Extractive Contest Design^††thanks: The author is grateful to Kazuo Yamaguchi for his valuable comments. This work was supported by JSPS KAKENHI Grant Number JP19K01563.

In the literature on contests, formalized by Tullock (1980), the design of contest success functions (CSFs) that maximize the aggregate effort of contestants is a key topic. Many papers have examined this issue. Maximization of aggregate effort provides CSFs with a positive foundation.¹¹1 Jia et al. (2013) referred to this as the optimally derived foundation, one among the four types of foundations. ^,222 Some papers have provided CSFs with axiomatic foundations (e.g., Skaperdas (1996) and Clark and Riis (1998)). In the rent-seeking interpretation, the contest designer (politician) intends to maximize the contestants’ efforts. This is because if the efforts are political contributions, he/she obtains monetary benefits from these efforts, and if the efforts are political lobbying, he/she flaunts his/her power through the efforts. Thus, he/she should determine the CSF as it maximizes the efforts.

This paper considers the design of CSFs that extract the contestants’ values. Most of the existing literature has investigated the maximization of aggregate effort in a class of CSFs that satisfy some restrictions. Owing to these restrictions, the aggregate effort is generically less than the highest value of the contestants, i.e., the optimal CSF does not extract contestants’ values. Instead, this paper considers the design of CSFs without restrictions and examines the extraction of contestants’ values.

We define extractiveness of CSFs. In a contest, contestants make an effort, and the winner of a prize is determined according to a probability distribution that depends on the efforts. A function that maps effort tuples to winning probability distributions is called a CSF. We consider the design of CSFs that extract the contestants’ prize values through the contestants’ efforts. We say that a CSF is extractive if under this CSF, there exists a Nash equilibrium such that the aggregate effort is equal to the maximum value (the maximum of the contestants’ prize values). We also say that a CSF is strictly extractive if this CSF is extractive, and under this CSF, in every Nash equilibrium, the aggregate effort is equal to the maximum value. We consider pure-strategy Nash equilibria.

We examine extractiveness of CSFs, focusing on whether the contestants’ prize values are common or private and whether they are observable or unobservable by the contest designer. First, we consider the case where these values are common. We consider the CSF such that the winning probabilities are proportional to the $\frac{a}{a-1}$th power of the efforts, where $a$ is in $\left\{2,3,\dots,n\right\}$ ($n$ is the number of contestants). We show that the CSF with $a=2$ is strictly extractive for all common values. Because this CSF does not depend on common values, this result holds regardless of whether they are observable or unobservable. For every common value, we also show that the CSF with $a\geq 3$ is extractive, but not strictly extractive. Thus, aggregate effort equivalence across Nash equilibria holds for $a=2$, but not for $a\geq 3$. Second, we consider the case where the values are private and observable. For every value tuple, we present an extractive CSF. For every value tuple, we show that there exists a strictly extractive CSF if and only if the number of contestants is greater than or equal to $3$ or the values are homogeneous. When strictly extractive CSFs exist, we present one of them. The aggregate effort equivalence across Nash equilibria holds if and only if the number of contestants is greater than or equal to $3$ or the values are homogeneous. Third, we consider the case where the values are private and unobservable. We show that there exists no CSF that is extractive for all value tuples. Therefore, observability matters in the private-value case, but not in the common-value case.

Several papers have presented CSFs that are reduced to CSFs extractive in the unobservable-common-value case. In the $2$-contestant case, Nti (2004) (Epstein et al. (2013); Ewerhart (2017a), resp.) presented a CSF that maximizes the aggregate effort in a class of CSFs (Section 4 (Subsection 4.2; Proposition 6, resp.)). In the $n$-contestant common-value case, Michaels (1988) did so (Subsection 2.1). Each CSF in Nti (2004), Epstein et al. (2013) and Ewerhart (2017a) in the common-value case (the CSF in Michaels (1988), resp.) is the CSF such that the winning probabilities are proportional to the $2$nd ($\frac{n}{n-1}$th, resp.) power of the efforts, and it is in the unobservable-common-value case because it does not depend on the common value. The CSF using the $2$nd power is strictly extractive in the $2$-contestant case. We show that this CSF is strictly extractive in the $n$-contestant case. The CSF using the $\frac{n}{n-1}$th power is extractive. We show that if $n\geq 3$, this CSF is not strictly extractive. Pérez-Castrillo and Verdier (1992) derived Nash equilibria under the CSF such that the winning probabilities are proportional to the $r$th power of the efforts in the $n$-contestant case (Proposition 4). The result of Pérez-Castrillo and Verdier (1992) implies that CSFs such that the winning probabilities are proportional to the $\frac{a}{a-1}$th power of the efforts ($2\leq a\leq n$) are extractive. We show that if $a\geq 3$, this CSF is not strictly extractive.

Several papers have presented CSFs that are extractive but not strictly extractive in the observable-value case. In the $2$-contestant common-value case, Glazer (1993) presented a CSF such that a certain contestant wins if his/her effort is equal to his/her value, and the other contestant wins otherwise (Subsection 3.1). In the $2$-contestant case (the $n$-contestant case, resp.), Nti (2004) (Franke et al. (2018), resp.) presented a CSF such that a contestant with the maximum value wins if his/her effort is greater than or equal to his/her value, and a contestant with the second highest value, which may be equal to the maximum value, wins otherwise (Proposition 2 (Proposition 4.7, resp.)). These CSFs are extractive. However, they are not strictly extractive, because there exists a Nash equilibrium such that every contestant’s effort is zero. Nti (2004) suggested that under a modified CSF such that the effort threshold is slightly lowered, the Nash equilibrium such that every contestant’s effort is zero is removed. However, under the modified CSF, in a unique Nash equilibrium, the aggregate effort is slightly smaller than the maximum value. Meanwhile, in the $3$-or-more-contestant or homogeneous-value case, we present strictly extractive CSFs. In any Nash equilibrium under such CSFs, the aggregate effort is exactly equal to the maximum value. In the $2$-heterogeneous-contestant case, we show that there exists no strictly extractive CSF. This implies that the contest designer cannot design a strictly extractive CSF even though he/she can fully use the information of the values.

Several papers have shown the extraction of values in mixed-strategy Nash equilibria. Hillman and Riley (1989) (Che and Gale (1998); Baye et al. (1993); Baye et al. (1996), resp.) showed that in the all-pay auction, if the highest two values are equal, the expected aggregate effort is equal to the maximum value in any mixed-strategy Nash equilibrium (Proposition 1; the second last paragraph in Section 3 (equation (9); Theorem 1, resp.)). Ewerhart (2017b) showed it in a modified all-pay auction (Proposition 1). Alcalde and Dahm (2010) showed that under CSFs that satisfy certain conditions, there exists a mixed-strategy Nash equilibrium such that if the highest two values are equal, the expected aggregate effort is equal to the maximum value (Theorem 3.2). We show the extraction of values in pure-strategy Nash equilibria, even if the highest two values are not equal.

Several papers have considered maximizing aggregate effort in a class of CSFs. CSFs with the following devices have been examined: concave technologies³³3 A concave technology is a concave function that transforms efforts. Winning probabilities are determined based on the transformed efforts. in the lottery contest (Dasgupta and Nti (1998)); concave technologies and power technologies⁴⁴4 A power technology is a power function that transforms efforts. in the lottery contest (Nti (2004)); power technologies in the lottery contest (Michaels (1988)); biases multiplying efforts in the lottery contest (Franke et al. (2013)); biases multiplying efforts with power technologies in the lottery contest and biases multiplying efforts in the all-pay auction (Epstein et al. (2013));⁵⁵5 Epstein et al. (2011) considered the same class of CSF, but a different objective of the contest designer, which is the weighted sum of the aggregate effort and welfare. biases multiplying efforts in the lottery contest and all-pay auction (Franke et al. (2014a)); head starts added to efforts in the lottery contest and all-pay auction (Franke et al. (2014b)); biases multiplying efforts given a power technology in the lottery contest (Ewerhart (2017a)); biases multiplying efforts and head starts added to efforts in the lottery contest and all-pay auction (Franke et al. (2018)). Fang (2002) compared the simple lottery contest with the simple all-pay auction. Owing to restrictions on the forms of CSFs, the maximized aggregate effort is not equal to the maximum value except for the aforementioned results. In our paper, because no restrictions are imposed on the forms of CSFs, the values are extracted.

Several papers have considered the aggregate effort under asymmetric information. In Kirkegaard (2012), Pérez-Castrillo and Wettstein (2016), Matros and Possajennikov (2016), Drugov and Ryvkin (2017) and Olszewski and Siegel (2020), the values or productivities of the efforts are private information. In our paper, contestants know the contestants’ values; the contest designer knows them in the observable-value case, but not in the unobservable-value case.

The contributions of this paper are as follows. In the observable-private-value case, we present strictly extractive CSFs in the $3$-or-more-contestant or homogeneous-value subcase, where we only use the pure-strategy Nash equilibria, whereas we show that there exists no strictly extractive CSF in the other subcase. In the unobservable-private-value case, we show that there exists no extractive CSF. In the common-value case, we show that the CSF with the $2$nd-power technology is strictly extractive in the multi-contestant contest, and the CSF with the $\frac{a}{a-1}$th-power technology ($a\geq 3$) is not strictly extractive. We demonstrate that for extractive or strictly extractive CSFs to exist, observability of values matters in the private-value case, but not in the common-value case. The framework in this paper could serve as a general framework for investigating the extraction of values in contests.

The remainder of this paper is organized as follows. Section 2 describes the model. Section 3 presents the results. Section 4 concludes the paper. The proofs of all the propositions are provided in the appendix.

For any sets $X$, $Y$ and $I$, any $f:X\to Y^{I}$ and any $x\in X$ and $i\in I$, let $f_{i}\left(x\right)$ be the value of $f\left(x\right)$ for $i$.

Let $N$ be a finite set such that $\left|N\right|\geq 2$: $N$ is the set of contestants. Let $n:=\left|N\right|$. Let $X:=\mathbb{R}_{\geq 0}^{N}$: $X$ is the set of tuples of contestants’ efforts. Let $\Delta:=\left\{p\in\mathbb{R}_{\geq 0}^{N}\mid\sum_{i\in N}p_{i}=1\right\}$: the set of tuples of contestants’ success probabilities (for any $p\in\Delta$ and any $i\in N$, $p_{i}$ is the probability of contestant $i$’s winning). Let $F:=\left\{f\mid f:X\to\Delta\right\}$: $F$ is the set of contest success functions (CSFs). Let $V:=\mathbb{R}_{>0}^{N}$: the set of tuples of contestants’ prize values. For any $f\in F$ and any $v\in V$, let $u^{fv}:X\to\mathbb{R}^{N}$ such that for any $x\in X$ and any $i\in N$, $u_{i}^{fv}\left(x\right)=f_{i}\left(x\right)v_{i}-x_{i}$: $u_{i}^{fv}\left(x\right)$ is contestant $i$’s utility from effort tuple $x$ ($f_{i}\left(x\right)v_{i}$ is the expected value that he/she obtains, and $x_{i}$ is the cost of his/her effort).

For any $f\in F$ and any $v\in V$, $\left(N,X,u^{fv}\right)$ is a strategic-form game: $N$ is the set of players, $X$ is the set of strategy tuples, and $u^{fv}$ is the function that maps each strategy tuple to its payoff tuple. For any $f\in F$ and any $v\in V$, let $E^{fv}$ be the set of pure-strategy Nash equilibria in $\left(N,X,u^{fv}\right)$. In the following, we refer to a pure-strategy Nash equilibrium simply as a Nash equilibrium.

Let $\hat{V}:=\left\{v\in V\mid\left(\forall i,j\in N\right)v_{i}=v_{j}\right\}$: the set of value tuples such that all contestants have a common value. For any $v\in V$, let $m^{v}:=\max_{i\in N}v_{i}$ and $M^{v}:=\arg\max_{i\in N}v_{i}$: $m^{v}$ is the maximum of contestants’ prize values, and $M^{v}$ is the set of contestants with the maximum value.

We refer to the case where the domain of the value tuples is $\hat{V}$ ($V$, resp.) as the common-value case (private-value case, resp.). We also refer to the case where value tuples are observable (unobservable, resp.) by the contest designer, i.e., CSFs can (cannot, resp.) depend on value tuples the observable-value case (unobservable-value case, resp.). We seek CSFs under which the equilibrium aggregate effort is equal to the maximum value in the common-value case, the observable-private-value case and the unobservable-private value case, respectively. We say that in the private-value case, if $v\in V$ satisfies that for any $i,j\in N$, $v_{i}=v_{j}$, we say that $v$ is homogeneous. The value tuples are observable by the contestants.

Table 1 summarizes the results of this paper, where $\phi^{\mathrm{E}}$ is a formula meaning that $f$ is extractive for $v$, and $\phi^{\mathrm{SE}}$ is a formula meaning that $f$ is strictly extractive for $v$. Whether the values are common or private is represented by whether the domain of the values is $\hat{V}$ or $V$. Whether the values are observable or unobservable is represented by whether the order of the quantifiers is $\left(\forall v\right)\left(\exists f\right)$ or $\left(\exists f\right)\left(\forall v\right)$. In the common-value case, regardless of the observability, there exist extractive and strictly extractive CSFs. In the observable-private-value case, there exists an extractive CSF, but there does not always exist a strictly extractive CSF. In the unobservable-private-value case, there exists neither extractive nor strictly extractive CSF. In the common-value case, we also present a class of extractive CSFs that can be used to control the number of active contestants.

In the unobservable-private-value case, there exists no extractive CSF. In such a case, it is necessary to derive CSFs that maximize the expectation of the aggregate effort under some belief on value tuples. For example, this problem is formalized as follows:

where $P$ is a cumulative distribution function on $V$ (the designer’s belief on value tuples).

In this paper, we consider pure strategies, but not mixed strategies. Propositions 3 and 4 also hold in mixed strategies because if there exists a pure-strategy Nash equilibrium, it is also a mixed-strategy Nash equilibrium. However, it is not clear whether Propositions 2, 5 and 6 hold in mixed strategies because there might exist mixed-strategy Nash equilibria other than pure-strategy Nash equilibria.