First-price winner-takes-all contests

Abstract

We introduce a unifying model for contests with perfect discrimination and show that it can be used to model many well-known economic situations, such as auctions, Bertrand competition, politically contestable rents and transfers, tax competition and litigation problems. Furthermore, we hope that the generality of our model can be used to study models of contests in settings in which they have not been applied yet. Our main result is a classification of the set of Nash equilibria of first-price winner-takes-all contests with complete information. Finally, we discuss the implications of our results in each one of the specific models.

Keywords:

• winner-takes-all contests
• first price
• perfect discrimination
• complete information
• auctions
• rent-seeking
• bertrand competition
• timing games
• tax competition
• litigation
• political campaigns

AMS Subject Classifications:

• 91A40
• 91A06
• 91A55

Acknowledgements

The author is indebted to Ron Siegel for his help through very enlightening discussions. The author is also grateful to Peter Borm, Luis Corchón, Diego Domínguez, Ehud Kalai, Jordi Massó, Miguel A. Meléndez-Jiménez, Henk Norde and Sergio Vicente for helpful discussions and also to the people at Northwestern University, where most of this research was carried out, for their hospitality. Moreover, the author acknowledges the financial support of the Spanish Ministry for Science and Innovation and FEDER through project ECO2008-03484-C02-02. This research has also been supported by a Marie Curie International Fellowship within the 6th European Community Framework Programme.

Notes

1. Moldovanu and Sela 25 make a detailed review of these connections in the introduction to their paper.

2. In 2, the authors do a similar exercise to the one we present here and formally establish the strategic equivalence of three models of contests that do not assume perfect discrimination. Also, refer to 10,30 for two papers without perfect discrimination (on the set of winners).

3. Indeed, Baye et al. 7 already pointed out that it would be interesting to extend their results to situations where such asymmetries in the returns for the efforts are present (yet, they do not assume complete information, being this extension far more difficult in their setting).

4. We are making the following abuse of notation. If M = +∞, then [m, M] ≔ [m, +∞), and the same for the [m _i , M _i ] intervals.

5. Note that the definition of the r _i functions implies that they are continuous and, moreover, it also implies that either M < ∞ and, for each i ∈ N, M _i  < ∞ or, for each i ∈ N, M = M _i  = ∞.

6. Here we even allow for situations in which players ‘lose less’ when they are tied than they would lose being alone. Note that this is the case, for instance, in first price auctions when players are tied at bids that exceed their valuations.

7. In BM, we assume that the demand and cost functions satisfy that is strictly decreasing in e. We briefly discuss about the necessity of this assumption in Section 6.

8. We have to mention a subtle technical detail. To rigorously define the Lebesgue-Stieltjes integrals with respect to the distribution functions we should integrate over ℝ. Hence, we should also define payoff functions over ℝ. But note that, no matter the extension of the u _i functions we consider, the integrals over [m _i , M _i ] remain the same (just because the support of the mixed strategies is restricted to [m _i , M _i ]). If we do not consider integrals defined over ℝ, then we might have problems when calculating expected payoffs of mixed strategies that put positive probability at m _i . Hence, we use ∫ u _i (G ₁, … , G _i−1, e, G _i+1, … , G _n )dG _i (e) to mean , where can be any arbitrarily chosen extension of u _i to ℝ.

9. Let f = ({b _i } _i∈N , {p _i } _i∈N , {T _i } _i∈N ) be a direct FP-WTA form such that, in C ^f , m ≠ 0. Then, we just need to consider the FP-WTA form , where each is defined by , and similarly for the and functions. The games C ^f and C ^f′ are completely equivalent from the strategic point of view.

10. This non-crossing property is quite standard in the literature on contests. In the recent years there have been some efforts trying to dispense with it in different settings; remarkably, Siegel 28,29 introduces a general model of contests, similar to the one we present here, and develops his analysis without assuming any kind of non-crossing property. Yet, he needs some other assumptions to characterize the set of equilibria in his setting.

11. Model MS might not meet the assumption if we allow simultaneously for different numbers of loyal consumers and different cost functions for the different firms.

12. The equilibria of the models discussed in this section are particular cases of Theorem 1 and are described by the expressions presented in the appendix.

13. With the exception of the possible existence in our general setting of the gap equilibria defined below, our classification is the extension of the one made in 6 to our general framework.

14. Siegel 28 shows an example of a ‘non-constructible’ equilibrium in which S(G ₁) coincides with the Cantor set.

15. Yet, one observation has to be made. Namely, for the specific parameter case ē₁ > ē₂ = ē₃, the extension of the closed form characterization in Baye et al. 6 might not completely characterize the set of equilibria. We refer the reader to the appendix for the details and discussion around this specific parameter configuration.

16. Now we do not try to give a characterization result as Theorem 1. This is because the set of Nash equilibria would depend on the specific configurations of the ē _i parameters, on the tie payoff functions, and on whether NO-CROSSING and NO-CROSSING* are met or not. Hence, a clean result as Theorem 1 is not possible here (at least we have not been able to find it).

17. Again, in BM, has to be strictly decreasing in e.

18. If there are no discontinuities in (e, e + ϵ], then a continuity argument for that interval does the job. If there are discontinuities of in (e, e + ϵ], then they make the function take even higher values at e + ϵ′.

19. Note that Equation (2) is consistent with the requirement ∏ _j≠1 G _j (e _G ) = 1 imposed by statement (i) of Lemma 3.

20. In the corresponding expressions for the equilibrium strategies when ē₁ = ē₂ included in Baye et al. 3,6 there is a minor typo. They wrote instead of in the expressions of the G _i functions.