Voluntary Disclosure in Asymmetric Contests

Abstract

This article studies the incentives for interim voluntary disclosure of verifiable information in probabilistic all-pay contests with two-sided incomplete information. Private information may concern marginal cost, valuations, and ability. Our main result says that, if the contest is uniformly asymmetric, then full revelation is the unique perfect Bayesian equilibrium outcome. This is so because the weakest type of the underdog reveals her type in an attempt to moderate the favourite, while the strongest type of the favourite tries to discourage the underdog—so that the contest unravels. This strong-form disclosure principle is robust with respect to correlation, partitional evidence, randomized disclosures, sequential moves, and continuous type spaces. Moreover, the assumption of uniform asymmetry is not needed when incomplete information is one-sided. However, the principle may break down when type distributions are too similar, contestants possess commitment power, or information is unverifiable. In fact, cheap talk will always be ignored, even if mediated by a trustworthy third party.

1 INTRODUCTION

On February 24, 2014, the concentration of Russian troops along the entire Ukrainian–Russian border became overwhelming. In a major demonstration of force and simultaneous preparation for invasion, the Kremlin had decided to concentrate in the Kyiv, Kharkiv, and Donetsk directions 38,000 men, 761 armed tanks, 2,200 armoured vehicles, 720 artillery systems and multiple rocket launchers, as well as up to 40 attack helicopters, 90 combat support helicopters, and 90 attack aircraft. In the Black Sea, 80 Russian warships were on combat duty. On that same day, the Russian Black Sea Fleet Commander had a conversation with the Ukraine Naval Forces Commander, advocating complete surrender and handing over of the Crimea. And indeed, Ukrainian resistance quickly ebbed away in the days to follow.¹

In this article, we extend the standard model of a probabilistic contest (Rosen, 1986; Dixit, 1987) by allowing for pre-play communication of verifiable information (Okuno-Fujiwara et al., 1990; van Zandt and Vives, 2007; Hagenbach et al., 2014). Contestants are assumed to possess private information regarding parameters indicative of their strength in the competition. In general, these parameters may concern marginal cost, valuations, and ability. Then, at the interim stage, i.e. after having observed her type but prior to the contest, each player may choose to disclose that information to her opponent. In this type of framework, we evaluate the incentives of players to voluntarily disclose their private information. Moreover, we characterize the perfect Bayesian equilibrium outcome of the resulting two-stage game. The focus of the analysis lies on contests that are uniformly asymmetric in the sense that one of the contestants is, subject to activity, interim always strictly more likely to win than the other.² To be on safe ground, we offer a condition on the primitives of the model that guarantees that the contest is uniformly asymmetric. While restrictive, that condition is consistent with heterogeneity in both valuations (e.g. Amann and Leininger, 1996; Maskin and Riley, 2000) and ability (e.g. O’Keeffe et al., 1984; Meyer, 1992; Franke et al., 2014). But, we will also delineate the scope of the strong-form disclosure principle in more general contests.

Our main result says that, provided that the contest is uniformly asymmetric, the only outcome of the revelation game consistent with the assumption of perfect Bayesian rationality is the one in which all the privately held information is unfolded prior to the contest. Thus, we find general conditions under which the disclosure principle in the strong form applies to a standard contest setting. This may be of interest because effort choices in probabilistic contests are neither strategic substitutes nor strategic complements, and consequently contests do not satisfy the usual conditions sufficient for the strong-form disclosure principle.³

There is a simple intuition for why contestants find it optimal to release hard evidence in a uniformly asymmetric contest. In view of the high effort expected from a favourite that is left to speculate about the underdog’s strength, the weakest type of the underdog has a strict incentive to self-disclose, so as to moderate the favourite. But then, the set of types deemed possible by the favourite in the absence of any evidence becomes smaller. Therefore, the weakest of the remaining types of the underdog will choose to disclose her private information as well. And so on. Thus, there is an unravelling on the underdog’s side. But in the resulting contest with one-sided incomplete information, the unravelling continues on the side of the favourite. Indeed, the respective strongest type of the favourite has a strict incentive to self-disclose, so as to discourage the underdog. Ultimately, full revelation of all private information isinevitable.⁴

This intuition is robust in a number of dimensions. First, the strong-form disclosure principle continues to be valid generically when types are correlated. The role of the weakest type of the underdog is then taken on by the lowest-bidding type in the contest. Second, the main result extends to partitional information releases and randomized signals. Next, we show that the principle applies likewise if disclosure decisions are taken in a sequential fashion, i.e. with either the favourite or the underdog moving first. Further, we allow for continuous type spaces. Finally, we show that, in the case of one-sided private information, the assumption of uniform asymmetry may be dropped without losing the conclusion. In that case, there is always one extremal type that strictly prefers to self-disclose. The order of the unravelling may then switch hence and forth, in a “bang-bang” fashion, between the remaining weak and strong types of the informed side.

The strong-form disclosure principle is, however, not universally valid. First, if a contest with two-sided incomplete information is not uniformly asymmetric, then it may happen that no type has an incentive to self-disclose. The reason is that contestants face countervailing incentives. While a relatively efficient type benefits from demoralizing an inefficient opponent, she simultaneously suffers from revealing her information to an opponent of comparable strength. Since the situation may be similar for a relatively inefficient type, full concealment can be an equilibrium. Second, full revelation is not a necessity if contestants possess commitment power. For example, Bayesian persuasion need not lead to full revelation even if the contest would otherwise unravel. Finally, the disclosure principle crucially depends on the assumption that private information is verifiable. In fact, as we show, unverifiable messages are necessarily ignored in any probabilistic contest, even in the presence of a trustworthymediator.

Regarding welfare, we point out that unrestricted communication in probabilistic contests may lead to a Pareto-inferior outcome for the contestants. Still, depending on the objective of the contest organizer, full revelation may be socially desirable and, in particular, the result of optimal information design.

Further illustrations. Our introductory example was taken from the area of military conflict and war. The following examples may serve as additional illustrations.

• •

  Public enforcement in the U.S. is characterized by a large disparity between the power of the state prosecutor and the typical criminal defendant (Lynch, 1998). Both parties possess verifiable information (Bibas, 2004). Even when plea bargaining is banned, it is common for the defendant to plead guilty to a subset of the charges (Weninger, 1987). Conversely, the prosecutor releases evidence to induce the defendant to confess (Petegorsky, 2012). As a result, only a small fraction of criminal cases go to trial.

• •

  Implicit or explicit threats are used to intimidate whistleblowers (Chassang and Padró I Miquel, 2019) and witnesses (Maynard, 1994).

• •

  In R&D and patent races, the frontrunner reveals research results and funding successes to discourage competitors (Baker and Mezzetti, 2005). But also laggards announce new products to influence the outcome of competition in their interest (Robertson et al., 1995).

• •

  In social conflict, the use of phenotype “indices” resolves or avoids physical conflicts in dyadic relationships (Hand, 1986). For conflicts arising within dominance or subordination relationships, signals tend to be placating or acquiescent. Within egalitarian or unresolved relationships, however, there are either no signals or signals indicate relative desire for an item on a case-by-case basis.

Related literature. The economics literature has a long tradition of studying incentives for the voluntary disclosure of private information. In seminal contributions, Grossman (1981) and Milgrom (1981) pointed out that, as a consequence of unravelling, sellers will it find hard to withhold verifiable information about the quality of their products. The underlying disclosure principle has since shaped the theoretical discussion about the pros and cons of disclosure regulation, as is reflected by a very large body of literature.⁵

Probabilistic contests of incomplete information have been studied for some time. Rosen (1986, fn. 7) still complained that “few analytical results” were available. Early papers include Linster (1993) and Baik and Shogren (1995). The general framework with one-sided and two-sided private valuations is due to Hurley and Shogren (1998a, 1998b). Wärneryd (2003) observed that the uninformed player in a common-value setting is more likely to win than the informed player. Malueg and Yates (2004) analysed a symmetric two-player Tullock contest with two equally likely and possibly correlated types. Schoonbeek and Winkel (2006) noted that individual types may remain inactive. General results on the existence and uniqueness of Bayesian equilibrium have been obtained by Ewerhart (2014), Einy et al. (2015), and Ewerhart and Quartieri (2020).

The present paper falls into the recent and quickly expanding literature concerned with the disclosure of verifiable information in contests.⁶ That literature has tended to focus on either ex ante voluntary disclosure, optimal disclosure policies, or interim voluntary disclosure.⁷ Ex ante voluntary disclosure in probabilistic contests has been studied by Denter et al. (2014), in particular. Assuming a probabilistic contest technology with one-sided incomplete information, they showed that a “laissez-faire” policy regarding the informed player’s ex ante disclosure decision leads to lower expected lobbying expenditures than a policy of mandatory disclosure.⁸ The second topic, optimal disclosure policies in contests, has recently seen a strong development. In particular, effort-maximizing disclosure policies have been characterized by Zhang and Zhou (2016) and Serena (2022) for probabilistic technologies, and by Fu et al. (2014), Chen et al. (2017), and Lu et al. (2018) for deterministic technologies.⁹

The present analysis is mainly concerned, however, with the third topic, i.e. the interim voluntary disclosure in contests. As far as we know, there is only one paper that has dealt with this issue on a comparable level of generality.¹⁰ Specifically, Kovenock et al. (2015) showed that, regardless of whether valuations are private or common, the interim information sharing game followed by an all-pay auction admits a perfect Bayesian equilibrium in which no player ever shares her private information. Instead of the all-pay auction, however, we consider a probabilistic contest. Overall, the review of the literature suggests that the specific research question pursued in the present paper, viz. the analysis of incentives for the interim voluntary disclosure of hard evidence in contests with probabilistic technologies and two-sided incomplete information, has not been addressed in prior work.¹¹

The remainder of this article is structured as follows. Section 2 introduces the set-up. The main result is stated in Section 3. Section 4 outlines the proof. Section 5 offers extensions, while Section 6 discusses limits to the scope of the disclosure principle. Section 7 concerns efficiency. Section 8 concludes. A Supplementary Appendix contains technical proofs and other material omitted from the body of the article.

2 SET-UP

Considered is an interaction over two stages, referred to as revelation stage and contest stage, respectively. The modelling follows the literature on pre-play communication (Okuno-Fujiwara et al., 1990; van Zandt and Vives, 2007; Hagenbach et al., 2014). We will start with the contest stage and continue backwards with the revelation stage.

2.1 The contest stage

Two players (or teams) $i=1,2$ exert effort at marginal cost $c_i>0$ so as to increase their respective odds of winning a contested prize. Player i values winning at $V_i$⁠, and losing at $L_i$⁠, where $V_i>L_i$⁠. Contestant i’s effort (or bid) is denoted by $x_i\ge 0$⁠. Following Rosen (1986), we assume that player i’s probability of winning against $j\ne i$ is given as

where ${\gamma }_i>0$ denotes i’s ability, while $h\equiv h(z)$ is a continuous production function that is twice continuously differentiable at positive bid levels, with $h(0)=0$⁠, $h^{\mathrm{\prime }}>0$⁠, and $h^{\mathrm{″}}\le 0$⁠.¹² Thus, player i’s expected payoff may be written as

where ${\theta }_i=(c_i,V_i,L_i,{\gamma }_i)$ denotes player i’s type. This set-up includes, as an important special case, the biased Tullock contest (Tullock, 1975; Leininger, 1993; Clark and Riis, 1998), where the production function is given by $h(z)=h^{\mathrm{TUL}}(z;r)\equiv z^r$ for some exogenous $r\in (0,1]$⁠. The lottery contest corresponds to the case $r=1$⁠.

For convenience, we will assume that each player i’s type is independently and discretely distributed—and that it concerns the marginal cost parameter $c_i$ only. As will be explained, the restriction to pure cost types is without loss of generality if either ability is publicly observable or the contest is of the Tullock form. Thus, player i’s type is assumed to be drawn from a probability distribution over the finite set $C_i=\{c_i^1,\dots ,c_i^{K_i}\}$⁠, where $K_i\ge 1$⁠, and

The symbol ${\underset{\_}{c}}_i$ denotes the most efficient, or strongest type, while ${\overline{c}}_i$ denotes the least efficient, or weakest type of player i. The ex ante probability of type $c_i^k$ is denoted by $q_i^k\equiv q_i(c_i^k)$⁠, for $k\in \{1,\dots ,K_i\}$⁠, with $q_i^k>0$⁠. Valuations will be normalized so that $V_i=1$ and $L_i=0$⁠, for $i\in \{1,2\}$⁠. We will also write ${\mathrm{\Pi }}_i(x_i,x_j;c_i)={\mathrm{\Pi }}_i(x_i,x_j;{\theta }_i)$⁠.

A bid schedule for player $i\in \{1,2\}$ is a mapping ${\xi }_i:C_i\to {\mathbb{R}}_{+}$⁠. The set of i’s bid schedules will be denoted as $X_i$⁠. A pair of bid schedules ${\xi }^{*}=({\xi }_1^{*},{\xi }_2^{*})\in X_1\times X_2$ is a Bayesian Nash equilibrium if, for any type $c_i\in C_i$ of any player $i\in \{1,2\}$⁠, the effort level $x_i={\xi }_i^{*}(c_i)$ maximizes type $c_i$’s expected payoff $E_{c_j}[{\mathrm{\Pi }}_i(x_i,{\xi }_j^{*}(c_j);c_i)]$⁠, where $E_{c_j}[.]$ denotes the expectation over the realizations of $c_j\in C_j$⁠, with $j\ne i$⁠. Following Schoonbeek and Winkel (2006), a type $c_i\in C_i$ with ${\xi }_i^{*}(c_i)>0$ (⁠${\xi }_i^{*}(c_i)=0$⁠) will be called active (inactive). As usual in this type of model, the discontinuity of the payoff functions at the origin implies that both players are necessarily active with positive probability.¹³ By the same token, at least one player is active with probability one.

 

Special notation will be used in the cases of complete and one-sided incomplete information. If $(c_1,c_2)=(c_1^{\circ },c_2^{\circ })$ is public information, then i’s equilibrium strategy will be written as $x_i^{\circ }=x_i^{\circ }(c_1^{\circ },c_2^{\circ })$⁠. Further, if player i’s type $c_i=c_i^{\mathrm{\#}}$ is public, while player j’s type, with $j\ne i$⁠, remains uncertain, then equilibrium strategies will be written as $x_i^{\mathrm{\#}}=x_i^{\mathrm{\#}}(c_i^{\mathrm{\#}})$ for player i and ${\xi }_j^{\mathrm{\#}}={\xi }_j^{\mathrm{\#}}(.;c_i^{\mathrm{\#}})$ for player j, so that ${\xi }_j^{\mathrm{\#}}(c_j;c_i^{\mathrm{\#}})$ is type $c_j$’s equilibrium effort.

2.2 The revelation stage

At a stage preceding the contest, players simultaneously and independently decide whether to disclose their respective type or not. Initially, it will be assumed that private information cannot be misrepresented. Further, we assume that the decision to self-disclose does not lead to any direct costs.¹⁵

In response to the observation of verifiable information, prior beliefs are updated according to Bayes’ rule whenever possible. One notes that off-equilibrium beliefs may arise, but only in the distinct case where a player chooses to conceal her private information even though the equilibrium strategy entails self-disclosure by all types of that player.¹⁶

In any case, the contest stage begins with a well-defined posterior belief${\mu }_i\in \mathrm{\Delta }(C_i)$ about each player $i\in \{1,2\}$⁠, where $\mathrm{\Delta }(C_i)=\{{\mu }_i:C_i\to [0,1]$ s.t. $\sum _{k=1}^{K_i}{\mu }_i(c_i^k)=1\}$⁠. Ignoring zero-probability types, a unique Bayesian equilibrium exists by Lemma 1. In particular, the expected continuation payoff from the contest stage is well-defined for any $c_i\in C_i$ and $i\in \{1,2\}$⁠.¹⁷ A (reduced-form) perfect Bayesian equilibrium consists of (i) a set $S_i\subseteq C_i$ of revealing types, for each player $i\in \{1,2\}$ and (ii) an off-equilibrium belief ${\mu }_i^0\in \mathrm{\Delta }(C_i)$ for any $i\in \{1,2\}$ with $S_i=C_i$⁠, such that $E_{c_j}[{\mathrm{\Pi }}_i(x_i^{\mathrm{\#}},{\xi }_j^{\mathrm{\#}}(c_j);c_i)]\ge E_{c_j}[{\mathrm{\Pi }}_i(x_i,{\xi }_j^{*}(c_j);c_i)]$⁠, for any (interim measurable) deviation $x_i\ge 0$ and type $c_i\in S_i$⁠, as well as $E_{c_j}[{\mathrm{\Pi }}_i({\xi }_i^{*}(c_i),{\xi }_j^{*}(c_j);c_i)]\ge E_{c_j}[{\mathrm{\Pi }}_i(x_i^{\mathrm{\#}},{\xi }_j^{\mathrm{\#}}(c_j);c_i)]$⁠, for any $c_i\in C_i\mathrm{\setminus }{S}_i$⁠. Here we dropped, for convenience, the reference to prior disclosure decisions in the notation of equilibrium bids and possible deviations.¹⁸

3 THE UNRAVELLING THEOREM

This section is central to our analysis. We start by defining what we call uniformly asymmetric contests. We then provide a sufficient condition for a contest to be uniformly asymmetric. Finally, we present the main result of this article.

3.1 Uniformly asymmetric contests

The focus of our analysis lies on probabilistic contests with two-sided incomplete information that satisfy the following definition.

 

Here, as usual, supp $({\mu }_i)=\{c_i\in C_i:{\mu }_i(c_i)>0\}$ denotes the support of player i’s posterior belief ${\mu }_i$⁠, for $i\in \{1,2\}$⁠. Thus, in a uniformly asymmetric contest, two properties hold regardless of posterior beliefs at the contest stage. First, Player 1 is active with probability one. Second, provided that Player 2 is also active with probability one, Player 1 is interim always (i.e. for all type realizations) more likely to win than Player 2.¹⁹

If the contest is of complete information (i.e. if $K_1=K_2=1$⁠), then being uniformly asymmetric is equivalent to what Dixit (1987) called an asymmetric contest. Correspondingly, we will henceforth refer to Player 1 alternatively as the favourite and to Player 2 as the underdog.

3.2 A sufficient condition

In this section, we derive a condition on the primitives of the model that is sufficient for a contest to be uniformly asymmetric. While the assumption is strong, it will allow us to capture a very clear and robust intuition.

 

Assumption 1 is a joint restriction on four parameters, each of which admits an intuitive interpretation.²¹ First, $\underset{\_}{\rho }$ measures the degree of noise in the contest technology, where a larger value corresponds to more noise. Second, σ captures Player 1’s resolve (Hurley and Shogren, 1998a, 1998b), i.e. Player 1’s worst-case relative cost position vis-a-vis Player 2. If $\sigma >1$⁠, then Player 1 is always more efficient than Player 2, whereas if $\sigma \le 1$⁠, then Player 1 may be weakly less efficient than Player 2. Third, ${\pi }_i\in (0,1]$ reflects the predictability of player i’s marginal cost, where the maximum value of one corresponds to complete information about $c_i$⁠. Fourth and finally, the net bias γ has an obvious interpretation, where $\gamma <1$⁠, for example, means that the contest technology is biased against Player 2.

When positive, the threshold value ${\gamma }^{*}$ is weakly declining in $\underset{\_}{\rho }$⁠, as well as strictly increasing in σ, ${\pi }_1$⁠, and ${\pi }_2$⁠. Thus, for any given net bias, the assumption is more likely to hold when there is less noise, Player 1’s resolve is larger, or marginal costs are more predictable. In particular, we see that, if Assumption 1 holds for a given contest, changes to the information structure caused by pre-play disclosure decisions cannot invalidate it. For instance, if either $C_1$ or $C_2$ is substituted by a non-empty subset, then σ, ${\pi }_1$⁠, and ${\pi }_2$ all rise weakly, so that the cut-off value for the bias, ${\gamma }^{*}$⁠, likewise rises weakly. Thus, if the assumption holds for type sets $C_1$ and $C_2$⁠, then it holds also for any pair of non-empty subsets. A similar remark applies to any updating of beliefs.²²

In the limit case of complete information and symmetric costs (i.e.${\underset{\_}{c}}_1={\overline{c}}_1={\underset{\_}{c}}_2={\overline{c}}_2$⁠), Assumption 1 says that the technology is biased against Player 2 (i.e.${\gamma }_2<{\gamma }_1$⁠). Further, the case of a biased contest with ex ante symmetric type distributions (i.e.${\underset{\_}{c}}_1={\underset{\_}{c}}_2\le {\overline{c}}_1={\overline{c}}_2$⁠), as discussed, e.g. by Drugov and Ryvkin (2017), is not generally excluded by Assumption 1.²³

Clearly, with Assumption 1 in place, Player 1 is in a quite strong position relative to Player 2. And indeed, as the following result shows, the assumption implies that the contest is uniformly asymmetric.

(Sufficient condition)

 

3.3 Main result

We will use the term full revelation to characterize the perfect Bayesian equilibrium, or the perfect Bayesian equilibrium outcome, in which all private information is revealed. The main result of the present paper is the following.

(Strong-form disclosure principle)

 

Theorem 1 states that the strong-form disclosure principle applies to any uniformly asymmetric contest.

It is not hard to see that self-disclosure by all types is an equilibrium. Indeed, it suffices to specify off-equilibrium beliefs so that a player that surprises her opponent by concealing her private information is understood to be the worst-case type, i.e. the type that no other type would like to masquerade as.²⁴ In our setting, the worst-case types are the most efficient underdog and the least efficient favourite, respectively. Provided that off-equilibrium beliefs are specified in that sceptical way (Milgrom, 2008), it is optimal for all types to stick to self-disclosure.

Note that uniqueness is claimed for the equilibrium outcome only. To reveal all private information, it suffices that, for each player, all types except one disclose their private information. However, that multiplicity of perfect Bayesian equilibria is trivial since it does not affect the outcome of the contest.

For auctions with interdependent valuations (Benoît and Dubra, 2006; Tan, 2016), incentives to reveal a private signal are typically strongest at the bottom of the signal support of the common-value component. For instance, a bidder in an auction that learns that a supposedly original painting is not authentic has an incentive to share that information with the other bidders. Thus, as in the present analysis, voluntary disclosure is likely to occur whenever it reduces the opponent’s incentives for bidding too aggressively.²⁵

4 UNDERSTANDING THE UNRAVELLING RESULT

In this section, we explain the mechanics underlying Theorem 1, dealing first with the underdog’s and, subsequently, with the favourite’s disclosure decision. The section ends with a discussion.

4.1 Benefits of self-disclosure for the underdog

We focus on the weakest type of the underdog, ${\overline{c}}_2$⁠, while assuming that there are at least two possible type realizations for $c_2$ (otherwise, there is nothing to show). Let ${\xi }^{*}=({\xi }_1^{*},{\xi }_2^{*})$ denote the equilibrium at the contest stage that results if ${\overline{c}}_2$ does not disclose her type. Then, the probability of winning and the expected payoff for ${\overline{c}}_2$ are given by $p_2^{*}({\overline{c}}_2)=E_{c_1}[p_2({\xi }_2^{*}({\overline{c}}_2),{\xi }_1^{*}(c_1))]$ and ${\mathrm{\Pi }}_2^{*}({\overline{c}}_2)=E_{c_1}[{\mathrm{\Pi }}_2({\xi }_2^{*}({\overline{c}}_2),{\xi }_1^{*}(c_1);{\overline{c}}_2)]$⁠, respectively. Similarly, let $({\xi }_1^{\mathrm{\#}},x_2^{\mathrm{\#}})$ denote the equilibrium in the contest with one-sided incomplete information that results if ${\overline{c}}_2$ reveals her type. Then, ${\overline{c}}_2$’s probability of winning and expected payoff are given by $p_2^{\mathrm{\#}}=E_{c_1}[p_2(x_2^{\mathrm{\#}},{\xi }_1^{\mathrm{\#}}(c_1))]$ and ${\mathrm{\Pi }}_2^{\mathrm{\#}}=E_{c_1}[{\mathrm{\Pi }}_2(x_2^{\mathrm{\#}},{\xi }_1^{\mathrm{\#}}(c_1);{\overline{c}}_2)]$⁠, respectively. The following result summarizes the comparative statics of the equilibrium at the contest stage with respect to ${\overline{c}}_2$’s disclosure decision.

(Self-disclosure by the weakest type of the underdog)

 

Thus, after revealing her relative weakness, the weakest type of the underdog behaves as if gaining confidence. She bids more aggressively and wins with a strictly higher probability. Moreover, the self-disclosure is always strictly beneficial for her.²⁶

The proof of Proposition 1 is based on the monotonicity properties of best response mappings in uniformly asymmetric contests. Let $(X_i,{⪰}_i)$ denote the set of player i’s bid schedules equipped with the product order.²⁷ Denote by $X_j^{*}\subseteq X_j$ the set of bid schedules ${\xi }_j$ for player $j\in \{1,2\}$ that admit a unique maximizer $x_i\equiv {\tilde{\beta }}_i({\xi }_j;c_i)\in {\mathbb{R}}_{+}$ of the expected payoff function $x_i\mapsto E_{c_j}[{\mathrm{\Pi }}_i(x_i,{\xi }_j(c_j);c_i)]$⁠, for any $c_i\in C_i$ with $i\ne j$⁠. Given ${\xi }_j\in X_j^{*}$⁠, the bid schedule ${\beta }_i({\xi }_j)={\tilde{\beta }}_i({\xi }_j;\cdot ):C_i\to {\mathbb{R}}_{+}$ will be called the best-response bid schedule against ${\xi }_j$⁠. As shown in the Supplementary Appendix, the best-response bid schedule ${\beta }_i({\xi }_j)$ is weakly declining in the type for any ${\xi }_j\in X_j^{*}$⁠, and strictly so at positive bid levels. Moreover, the thereby defined best-response mapping${\beta }_i:X_j^{*}\to X_i$ satisfies monotonicity properties under suitable domain restrictions.²⁸

The fact that ${\overline{c}}_2$ raises her effort after self-disclosure is crucial. To understand this point, suppose that, instead of strictly raising her effort, ${\overline{c}}_2$ were to weakly lower her effort after disclosure, i.e.$x_2^{\mathrm{\#}}\le {\xi }_2^{*}({\overline{c}}_2)$⁠, as illustrated on the right-hand side of Figure 1 for the strict case. Consider now the flat bid schedule ${\psi }_2(x_2^{\mathrm{\#}})\in X_2$ that prescribes an effort of $x_2^{\mathrm{\#}}$ for each $c_2\in C_2$⁠. Then, since there are at least two types in $C_2$⁠, and since the equilibrium bid schedule ${\xi }_2^{*}$ is strictly declining at positive bid levels (also recalling that ${\xi }_2^{*}\equiv 0$ is not feasible), we get ${\xi }_2^{*}\succ {\psi }_2(x_2^{\mathrm{\#}})$⁠. From the strict monotonicity of Player 1’s best-response mapping, checking domain conditions, we therefore obtain ${\xi }_1^{*}={\beta }_1({\xi }_2^{*})\succ {\beta }_1({\psi }_2(x_2^{\mathrm{\#}}))={\xi }_1^{\mathrm{\#}}$⁠, as shown on the left-hand side of Figure 1. Applying now the strictly declining best-response mapping of ${\overline{c}}_2$⁠, checking domain conditions also here, one arrives at ${\xi }_2^{*}({\overline{c}}_2)={\tilde{\beta }}_2({\xi }_1^{*};{\overline{c}}_2)<{\tilde{\beta }}_2({\xi }_1^{\mathrm{\#}};{\overline{c}}_2)=x_2^{\mathrm{\#}}$⁠, which yields the desired contradiction. Thus, the weakest type of the underdog indeed raises her bid after self-disclosure.

Self-disclosure raises also the probability of winning for ${\overline{c}}_2$⁠. This follows from what we call Stackelberg monotonicity in the complete-information model. By this, we mean that an increase of player i’s bid, subject to an optimal response by the opponent j, always raises player i’s winning probability (and strictly so in the interior). Intuitively, a higher effort is rewarded in terms of a higher winning probability.²⁹ Applied to the present situation, this says that a Stackelberg-leading underdog that raises her bid from $x_2={\xi }_2^{*}({\overline{c}}_2)$ to $x_2^{\mathrm{\#}}$ strictly increases her probability of winning against any best-responding type $c_1$⁠. But $c_1$’s best response to $x_2={\xi }_2^{*}({\overline{c}}_2)$ is already weakly lower than ${\xi }_1^{*}(c_1)$⁠. It follows that, indeed, the probability of winning for the weakest type of the underdog against any $c_1$ increases strictly from her self-disclosure.

In a final step, it is shown that the weakest type of the underdog has a strict incentive to self-disclose. The proof we managed to come up with exploits type ${\overline{c}}_2$’s first-order condition to rewrite her expected payoff from the contest as a monotone function of ex post winning probabilities and bids. For instance, in the Tullock contest with parameter r, type ${\overline{c}}_2$’s equilibrium payoff with and without disclosure may be represented as

respectively. Given parts (i) and (ii) of Proposition 1, this suffices to prove the claim.

4.2 Benefits of self-disclosure for the favourite

Repeated application of Proposition 1 shows that the underdog’s side of the contest equilibrium unravels. Let $c_2^{\mathrm{\#}}$ denote the commonly known cost type of the underdog. Assuming that there are at least two type realizations for $c_1$⁠, we will now study the disclosure decision of the strongest type of the favourite, ${\underset{\_}{c}}_1$⁠.

If type ${\underset{\_}{c}}_1$ decides to conceal her private information, then the ensuing contest is one of one-sided incomplete information, with equilibrium efforts ${\xi }_1^{\mathrm{\#}}({\underset{\_}{c}}_1)\equiv {\xi }_1^{\mathrm{\#}}({\underset{\_}{c}}_1;c_2^{\mathrm{\#}})$ and $x_2^{\mathrm{\#}}\equiv x_2^{\mathrm{\#}}(c_2^{\mathrm{\#}})$⁠. Type ${\underset{\_}{c}}_1$’s probability of winning and expected payoff are consequently given by $p_1^{\mathrm{\#}}=p_1({\xi }_1^{\mathrm{\#}}({\underset{\_}{c}}_1),x_2^{\mathrm{\#}})$ and ${\mathrm{\Pi }}_1^{\mathrm{\#}}={\mathrm{\Pi }}_1({\xi }_1^{\mathrm{\#}}({\underset{\_}{c}}_1),x_2^{\mathrm{\#}};{\underset{\_}{c}}_1)$⁠, respectively. If, however, type ${\underset{\_}{c}}_1$ decides to disclose her private information (assuming, w.l.o.g., non-disclosure for the other types), then the contest is one of complete information, with equilibrium efforts $x_i^{\circ }\equiv x_i^{\circ }({\underset{\_}{c}}_1,c_2^{\mathrm{\#}})$⁠, for $i=1,2$⁠. In that case, type ${\underset{\_}{c}}_1$’s probability of winning and expected payoff are given by $p_1^{\circ }=p_1(x_1^{\circ },x_2^{\circ })$ and ${\mathrm{\Pi }}_1^{\circ }={\mathrm{\Pi }}_1(x_1^{\circ },x_2^{\circ };{\underset{\_}{c}}_1)$⁠, respectively. The following result summarizes the comparative statics of the one-sided incomplete-information contest with respect to a revelation by ${\underset{\_}{c}}_1$⁠.

(Self-disclosure by the strongest type of the favourite)

 

Thus, if the type of the underdog is public, then the self-revelation by the strongest type of the favourite discourages the underdog. As a result, the strongest type of the favourite exerts a lower effort, but still wins with higher probability. While the proof of Proposition 2 employs the same methods that have been used before, the argument is, of course, much simpler in this case.³⁰

An iterated application of Proposition 2 implies that also the favourite’s side unravels. Thus, in a uniformly asymmetric contest, full revelation is the only outcome consistent with the assumption of perfect Bayesian rationality. But, as already discussed, self-disclosure by all types of both players is indeed a perfect Bayesian equilibrium, which yields the conclusion of Theorem 1.

4.3 Discussion: dominance and defiance

As mentioned in Section 1, the reason why the proof of Theorem 1 is not as straightforward as one might expect is that, in general, a unilateral disclosure of some type may cause some types of the opponent to raise their bids. For intuition, note that there are two countervailing effects. On the one hand, following the self-disclosure by ${\overline{c}}_2$⁠, say, the favourite’s belief collapses, inducing her to lower her bid. On the other hand, ${\overline{c}}_2$ raises her bid, which induces the favourite to do the same. As a result, the overall effect of the underdog’s self-disclosure on the bid of a given type of the favourite is ambiguous. The situation is similar for the underdog who, under two-sided incomplete information, may either drop out, lower her bid, or raise her bid in response to the favourite’s self-disclosure. In the Supplementary Appendix, we illustrate dominant and defiant reactions to self-disclosure using numerical examples and relate those anomalies to general instability properties of probabilistic contests (Wärneryd, 2018).³¹

5 EXTENSIONS

In this section, we discuss a variety of extensions of Theorem 1. To keep the exposition as non-technical as possible, most of the formal results and derivations underlying the discussion have been moved to the Supplementary Appendix.

5.1 Correlated types

By continuity, Theorem 1 is robust with respect to the introduction of any sufficiently small correlation between contestants’ types. But suppose that correlation is more substantial. Then, negative correlation renders stronger types of the underdog more optimistic, and the proof of the crucial Proposition 1 goes through as before. Positive or general correlation, however, may render stronger types of the underdog more pessimistic. In that case, some type of the underdog other than ${\overline{c}}_2$⁠, say ${\hat{c}}_2$⁠, may submit the lowest bid in the contest. Then, the argument underlying Proposition 1 will generically go through with ${\hat{c}}_2$ replacing ${\overline{c}}_2$⁠. The genericity assumption is needed because, without it, positive correlation might induce all types of the underdog to choose the same bid, so that self-disclosure becomes useless. Thus, straightforward variants of Theorem 1 hold for sufficiently small correlation, negative correlation, and generically for any type of correlation.

5.2 Partitional disclosures

In the main analysis, we assumed that disclosure is “all-or-nothing”. However, in many cases, contestants may have more control over the information they choose to disclose than what has been assumed so far. In a model with pre-play partitional disclosure of the state space, Hagenbach et al. (2014) identified necessary and sufficient conditions for the existence of a fully revealing sequential equilibrium with “extremal” off-equilibrium beliefs that implements a given Nash equilibrium action profile on and off the equilibrium path. Our main result continues to hold if contestants’ message correspondences each contain an evidence base. For example, the disclosure decision might alternatively establish an upper (lower) bound for the favourite’s (the underdog’s) cost parameter. It should be immediate to see that the unravelling argument underlying Theorem 1 extends along these lines to the more general framework of partitional disclosures.³²

5.3 Randomized revelations

Allowing for randomized revelations does not change the conclusion of Theorem 1. Indeed, a type’s randomized decision regarding self-disclosure cannot be more profitable than a pure decision. Therefore, full revelation remains an equilibrium outcome. But, we also note that the unravelling is inevitable when players may use randomized revelations because the self-disclosure of the relevant extremal type remains strictly optimal regardless of conditional type distributions.

5.4 Sequential moves

Theorem 1 continues to hold when the revelation stage is replaced by a sequential-move game in which the disclosure decision is made first by the favourite and then, after observation, by the underdog. Intuitively, in any equilibrium in which two or more types of the favourite pool with positive probability in the same information set, the underdog’s type will be revealed by Proposition 1. Anticipating this, the strongest type of the favourite strictly prefers to self-disclose instead of pooling with any weaker type. But also in the case where the underdog moves first, we show in the Supplementary Appendix that full revelation remains the unique perfect Bayesian equilibrium outcome in the lottery contest.³³

5.5 One-sided incomplete information

If just one of the contestants is privately informed, then the assumption of uniform asymmetry is typically not needed to obtain full revelation as the unique equilibrium outcome.

(One-sided incomplete information)

 

While the result holds generally, the intuition for the proof is most transparent in the special case of the Tullock contest. Two observations are important, both of which are derived from the first-order conditions. First, in equilibrium, the expenses of the uninformed contestant, $c_i^{\mathrm{\#}}{x}_i^{\mathrm{\#}}$⁠, correspond precisely to the expected expenses of the informed contestant, $E_{c_j}[c_j{\xi }_j^{\mathrm{\#}}(c_j)]$⁠. Second, the type-specific expenses of the informed contestant, $c_j{\xi }_j^{\mathrm{\#}}(c_j)$⁠, are strictly hump-shaped as a function of the type. Combining these observations, there is always at least one extremal type of the informed contestant, either ${\underset{\_}{c}}_j$ or ${\overline{c}}_j$⁠, such that making that type marginally more likely lowers the expected expenses of the uninformed contestant. Which of the two extremal types has this property depends only on the three parameters ${\underset{\_}{c}}_j$⁠, ${\overline{c}}_j$⁠, and $c_i^{\mathrm{\#}}$⁠, regardless of the distribution of probabilities. Specifically, the efficient type ${\underset{\_}{c}}_j$ has a strict incentive to self-disclose if $c_i^{\mathrm{\#}}>\sqrt{{\underset{\_}{c}}_j{\overline{c}}_j}$⁠, while the inefficient type ${\overline{c}}_j$ has a strict incentive to self-disclose if $c_i^{\mathrm{\#}}<\sqrt{{\underset{\_}{c}}_j{\overline{c}}_j}$⁠. Thus, replacing the assumption of uniform asymmetry by one-sided incomplete information, there is an additional twist to the logic of the unravelling argument. Rather than following the linear ordering of types on each side of the contest, the unravelling may now follow a “bang-bang” order, in the sense that extremal efficient and extremal inefficient types alternatingly find it strictly optimal to self-disclose conditional on hypothesized prior disclosures.³⁴

5.6 Continuous type distributions

Benoît and Dubra (2006) have derived a general unravelling result for auctions and other Bayesian games that allows for multiple players and metric type spaces. That result may be used to extend Theorem 1 to the case of continuous type distributions.³⁵ Suppose that for $i\in \{1,2\}$⁠, player i’s marginal cost is drawn from an interval $[{\underset{\_}{c}}_i,{\overline{c}}_i]$⁠, with $0<{\underset{\_}{c}}_i<{\overline{c}}_i$⁠, according to some continuous distribution function $F_i$⁠. Both Definition 1 and Lemma 2 extend to this case, provided that the probability ranking property (4) is required for any pair of cost realizations in the support of players’ posterior beliefs. Considering now a uniformly asymmetric lottery contest with independent types, contestants’ types are almost surely revealed in any perfect Bayesian equilibrium.

5.7 Private information about valuations and ability

We assumed above that private information concerns marginal cost only. To accommodate more general forms of uncertainty, suppose that player i’s private information is instead summarized in the vector ${\theta }_i=(c_i,V_i,L_i,{\gamma }_i)$⁠, where the components satisfy the same restrictions as before.

 

Part (i) captures the (well-known) equivalence between uncertainty about marginal cost and valuations. Part (ii) says that, in the case of the Tullock contest, our analysis covers multi-dimensional uncertainty without restriction. This is an important point because, as noted by a referee, in many real-world contests, verifiable evidence does not concern contestants’ preferences but rather the likelihood to win.

6 LIMITS OF THE SCOPE OF THE DISCLOSURE PRINCIPLE

The strong-form disclosure principle is not universally valid in probabilistic contests. As will be shown in this section, the principle may break down if the contest is not uniformly asymmetric or if contestants have commitment power. Moreover, the principle never holds in probabilistic contests if information is unverifiable.

6.1 Contests that are not uniformly asymmetric

The conclusion of Theorem 1 may fail if the contest is not uniformly asymmetric. In line with the intuition provided in Section 1, we outline below a numerical example of a probabilistic contest that is not uniformly asymmetric and in which the perfect Bayesian equilibrium outcome need not be fully revealing. In view of Theorem 2, any counterexample of this sort necessarily features two-sided asymmetric information.

(Countervailing incentives)

 

Thus, the assumption of uniform asymmetry cannot be easily dropped without losing the strong-form disclosure principle in probabilistic contests with two-sided incomplete information.³⁶

6.2 Commitment power and Bayesian persuasion

Self-disclosure of one type need not be in the interest of other types. If, for example, the strongest type of the favourite reveals her private information in a uniformly asymmetric contest, then the competition for the remaining types of the favourite will typically become tougher. Thus, a contestant may be worse off as a result of voluntary disclosure. Following Kamenica and Gentzkow (2011), one may assume that each contestant possesses commitment power that allows her to follow a communication strategy optimized from an ex ante perspective. As we show in the Supplementary Appendix, however, a contestant’s optimal Bayesian persuasion strategy may take different forms, including not only full concealment but also full disclosure (in which case it may be redundant), or even the use of randomized signals. For example, given precommitment to a signal, an efficient type of the underdog may occasionally pool with an inefficient type. Conversely, a contestant might seek ways to avoid receiving messages by shutting down communication channels.

6.3 Cheap talk

In Crawford and Sobel’s (1982) model of cheap talk,³⁷ pre-play messages released at the interim stage are assumed to be unverifiable.³⁸ More generally, in a communication equilibrium (Myerson, 1982), each player reports private information to a mediator in an unverifiable way. Having received the reports, the mediator follows her precommitted instructions and releases a recommendation to each of the players via a bilateral communication channel. Then, based on the recommendation and her type, each player chooses a strategy. For example, the mediator may be precommitted to relay the reports unchanged to the respective other party, which would correspond to cheap talk. In the case of the two-player all-pay auction, Pavlov (2013) identified assumptions under which every communication equilibrium is interim payoff equivalent to the Bayesian equilibrium. The following result is an analogous observation for probabilistic contests.

(Babbling)

 

Thus, cheap talk in probabilistic contests is necessarily ineffective, even if intermediated by a trustworthy third party. If the contest were a zero-sum game, then this observation would be obvious. Indeed, as noted by Farrell (1985), a player should be very suspicious to make use of information provided by another player with completely opposite preferences. However, probabilistic contests are not zero-sum, but only strategically equivalent to a zero-sum game (Moulin and Vial, 1978). To see this, one notes that adding the expenses of a player to the other player’s payoff function does not change marginal incentives. Therefore, in principle, there might be communication strategies mutually beneficial for both players to select Pareto superior outcomes. However, the Bayesian equilibrium is unique by Lemma 1, which turns out to imply that unverifiable communication is ineffective.³⁹

7 EFFICIENCY

What are the welfare implications of information disclosure in probabilistic contests? In the Supplementary Appendix, we show with the help of an example that, in the absence of commitment power on the part of the contestants, the unravelling may lead into a “disclosure trap”. By this, we mean an outcome in which the ex ante expected payoff for both contestants is strictly lower than under mandatory concealment. Thus, in contrast to the more common situation in which the receiver in a persuasion game, such as an employer, a consumer, or a health insurer, tends to benefit from the unravelling, this need not be the case in a contest. However, many real-world contests have informational and allocational externalities on third parties. It may, therefore, be short-sighted to limit the welfare discussion solely to the expected payoffs of the contestants. For example, if contests are organized to maximize total expected expenses, and if information disclosures lower expected payoffs by tightening the competition, then that may well be desirable from the organizer’s point of view.⁴⁰

8 CONCLUSION

In this article, we have identified general and robust conditions under which a probabilistic contest with verifiable pre-play communication admits full disclosure as the unique perfect Bayesian equilibrium outcome. Given that the usual assumptions for the uniqueness of the fully revealing equilibrium outcome (Milgrom, 1981; Okuno-Fujiwara et al., 1990; Seidmann and Winter, 1997; van Zandt and Vives, 2007) fail to hold for contests, our results mean an extension of existing theory. In particular, the strong-form disclosure principle is more general than previously perceived. In addition, the analysis has formalized several intuitive concepts for which, to our knowledge, a flexible and all-encompassing framework in the realm of contest theory has been lacking so far. In sum, the analysis sheds further light on the incentives for communication of both verifiable and unverifiable information in competitive situations. This is not only desirable from the perspective of economic theory but also holds the potential of facilitating both the mitigation of harmful conflict and the efficient design of real-world contests.⁴¹

Acknowledgments

Valuable comments by the Editor (Christian Hellwig) and two anonymous referees helped us to improve the article. This work has further benefited from conversations with David Austen-Smith, Mikhail Drugov, Jörg Franke, Dan Kovenock, Wolfgang Leininger, Jingfeng Lu, Meg Mayer, Alessandro Pavan, Dmitry Ryvkin, Tarun Sabarwal, Aner Sela, Curtis Taylor, Karl Wärneryd, Juuso Välimäki, Xavier Vives, and Junjie Zhou. The article has been presented at the Oligo Workshop in Piraeus, at PET in Hue, Vietnam, at the SAET conference in Taipei, as well as to seminar audiences in Bath, Zurich, and at UCLA.

Supplementary Data

Supplementary data are available at Review of Economic Studies online.

References

Footnotes

Author notes