Skip to main content
Log in

Contests with dominant strategies

  • Research Article
  • Published:
Economic Theory Aims and scope Submit manuscript

Abstract

In this paper, we first characterize the class of contest success functions (CSFs) that yield contests with equilibria in dominant strategies. Then we study the optimal contest as a decision problem under uncertainty. We consider the classical criteria of Wald, Laplace, Hurwicz and Savage. We find that the CSF that maximizes aggregate effort under these criteria has the form of an additively separable cutoff CSF.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. For the complexity of computing Nash equilibrium, see e.g., Daskalakis et al. (2009). These problems might explain why Nash equilibrium predictions do not always fare well in experiments, see e.g., Dechenaux et al. (2015).

  2. The assumption of a common prior is specially problematic in contests like trials, awarding scholarships or job applications.

  3. Other notions of robustness have appeared in the literature. Corchon and Ortuño-Ortin (1994) assume that agents have complete information about a part of the environment and look for a mechanism yielding socially optimal allocations in Uniform Nash Equilibrium and Bayesian Equilibrium for any possible prior.

  4. The only papers that we are aware of that take a similar approach to ours is by Bergemann and Schlag (2011) on the design of monopoly pricing and by Rezai and van der Ploeg (2017) on the design of climate policy.

  5. In the Minimax Regret case, we assume that the optimal CSF is of a cutoff type, which is not the case in the other three criteria where cutoff mechanism emerges as the optimal ones.

  6. In the standard Bayesian setting agents know the domain of preferences, a vector of prior probabilities of occurrence of the valuations of other agents and the statistical distribution of other agents’ valuations. In our setting, there is no knowledge of both priors and the probability distribution.

  7. We refer to dominant strategies in the weak sense.

  8. As far as we know, the first paper using differentiability to find mechanisms with dominant strategies is Laffont and Maskin (1980).

  9. Given that \(u_{i}(0,{\mathbf {g}}_{-i}\mathbf {)}\ge 0\) and \(u_{i}(g_{i}, {\mathbf {g}}_{-i}\mathbf {)}\le 0\) for all \(g_{i}\ge v_{i},\) interiority is a valid assumption as long as \(\frac{\partial p(_{0},{\mathbf {g}}_{-i}\mathbf {)} }{\partial g_{i}}v_{i}>1\).

  10. The revelation principle has been applied to crowdsourcing contests by Chawla et al. (2019). In these contests, firms use inputs obtained from Internet users to build projects. There are similarities and differences of this paper with our approach. Among the former, they consider anonymous mechanisms and study the optimal prior independent contest. Among the latter, they assume that the planner is only interested in the submission with the highest quality. They use Bayesian equilibrium as equilibrium concept and their setting is an all pay auction.

  11. There is a literature evolving from the paper by Myerson (1983) which assumes that the principal has private information. This literature is important but it does not form the backbone of mechanism design. More references on this topic can be found in https://sites.google.com/site/tmylovanov/informed-principal.

  12. We note that our result here is driven by the maximin criterion which forces the planner to focus on the worst possible event and not by the solution concept, i.e., equilibrium in dominant strategies.

  13. For \(\alpha =1\) this reduces to the Maximin criterion.

  14. We can partition the space of valuations in this way because a monotone function has at most a countable number of discontinuities.

References

  • Barberá, S., Pattanaik, P.: Falmagne and the rationalizability of stochastic choices in terms of random orderings. Econometrica 54, 707–715 (1986)

    Article  Google Scholar 

  • Bergemann, D., Schlag, K.: Robust monopoly pricing. J. Econ. Theory 146, 2527–2543 (2011)

    Article  Google Scholar 

  • Bergemann, D., Morris, S.: Robust Mechanism Design: The Role of Private Information and Higher Order Beliefs, vol. 8318. World Scientific Publishing, Singapore (2012)

    Book  Google Scholar 

  • Bossert, W., Pattanaik, P.K., Xu, Y.: Choice under complete uncertainty: axiomatic characterizations of some decision rules. Econ. Theory 16, 295–312 (2000). https://doi.org/10.1007/PL00004080

    Article  Google Scholar 

  • Chawla, S., Hartline, jD, Sivan, B.: Optimal crowdsourcing contests. Games Econ. Behav. 113, 80–96 (2019)

    Article  Google Scholar 

  • Corchon, L., Ortuño-Ortin, I.: Robust implementation under alternative information structures. Econ. Des. 1, 159–171 (1994)

    Google Scholar 

  • Cubel, M., Sanchez-Pages, S.: An axiomatization of difference-form contest success functions. J. Econ. Behav. Organ. 131, 92–105 (2016)

    Article  Google Scholar 

  • Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a nash equilibrium. SIAM J. Comput. 39, 195–259 (2009)

    Article  Google Scholar 

  • Dechenaux, E., Kovenock, D., Sheremeta, R.: A survey of experimental research on contests, all-pay auctions and tournaments. Exp. Econ. 18, 609–669 (2015)

    Article  Google Scholar 

  • Gill, D., Prowse, V.: A structural analysis of disappointment aversion in real effort competion. Am. Econ. Rev. 102, 469–503 (2012)

    Article  Google Scholar 

  • Jia, H., Skaperdas, S., Vaidya, S.: Contest functions: theoretical foundations and issues in estimation. Int. J. Ind. Organ. 31, 211–22 (2013)

    Article  Google Scholar 

  • Laffont, J.J., Maskin, E.: A differential approach to dominant strategy mechanisms. Econometrica 48, 1507–1520 (1980)

    Article  Google Scholar 

  • Maskin, E., Sjöström, T.: Implementation theory. In: Arrow, K., Sen, A., Suzumura, K. (eds.) Handbook of Social Choice Theory, vol. I, pp. 237–288. Elsevier, Amsterdam (2002)

    Google Scholar 

  • Milnor, J.: Games against nature. In: Thrall, R., Coombs, C., Davis, R. (eds.) Decision Processes, pp. 49–61. Wiley, New York (1954)

    Google Scholar 

  • Myerson, R.: Optimal auction design. Math. Oper. Res. 6, 58–73 (1981)

    Article  Google Scholar 

  • Myerson, R.: Mechanism design by an informed principal. Econometrica 51, 1767–1797 (1983)

    Article  Google Scholar 

  • Polishchuk, L., Tonis, A.: Endogenous contest success functions: a mechanism design approach. Econ. Theory 52, 271–297 (2013). https://doi.org/10.1007/s00199-011-0622-x

    Article  Google Scholar 

  • Renou, L., Schlag, K.: Minimax regret and strategic uncertainty. J. Econ. Theory 145, 264–286 (2010)

    Article  Google Scholar 

  • Rezai, A., van der Ploeg, F.: Climate policies under climate model uncertainty: max–min and min–max regret uncertainty. Energy Econ. 68, 4–16 (2017)

    Article  Google Scholar 

  • Serena, M.: Quality contests. Eur. J. Polit. Econ. 46, 15–25 (2017)

    Article  Google Scholar 

  • Skaperdas, S., Vaidya, S.: Persuasion as a contest. Econ. Theory 51, 465–486 (2012). https://doi.org/10.1007/s00199-009-0497-2

    Article  Google Scholar 

  • Skaperdas, S., Toukanb, A., Vaidya, S.: Difference-form Persuasion contests. J. Public Econ. Theory 18, 882–909 (2016)

    Article  Google Scholar 

  • Wilson, R.: Game-theoretic approaches to trading processes. In: Bewley, T. (ed.) Advances in Economic Theory: Fifth World Congress, pp. 33–77. Cambridge University Press, Cambridge (1987)

    Chapter  Google Scholar 

Download references

Acknowledgements

We are grateful to Z. Méder, S. Heinsalu, N. Kudryashova, J. Rueda-Llano, H. Sabourian, B. Schipper, the associate editor and four anonymous referees for very helpful comments. Thanks to the MOMA network under the project ECO2014-57673-REDT for financial support. The first author acknowledges financial support from ECO2014 53051, SGR2014-515 and PROMETEO/2013/037. The second author acknowledges financial support from ECO2014_57442_P, ECO2017_87769_P, MDM 2014-0431 and S2015/HUM-3444.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carmen Beviá.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

The proof of Lemma 1 uses two more lemmas. The first one is a direct adaptation to our framework of Lemma 2 in Myerson (1981). We present here the statement of this lemma without proof. The second lemma is an auxiliary technical result.

Lemma 3

(Myerson 1981) A revelation contest \((\mathbf {P,G})\) is IC and IR if and only if

  1. (1)

    For all \({\mathbf {V}}_{-i}\in [0,{\bar{V}}]^{n-1},\) if \(s_{i}\le t_{i},\) \(s_{i},t_{i}\in [0,{\bar{V}}]\) then \(P(s_{i},{\mathbf {V}} _{-i})\le P(t_{i},{\mathbf {V}}_{-i}).\)

  2. (2)

    \(u(s_{i},{\mathbf {V}}_{-i}\mid s_{i})=u(0,{\mathbf {V}}_{-i}\mid 0)+\int _{0}^{s_{i}}P(s,{\mathbf {V}}_{-i})\) \(\mathrm{d}s.\)

  3. (3)

    \(u(0,{\mathbf {V}}_{-i}\mid 0)\ge 0\) for all \({\mathbf {V}}_{-i}\in [0, {\bar{V}}]^{n-1}.\)

Lemma 4

Let \(\phi :[0,{\bar{V}}]\rightarrow {\mathbb {R}} \) and \(\psi :[0,{\bar{V}}]^{n-1}\rightarrow {\mathbb {R}} \) be such that \(\phi (0)=0,\) \(\psi ({\mathbf {0}}_{-i})=0\) for all i,  and \( \sum _{i=1}^{n}\phi (V_{i})=-\sum _{i=1}^{n}\psi ({\mathbf {V}}_{-i}).\) Then, \( \psi ({\mathbf {V}}_{-i})=-\frac{1}{n-1}\sum _{j\ne i}\phi (V_{j}).\)

Proof

Let \((V_{i},{\mathbf {0}}_{-i}),\) then

$$\begin{aligned} \sum _{i=1}^{n}\phi (V_{i})=\phi (V_{i})=-\sum _{j=1,j\ne i}^{n}\psi (V_{i}, {\mathbf {0}}_{-i-j})=-(n-1)\psi (V_{i},0,0,..^{(n-2)}.,0). \end{aligned}$$

Let \((V_{i},V_{j},{\mathbf {0}}_{-i-j}),\) then

$$\begin{aligned} \sum _{i=1}^{n}\phi (V_{i})= & {} \phi (V_{i})+\phi (V_{j})\\= & {} -\psi (V_{i},{\mathbf {0}} _{-i-j})-\psi (V_{j},{\mathbf {0}}_{-i-j})-\sum _{k=1,k\ne i,j}^{n}\psi (V_{i},V_{j},{\mathbf {0}}_{-i-j-k}). \end{aligned}$$

Since \(-\psi (V_{i},{\mathbf {0}}_{-i-j})=\frac{1}{n-1}\phi (V_{i}),\) and \( -\psi (V_{j},{\mathbf {0}}_{-i-j})=\frac{1}{n-1}\phi (V_{j}),\)

$$\begin{aligned} \phi (V_{i})+\phi (V_{j})=\frac{1}{n-1}\phi (V_{i})+\frac{1}{n-1}\phi (V_{j})-(n-2)\psi (V_{i},V_{j},0,0,..^{(n-3)}.,0)). \end{aligned}$$

Thus, \(\frac{1}{n-1}(\phi (V_{i})+\phi (V_{j}))=-\psi (V_{i},V_{j},0,0,..^{(n-3)}.,0))\). Applying an induction argument, we get that \(\frac{1}{n-1}\sum _{j\ne i}\phi (V_{j})=-\psi ({\mathbf {V}}_{-i}).\) \(\square \)

Proof of Lemma 1

If the revelation contest \((\mathbf {P,G})\) is incentive compatible and individually rational, it satisfies (1), (2) and (3) in Lemma 3. First of all, note that condition (2) in Lemma 3 implies that

$$\begin{aligned} G(V_{i})=V_{i}P(V_{i},{\mathbf {V}}_{-i})+G(0)-\int _{0}^{V_{i}}P(s, {\mathbf {V}}_{-i})\mathrm{d}s. \end{aligned}$$
(5.1)

Let us see that \(G(\cdot )\) is increasing. By (1) in Lemma 3, \( P(V_{i},{\mathbf {V}}_{-i})\) is increasing in \(V_{i}.\) Suppose that there is a pair \(V_{i},V_{i}^{\prime }\) such that \(V_{i}^{\prime }>V_{i}\) but \( G(V_{i})>G(V_{i}^{\prime })\). Thus \(P(V_{i}^{\prime },{\mathbf {V}}_{-i})\ge P(V_{i},{\mathbf {V}}_{-i})\) and by incentive compatibility,

$$\begin{aligned} P(V_{i},{\mathbf {V}}_{-i})V_{i}-G(V_{i})\ge & {} P(V_{i}^{\prime }, {\mathbf {V}}_{-i})V_{i}-G(V_{i}^{\prime })\text { or equivalently, } \\ (P(V_{i},{\mathbf {V}}_{-i})-P(V_{i}^{\prime },{\mathbf {V}}_{-i}))V_{i}\ge & {} G(V_{i})-G(V_{i}^{\prime })>0 \end{aligned}$$

which is a contradiction. Thus if \(P(V_{i},{\mathbf {V}}_{-i})\) is increasing in \(V_{i},\) \(G(V_{i})\) is increasing in \(V_{i}\) too. Given (5.1),

$$\begin{aligned} V_{i}(P(V_{i},{\mathbf {V}}_{-i})-P(V_{i},{\mathbf {V}}_{-i}^{\prime }))=\int _{0}^{V_{i}}(P(s,{\mathbf {V}}_{-i})-P(s,{\mathbf {V}}_{-i}^{\prime }))\mathrm{d}s, \end{aligned}$$

which implies that \(P\, \ \)is separable, that is:

$$\begin{aligned} P(V_{i},{\mathbf {V}}_{-i})=F(V_{i})+H({\mathbf {V}}_{-i}). \end{aligned}$$

Let \(\phi (V_{i})=F(V_{i})-F(0),\) and \(\psi ({\mathbf {V}}_{-i})=H({\mathbf {V}} _{-i})-H({\mathbf {0}}_{-i})\) and let \(A=F(0)+H({\mathbf {0}}_{-i}).\) Thus, \( P(V_{i},{\mathbf {V}}_{-i})=\phi (V_{i})+\psi ({\mathbf {V}}_{-i})+A.\) Note that \( \phi (0)=0\) and \(\psi ({\mathbf {0}}_{-i})=0\), therefore \(P({\mathbf {0}})=A\) for all i. Thus, \(A=1/n.\) Given that \(\sum _{i=1}^{n}P(V_{i},{\mathbf {V}} _{-i})=1, \) \(\sum _{i=1}^{n}\phi (V_{i})=-\sum _{i=1}^{n}\psi ({\mathbf {V}} _{-i}).\) By Lemma  4 using this equality and playing with vectors of valuations of the form \((V_{i},{\mathbf {0}}_{-i}),\) \((V_{i},V_{j}, {\mathbf {0}}_{-i-j})\) and so on, \(\psi ({\mathbf {V}}_{-i})=-\frac{1}{n-1} \sum _{j\ne i}\phi (V_{j}).\) Thus, \(P(V_{i},{\mathbf {V}}_{-i})=\phi (V_{i})- \frac{1}{n-1}\sum _{j\ne i}\phi (V_{j})+\frac{1}{n}.\) By Lemma 3, \(P(\cdot )\) is increasing in \(V_{i},\) thus \(\phi (\cdot )\) is increasing, and since \(P(\cdot )\) is a probability distribution, \(0\le \phi (V_{j})\le 1/n\). By point (3) in Lemma 3, \(u(0,{\mathbf {V}}_{-i}\mid 0)\ge 0\) which implies that \(G(0)=0.\) By (5.1) and point (1)

$$\begin{aligned} G(V_{i})=V_{i}\phi (V_{i})-\int _{0}^{V_{i}}\phi (s)\mathrm{d}s. \end{aligned}$$

Finally, let us see that a revelation contest \((\mathbf {P,G})\) that satisfies (1) and (2) in the statement of this lemma is incentive compatible and individually rational. Let us see first that \((\mathbf {P,G})\) is individually rational, that is, \(P(V_{i},{\mathbf {V}}_{-i})\) \( V_{i}-G(V_{i})\ge 0.\, \ \)Given that \(0\le \phi (V_{i})\le 1/n\, \) for all \(V_{i},\) \(P(V_{i},{\mathbf {V}}_{-i})\) \(V_{i}-G(V_{i})\ge \int _{0}^{V_{i}}\phi (s)\) \(\mathrm{d}s\ge 0.\) In order to prove that \((\mathbf {P,G})\) is incentive compatible, we have to show that for each individual with valuation \(v_{i},\)

$$\begin{aligned} v_{i}P_{i}(v_{i},{\mathbf {V}}_{-i})-G_{i}(v_{i})\ge v_{i} P_{i}(V_{i},{\mathbf {V}}_{-i})-G_{i}(V_{i}),\forall (V_{i},{\mathbf {V}} _{-i}), \end{aligned}$$

which is equivalent to proving that

$$\begin{aligned} \phi (V_{i})(V_{i}-v_{i})\ge \int _{0}^{V_{i}}\phi (s) \mathrm{d}s-\int _{0}^{v_{i}}\phi (s)\mathrm{d}s. \end{aligned}$$
(5.2)

If \(V_{i}>v_{i},\)

$$\begin{aligned} \int _{0}^{V_{i}}\phi (s)\mathrm{d}s-\int _{0}^{v_{i}}\phi (s) \mathrm{d}s=\int _{v_{i}}^{V_{i}}\phi (s)\mathrm{d}s, \end{aligned}$$

and since \(\phi \) is increasing, for all \(s\in [v_{i},V_{i}],\) \(\phi (s)\le \phi (V_{i}).\) Therefore,

$$\begin{aligned} \int _{v_{i}}^{V_{i}}\phi (s)\mathrm{d}s\le \phi (V_{i})(V_{i}-v_{i}). \end{aligned}$$

If \(V_{i}<v_{i},\)

$$\begin{aligned} \int _{0}^{V_{i}}\phi (s)\mathrm{d}s-\int _{0}^{v_{i}}\phi (s) \mathrm{d}s=-\int _{V_{i}}^{v_{i}}\phi (s)\mathrm{d}s \end{aligned}$$

and since \(\phi \) is increasing, for all \(s\in [V_{i},v_{i}],\) \(\phi (s)\ge \phi (V_{i}).\) Therefore,

$$\begin{aligned} -\int _{V_{i}}^{v_{i}}\phi (s)\mathrm{d}s\le -\phi (V_{i})(v_{i}-V_{i})=\phi (V_{i})(V_{i}-v_{i}). \end{aligned}$$

Thus, in both cases (5.2) holds and therefore the revelation contest \(( \mathbf {P,G})\) is incentive compatible. \(\square \)

Proof of Proposition 1

If a CSF yields contests with an equilibrium in dominant strategies, it has an associated revelation contest, \(({\mathbf {P}},{\mathbf {G}}),\) which is incentive compatible and individually rational. By Lemma 1, \((\mathbf {P,G)}\) is such that \( P(V_{i},{\mathbf {V}}_{-i})=\phi (V_{i})-\frac{1}{n-1}\sum _{j\ne i}\phi (V_{j})+\frac{1}{n}\), where \(\phi :[0,{\bar{V}}]\rightarrow {\mathbb {R}} _{+}\) is an increasing function that satisfies \(\phi (0)=0\) and \(\phi ({\bar{V}})\le \frac{1}{n}.\) Consider a partition of \([0,{\bar{V}}]\) in subintervals, such that, in each of the intervals of the partition, either \(\phi (\cdot )\) is strictly increasing and continuous, or it is constant.Footnote 14 Since \(G(\cdot )\) is given by (2) in Lemma 1, in each of the intervals of the partition, \(G(\cdot )\) is strictly increasing and continuous, or it is constant.

If the range of \(G(\cdot )\) is \([0,G({\bar{V}})],\) the definition of \(\varphi (\cdot )\) is an easy task immediate from \(\phi (\cdot )\). That is, for all \( g_{i}\in [0,G({\bar{V}})],\) there is a \(V_{i}\in \) \([0,{\bar{V}}]\) such that \(G(V_{i})=g_{i}\) and we can define \(\varphi (g_{i})=\phi (V_{i}).\) But the CSF should be defined for all \(g_{i}\in {\mathbb {R}} _{+},\) so the domain of \(g_{i}\) also includes \((G({\bar{V}}),\infty ).\) We have to extend \(\varphi (\cdot )\) in such a way that \(g_{i}=G(v_{i})\) is a dominant strategy for each agent i. We only need to define \(\varphi (g_{i})=\phi ({\bar{V}})\) for all \(g_{i}\in (G({\bar{V}}),\infty ).\) Clearly, with this extension no agent will make more effort than \(G({\bar{V}})\). Thus, the CSF of the form (2.2) defined in this way produces the same dominant strategy profile as the initial game. If the range is not \([0,G( {\bar{V}})],\) is because \(\phi (\cdot )\) is constant in all the domain, or \( \phi (\cdot )\) is not a continuous function. In the first case, \(\phi (V_{i})=k\) for all \(V_{i}\in [0,{\bar{V}}],\) define \(\varphi (g_{i})=k\) for all \(g_{i}.\) In the second case, the discontinuities of \(\phi (\cdot )\) imply that \(G(\cdot )\) is not a continuous function either. The range is included in \([0,G({\bar{V}})]\) but it has holes. For all \(g_{i}\) in the range and in \((G({\bar{V}}),\infty )\) we define \(\varphi (\cdot )\) as before. It only remains to define \(\varphi (\cdot )\) at all the possible efforts, \( g_{i},\) in the holes of \([0,G({\bar{V}})]\) produced by the discontinuities. The holes which are not in the range are of the form \([G^{m},G^{m+1})\) or \( (G^{l},G^{l+1}].\) Consider first a hole of the form \([G^{m},G^{m+1}).~\)We can associate this hole with two intervals in the domain: \([V^{m-1},V^{m})\) and \([V^{m},V^{m+1}]\) with the following properties: \(G(V^{m})=G^{m+1};\) and \(G^{m}=\lim _{k\rightarrow m}G(V^{k})\,\) with \(V^{k}\in [V^{m-1},V^{m})\) (at \(V^{m-1}\) and at \(V^{m+1}\) the interval could be open or closed, but this is irrelevant now for the present analysis). For all \( g_{i}\in [G^{m},G^{m+1}),\) \(\varphi (g_{i})=\lim _{k\rightarrow m}\phi (V^{k})\) with \(V^{k}\in [V^{m-1},V^{m})\). For the hole of the form \((G^{l},G^{l+1}]\), we can associate two intervals in the domain: \( [V^{l-1},V^{l}]\) and \((V^{l},V^{l+1}]\) with the following properties \( G(V^{l})=G^{l},\) and \(G_{l+1}=\) \(\lim _{k\rightarrow l}G(V^{k})\) with \( V^{k}\in \) \((V^{l},V^{l+1}]\) (at \(V^{l-1}\) and at \(V^{l+1}\) the interval could be open or closed, but it is irrelevant now for this analysis).

We already know that an agent with valuation \(v_{i}\) has a payoff at \( g_{i}=G(v_{i})\) that is greater than or equal to the payoff at any other possible effort in the range of \(G(\cdot )\) or at any effort in \((G({\bar{V}} ),\infty ).\) It remains to prove that the payoff is also greater than or equal to any possible effort in the holes. For that, it is sufficient to prove first that an agent with valuation \(V^{m}\) prefers to choose the effort \(G(V^{m})=G^{m+1}\) rather than any other effort \(g\in [G^{m},G^{m+1}).\) Since in all this interval the probability of winning is the same (given the definition of \(\varphi (\cdot )),\) the greater payoff in this interval is obtained at \(g=G^{m}.\) Note that \([G(V^{m-1}),G^{m})\) is in the range of \(G(\cdot ).\) Thus, for any given vector of effort for the other agents \((g_{-i}),\) and for all \(g\in [G(V^{m-1}),G^{m}),\)

$$\begin{aligned} \left( \varphi (G^{m+1})-\frac{1}{n-1}\sum _{j\ne i}\varphi (g_{j})+\frac{1}{n}\right) V^{m}-G^{m+1}\ge (\varphi (g)-\frac{1}{n-1}\sum _{j\ne i}\varphi (g_{j})+\frac{1}{n})V^{m}-g; \end{aligned}$$

or equivalently

$$\begin{aligned} \varphi (G^{m+1})V^{m}-G^{m+1}\ge \varphi (g)V^{m}-g \end{aligned}$$

Since for all \(g\in [G(V^{m-1}),G^{m})\,\) and this interval is in the range, there is \(V^{k}\in \) \([V^{m-1},V^{m})\) such that \(G(V^{k})=g.\) Thus,

$$\begin{aligned} \phi (V^{m})V^{m}-G(V^{m})\ge \phi (V^{k})V^{m}-G(V^{k}), \end{aligned}$$

and therefore

$$\begin{aligned} \phi (V^{m})V^{m}-G(V^{m})\ge \lim _{k\rightarrow m}\phi (V^{k}) V^{m}-\lim _{k\rightarrow m}G(V^{k})=\lim _{k\rightarrow m}\phi (V^{k}) V^{m}-G^{m}. \end{aligned}$$

Thus, it is a dominant strategy for an agent with valuation \(V^{m}\) to choose effort \(g=G(V^{m}).\)

Finally, it remains to prove that an agent with valuation \(V^{l}\) prefers to choose the effort \(G(V^{l})=G^{l}\) to any other effort \(g\in (G^{l},G^{l+1}]. \) We omit this part since it is basically a replica of the above argument. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beviá, C., Corchón, L. Contests with dominant strategies. Econ Theory 74, 1–19 (2022). https://doi.org/10.1007/s00199-019-01226-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00199-019-01226-3

Keywords

JEL classification

Navigation