Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players

I characterize the optimal accuracy level r of an unbiased Tullock contest be-tween two players with heterogeneous prize valuations. The designer maximizes the winning probability of the strong player or the winner’s expected valuation by choosing a contest with an all-pay auction equilibrium ( r ≥ 2). By contrast, if she aims at maximizing the expected aggregate eﬀort or the winner’s expected eﬀort, she will choose a contest with a pure-strategy equilibrium, and the optimal accuracy level r < 2 decreases in the players’ heterogeneity. Finally, a contest designer who faces a tradeoﬀ between selection quality and minimum (maximum) eﬀort will never (may) chose a contest with a semi-mixed equilibrium

1 Introduction I characterize the optimal accuracy level, sometimes also referred to as decisiveness parameter or discriminatory power, of an unbiased Tullock contest between two players with heterogeneous prize valuations under different objectives.As the accuracy level affects efforts, winning probabilities, and payoffs, it is an important tool for designing a contest, particularly when an explicit bias or affirmative action is not feasible.Real world examples are countless and range from defining the type and size of the jury in litigation (number of jurors/judges) to specifying the rules in sports like car racing (technical limitations), table tennis (size of the ball), or soccer (tie breaking regulations).
The analysis thus contributes to the large literature on (optimal) contest design.More specifically, it complements the articles that emphasize the role of the accuracy of the contest success function, e.g., Nti (2004), Alcalde and Dahm (2010), Wang (2010), and, in particular, Ewerhart (2017) who provides a revenue ranking for optimally biased contests.
The paper is organized as follows.Section 2 introduces the formal set-up.In Sections 3 and 4, I examine the optimal accuracy level under the assumptions that the designer maximizes the winning probability of the strong player, the winner's expected valuation, the expected aggregate effort, and the winner's expected effort, respectively.Section 5 discusses the optimal solution to different tradeoffs between selection quality and effort.Section 6 concludes.

Set-up and Notation
I consider the standard model of a Tullock contest (Tullock, 1980) between two players with linear effort costs and use the same notation as Ewerhart (2017).Player i's probability of winning is where x i denotes the effort of player i ∈ {1, 2} and r ≥ 0 describes the accuracy level of the contest.1 Player i ∈ {1, 2} chooses x i to maximize the payoff Π i = p i V i − x i , where the players' valuations for the prize are normalized to V 1 = 1 and V 2 = ω ∈ (0, 1).I thus refer to player 1 (2) as the strong (weak) player.
Propositions 1 -4 in Ewerhart (2017) show that, for any given ω ∈ (0, 1), there is a unique Nash equilibrium, which is in pure strategies, if 0 ≤ r ≤ r, there is a unique Nash equilibrium, which is in semi-mixed strategies, if r < r ≤ 2, any Nash equilibrium is an all-pay auction equilibrium2 in mixed strategies if 2 < r, where r is an implicit function of ω defined by Where appropriate, I mark equilibrium values with an asterisk.

Maximization of Selection Quality
I first consider different objectives associated with the selection quality of the contest.

Maximization of the Winner's Expected Valuation
Since the winner's expected equilibrium valuation equals and 1 − ω > 0 for all ω ∈ (0, 1), a contest that maximizes the strong player's winning probability also maximizes the winner's expected valuation.
Proposition 2. For any ω ∈ (0, 1), the designer maximizes the winner's expected valuation by choosing any contest with an all-pay auction equilibrium (r ≥ 2).

Effort Maximization
I now consider different objectives associated with effort maximization.

Maximization of Aggregate Effort
For any ω ∈ (0, 1), Nti (2004) determines the accuracy level r that maximizes aggregate effort in the range of pure strategy equilibria, i.e., under the constraint r ≤ r.Alcalde and Dahm (2010) show that for any r ≥ 2, there exists an all-pay auction equilibrium, and for any r > 2, any equilibrium is an all-pay auction equilibrium.Epstein et al. (2013) show that, for any ω ∈ (0, 1), the accuracy level r that maximizes aggregate effort in the range of pure strategy equilibria also leads to a higher aggregate effort than an all-pay auction (r ≥ 2).Wang (2010) determines a semi-mixed equilibrium for all r < r ≤ 2 and shows that, within this class of equilibria, the aggregate equilibrium effort R decreases in the accuracy level r for any ω ∈ (0, 1), i.e., dR/dr < 0 if r < r ≤ 2. Finally, Ewerhart (2017) shows that for any r ≤ 2 the equilibrium is unique.Together, these results allow for a unique identification of the optimal accuracy level.More explicitly, Ewerhart (2017, Table 1) shows that for any ω ∈ (0, 1), aggregate equilibrium effort R is a continuous function of r.This implies that, for any ω ∈ (0, 1), the optimal accuracy level must satisfy r ≤ r.It thus coincides with the optimal accuracy level within the region of pure-strategy equilibria as characterized by Nti (2004).I briefly summarize his analysis and add an exact equation for the threshold he approximates (cf.Nti, 2004, Table 1 and Proposition 3).
Proposition 3.For any ω ∈ (0, 1), aggregate effort is an inverted U-shaped function of the accuracy level.The designer maximizes aggregate effort by choosing a contest with a pure-strategy equilibrium.The optimal accuracy level equals r = min{r A , r} and decreases as the players' heterogeneity increases: dr/dω > 0.3 It is straightforward to show that f is strictly increasing for all r A ∈ (1, 2) and has a unique root which I denote by rA .Therefore, r A < r if and only if r A < rA or, equivalently, ω < ωA , where ωA := (r A − 1) 1/r A and rA + (2 − rA ) ln(r A − 1) = 0. (3) Corollary 1.The designer maximizes aggregate effort by choosing Figure 1 illustrates Proposition 3 and Corollary 1.The solid (dotted) curve depicts r A (r) as a function of ω.The curves intersect at some point A ≈ (0.2804; 1.2137) to the left (right) of which the optimal accuracy level is unconstrained (constrained).

Maximization of the Winner's Expected Effort
Straightforward calculations show that, for all ω ∈ (0, 1), the winner's expected equilibrium effort EX W = p 1 x 1 + p 2 x 2 is also a continuous function of r with dEX W /dr < 0 for r < r ≤ 2. Again, these observations imply that, for any ω ∈ (0, 1), the optimal accuracy level must satisfy r ≤ r and thus coincides with the optimal accuracy level within the region of pure-strategy equilibria.
For any ω ∈ (0, 1), the optimal accuracy level r maximizes the winner's expected equilibrium effort EX W = rω r (1+ω r+1 ) (1+ω r ) 3 subject to the constraint that r ≤ r.The first order condition dEX W /dr = ω r (1+ω r ) 4 G(ω, r) = 0 for an unconstrained maximizer r B implies Proposition 4. For any ω ∈ (0, 1), the winner's expected effort is an inverted U-shaped function of the accuracy level.The designer maximizes the winner's expected effort by choosing a contest with a pure-strategy equilibrium.The optimal accuracy level equals r = min{r B , r}.
Moreover, numerical approximations suggest dr B /dω = − ∂G/∂ω ∂G/∂r B > 0 for all ω ∈ (0, 1), i.e., the optimal accuracy level decreases as the players' heterogeneity increases. Inserting One can show that g is strictly increasing4 and has a unique root which I denote by rB .Therefore, r B < r if and only if r B < rB or, equivalently, ω < ωB , where ωB := (r B − 1) 1/r B and g(r B ) = 0. (5) Corollary 2. The designer maximizes the winner's expected effort by choosing Figure 1 illustrates Proposition 4 and its Corollary.The dashed (dotted) curve depicts r B (r) as a function of ω.The curves intersect at some point B ≈ (0.2337; 1.1799) to the left (right) of which the optimal accuracy level is unconstrained (constrained).While maximizing the aggregate effort is equivalent to maximizing the players' average effort (with equal weights), maximizing the winner's expected effort is equivalent to maximizing the players' weighted average effort with a higher equilibrium weight p 1 > p 2 on the stronger player.Intuitively, the solution to this problem is thus a compromise between the maximization of aggregate effort and the maximization of the strong player's winning probability.As a result, r B ≥ r A for all ω ∈ (0, 1).Hence, the range of heterogeneities ω for which r B is constrained by r must be larger than that for which r A is constrained by r, i.e., 0 < ωB < ωA .
In the next section, I characterize the optimal compromise between conflicting objectives more generally.

Conflicting Objectives
Contest designers often have multiple objectives which may conflict.During a pre-election, for example, a political party tries to select the best candidate but, at the same time, limit pre-election efforts in order to save resources for the main election campaign (Bruckner and Sahm, 2022).By contrast, the organizer of a qualifying competition tries to select the best athlete and provoke as much effort as possible because a highly intense competition attracts more attention from spectators and sponsors.

Tradeoff between Selection Quality and Minimum Effort
Obviously, the contest that minimizes aggregate effort is purely random: an accuracy level of r = 0 leads to zero efforts.The previous analysis thus suggests that a designer who optimally solves a tradeoff between selection quality and minimum aggregate effort (rent dissipation) will never choose a contest with a semi-mixed equilibrium because, in this range, an increasing accuracy implies both, better selection and lower efforts.More precisely, for any ω ∈ (0, 1), he will choose an all-pay auction (r * ≥ 2) if and only if he puts sufficiently much weight on selection quality.Otherwise, he will choose an accuracy level r * < min{r A , r} that leads to a pure-strategy equilibrium.A smaller upper-bound for the optimal r * is then given by the (smallest) accuracy level r that equates the aggregate effort in the pure-strategy equilibrium and the expected aggregate effort of the all-pay auction equilibrium:

Tradeoff between Selection Quality and Maximum Effort
By contrast, a designer who optimally solves a tradeoff between selection quality and maximum aggregate effort will always choose an accuracy level r that is larger than the one that maximizes aggregate effort.In particular, he may choose a contest with a semimixed equilibrium (if he puts sufficiently much weight on selection quality), and will definitely chose an accuracy level r ≥ r if ωA ≤ ω < 1 (see Corollary 1).

Conclusion
I have examined the optimal accuracy level r of an unbiased Tullock contest between two players with heterogeneous prize valuations under different objectives.The designer maximizes the winning probability of the strong player or the winner's expected valuation by choosing a contest with an all-pay auction equilibrium (r ≥ 2).By contrast, if she aims at maximizing the expected aggregate effort or the winner's expected effort, she will choose a contest with a pure-strategy equilibrium, and the optimal accuracy level r < 2 decreases in the players' heterogeneity.Finally, a contest designer who faces a tradeoff between selection quality and minimum (maximum) effort will never (may) chose a contest with a semi-mixed equilibrium.

Figure 1 :
Figure 1: Optimal accuracy level and heterogeneity