# Heterogeneity and Number of Players in Rent-Seeking, Innovation, and Patent-Race Games

## Abstract

Many economists have studied rent-seeking contests, innovation tournaments, and patent-race games independently. These three seemingly different games are known to be strategically equivalent under some reasonable assumptions. In these classic games, it is assumed that the value of a prize, i.e. the gain from rent-seeking, achieving an innovation, or obtaining a patent, is exogenously given and does not depend on the number of players, so that an increase in the number of players decreases the winning rate of each player. However, if players engage in R&D and then set quantities à la Cournot, the value of the prize in general depends on the number of players. In this paper, we set up a model with one efficient player and identical inefficient players to analyze how an increase in heterogeneity among players or the number of players changes the wining rate of the efficient player. One of the main results is that if the number of players is larger than some critical value, which can be less than two, an increase in the number of inefficient players always increases the winning rate of the efficient player.

## Keywords

Marginal Cost Cost Difference Investment Level Investment Incentive Unit Production Cost## Notes

### Acknowledgements

This research was in part supported by JSPS KAKENHI (Grants-in-Aid for Scientific Research) Grant Number 19530151. An earlier version of this paper was presented at PET10 (the 9th annual conference of the association for public economic theory) held in Istanbul, 2010. We would like to thank a referee of this Festschrift and the participants in PET10 and the workshop held at Niigata University for many helpful comments and suggestions. Only the authors are responsible for any remaining errors and omissions.

## References

- Baye MR, Hoppe HC (2003) The strategic equivalence of rent-seeking, innovation, and patent-race games. Games Econ Behav 44:217–226CrossRefGoogle Scholar
- Dasgupta P, Stiglitz J (1980) Uncertainty, industrial structure, and the speed of R&D. Bell J Econ 11:1–28CrossRefGoogle Scholar
- Ishida J, Matsumura T, Matsushima N (2011) Market competition, R&D and firm profits in asymmetric oligopoly. J Ind Econ 59:484–505CrossRefGoogle Scholar
- Kooreman P, Schoonbeek L (1997) The specification of the probability functions in Tullock’s rent-seeking contests. Econ Lett 56:59–61CrossRefGoogle Scholar
- Loury GC (1979) Market structure and innovation. Q J Econ 93:395–410CrossRefGoogle Scholar
- Perez-Castrillo JD, Verdier T (1992) A general analysis of rent-seeking games. Public Choice 73:335–350CrossRefGoogle Scholar
- Skaperdas S (1997) Contest success functions. Econ Theory 7:283–290CrossRefGoogle Scholar
- Szidarovszky F, Okuguchi K (1997) On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games Econ Behav 18:135–140CrossRefGoogle Scholar
- Tullock G (1980) Efficient rent-seeking. In: Buchanan JM, Tollison RD, Tullock G (eds) Toward a theory of the rent-seeking society. Texas A&M Press, College StationGoogle Scholar
- Yamazaki T (2008) On the existence and uniqueness of pure-strategy nash equilibrium in asymmetric rent-seeking contests. J Public Econ Theory 10:317–327CrossRefGoogle Scholar