Abstract
We study two-stage elimination Tullock contests. In the first stage all the players compete against each other; then some advance to the second stage while the others are removed. The finalists compete against each other in the second stage, and one of them wins the prize. To maximize the expected total effort, the designer can give a head start to the winner of the first stage when he competes against the other finalists in the second stage. We show that the optimal head start, independent of the number of finalists, always increases the players’ expected total effort. We also show how the number of players and finalists affect the value of the optimal head start.
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Notes
A number of studies provided axiomatic justification for the Tullock contest (see, for example, Skaperdas 1996; Clark and Riis 1998). Baye and Hoppe (2003) have identified conditions under which a variety of rent-seeking contests, innovation tournaments, and patent-race games are strategically equivalent to the Tullock contest.
The incentive for the players who advance from the first stage to the second could also be endogenous since a win in the first stage may give an advantage in the second one.
It can be shown that it is optimal to award the entire prize sum (which is equal to 1) to the winner in the second stage instead of splitting the entire prize into two prizes, one for the finalist with the highest effort and the other for the second finalist.
Our results are robust for other forms of the head start. If , for example, we assume that player i’s probability to win the first prize is \(\frac{(y_{i})^{\beta }}{(y_{i})^{\beta }+y_{j}}\) , the results will remain robust.
It can be easily verified that the S.O.C. for this maximization problem is satisfied.
It can be easily shown that the S.O.C. is satisfied for this value of the optimal head start.
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Appendix
Appendix
1.1 Proof of Proposition 1
The F.O.C. of the maximization problem (5) is
By symmetry, we obtain
Thus, the subgame perfect equilibrium strategy in the first stage is
1.2 Proof of Proposition 3
By symmetry, the F.O.C. of the maximization problem (12) yields
Thus, the subgame perfect equilibrium strategy in the first stage is
1.3 Proof of Proposition 4
The F.O.C. of the maximization of the total effort given by (13) is
By some calculations we have
When \(\alpha \) approaches one we obtain that
Since
We obtain,
Therefore, the head start \(\alpha >1\), independent of the number of the finalists k, increases the players’ total effort.
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Cohen, N., Maor, G. & Sela, A. Two-stage elimination contests with optimal head starts. Rev Econ Design 22, 177–192 (2018). https://doi.org/10.1007/s10058-018-0216-1
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DOI: https://doi.org/10.1007/s10058-018-0216-1