The Power of Passivity in the Hirshleifer Contest Under Small Noise

Abstract

Hirshleifer’s difference-form contest technology is a useful tool in the study of a class of conflict, especially military combats. We aim to highlight an important feature that the Hirshleifer contest model distinctively has, namely passivity (bidding zero effort) may stand as an effective choice in conflict even when the contest is highly deterministic (i.e., with small noise). For that purpose, we establish two propositions on the contest with $n\ge 2$ risk-neutral contestants under small noise. The first proposition states that every contestant bids arbitrarily close to zero (if not bidding zero with positive probability at all) under sufficiently small noise. The second proposition, more strikingly, states that every contestant either bids arbitrarily close to the second-highest valuation (among all the contestants’ valuations), or simply remains passive with certainty under any sufficiently small noise. We further show that the first proposition holds for the contest between risk-averse individuals endowed with constant absolute risk aversion as well, and illustrate by an example how quickly polarization in bidding among contestants, as is predicted by the propositions, may emerge as the noise of the contest abates. These results help pave the way toward a complete characterization of the difference-form contest.

1. Introduction

Contest theory has proved to be useful in the study of conflict, be it political rent-seeking, military combats, sports games, lawsuits, labor union’s strikes, R&D competition, etc. (refer to, e.g., Konrad (2009) for an introduction, especially on dynamic contests; Beviá and Corchόn (2024) for a postgraduate-level textbook; Corchόn and Serena (2018) and Kimbrough et al. (2020) for two recent literature reviews). Among the contest technologies that have been extensively employed in applied analysis stand two canonical models, respectively represented by Gordon Tullock’s (1980) ratio-form and Jack Hirshleifer’s (1988, 1989) difference-form contest success functions (CSFs). Both forms of the contest can be derived axiomatically (refer to, e.g., Skaperdas, 1996), or stochastically (see chapter 10 in Hirshleifer and Riley (1992) on the special case of lottery and Jia’s (2008) extension that allows for any power functional form of the Tullock contest, and McFadden’s (1974) estimation of the logit model that can serve as a stochastic derivation of the Hirshleifer CSF).

Compared to Tullock’s ratio-form contest technology that has drawn a great deal of attention, the Hirshleifer-type contest technology, in part due to analytical challenges,1 was rigorously analyzed only recently (Baik, 1998; Che & Gale, 2000; Alcalde & Dahm, 2007; Beviá & Corchόn, 2015; Cubel & Sanchez-Pages, 2016; Ewerhart & Sun, 2018, 2024; Levine & Mattozzi, 2022). It is worth recalling that Hirshleifer deliberately developed his very difference-form contest model to address two important drawbacks of the Tullockian technology. First and foremost, in the Tullock ratio-form contest, any contestant who bids zero effort would lose the game with certainty when any opponent commits an extremely small amount of resources. This stands in striking contrast with many actual conflicts. One strong case in point is a widely known observation in military history, which is perhaps best summarized by Sun Tzu’s famous remark that the supreme art of war is to subdue the enemy without fighting, or von Clausewitz’s widely quoted advice that defense is the stronger form of fighting than attack. Indeed, episodes that align with this insight abound. A good number of eras over the history of humanity are actually defense-dominated in military conflicts, ranging from medieval Europe, part of the civil war of the United States, to World War I (Hirshleifer, 2000, p. 786). Moreover, the power of inaction continued to be demonstrated in the modern wars with tanks and airplanes as well (see, e.g., Lawrence, 2017, Chapters 2 and 3). At times, it is reasonable to surrender rather than resist in military conflict between nations (Hirshleifer, 1989, p. 103).2 Secondly, the Tullock CSF fails to display increasing returns to effort up to the inflection point at equal resource commitments, another important observation often found in conflict. In particular, when one side is slightly inferior to its opponent in terms of committed resources, the gain of bidding more to be slightly superior would be quite remarkable. But the Tullock ratio-form contest technology is silent on this salient phenomenon, which is well known to researchers of military conflict. A good example is Dupuy’s (1987, Chapter 11) meticulous research of the combats between the German and Allied forces in World War II, which, as Hirshleifer (1989, p. 111; 2000, p. 778) points out, is equivalent to a game-theoretical analysis based on the difference-form contest.

Among the existing literature on difference-form contests, Ewerhart and Sun (2024), hereafter referred to as ES 2024, investigate the equilibrium set of the multiplayer Hirshleifer contest, obtaining a number of novel results. But the analysis made in the said paper on the effectiveness of passivity is far from complete. The present piece is therefore motivated to explore how far we may go along this Hirshleiferian line of inquiry of conflict. We establish two new results on the equilibria of the multiplayer Hirshleifer contest under small noise, and thereby further illuminate the power of passivity in the contest. The first proposition says that every contestant bids arbitrarily close to zero (if not bidding zero at all) with positive probability. The second proposition, perhaps a much more interesting result, says that each contestant either bids arbitrarily close to the second-highest valuation (among all the contestants’ valuations) or simply remains passive (bidding zero) with certainty when the noise of the contest is small enough (it does not need to vanish). It is worth emphasizing that these two propositions simultaneously apply to any equilibrium under small noise, a point that has significant implications (refer to Corollary 1 in Section 3, below).

As is noted in several existing studies on contests (e.g., Che & Gale, 2000; Ewerhart, 2017; Levine & Mattozzi, 2022), the all-pay auction turns out to be robust in the sense that, under certain circumstances, the equilibrium set or/and the equilibrium payoffs in some contests under vanishing noise converge to those in the all-pay auction. Che and Gale (2000) develop a two-player difference-form contest, which can hardly be extended to a contest with more than two contestants, and prove the uniform convergence of equilibrium bids as the noise vanishes, while Levine and Mattozzi (2022) characterize the limiting behavior of the two-player Tullock contest. The equilibrium set in the all-pay auction with multiple players, however, is strikingly complex (Baye et al., 1996), much more complicated compared to the auction with only two bidders (Hillman & Riley, 1989). It remains unclear which part of the equilibrium set under the all-pay auction stands as the limit for any difference- or ratio-form contest with multiple contestants under vanishing noise. Relatedly, virtually nothing is known yet about the convergence process of the equilibrium set when there are multiple constantants. The results we obtain in the present study help pave the way for exploring these problems.

We shall describe the model and make some technical preparations in the next section. We then state and briefly elaborate on our main results in Section 3, relegating rigorous proof of these results to Appendix A. In Section 4, we offer two illustrative examples. The first one demonstrates how fast the strategy of a contestant, even if she is as capable as anyone of bidding aggressively, may converge to passivity (bidding zero with certainty) as the noise of the contest becomes small enough. The second one shows that part of our main results, namely Proposition 1, is robust with respect to some variation in risk attitude of the contestants, while whether Proposition 2 remains valid as well is unknown. Section 5 concludes with remarks.

2. Set-Up and Preparations

We now describe the multiplayer Hirshleifer contest. Denote by $N\equiv \left\{1,\dots ,n\right\}$ where $n\ge 2$, the set of all the contestants with valuations ${\displaystyle V_1}\ge {\displaystyle V_2}\ge \cdots \ge {\displaystyle V_n}>0$, and by ${\displaystyle x_i}$, the effort made by any $i\in N$. Any $i\in N$ wins the prize with probability ${\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right)\equiv {\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{1,}}\dots ,{\displaystyle x_{i-1,}}{\displaystyle x_{i+1,}}\dots ,{\displaystyle x_n}\right)={\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\alpha \left({\displaystyle x_k}-{\displaystyle x_i}\right)\right)}}$, where $\alpha \ge 0$ is the decisiveness parameter. The greater the value of $\alpha$, the less noisy the contest is, and in the limit of $\alpha \to \infty$, the contest becomes an all-pay auction in which the highest bidder wins with certainty (Baye et al., 1996). The cost of bidding is identical to the effort exerted. Every contestant is risk-neutral and therefore receives a payoff ${\displaystyle {\pi }_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right)={\displaystyle V_i}{\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right)-{\displaystyle x_i},$ under effort profile $\left({\displaystyle x_i},{\displaystyle x_{-i}}\right), \forall i\in N.$

At any $\alpha >0$, a standard argument invoking Glicksberg’s (1952) fixed point theorem suffices to prove the existence of an equilibrium in mixed strategies, denoted as ${\mu }^{\ast }=({\mu }_1^{\ast },\dots ,{\mu }_n^{\ast })$ (refer to, for instance, Lemma 1 in Ewerhart and Sun (2024)). To ease exposition, we shall restate two lemmas that have already been stated in the existing studies as the starting point of the present analysis.

Lemma 1.

At any $\alpha >0$,

(i) 

Theequilibrium strategy of any contestant $i\in N$ randomizes over a finite number of bids, denoted as $y_i^{\left(1\right)}(\alpha )>\cdots >y_i^{\left({\displaystyle L_i}\right)}(\alpha )\ge 0$, where ${\displaystyle L_i}={\displaystyle L_i}\left(\alpha \right)<\infty$;

(ii) 

Zero is contained in the support of the equilibrium strategy for at least $\left(n-1\right)$ contestants, and each of them receives a payoff no greater than ${\displaystyle \frac{n}{(n-1)\alpha }}$.

Part (i) of the lemma follows from analyticity of the payoff functions on an open neighborhood and compactness of the set of possible effort (Ewerhart & Sun, 2018, Lemma 1). A technique that invokes Cauchy’s theorem on one-variable complex functions to establish discreteness of the support of the equilibrium strategy for analytic payoff functions was first introduced into the literature of contest theory in Ewerhart (2015, Theorem 3.2), and was then used in Levine and Mattozzi (2022) and Ewerhart and Sun (2024), among a few other studies. An alternative real-analysis approach to proving that the equilibrium a symmetric Hirshleifer contest admits has a finite support is found in Sun (2017). Part (ii) combines Lemmas 2 and A.1 in Ewerhart and Sun (2024).

For the Hirshleifer contest under vanishing noise, it has been shown that the payoff for all but the single highest-valuation contestant (in the case that ${\displaystyle V_1}>{\displaystyle V_2}$) approaches zero, and that everyone bids up to ${\displaystyle V_2}$ or ultimately becomes inactive. It is formally stated as,

Lemma 2

(Ewerhart & Sun, 2024, Proposition 7). Let $\epsilon$ be a small positive number. For sufficiently large $\alpha$,

(i) 

At least two constants, including contestant one if ${\displaystyle V_1}>{\displaystyle V_2}$, bid more than ${\displaystyle V_2}-\epsilon$ with positive probability;

(ii) 

$\left|{\Pi }_1^{\ast }-({\displaystyle V_1}-{\displaystyle V_2})\right|<\epsilon , \forall i\in N\backslash \left\{1\right\},{\Pi }_i^{\ast }<\epsilon$;

(iii) 

No contestant $i$ such that $y_i^{\left(1\right)}\left(\alpha \right)\in \left[\epsilon ,{\displaystyle V_2}-\epsilon \right];$

(iv) 

If ${\displaystyle V_1}={\displaystyle V_2}$, ${\mu }_i^{\ast }\left(\left\{0\right\}\right)<\epsilon$ for at least two contestants $i\in N$.

We now introduce a new and technically useful lemma. For any contestant $i$, let ${\displaystyle E_{{\mu }_{-i}^{\ast }}}\left[{\displaystyle p_i}\left(x,{\displaystyle x_{-i}}\right)\right]$ be the probability with which $i$ wins when bidding $x$ against any possible ${\displaystyle x_{-i}}$ under equilibrium ${\mu }^{\ast }={\mu }^{\ast }\left(\alpha \right)$ (we routinely suppress dependence on $\alpha$ wherever convenient). As usual, let $O\left(x\right)$ represent a (positive) infinitesimal that approaches zero at the same order as does $x$ when $x$ approaches zero.

Lemma 3.

$\forall i\in N$, for any ${\displaystyle y_i}\in supp\left\{{\mu }_i^{\ast }\left(\alpha \right)\right\}$, the following holds when $\alpha$ is sufficiently large.3

(i) 

${\displaystyle E_{{\mu }_{-i}^{\ast }}}\left[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)\right]\ge \left[1-O\left(exp(-{\displaystyle \frac{1}{2}}\sqrt{\alpha })\right)\right]{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right[0,{\displaystyle y_i}+{\displaystyle \frac{1}{2\sqrt{\alpha }}}\left]\right)$;

(ii) 

If there exists some $b>0$, ${\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)\ge b$ for any large $\alpha$, then

$${\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)<\left[1+O\left(\mathit{exp}\left(-\sqrt{\alpha }\right)\right)\right]{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\left[0,{\displaystyle y_i}+{\displaystyle \frac{2}{\sqrt{\alpha }}}\right]\right).$$

Proof. 

(i) By Lemma 1(i), at any $\alpha >0$ the equilibrium support of each contestant is finite. Therefore,

$$\begin{array}{l}{\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)\\ \ge {\displaystyle {\sum }_{\forall k\ne i,{\displaystyle x_k}\le {\displaystyle y_i}+{\displaystyle \frac{1}{2\sqrt{\alpha }}}}}{\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\alpha \right({\displaystyle x_k}-{\displaystyle y_i}-\frac{1}{\sqrt{\alpha }}\left)\right)}}{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)\\ \ge {\displaystyle {\sum }_{\forall k\ne i,{\displaystyle x_k}\le {\displaystyle y_i}+{\displaystyle \frac{1}{2\sqrt{\alpha }}}}}{\displaystyle \frac{1}{1+\left(n-1\right)\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{1}{2}\sqrt{\alpha })}}{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)\\ ={\displaystyle {\sum }_{\forall k\ne i,{\displaystyle x_k}\le {\displaystyle y_i}+{\displaystyle \frac{1}{2\sqrt{\alpha }}}}}[1-{\displaystyle \frac{\left(n-1\right)\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{1}{2}\sqrt{\alpha })}{1+\left(n-1\right)\mathrm{e}\mathrm{x}\mathrm{p}(-\frac{1}{2}\sqrt{\alpha })}}]{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)\\ =\left[1-O\left(\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{1}{2}}\sqrt{\alpha })\right)\right]{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right[0,{\displaystyle y_i}+{\displaystyle \frac{1}{2\sqrt{\alpha }}}\left]\right).\end{array}$$

(ii) First note

$$\begin{array}{l}{\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)\\ ={\displaystyle {\sum }_{\forall k\ne i,{\displaystyle x_k}\le {\displaystyle y_i}+{\displaystyle \frac{2}{\sqrt{\alpha }}}}}{\displaystyle \frac{1}{1+{\displaystyle \sum _{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\alpha \right({\displaystyle x_k}-{\displaystyle y_i}-\frac{1}{\sqrt{\alpha }}\left)\right)}}{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)\\ +{\displaystyle {\sum }_{\exists k\ne i,{\displaystyle x_k}>{\displaystyle y_i}+{\displaystyle \frac{2}{\sqrt{\alpha }}}}}{\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\alpha \right({\displaystyle x_k}-{\displaystyle y_i}-\frac{1}{\sqrt{\alpha }}\left)\right)}}{\displaystyle \prod _{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)\\ <{\displaystyle {\sum }_{\forall k\ne i,{\displaystyle x_k}\le {\displaystyle y_i}+{\displaystyle \frac{2}{\sqrt{\alpha }}}}}{\displaystyle \prod _{k\ne i}}{\mu }_k^{\ast }\left(\right\{{\displaystyle x_k}\left\}\right)+O\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\sqrt{\alpha })\right).\end{array}$$

In light of the fact that ${\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)\ge b>0$, we then obtain

$${\displaystyle E_{{\mu }_{-i}^{\ast }}}[{\displaystyle p_i}\left({\displaystyle y_i}+{\displaystyle \frac{1}{\sqrt{\alpha }}},{\displaystyle x_{-i}}\right)<\left[1+O\left(\mathrm{e}\mathrm{x}\mathrm{p}(-\sqrt{\alpha })\right)\right]{\displaystyle {\prod }_{k\ne i}}{\mu }_k^{\ast }\left(\right[0,{\displaystyle y_i}+{\displaystyle \frac{2}{\sqrt{\alpha }}}\left]\right).$$

□

3. The Main Results

At any $\alpha >0$, by Lemma 1(i), the support of any contestant’s strategy has a finite number of elements. We first establish a result on the lower bound of the support of any contestant.

Proposition 1.

In the Hirshleifer contest between $n\ge 2$ risk-neutral contestants with linear effort-cost functions and heterogenous valuations of the prize, for any contestant $i,{\displaystyle \underset{\alpha \to \infty }{\mathit{lim}\mathit{inf}}}\left\{supp\left\{{\mu }_i^{\ast }\right(\alpha \left)\right\}\right\}=0$.

In the light of Lemma 1(ii), the message conveyed by the proposition is that at least $n-1$ contestants bid zero with positive probability at any $\alpha >0$, and that the other one, if any, who does not bid arbitrarily close to zero for sufficiently large $\alpha$. A rigorous proof of the proposition is found in Appendix A. The proof proceeds by negation, starting from the assumption, without loss of generality, that there exists contestant $j$ who always bids at minimum ${\displaystyle \underset{\_}{{\displaystyle y_j}}}\left(\alpha \right)\equiv min\left\{supp\left\{{\mu }_j^{\ast }\left(\alpha \right)\right\}\right\}=y_j^{\left({\displaystyle L_j}\right)}(\alpha )$ that is above zero and approaches some $b>0$ as $\alpha \to \infty$ in each of the two possible scenarios, namely ${\displaystyle V_1}={\displaystyle V_2}$ and ${\displaystyle V_1}>{\displaystyle V_2}.$ The intuition of the argument for the case of ${\displaystyle V_1}={\displaystyle V_2}$ is as follows. There then exists contestant $i\ne j$ who bids zero with an arbitrarily small probability as $\alpha \to \infty ,$ by Lemma 2(iv). But no contestant bids positive weakly below ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )$, by the second-order condition at any interior bid. The reason that contestant $j$ always bids close to $b>0$ for any $\alpha$ large enough is that each contestant $i\ne j$ necessarily bids around ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )$ with a substantially large probability. That in turn implies that contestant $j$ can then overwhelmingly subdue all the other contestants with a much higher probability to win the prize and marginally higher effort-cost by bidding ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )+{\displaystyle \frac{1}{\sqrt{\alpha }}}$ than bidding ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )$ at any large $\alpha ,$ contradicting ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )\in supp\left\{{\mu }_j^{\ast }\left(\alpha \right)\right\}$. The argument for the case of ${\displaystyle V_1}>{\displaystyle V_2}$ can be likewise understood intuitively as follows. By Lemmas 1(ii) and 2(ii), the always active contestant (i.e., anyone who bids zero effort with probability zero) must be contestant one. In this case, any contestant $i\ne 1$ bids zero with a probability above a certain positive number. Otherwise, each of them necessarily bids around ${\displaystyle \underset{\_}{{\displaystyle y_1}}}(\alpha )\equiv min\left\{supp\left\{{\mu }_1^{\ast }(\alpha )\right\}\right\}$ with a substantial probability so that bidding ${\displaystyle \underset{\_}{{\displaystyle y_1}}}(\alpha )$ guarantees contestant one a payoff approaching ${\displaystyle V_1}-{\displaystyle V_2}$ (by Lemma 2(ii) again), which in turn implies that, by bidding slightly above ${\displaystyle \underset{\_}{{\displaystyle y_1}}}\left(\alpha \right)$, contestant one would receive a strictly greater payoff than bidding ${\displaystyle \underset{\_}{{\displaystyle y_1}}}\left(\alpha \right)$, an impossibility. Note that ${\displaystyle \underset{\_}{{\displaystyle y_1}}}(\alpha )$ is close to $b$ for large $\alpha$. It then follows that bidding $b/2$ would be a strictly better option for contestant one, in conflict with ${\displaystyle \underset{\_}{{\displaystyle y_1}}}\left(\alpha \right)\in supp\left\{{\mu }_1^{\ast }\left(\alpha \right)\right\}$. It is worth emphasizing that both the above explanation and the rigorous proof offered in Appendix A apply to any equilibrium of the contest.

Proposition 1 establishes that the lower bound of the support of every contestant is zero when the noise of the Hirshleifer contest vanishes, consistent with Lemma 2 in Baye et al. (1996, p. 299) on the equilibria of the all-pay auction. We now sharply strengthen claim (iii) in Lemma 2 on the upper bound of any contestant’s support under small noise.

Proposition 2.

In the Hirshleifer contest between $n\ge 2$ risk-neutral contestants with linear effort-cost functions and heterogenous valuations of the prize, for any contestant $i$, either $y_i^{\left(1\right)}\left(\alpha \right)\to {\displaystyle V_2} as \alpha \to \infty$, or $y_i^{\left(1\right)}\left(\alpha \right)=0$ at any sufficiently large $\alpha$.

That is, polarization even among homogenous contestants emerges once $\alpha$ reaches some finite threshold: any contestant either bids aggressively or drops out. When $n=2,$ the proposition directly follows from Lemma 2(i). The proof of the proposition for any $n\ge 3$, relegated to Appendix A, proceeds by negation. Starting with the assumption that there exists some contestant $j$ whose highest possible bid $y_j^{\left(1\right)}$ is arbitrarily small but always remains positive as $\alpha \to \infty$, we demonstrate that this assumption necessarily results in an absurdity. The reasoning is more involved than what is used to establish Proposition 1, but the roadmap and intuition of the argument can be summarized as follows.

We first take care of the case of ${\displaystyle V_1}={\displaystyle V_2}.$ That arbitrarily small $y_j^{\left(1\right)}\left(\alpha \right)>0$ holds true at any $\alpha$ for contestant $j$ implies the existence of some contestant $i\ne j$ who bids zero with positive probability but bids around or below $y_j^{\left(1\right)}$ with a vanishing probability. By bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-i}^{\ast }$, therefore, contestant $i$ weakly subdues contestant $j$, while the latter, when bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-j}^{\ast }$, is subdued by the former almost surely (with a probability approaching one as $\alpha \to \infty$). As such, by bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-i}^{\ast },$ contestant $i$ would win with a probability that is enormously larger than the probability with which contestant $j$ wins when bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-j}^{\ast }$. As a consequence, bidding ${\mathrm{y}}_j^{\left(1\right)},$ relative to bidding zero, against ${\mu }_{-i}^{\ast }$ by contestant $i$ entails a vanishing increase in effort-cost, but brings about a substantial increase in gross benefit. That is, ${\mathrm{y}}_j^{\left(1\right)}$ strictly dominates zero for contestant $i$, hence in conflict with the equilibrium property of the zero bid for the contestant.

We then come to grips with the more challenging case ${\displaystyle V_1}>{\displaystyle V_2}.$ Clearly, in the light of Lemmas 1(ii) and 2(ii), contestant one never bids zero for $\alpha$ large; that is, $j\ne 1.$ We shall argue that there exists a contestant whose valuation of the prize equals ${\displaystyle V_2}$ and who bids zero with positive probability, on the one hand, and bidding $y_j^{\left(1\right)}\left(\alpha \right)>0$ strictly dominates bidding zero for $\alpha$ large enough, on the other hand, hence a contradiction. The argument proceeds in a few steps. We first establish a preparatory lemma, Lemma A1, in step 1. We show that, in spite of the fact that contestant one’s minimal possible bid ${\displaystyle \underset{\_}{{\displaystyle y_1}}}(\alpha )\equiv min\left\{supp\left\{{\mu }_1^{\ast }\left(\alpha \right)\right\}\right\}>0$ is below $y_j^{\left(1\right)}\left(\alpha \right)$, $\alpha {\displaystyle \underset{\_}{{\displaystyle y_1}}}(\alpha )$ is unbounded as $\alpha$ grows, while everyone else bids zero with a probability that is bounded away from zero but virtually does not bid any positive effort around or below $y_j^{\left(1\right)}\left(\alpha \right).$ We also observe that there exists a contestant $i\ne j$ such that ${\displaystyle V_i}={\displaystyle V_2}$ who bids aggressively (say above ${\displaystyle V_2}/2)$ with a probability bounded away from zero. We are then enabled to demonstrate in steps 2 and 3 that, by bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-i}^{\ast },$ contestant $i$ wins with a probability that is not only substantially larger than the probability with which contestant $j$ wins when bidding ${\mathrm{y}}_j^{\left(1\right)}$ against ${\mu }_{-j}^{\ast }$ (refer to inequality (A21) in Appendix A), but incomparably larger than the probability with which contestant $i$ wins when bidding zero against ${\mu }_{-i}^{\ast }$ (refer to inequality (A22)). To understand the latter observation, one needs to notice the following technically useful point: against any possible ${\displaystyle x_{-i}}$ in equilibrium, the probability with which contestant $i$ wins when bidding zero is ${\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\displaystyle x_k})}}$, while the probability of winning if contestant $i$ bids ${\mathrm{y}}_j^{\left(1\right)}$ instead is ${\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha ({\displaystyle x_k}-y_j^{\left(1\right)}))}}.$ Noting that ${\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\displaystyle x_k})>\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\displaystyle \underset{\_}{{\displaystyle y_1}}})$ for any $\alpha$, therefore, the ratio of the former probability to the latter probability equals ${\displaystyle \frac{1}{1+{\displaystyle {\sum }_{k\ne i}}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\displaystyle x_k})}}\cdot {\displaystyle \frac{\exp \left(\alpha y_j^{\left(1\right)}\right)+{\displaystyle {\sum }_{k\ne i}}\exp \left(\alpha {\displaystyle x_k}\right)}{\exp \left(\alpha y_j^{\left(1\right)}\right)}}<{\displaystyle \frac{1}{{\displaystyle {\sum }_{k\ne i}}\exp \left(\alpha {\displaystyle x_k}\right)}}\cdot {\displaystyle \frac{\exp \left(\alpha y_j^{\left(1\right)}\right)+{\displaystyle {\sum }_{k\ne i}}\exp \left(\alpha {\displaystyle x_k}\right)}{\exp \left(\alpha y_j^{\left(1\right)}\right)}}={\displaystyle \frac{1}{{\displaystyle {\sum }_{k\ne i}}\exp \left(\alpha {\displaystyle x_k}\right)}}+{\displaystyle \frac{1}{\exp \left(\alpha y_j^{\left(1\right)}\right)}}<{\displaystyle \frac{2}{\mathrm{e}\mathrm{x}\mathrm{p}(\alpha {\displaystyle \underset{\_}{{\displaystyle y_1}}})}}\to 0$ as $\alpha \to \infty .$ Using the equilibrium property of bid ${\mathrm{y}}_j^{\left(1\right)}$ by contestant $j$, we can then finally show in step 4 that bidding ${\mathrm{y}}_j^{\left(1\right)},$ relative to bidding zero, against ${\mu }_{-i}^{\ast }$ by contestant $i$ entails slightly more effort-cost but results in a larger gain in gross benefit. Therefore, ${\mathrm{y}}_j^{\left(1\right)}$ strictly dominates zero for contestant $i$, contradicting $0\in supp\left\{{\mu }_i^{\ast }\right\}$.

As consequences of Propositions 1 and 2, we have the following observation (a rigorous proof is found in Appendix A).

Corollary 1.

In the Hirshleifer contest between $n\ge 2$ risk-neutral contestants with the same linear effort-cost function, and, respectively, with valuations ${\displaystyle V_1}\ge {\displaystyle V_2}=\cdots ={\displaystyle V_m}>{\displaystyle V_{m+1}}\ge \cdots \ge {\displaystyle V_n}>0$ (let ${\displaystyle V_{m+1}}=0$ if $m=n),$

(i) 

For any $b\in \left(0,1\right)$, there exists ${\alpha }^{\#}={\alpha }^{\#}(b,{\displaystyle V_2},n,m)$ such that at any $\alpha >{\alpha }^{\#}$, no less than two contestants $i\in N$ among the $m$ top ones bid $y_i^{\left(1\right)}\left(\alpha \right)>b{\displaystyle V_2}$, and all the others, including each contestant $j>m$ if any, bid zero with certainty;

(ii) 

When ${\displaystyle V_1}={\displaystyle V_2}$, at least $m-1$ contestants among the top $m$ ones bid zero with positive probability at any $\alpha >0$, and the only one, if any, who does not do so bids arbitrarily close to zero as $\alpha$ grows unboundedly;

(iii) 

When${\displaystyle V_1}>{\displaystyle V_2}$, contestant one’s highest bid is arbitrarily close to ${\displaystyle V_2}$ and the lowest bid, which remains positive at any finite $\alpha$, is arbitrarily close to zero as $\alpha$ grows unboundedly, while any contestant $i\ne 1$ bids zero with ${\mu }_i^{\ast }\left(\right\{0\left\}\right)$ such that ${\displaystyle {\prod }_{k\ne 1}}{\mu }_k^{\ast }\left(\right\{0\left\}\right)$ is arbitrarily close to ${\displaystyle \frac{{\displaystyle V_1}-{\displaystyle V_2}}{{\displaystyle V_1}}}$ as $\alpha$ grows unboundedly.

In the case of ${\displaystyle V_1}={\displaystyle V_2}=V,$ by part (i) of the corollary, every contestant either remains passive (bidding zero with probability one) or is very active (bidding arbitrarily close to $V$ with some positive probability) as the noise of the contest becomes small enough. Such polarization in bidding among the contestants with equal and top-level valuation under small noise is, in and of itself, of interest. For instance, the expected payoffs could differ among ex ante identical contestants. In the case of ${\displaystyle V_1}>{\displaystyle V_2},$ while any contestant whose valuation is strictly below ${\displaystyle V_2}$ completely drops out when $\alpha >{\alpha }^{\#}$, each of those of valuation ${\displaystyle V_2}$ bids zero with a significantly large probability that is at least around ${\displaystyle \frac{{\displaystyle V_1}-{\displaystyle V_2}}{{\displaystyle V_1}}}$ as $\alpha$ grows unbounded, by part (ii) of the corollary. This aligns with one well-known observation on the all-pay auction with ${\displaystyle V_1}>{\displaystyle V_2}$ (refer to part A of Theorem 2 in Baye et al., 1996).

4. Illustrative Examples: Quick Convergence to Passivity and Passivity Under Risk Aversion

We offer two illustrative examples, as described in the above section title.

Example 1.

How quickly is the convergence to polarized bidding among contestants?

Consider the Hirshleifer contest between three homogeneous constants, all with the same valuation of the prize, normalized to one. Since at least two contestants, say contestants 1 and 2, become aggressive in bidding as the noise of the contest abates to a certain extent (refer to Lemma 2(i) and Corollary 1(i)), it is of interest to explore how quickly passivity emerges as the equilibrium strategy for some contestants over the course. For that purpose, we focus on such possible semi-symmetric equilibria in which contestants 1 and 2 play the same strategy, while contestant 3 behaves differently. It goes without saying that such a symmetric contest often admits multiple equilibria, mostly being asymmetric (refer to Section 4.1 in Ewerhart and Sun (2024)). What is of particular interest is the possible scenario in which both contestants 1 and 2 aggressively bid in the sense that each possibly bids more than a half of the prize and therefore ex post rent overdissipation (the total bid by all exceeds the value of the prize) occurs with positive probability (Baye et al., 1999), while contestant 3 is passive.

It is already shown via numerical simulation in Ewerhart and Sun (2024, Section 4.2, Example 2) that when $\alpha \in (4.12, 7.02)$, for the three-player contest described above, there exists such an equilibrium that two constants randomize over zero and a positive bid while the other one plays a zero-bid pure strategy. As an extension, we used Mathematica 14.0 to make simulation allowing for a larger range of the parameter. We found that, when $\alpha \in \left(7.02, 9.20\right),$ at equilibrium the strategy of contestant 3, against two aggressive opponents both of whom individually randomize over zero and two positive bids, is being passive. For instance, when α = 8.12, each of constants 1 and 2 bids 0.628, 0.313 and zero respectively with probability 0.416, 0.203 and 0.381, and contestant 3 bids zero with probability one. That is, when α = 8.12, the chance of rent overdissipation is already around 17%, while passivity remains the equilibrium strategy for the other individual. In Figure 1, we illustrate the expected payoff of contestant 3 against two aggressive opponents for any $\alpha$ over the interval $(7.02, 9.20)$. Passivity clearly stands as the best reply for the contestant. When $\alpha$ is even larger, passivity appears to be the equilibrium strategy for one contestant against two aggressive opponents with a larger support that contains zero and 3 or more positive bids.

What can we learn from this simple example? Perhaps not much, except that polarization in bidding among contestants (even if ex ante identical) can emerge surprisingly quickly. In other words, the strong qualification in the statement of Proposition 2 on the noise, viz. “sufficiently large $\alpha$”, might better be loosely interpreted as a condition of “(for) not too small $\alpha$”, to make the insight that passivity could be a powerful choice even sharper. That said, the example only serves an illustrative purpose, and therefore we cannot go too far here to risk overinterpretation.

Example 2.

Passivity in the Hirshleifer contest with risk averse contestants.

Our analysis has thus far been confined to the Hirshleifer contest with risk-neutral contestants. We now consider a Hirshleifer contest, which is the same as that described in Section 2 except the contestants’ risk attitude. We assume that all contestants are risk averse, and everyone has a constant measure of absolute risk aversion ${\displaystyle {\mathsf{\gamma }}_i}, i\in N.$ Therefore, any contestant $i$’s preference can be described by a CARA utility function ${\displaystyle {\mathrm{u}}_i}\left(\mathrm{c}\right)=1-\exp \left(-{\displaystyle {\mathsf{\gamma }}_i}\mathrm{c}\right),$ where $\mathrm{c}$ is consumption, and parameter ${\displaystyle {\mathsf{\gamma }}_i}>0$ is the so-called Arrow-Pratt measure that could differ across the contestants. Denote by ${\displaystyle W_i}$ the endowment of any player $i.$ The expected utility of any contestant $i$ then equals ${\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right){\displaystyle {\mathrm{u}}_i}\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right)+\left(1-{\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right)\right){\displaystyle {\mathrm{u}}_i}\left({\displaystyle W_i}-{\displaystyle x_i}\right)$ when biding ${\displaystyle x_i}$ against a bid profile ${\displaystyle x_{-i}}$ by the other contestants. At any $\alpha >0,$ a standard argument would do for establishing the existence of an equilibrium in mixed strategies, still denoted as ${\mu }^{\ast }=\left({\mu }_1^{\ast },\dots ,{\mu }_n^{\ast }\right).$ At equilibrium, the expected payoff of contestant $i$ when bidding ${\displaystyle x_i}\in supp\left\{{\mu }_i^{\ast }\right\}$ against ${\mu }_{-\mathrm{i}}^{\ast }$ equals ${\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\Pi }_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]={\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right){\displaystyle {\mathrm{u}}_i}\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right)+\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right){\displaystyle {\mathrm{u}}_i}\left({\displaystyle W_i}-{\displaystyle x_i}\right)\right].$ By the analytical techniques described in Section 2, one finds that the support of any contestant’s equilibrium strategy is a finite set. The equilibrium payoff, therefore, assumes a polylinear form in probabilities with a finite number of terms.

Based on the above observations, we show that the first half of Lemma 1(ii) holds still, i.e., zero is contained in the support of at least $n-1$ contestants. The argument runs as follows. Take any positive element ${\displaystyle x_i}\in supp\left\{{\mu }_j^{\ast }\left(\alpha \right)\right\},$ and let $∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right)\equiv {\displaystyle {\mathrm{u}}_i}\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right)-{\displaystyle {\mathrm{u}}_i}\left({\displaystyle W_i}-{\displaystyle x_i}\right).$ Recall that for the Hirshleifer CSF, ${\displaystyle \frac{\partial }{\partial {\displaystyle x_i}}}{\displaystyle p_i}\left({\displaystyle x_i},{\displaystyle x_{-i}}\right)=\alpha {\displaystyle p_i}\left(1-{\displaystyle p_i}\right)$ for any bid ${\displaystyle x_i}$ against a profile ${\displaystyle x_{-i}}$ by the other contestants. In the light of the above observation that ${\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\Pi }_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]$ assumes a polylinear form in probabilities with a finite number of terms under ${\mu }_{-\mathrm{i}}^{\ast },$ we can exchange differentiation and integration to compute ${\displaystyle \frac{\partial }{\partial {\displaystyle x_i}}}{\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathsf{\pi }}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right].$ The first-order condition at ${\displaystyle x_i}>0$ for contestant $i$ turns out to be ${\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\alpha {\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right)-[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)u_i^{\prime }\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right)+\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)u_i^{\prime }\left({\displaystyle W_i}-{\displaystyle x_i}\right)]\right]=0$, i.e.,

$$\begin{array}{l}\alpha ∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ =u_i^{\prime }\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]+u_i^{\prime }\left({\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right].\end{array}$$

(1)

Noting ${\displaystyle \frac{\partial ∆{\displaystyle {\mathrm{u}}_i}}{\partial x}}=u_i^{\prime }\left({\displaystyle W_i}-{\displaystyle x_i}\right)-u_i^{\prime }\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right),u_i^{″}(c)=-{\displaystyle {\mathsf{\gamma }}_i}{u}_i^{\prime }(c)$ and $u_i^{\prime }(c)=-{\displaystyle {\mathsf{\gamma }}_i}[{\displaystyle {\mathrm{u}}_i}\left(c\right)-1]$ for any consumption $c,$ we likewise compute the second-order condition,

$$\begin{array}{l}{\displaystyle \frac{{\partial }^2}{\partial x_i^2}}{\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathsf{\pi }}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]\\ ={\alpha }^2∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\left(1-2{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right){\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ +2\alpha {\displaystyle {\mathsf{\gamma }}_i}∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ -{\displaystyle {\mathsf{\gamma }}_i}{u}_i^{\prime }\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]-{\displaystyle {\mathsf{\gamma }}_i}{u}_i^{\prime }\left({\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ ={\alpha }^2∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\left(1-2{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right){\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ +{\displaystyle {\mathsf{\gamma }}_i}\{2\alpha ∆{\displaystyle {\mathrm{u}}_i}({\displaystyle x_i}){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\\ -u_i^{\prime }\left({\displaystyle V_i}+{\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right]-u_i^{\prime }\left({\displaystyle W_i}-{\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]\}\\ <0.\end{array}$$

But what is in braces in the second last term, by FOC (Equation (1)), equals $\alpha ∆{\displaystyle {\mathrm{u}}_i}\left({\displaystyle x_i}\right){\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]>0.$ Therefore,

$${\displaystyle E_{{\mu }_{-\mathrm{i}}^{\ast }}}\left[\left(1-2{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right){\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\left(1-{\displaystyle {\mathrm{p}}_i}\left({\displaystyle x_i},{\displaystyle x_{-\mathrm{i}}}\right)\right)\right]<0.$$

Should there be two contestants $i$ and $j$ who never bid zero at some $\alpha >0,$ we may assume, without loss of generality, that the smallest element of contestant $i^{\prime }s$ support, denoted as ${\displaystyle \underset{\_}{{\displaystyle y_i}}},$ is the minimum among all the positive elements of ${\displaystyle {\bigcup }_{k\in N}}supp\left\{{\mu }_k^{\ast }\left(\alpha \right)\right\}.$ Hence, ${\displaystyle \underset{\_}{{\displaystyle y_i}}}\le {\displaystyle \underset{\_}{{\displaystyle y_j}}},$ where ${\displaystyle \underset{\_}{{\displaystyle y_j}}}$ is the smallest number among $supp\left\{{\mu }_j^{\ast }\left(\alpha \right)\right\}$. Then, against any possible bid profile ${\displaystyle x_{-i}}$ by the other contestants, the probability of wining the prize by contestant $i$ when bidding ${\displaystyle \underset{\_}{{\displaystyle y_i}}}$ is at most a half. The probability equals a half if and only if contestant $j$ plays a pure strategy that equals ${\displaystyle \underset{\_}{{\displaystyle y_i}}}$ and $n=2.$ But that is impossible, since the SOC, by the above reasoning, then turns out to be (noting ${\displaystyle {\mathrm{p}}_j}({\displaystyle \underset{\_}{{\displaystyle y_i}}},{\displaystyle \underset{\_}{{\displaystyle y_j}}})=1/2$), ${\displaystyle \frac{{\partial }^2}{\partial y_i^2}}{\displaystyle {\pi }_i}({\displaystyle \underset{\_}{{\displaystyle y_i}}},{\displaystyle \underset{\_}{{\displaystyle y_j}}})={\displaystyle \frac{\mathsf{\gamma }\alpha ∆u\left({\displaystyle \underset{\_}{{\displaystyle y_i}}}\right)}{2}}>0.$ We are ready to show that at least $n-1$ contestants bid zero with positive probability at any $\alpha >0.$ Suppose otherwise. For the contestant who is always active (never bids zero) and whose minimal element among his support is no greater than the minimal bid of any other always active contestant, by the above nalysis, SOC at his minimal bid does not hold. That is, the first half of part (ii) in Lemma 1 holds in the case that each contestant has a constant Arrow–Pratt measure of absolute risk aversion (it is perhaps worth emphasizing again that the claim allows for Arrow–Pratt measures that are different cross contestants).

We now outline, without providing algebraic details (available upon request from the author) to save space, an argument for that Proposition 1 holds true for the Hirshleifer contest between risk-averse contestants with CARA preferences.4 We only need to appropriately modify the proof by negation for the contest with risk-neutral contestants. Suppose that there exists contestant $i$ who always bids at minimum positive ${\displaystyle \underset{\_}{{\displaystyle y_j}}}\left(\alpha \right)\equiv min\left\{supp\left\{{\mu }_j^{\ast }\left(\alpha \right)\right\}\right\}$ that approaches some $b>0$ as $\alpha \to \infty$. No contestant bids positive weakly below ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha ),$ by the second-order condition at any interior bid as is shown in the reasoning above. Therefore, the reason that contestant $j$ always bids close to $b>0$ for any $\alpha$ large enough is that each contestant $i\ne j$ necessarily bids zero or slightly above ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )$ with a substantially large probability. That in turn implies that contestant $j$ can then substantially raise the probability of wining when against some ${\displaystyle x_{-\mathrm{j}}}$ under ${\mu }_{-\mathrm{j}}^{\ast }$, thereby raising the value of ${\displaystyle {\mathrm{p}}_j}\left({\displaystyle x_j},{\displaystyle x_{-\mathrm{j}}}\right)∆{\displaystyle {\mathrm{u}}_j}\left({\displaystyle x_j}\right),$ at the expense of a marginal reduction in ${\displaystyle {\mathrm{u}}_j}\left({\displaystyle W_j}-{\displaystyle x_j}\right),$ by bidding ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )+{\displaystyle \frac{1}{\sqrt{\alpha }}}$ rather than bidding ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )$ at any large $\alpha .$5 It then results in an increase in ${\displaystyle E_{{\mu }_{-\mathrm{j}}^{\ast }}}\left[{\displaystyle {\mathrm{p}}_j}\left({\displaystyle x_j},{\displaystyle x_{-\mathrm{j}}}\right){\displaystyle {\mathrm{u}}_j}\left({\displaystyle V_j}+{\displaystyle W_j}-{\displaystyle x_j}\right)+\left(1-{\displaystyle {\mathrm{p}}_j}\left({\displaystyle x_j},{\displaystyle x_{-\mathrm{j}}}\right)\right){\displaystyle {\mathrm{u}}_j}\left({\displaystyle W_j}-{\displaystyle x_j}\right)\right],$ contradicting ${\displaystyle \underset{\_}{{\displaystyle y_j}}}(\alpha )\in supp\{{\mu }_j^{\ast }\left(\alpha \right)\}.$ As far as Proposition 2 is concerned, for the contest with two contestants, it is not difficult to see that both bid at most close to ${\displaystyle V_2}$ if ${\displaystyle W_i}>{\displaystyle V_i}$ for each contest $i=\mathrm{1,2}$, as $\alpha$ grows unboundedly. When $n\ge 3,$ however, we have not been able to rigorously prove, or falsify by a counterexample, Proposition 2 for risk-averse contestants with CARA preferences.

5. Concluding Remarks

Our analysis depicts a polarized picture of the equilibrium strategy profiles in the Hirshleifer contest when the noise of the contest is sufficiently small: any contestant either bids aggressively with positive probability, or simply drops out. Indeed, the power of passivity figures prominently in a class of conflict, a theme that Hirshleifer powerfully expounded over decades of inquiry into what he refers to as “the dark side of the force” (Hirshleifer, 2001). We further formalize the idea, showing that ex post polarization in bidding among (possibly ex ante identical) contestants emerges as the noise of the contest abates: some contestants bid widely, ranging from zero (or very close to zero) to some level of effort close to the second highest valuation, while the rest play a pure strategy of passivity as the value parameter $\alpha$ reaches some finite number.

We speculate that Proposition 1 and, more significantly, Proposition 2, hold not only for Hirshleifer contests but for quite some other difference-form contests as well. To rigorously prove the propositions, especially Proposition 2, for a class of difference-form contests, however, appears challenging. Recall that the most difficult part in our proof by negation of Proposition 2 is the argument used to deal with the case ${\displaystyle V_1}>{\displaystyle V_2}$ (refer to the Appendix A). Indeed, much effort is made there in establishing inequalities (A21) and (A22) in order to demonstrate that ${\mathrm{y}}_j^{\left(1\right)}$ strictly dominates zero for some contestant $i$ who bids zero at equilibrium, hence a contradiction. In deriving these inequalities, we invoke several technically useful results on the Hirshleifer contest technology, either borrowed from other work (Lemmas 1 and 2) or established in the present study (Lemmas 3 and A1). To follow more or less the same argument to prove Proposition 2 for a class of difference-form contests, therefore, necessitates verifying a number of similar claims, a task that turns out to be difficult. Alternatively, one may develop an argument radically differing from the one used in the present analysis. Moreover, it is as tempting to explore whether the proposition holds for multiplayer contests with multiplicative noise, such as the Tullock contest. Due to a remarkable lack of understanding of the equilibrium set of such contests under vanishing noise in the literature, however, this open question appears to be even harder.

Another valuable avenue of further exploration is how risk attitude shapes the equilibrium in difference-form contests, an entirely unchartered territory in contest theory as far as we know. We demonstrate in the preceding section that, should all contestants be risk averse and endowed with CARA utility functions, our Proposition 1 holds still. However, whether Proposition 2 holds is an open question. Indeed, for the Hirshleifer contest between any number of risk-averse contestants (not necessarily with some CARA preference), it remains unknown whether zero is contained in the equilibrium supports of at least $n-1$ contestants at any $\alpha >0$, a crucial step for establishing Propositions 1 and 2 for risk-neutral Hirshleifer contestants. To the best of our knowledge, no serious work has been done on implications of risk attitude for the equilibrium set in difference-form contests, a topic that appears significantly rewarding, especially in the light of the explanatory power of such CSF in the study of war. We, therefore, leave for future research the problem of how risk attitude shapes the equilibrium set at any level of the noise of the contest and convergence of the set as the noise abates and ultimately vanishes.