. (2005). Asymmetric contests with general technologies.
*Economic Theory*,*26*(4), 923–946.CrossRefGoogle Scholar - Cornes, R., & Hartley, R. (2012). Risk aversion in symmetric and asymmetric contests.
*Economic Theory*,*51*(2), 247–275.CrossRefGoogle Scholar - Dixit, A. (1987). Strategic behavior in contests.
*American Economic Review*,*77*(5), 891–898.Google Scholar - Fan, K. (1952). Fixed-point and minimax theorems in locally convex topological linear spaces.
*Proceedings of the National Academy of Sciences of the United States of America*,*38*(2), 121.CrossRefGoogle Scholar - Forrest, D., & Simmons, R. (2002). Outcome uncertainty and attendance demand for sport: the case of English soccer.
*Journal of the Royal Statistical Society, Series D (The Statisticial)*,*51*, 229–241.CrossRefGoogle Scholar - Froeb, L. M., & Kobayashi, B. H. (1996). Naive, biased, yet Bayesian: can juries interpret selectively produced evidence?
*Journal of Law, Economics and Organization*,*12*(1), 257–276.CrossRefGoogle Scholar - Garfinkel, M. R., & Skaperdas, S. (2000). Conflict without misperceptions and incomplete information: How the future matters.
*Journal of Conflict Resolution*,*44*, 793–807.CrossRefGoogle Scholar - Glicksberg, I. L. (1952). A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points.
*Proceedings of the American Mathematical Society*,*3*(1), 170–174.Google Scholar - Hirai, S. (2012). Existence and uniqueness of pure Nash equilibrium in asymmetric contests with endogenous prizes.
*International Game Theory Review*,*32*(4), 2744–2751.Google Scholar - Hirai, S., & Szidarovszky, F. (2013). Existence and uniqueness of equilibrium in asymmetric contests with endogenous prizes.
*International Game Theory Review*,*15*(1), 1–9.CrossRefGoogle Scholar - Hirshleifer, J. (1995). Theorizing about conflict. In: Hartley, K., & Sandler, T. (Eds.),
*Handbook of defense economics*(vol. 1, pp 165–189). North-Holland: Department of Economics, University of California.Google Scholar - Hirshleifer, J. (1989). Conflict and rent-seeking success functions: Ratio vs. difference models of relative success.
*Public Choice*,*63*(2), 101–112.CrossRefGoogle Scholar - Hirshleifer, J. (2001).
*The dark side of the force: Economic foundations of conflict theory*. Cambridge: Cambridge University Press.Google Scholar - Kakutani, S. (1941). A generalization of Brouwer’s fixed point theorem.
*Duke Mathematical Journal*,*8*, 457–458.CrossRefGoogle Scholar - Konrad, K. A. (2009).
*Strategy and dynamics in contests*. New York: Oxford University Press.Google Scholar - Nti, K. O. (1997). Comparative statics of contests and rent-seeking games.
*International Economic Review*,*38*(1), 43–59.CrossRefGoogle Scholar - Owen, P. D., & King, N. (2015). Competitive balance measures in sports leagues: The effects of variation in season length.
*Economic Inquiry*,*53*(1), 731–744.CrossRefGoogle Scholar - Ridlon, R. (2016). Does manufacturer advertising crowd-in or crowd-out retailer advertising? An application of an endogenous prize contest with asymmetric players.
*Southern Economic Journal*,*83*(2), 364–379.CrossRefGoogle Scholar - Sanders, S., & Walia, B. (2014). Endogenous destruction in a model of armed conflict: Implications for conflict intensity, welfare, and third-party intervention.
*Journal of Public Economic Theory*,*16*(4), 606–619.CrossRefGoogle Scholar - Shaffer, S. (2006). War, labor tournaments, and contest payoffs.
*Economics Letters*,*92*(2), 250–255.CrossRefGoogle Scholar - Siegel, R. (2014). Contests with productive effort.
*International Journal of Game Theory*,*43*(3), 515–523.CrossRefGoogle Scholar - Skaperdas, S., & Gan, L. (1995). Risk aversion in contests.
*Economic Journal*,*105*(431), 951–962.CrossRefGoogle Scholar - Smith, A. C., Houser, D., Leeson, P. T., & Ostad, R. (2014). The costs of conflict.
*Journal of Economic Behavior and Organization*,*97*, 61–71.CrossRefGoogle Scholar - Szidarovszky, F., & Okuguchi, K. (1997). On the existence and uniqueness of pure Nash equilibrium in rent-seeking games.
*Games and Economic Behavior*,*18*(1), 135–140.CrossRefGoogle Scholar - Szymanski, S. (2003). The economic design of sporting contests.
*Journal of Economic Literature*,*41*, 1137–1187.CrossRefGoogle Scholar - Tullock, G. (1980). Efficient rent seeking. In James Buchanan, Roger Tollison, & Gordon Tullock (Eds.),
*Toward a Theory of the Rent-Seeking Society*(pp. 97–112). College Station: Texas A&M University Press.Google Scholar