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Nested Tullock contests with nonmonotone prizes

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Abstract

This paper demonstrates the possibility of a symmetric “binary-action mixed-strategy equilibrium” in the nested Tullock contest model (Clark and Riis in Public Choice 87:177–184, 1996; Eur J Polit Econ 14(4):605–625, 1998b) with multiple nonmonotone prizes. In this symmetric equilibrium, every player adopts the same mixed strategy: each exerts zero effort with some probability and a constant positive effort otherwise. This new type of equilibrium can coexist with the pure-strategy equilibria established in the literature; it may exist even when those pure-strategy equilibria do not. The coexisting (mixed and pure-strategy) equilibria may induce different levels of effort supply.

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Notes

  1. Under current NBA rules, the winning chance of getting the top pick for each of the bottom 3 teams is 14%, and the winning chance goes down to 0.5% for the top team (of the 14 non-playoff teams).

  2. This differs from the semi-pure equilibrium in the literature, in which some players adopt a pure strategy and others a mixed one (see, for example, Chowdhury et al. 2016).

  3. Although we are unable to derive explicit equilibrium solutions when \(N\ge 4\) or \(r\ne 1\), we can verify the existence of the binary-mixed equilibrium and fully characterize the equilibrium when it exists, once explicit values of N and r are given.

  4. See Appendix (Full surplus extraction with nonmonotone prizes) for a complete analysis of a three-player contest model.

  5. This is because in such an SPSE (with nonmonotone prizes), the maximum level of total effort is induced for certain; while any mixed-strategy equilibrium can at most induce the same level of total effort in expectation, as each player does not play a pure strategy in equilibrium.

  6. Many studies, such as Glazer and Hassin (1988), Clark and Riis (1998a), Barut and Kovenock (1998), Moldovanu and Sela (2001) and Siegel (2009), have examined different types of multi-prize perfectly discriminating contests (i.e., all-pay auctions).

  7. Lu and Wang (2015) provide an axiomatic foundation for the multi-prize nested contest model of Clark and Riis (1996, 1998b).

  8. Fu et al. (2015) study contests with endogenous entry. Applying Dasgupta and Maskin (1986), they establish the existence of a symmetric mixed-strategy equilibrium in which a player pays an entry fee and enters the contest with a probability, and stays out of the contest otherwise.

  9. In such a mixed-strategy equilibrium with two asymmetric players, it is common to see that the strong player adopts a pure strategy (always makes a positive effort) and the weak player plays a mixed strategy (only exerts positive effort with a probability).

  10. In the original model of Clark and Riis (1998b), there are K \((<N)\) prizes, denoted by \(\{V_{1},V_{2},\ldots ,V_{K}\}\), which is equivalent to our N-prize specification with \(V_{K+1}=V_{K+2}=\cdots =V_{N}=0\).

  11. Proposition 1 indicates that positive-effort SPSE can exist under nonmonotone prizes.

  12. See Footnote 6 for a detailed explanation.

  13. In the Appendix, one can see that Eqs. (3) and (4) are derived from the indifference condition (between positive effort and zero effort) and the first-order condition, respectively.

  14. We also show that a symmetric mixed-strategy equilibrium in which each player randomizes over three positive effort levels does not exist. The detailed proof is available upon request.

  15. The reasoning behind this result is that bidding \(\varepsilon\) rather than 0 would increase the player’s probability of winning \(V_{2}=0\) and decrease her probability of winning \(V_{3}=1.1\) significantly, which thus lowers her expected payoff in a significant manner.

  16. The two theorems in this section show that in a three-player lottery contest, with nonmonotone prizes, a binary-mixed equilibrium cannot coexist with an SPSE.

  17. Note that \(r=1\) in Examples 1 and 2, while \(r=1/2<1\) in Example 3.

  18. Note that superscripts CR and BM stand for the SPSE of Clark and Riis (CR) and the binary-mixed (BM) equilibrium proposed in this paper, respectively.

  19. The reverse is also possible, which depends on the prize structure and the value of the discriminatory power.

  20. Note that when \(V_{N}>\frac{1}{N}\left( \sum \nolimits _{k=1}^{N}V_{k}-Z\right)\) and \(V_{1}>\frac{1}{N-1} \sum _{k=2}^{N}V_{k}\), \(TE^{CR}\) can still be positive.

  21. Note that using Eq. (4) when \(p^{*}=1\), we can derive that \(Nx^{*}=r\sum \nolimits _{k=1}^{N}\left[ \left( 1-\sum \nolimits _{h=0}^{k-1} \frac{1}{N-h}\right) V_{k}\right]\), which coincides with the expression of \(TE^{CR}\) given by (2).

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Authors and Affiliations

  1. Department of Economics, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260, Singapore

    Jingfeng Lu

  2. School of Economics, Shandong University, 27 Shanda Nanlu, Jinan, 250100, China

    Zhewei Wang & Lixue Zhou

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  1. Jingfeng Lu
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We are grateful to editor Vijay Krishna, an associate editor, and two anonymous reviewers for their helpful comments and suggestions that significantly improved the quality of the paper. Jingfeng Lu gratefully acknowledges financial support from the MOE of Singapore (Grant No.: R122-000-298-115). Zhewei Wang gratefully acknowledges financial support from NSFC (Grant No.: 71973084), Taishan Scholars Program of Shandong Province, and Shandong University.

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Lu, J., Wang, Z. & Zhou, L. Nested Tullock contests with nonmonotone prizes. Int J Game Theory 52, 303–332 (2023). https://doi.org/10.1007/s00182-022-00820-5

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