Abstract
In this paper, we accommodate the costly entry of contestants and examine an information design problem when the organizer can decide how to generate contestants’ private information. The information designer should take into account both ex ante entry incentives and post-entry effort elicitation. We show that no transparency (full transparency) induces greater expected aggregate effort if the entry cost is lower (higher) than a threshold. We further consider randomized disclosure policies and identify the optimal degree of transparency, which increases with the entry cost to attract entry. In particular, depending on the entry cost, diverse randomized disclosure policies could be optimal. Our results indicate that endogenous participation plays a crucial role in the design of information revelation.


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Notes
Refer to Konrad (2009) for a comprehensive review of the contest literature.
For example, winning firms are sometimes asked to reveal technologies or programming steps to the public, and such requirements may differentially impact firms’ profits.
For example, science competitions such as the Google Science Fair and Intel International Science and Engineering Fair (Intel ISEF).
We also consider the free-entry case as a benchmark.
With full transparency, each contestant knows his/her valuation of the prize but not that of his/her opponent.
Refer to Bergemann and Morris (2019) for a more complete review of the literature on information design.
We briefly discuss the case in which participants observe the number of entrants in footnote 16 on page 13.
For example, an organizer could control the level of transparency in job descriptions, restrictions on use of research grants, and other terms and conditions attached to a prize.
Under full transparency, a participating contestant only observes his own valuation, rather than that of his opponent.
Refer to Lu et al. (2018).
Standard argument yields the result: A low-type does not exert positive effort when his value equals 0, while a high-type would be indifferent between any two bids in [0, \(pv_{H}]\).
Thanks to a referee for pointing out this issue on the allocation efficiency.
When only one contestant participates, the participant wins the prize automatically. When both contestants enter, under no transparency, both participants compete in an all-pay contest with common value v, and their effort strategy is uniform over [0, v]; under full transparency, both compete in an all-pay contest with incomplete information and the equilibrium bidding strategy is given by (1) and (2 ). One can verify that the expected aggregate efforts remain the same as in Eqs. (4) and (5 ).
Under full transparency, a contestant observes his own valuation, but not that of his opponent.
In particular, when \(c=0\), we have \(\alpha ^{*}(c)|_{c=0}=\frac{c}{ p(1-p)(v_{H}-v_{L})}|_{c=0}=0\), i.e., no transparency is the effort-maximizing policy, which is consistent with the result of Proposition 1.
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Acknowledgements
I am grateful to Jingfeng Lu, Satoru Takahashi, and Jun Zhang for their helpful discussions and insightful comments. I thank Yongmin Chen, Jianpei Li, Junjie Zhou and other participants in 2018 International Conference on Economic Theory and Applications and 2019 UIBE workshop for their helpful feedbacks. The paper has benefited from the comments of multiple anonymous referees, associate editor, and co-editor. I gratefully acknowledge financial support from the National Natural Science Foundation of China under Grant no. 71803019 and 72273063. Any remaining errors are mine.
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Appendices
Appendix
Proof of Lemma 3
For a contestant i, suppose that his opponent participates with probability \(q^{\prime \prime }\), \(i^{\prime }s\) ex ante expected payoff from participation equals
where \(u_{i}(v_{i}=v_{H})\) is the interim payoff of a \(v_{H}\)-type, which equals \((1-p)(v_{H}-v_{L})\) by (3), and \(u_{i}(v_{i}=v_{L})\) is the interim payoff of a \(v_{L}\)-type, which equals 0, by Lemma 1.
A contestant would be indifferent between participating and staying out:
whereby we solve for the equilibrium entry probability \(q^{\prime \prime }=\min \left\{\frac{v-c}{v-p(1-p)(v_{H}-v_{L})},1\right\}\in [0,1]\).
Under full transparency, after \(i^{\prime }s\) entry, i observes his valuation \(v_{i}\), which could be \(v_{H}\) or \(v_{L}\). Note that i will earn \((1-q^{\prime \prime })v_{i}\) from participation, regardless of effort, since his opponent will be absent with probability \((1-q^{\prime \prime })\). In fact, under full transparency, this contest game is equivalent to an incomplete-information all-pay auction with the adjusted valuations \(v_{H}^{\prime \prime }=q^{\prime \prime }v_{H}\) and \(v_{L}^{\prime \prime }=q^{\prime \prime }v_{L}\) and probability p. Therefore, by Lemma 1, the equilibrium strategy is given by
Based on the equilibrium characterization, we calculate the sum of contestants’ equilibrium efforts as follows.
\(\square\)
Proof of Proposition 2
By Lemma 3, \(TE_{Full}(c)=(q^{\prime \prime })^{2}\left[ p^{2}v_{H}+(1-p^{2})v_{L}\right]\), where \(q^{\prime \prime }=\min \{\frac{ v-c}{v-p(1-p)(v_{H}-v_{L})},1\}\). To show the result, we consider \([v-p(1-p)(v_{H}-v_{L})]^{2}\le v[p^{2}v_{H}+(1-p^{2})v_{L}]\), where
since \(p^{2}v_{H}+(1-p^{2})v_{L}<v=pv_{H}+(1-p)v_{L}\).
Case 1: If \(c\in [v-\{v[p^{2}v_{H}+(1-p^{2})v_{L}]\}^{\frac{1}{2}}\), \(p(1-p)(v_{H}-v_{L}))\), we have \(\frac{v-c}{v-p(1-p)(v_{H}-v_{L})}>1\), \(q^{\prime \prime }=1\), and \(TE_{Full}(c)=[p^{2}v_{H}+(1-p^{2})v_{L}]\). By Lemma 2, \(TE_{No}(c)=(1-\frac{c}{v})^{2}v\). Therefore, \(TE_{Full}(c)\ge TE_{No}(c)\) holds if and only if
i.e.,
Case 2: If \(c\in [p(1-p)(v_{H}-v_{L})\), v], we have \(\frac{v-c}{ v-p(1-p)(v_{H}-v_{L})}\le 1\), \(q^{\prime \prime }=\frac{v-c}{ v-p(1-p)(v_{H}-v_{L})}\), and \(TE_{Full}(c)=(\frac{v-c}{v-p(1-p)(v_{H}-v_{L})} )^{2}[p^{2}v_{H}+(1-p^{2})v_{L}]\). By Lemma 2, \(TE_{No}(c)=(1-\frac{c}{v})^{2}v\). \(TE_{Full}(c)\ge TE_{No}(c)\) holds if and only if
i.e.,
Since \(v-p(1-p)(v_{H}-v_{L})=p^{2}v_{H}+(1-p^{2})v_{L}\), the above inequality holds if and only if
which holds, whenever \(p\in (0,1)\). \(\square\)
Proof of Lemma 4
For a randomized disclosure policy with degree of transparency \(\alpha \in [0,1]\), a contestant’s ex ante expected payoff is simply a linear combination of the two payoffs resulting from full transparency and no transparency, which is \(\alpha p(1-p)(v_{H}-v_{L})\). Recall that with probability \(\alpha\), contestants are fully informed, and with probability \(1-\alpha\), contestants are fully uninformed.
A contestant would be indifferent between participating and staying out:
where \(q_{\alpha }\) is the entry probability of the other contestant. Therefore, the equilibrium entry probability \(q_{\alpha }=\min \left\{\frac{v-c}{ v-\alpha p(1-p)(v_{H}-v_{L})},1\right\}\), as \(q_{\alpha }\in [0,1]\).
After the entry stage, information about the prize will be disclosed according to the \(\alpha\)-degree randomized policy. When contestants are perfectly uninformed, analogous to the proof of Lemma 2, a participating contestant will exert effort following the uniform distribution over \([0,q_{\alpha }v]\), i.e., \(F(x|v)=\frac{x}{q_{\alpha }v} ,\forall x\in [0,q_{\alpha }v]\). The resulting aggregate effort equals \(2q_{\alpha }[\frac{1}{2}q_{\alpha }v]=(q_{\alpha })^{2}v\). Analogously, when contestants are perfectly informed, depending on his/her valuation, a participating contestant will exert effort following the distributions below:
where \(v_{H}(\alpha )=q_{\alpha }v_{H}\) and \(v_{L}(\alpha )=q_{\alpha }v_{L}\) . The resulting aggregate effort equals \(q_{\alpha }[p^{2}v_{H}(\alpha )+(1-p^{2})v_{L}(\alpha )]=(q_{\alpha })^{2}[p^{2}v_{H}+(1-p^{2})v_{L}]\).
To summarize, for an \(\alpha\)-degree randomized policy, with probability \(\alpha\), contestants are perfectly informed, and the resulting aggregate effort equals \((q_{\alpha })^{2}[p^{2}v_{H}+(1-p^{2})v_{L}]\); and with probability \(1-\alpha\), contestants are perfectly uninformed, and the resulting aggregate effort equals \((q_{\alpha })^{2}v\). Therefore, the ex ante expected aggregate effort resulting from an \(\alpha\)-degree disclosure policy equals
\(\square\)
Proof of Theorem 1
For each \(c\in [0,v)\), to determine the optimal \(\alpha \in [0,1]\), we consider two cases: \(c\ge {\widehat{c}}(\alpha )\) and \(c\le {\widehat{c}}(\alpha )\), where \({\widehat{c}}(\alpha )=\alpha p(1-p)(v_{H}-v_{L})\). Note that \(c\ge {\widehat{c}}(\alpha )\) if and only if \(\alpha \le \frac{c}{p(1-p)(v_{H}-v_{L})}\), we thus consider two cases: Case 1 where \(\alpha \le \frac{c}{p(1-p)(v_{H}-v_{L})}\) and Case 2 where \(\alpha \ge \frac{c}{p(1-p)(v_{H}-v_{L})}\) as follows.
Case 1: For \(\alpha \le \frac{c}{p(1-p)(v_{H}-v_{L})}\), \(TE^{\alpha }(c)= \frac{(v-c)^{2}}{v-\alpha p(1-p)(v_{H}-v_{L})}\), which increases with \(\alpha\). Therefore, the constrained optimum \(\alpha ^{*}=\min \{\frac{c }{p(1-p)(v_{H}-v_{L})},1\}\) in Case 1.
Case 2: For \(\alpha \ge \frac{c}{p(1-p)(v_{H}-v_{L})}\), since \(v=pv_{H}+(1-p)v_{L}\),
which decreases with \(\alpha\), as \(v:=pv_{H}+(1-p)v_{L}>p^{2}v_{H}+(1-p^{2})v_{L}\). Therefore, the constrained optimum \(\alpha ^{*}=\frac{c}{p(1-p)(v_{H}-v_{L})}\) in Case 2.
Combining the two cases, \(\alpha ^{*}(c)=\min \left\{\frac{c}{ p(1-p)(v_{H}-v_{L})},1\right\}\) for each \(c\in [0,v)\). \(\square\)
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Feng, X. Information disclosure in all-pay contests with costly entry. Int J Game Theory 52, 401–421 (2023). https://doi.org/10.1007/s00182-022-00822-3
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DOI: https://doi.org/10.1007/s00182-022-00822-3