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.
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Notes
The assumption of a common prior is specially problematic in contests like trials, awarding scholarships or job applications.
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.
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.
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.
We refer to dominant strategies in the weak sense.
As far as we know, the first paper using differentiability to find mechanisms with dominant strategies is Laffont and Maskin (1980).
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 \(\frac{\partial p(_{0},{\mathbf {g}}_{-i}\mathbf {)} }{\partial g_{i}}v_{i}>1\).
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.
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.
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.
For \(\alpha =1\) this reduces to the Maximin criterion.
We can partition the space of valuations in this way because a monotone function has at most a countable number of discontinuities.
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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.
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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)
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)
\(u(s_{i},{\mathbf {V}}_{-i}\mid s_{i})=u(0,{\mathbf {V}}_{-i}\mid 0)+\int _{0}^{s_{i}}P(s,{\mathbf {V}}_{-i})\) \(\mathrm{d}s.\)
-
(3)
\(u(0,{\mathbf {V}}_{-i}\mid 0)\ge 0\) 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
Let \((V_{i},V_{j},{\mathbf {0}}_{-i-j}),\) then
Since \(-\psi (V_{i},{\mathbf {0}}_{-i-j})=\frac{1}{n-1}\phi (V_{i}),\) and \( -\psi (V_{j},{\mathbf {0}}_{-i-j})=\frac{1}{n-1}\phi (V_{j}),\)
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
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,
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),
which implies that \(P\, \ \)is separable, that is:
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)
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 \(v_{i},\)
which is equivalent to proving that
If \(V_{i}>v_{i},\)
and since \(\phi \) is increasing, for all \(s\in [v_{i},V_{i}],\) \(\phi (s)\le \phi (V_{i}).\) Therefore,
If \(V_{i}<v_{i},\)
and since \(\phi \) is increasing, for all \(s\in [V_{i},v_{i}],\) \(\phi (s)\ge \phi (V_{i}).\) Therefore,
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 \(g\in [G(V^{m-1}),G^{m}),\)
or equivalently
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,
and therefore
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. \(\square \)
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Beviá, C., Corchón, L. Contests with dominant strategies. Econ Theory 74, 1–19 (2022). https://doi.org/10.1007/s00199-019-01226-3
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DOI: https://doi.org/10.1007/s00199-019-01226-3