Contests with delegation and initial probabilities of winning

ABSTRACT

We study a two-player contest in which each player is endowed with an initial probability of winning the prize and hires a delegate who expends effort on each player’s behalf. We show that (i) the players’ equilibrium delegation contracts are monotonic in the initial probabilities of winning and the impact parameter; (ii) the delegates’ equilibrium effort levels are monotonic in the impact parameter but not in the initial probabilities of winning; and (iii) a contest designer may attain various objectives of contest design by choosing a proper value of the impact parameter.

KEYWORDS:

• Contest
• delegation
• initial probability of winning
• contest design

JEL CLASSIFICATION:

• C72
• D72

Acknowledgments

This paper is based on the third chapter of my Ph.D. dissertation at Sungkyunkwan University. The supervision of Kyung Hwan Baik is greatly appreciated.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Several works consider players’ decisions on delegation and their delegation contracts in contests. See, among others, Baik and Kim (1997), Wärneryd (2000), Schoonbeek (2002, 2017), and Baik (2007).

² This contest success function has some desirable properties. Among others, given $m_i\ne 0$, player $\mathit{i}$ wins the prize with a positive probability, although delegate $\mathit{i}$‘s effort level is zero.

³ In the context of civil litigation, each player’s initial probability of winning corresponds to the objective merits of the case (see, e.g. Farmer and Pecorino 1999). Then, $m_i>m_j$, in our context, implies that the objective merits of the case favour litigant $\mathit{i}$ before a litigation contest. Also, note that the impact parameter represents how a judge uses the objective merits of the case when she reaches a verdict.

⁴ In the Appendix, we prove that equation (10)(10) $\left(1-\mathit{\lambda }+\mathit{m\lambda }\right)k^3+\left(2-3\mathit{\lambda }+3\mathit{m\lambda }\right)k^2+\left(3\mathit{m\lambda }-2\right)\mathit{k}+\mathit{m\lambda }-1=0.$(10) has a unique positive real root. Also, we use the software Mathematica to obtain the unknown $\mathit{k}$.

⁵ Each player’s final probability of winning and the expected payoffs of the players and their delegates, in equilibrium, are omitted because of space concerns.

⁶ The complexity of the equilibrium outcomes makes it difficult to prove the results in Propositions 1 and 2. To deal with this intractability, we graphically show that all the results described in Propositions 1 and 2 hold. We do not present the graphs for the brevity of the paper, but they are available upon request.

⁷ Note that these criteria may include how much weight the designer places on the players’ initial probabilities of winning when she chooses the winning player.

Additional information

Funding

This research was supported by the BK21 FOUR (Fostering Outstanding Universities for Research) funded by the Ministry of Education (MOE, Korea) and National Research Foundation of Korea (NRF).