European Journal of Political Economy 26 (2010) 558–567

Contents lists available at ScienceDirect

European Journal of Political Economy

j ourna l homepage: www.e lsev ie r.com/ locate /e jpe

Contests with private costs: Beyond two players

Dmitry Ryvkin⁎Department of Economics, Florida State University, Tallahassee, FL 32306-2180, USA

a r t i c l e i n f o

⁎ Tel.: +1 850 644 7209; fax: +1 850 644 4535.E-mail address: dryvkin@fsu.edu.

1 With rent seeking defined as a political economyrents through political institutions, such as lobbying (eto the political economy domain. For recent reviews(2007).

0176-2680/$ – see front matter © 2010 Elsevier B.V.doi:10.1016/j.ejpoleco.2010.09.001

a b s t r a c t

Article history:Received 22 March 2010Received in revised form 1 September 2010Accepted 2 September 2010Available online 9 September 2010

It was shown previously that in the symmetric contest game of two players, equilibriumbidding is lower in the case of private information than in the case of public information aboutthe players' costs. I consider symmetric contests of an arbitrary number of players withcontinuously distributed private costs and discuss the existence and properties of equilibriumbidding functions. I show that with more than two players the relationship betweenequilibrium bids in the cases of public and private information is no longer universal. Whilehigh-cost players still bid less in the private information case, relatively low-cost players maybid above or below their corresponding public information bids.

© 2010 Elsevier B.V. All rights reserved.

JEL classification:D72C72

Keywords:ContestPrivate informationBidding

1. Introduction

In contests, players compete for a valuable prize by expending costly effort or other resources. Such competition occurs in avariety of contexts, including rent seeking (e.g., Krueger, 1974; Aidt and Hillman, 2008), warfare conflicts (e.g., Amegashie andKutsoati, 2007; Chang et al., 2007), R&D competition (e.g., Taylor, 1995), sports (e.g., Szymanski, 2003; Kräkel, 2007; Berentsenet al., 2008), and the labor market (e.g., Lazear, 1999).1

Most of the literature on contests, starting from Tullock (1980), assumes a probabilistic law of winning whereby each player'seffort positively affects his or her chances of securing the prize, but exerting the highest effort does not guarantee winning. Suchcontests are often referred to as imperfectly discriminating (Hillman and Riley, 1989). Conceptually, contests are in many wayssimilar to auctions (see, e.g., Klemperer, 2004). From the organizer's perspective, both can be thought of as revenue-generatingmechanisms; from the participants' perspective, given the uncertainty about winning both, bidding behavior should bequalitatively similar. In fact, in the limit, if the player with the highest effort wins with certainty, contests are equivalent to all-payauctions (sometimes referred to as perfectly discriminating contests, see Hillman and Riley, 1989) — a widely used model ofcontests that avoids the probabilistic winning issue (see, e.g., Baye et al., 1996; Anderson et al., 1998; Moldovanu and Sela, 2001;Noussair and Silver, 2006; Cohen and Sela, 2008).

In auctions, in the classic independent private values (IPV) formulation, each player's strategy is his or her bidding function thatdescribes how much a player bids as a function of his or her private valuation of the auctioned object. In a similar, IPV-like,

concept (Congleton et al., 2008), rent seeking contests are activities involving competitive assignment o.g., Pecorino, 2010) or political campaigning (e.g., Leigh, 2008). Contest situations, however, are not limitedof the contest literature, see, e.g., Lockard and Tullock (2001), Garfinkel and Skaperdas (2007), Corchón

All rights reserved.

f

559D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

formulation of a symmetric contest game of incomplete information, a player's bidding function would describe the player's levelof effort as a function of his or her private “ability” (measured, for example, as the inverse marginal cost of effort). The distributionof ability in the population of contestants would be common knowledge, similar to the distribution of valuations in auctions.

Equilibrium bidding function is the basic characteristic of auctions that is well-understood for many auction formats, includingthe all-pay auction, especially in the symmetric case (see, e.g., Klemperer, 2004). At the same time, little is known about theproperties, and even existence, of an equilibrium bidding function in contests (not counting the all-pay auction) in the IPV-likeformulation with continuous distributions of players' abilities, even in the simplest symmetric case.

In the IPV formulation, each player faces uncertainty regarding other players' types. In classic auctions, this is the only source ofuncertainty; in contests, however, the probabilistic winning mechanism creates additional uncertainty, and combining the two isdifficult theoretically. Most of the existing literature on rent seeking avoids this issue by dealing only with the case of completeinformation, in which it is assumed that players' abilities (costs of effort) are common knowledge (for most general treatment, see,e.g., Cornes and Hartley, 2005; Malueg and Yates, 2006). Recently, however, there has also been a growing interest in contests withincomplete information as a more realistic model for applications. For example, Hurley and Shogren (1998a), Malueg and Yates(2004), and Sui (2009) consider models with two-sided incomplete information, in which players have only few (two or three)possible prize valuations (costs of effort)— a simplification that renders the double uncertainty problem tractable in some specialcases. A number of authors analyzed contests with asymmetric information where one player is not informed about the other'svaluation (Harstad, 1995; Hurley and Shogren, 1998a,b; Schoonbeek andWinkel, 2006; Pogrebna, 2008), or the common value ofthe prize (Wärneryd, 2003).

Finally, Fey (2008) proved the existence of, and characterized numerically, the symmetric equilibrium bidding function in anIPV-like formulation of the Tullock (1980) contest model with two players and uniformly distributed marginal costs of effort. Inthe concluding section of his paper, Fey (2008) discusses possible extensions of his work and notes that, “First, the model can bemade more general by allowing for more general distributions of costs and more than two players. Fortunately, it appears that inboth cases, the line of proof [...] should go through with only minor modifications. [...] A second line of work involves generalizingthe contest success function [...]. It is likely this will be a more difficult task to accomplish.”

One of the important findings in the literature on contests with private information is that equilibrium effort tends to be lowerthan in the corresponding symmetric contests with public information (Hurley and Shogren, 1998a; Fey, 2008). This result,however, was only demonstrated in special cases for contests of two players. For continuously distributed costs (Fey, 2008), theresult was only shown numerically. It is, thus, an open question to what extent this result generalizes to contests of more than twoplayers and how general it is even for two players for different distributions of costs and/or contest success functions.

In this paper, I study symmetric contests with incomplete information in a setting similar to Fey (2008), and extend his workalong the dimensions mentioned in the quote above. Thus, our model allows for arbitrary continuous distributions of costs, morethan two players, and for arbitrary contest success functions satisfying certain symmetry and independence assumptions(Skaperdas, 1996). There are two main results. First, I provide a proof for the existence of a symmetric equilibrium in this class ofcontest models. The proof follows the same logic as in Fey (2008), whose method turns out to be generalizable in a relativelystraightforward manner. Second, I show that the inequality in equilibrium effort between the public and private information casesholds generally for two-player contests but does not generalize to contests of more than two players.

The rest of the paper is organized as follows. Section 2 describes the model and theoretical results. In Section 3, I present theresults of numerical calculations of the equilibrium bidding function for several parameterizations. Section 4 contains a discussionand concluding remarks.

2. The model and main results

There are N risk-neutral players indexed by i=1,…,N. The players compete in a contest by choosing effort levels ei≥0.2 Playeri's probability of winning the contest, given the vector of all players' efforts, e = e1;…; eNð Þ, is given by the contest success function(CSF)

2 The3 The

objectivquasico

P ið Þeð Þ = h eið Þ

∑Nj = 1 h ej

� � : ð1Þ

Here, h(⋅) is a nonnegative, strictly increasing, concave, and twice continuously differentiable function, with h′N0, h″≤0.3 It isassumed in Eq. (1) that ∑ j=1

N h(ej)≠0; otherwise, P(i)=1/N.Contests with a CSF of the form (1) have been analyzed by a number of authors (see, e.g., Szidarovszky and Okuguchi, 1997;

Cornes and Hartley, 2005). It is the most general symmetric CSF satisfying the independence assumptions identified by Skaperdas(1996), and thus represents a class of CSFs that covers most of the contest literature starting with Tullock (1980) (see, e.g., reviewsby Lockard and Tullock, 2001; Garfinkel and Skaperdas, 2007; Corchón, 2007).

se can also be interpreted as investment levels, campaign contributions, number of troops, etc., depending on the context.assumption of concavity of h(⋅) is not the weakest possible assumption for the equilibrium existence result to hold. It guarantees the concavity of thee function in the maximization problem for the best response, π1(y1,c1), Eq. (4) below. In fact, it is sufficient to require that function π1(y1,c1) bencave, which may be the case even if function h(⋅) is not concave. Such a formulation is less transparent, though.

560 D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

Each player i has a private constant marginal cost of effort, ciN0. The values of ci for different players are drawn independentlyfrom the same distribution with probability density function (pdf) f(⋅) and corresponding cumulative density function (cdf) F(⋅).Assume that f(⋅) is continuous and has a compact support bounded away from zero. Similar to Fey (2008), and without loss ofgenerality, let the support of f(⋅) be D=[c0,c0+1], with minimal marginal cost c0N0.

The winner's prize in the contest is normalized to one. Thus, player i's expected payoff conditional on the vector of effort levelsis4

where

Here,

4 Aspurposevaluatio

πi ci jeð Þ = h eið Þ∑N

j = 1 h ej� �−ciei: ð2Þ

Similar to Cornes and Hartley (2005), it is convenient to change variables by introducing inputs, yi=h(ei). Since, due to the factthat h(⋅) is a strictly increasing function, there is a one-to-one correspondence between efforts and inputs, it is possible to build theentire theory only for inputs, and go back to efforts at the very end, if necessary. Thus, suppose that players choose inputsy = y1;…; yNð Þ. Let g(⋅) denote the function inverse to h(⋅). Note that g′N0 and g ″≥0, i.e. g(⋅) is strictly increasing and convex.Player i's payoff can be re-written in terms of inputs as

πi ci jyð Þ = yi∑N

j = 1 yj−cig yið Þ: ð3Þ

In the private information setting, each player only observes his or her own cost ci. Thus, player i's strategy is a bidding functionσi :D→R+ that assigns an input yi=σi(ci) to any cost ci∈D. Assume that players j=2,…,N choose their inputs according to biddingfunctions yj=σj(cj), and let σ−1(⋅)=(σ2(⋅),…,σN(⋅)) denote the vector of bidding functions of all players except player 1. Thenplayer 1's expected payoff, given his or her cost c1 and input y1, is

π1 c1; y1ð Þ = Φ y1;σ−1½ �−c1g y1ð Þ; ð4Þ

operator Φ is defined as

Φ y1;σ−1½ � = ∫DN−1 ∏j≥2

dF tj� �" #

y1y1 + ∑j≥2σ j tj

� � : ð5Þ

By choosing y1=0 player 1 can always guarantee herself a payoff of zero. Similar to Fey (2008), note that Φ[y1,σ−1] isbounded by 1, therefore player 1's optimal input will not exceed h(1/c1). Let E=h(1/c0) denote the (finite) upper bound ofpossible input levels of all players.

Player 1's best response to σ−1(⋅) is the solution to problem maxy1≥0π1(c1,y1). Thus, the best response is given by

b1 c1;σ−1½ � = 0; if Φ1 0;σ−1½ �bc1g ′ 0ð Þy1 such that Φ1 y1;σ−1½ � = c1g ′ y1ð Þ; otherwise

�ð6Þ

Φ1 y1;σ−1½ �≡∂Φ y1;σ−1½ �∂y1

= ∫DN−1 ∏j≥2

dF tj� �" # ∑j≥2σ j tj

� �y1 + ∑j≥2σ j tj

� �h i2 : ð7Þ

Eq. (6) implicitly defines player 1's best response operator b1[c1 ;σ−1]. Provided functions σj, j≥2, are continuous, it is easy tosee that the best response is continuous in c1. Note also that g ′(⋅) is an increasing function, whereas Φ1[y1 ;σ−1] is strictlydecreasing in y1, therefore b1[c1 ;σ−1] is strictly decreasing in c1 as long as b1[c1 ;σ−1]N0.

Our goal is to study the symmetric equilibrium, therefore assume that all players j=2,…,N choose inputs according to thesame bidding function σ(⋅) and define the symmetrized best response operator T [c ;σ]≡b1[c ;σ(⋅),…,σ(⋅)]. The symmetric Nashequilibrium bidding function, b*(c), is a solution of equation b(c)=T [c ;b(⋅)], i.e., it is a fixed point of operator T. The followingproposition establishes the existence of at least one such fixed point, and hence of at least one symmetric equilibrium biddingfunction.

seen from Eq. (2), a mathematically equivalent formulation of the game is a contest with heterogeneous prize valuations vi=1/ci. Although for thes of this paper it is mathematically convenient to formulate the problem in terms of cost parameters ci, all the results can in the end be recast in terms ons vi, if necessary.

f

561D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

Proposition 2.1. In the private information contest defined above, there exists a symmetric (Bayesian) Nash equilibrium biddingfunction b*(c).

Thedetails of theproof arepresented in theAppendix. The ideaof theproof is thesameas inFey(2008),withmodificationsallowingusto extend the proof to cover arbitrary continuous distributions of costs, arbitrary numbers of players, and arbitrary CSFs of the form (1).

Of interest is the comparison of equilibrium bidding function b*(c) to the symmetric equilibrium bid in the public informationcase, b0(c). The latter satisfies the symmetrized first-order condition for maximization of πi ci jyð Þ, Eq. (3), over yiwith ci=c:

Fig. 1. B(crossesdifferen

N−1N2b0 cð Þ = cg ′ b0 cð Þð Þ: ð8Þ

The following proposition summarizes the comparison.

Proposition 2.2. In the private information contest defined above, the equilibrium bidding function, b*(c), has the following properties.

(i) For N=2, b*(c)bb0(c) for all c∈D; for NN2, however, this is not true in general;(ii) Let Pc = maxc∈Dc. Then there is �N0 such that b*(c)bb0(c) for all c∈ Pc−�;

Pc½ �.The proof is in the Appendix. Part (i) of Proposition 2.2 states that in contests of two players equilibrium effort is always lower

in the private information case than in the corresponding symmetric public information case. For more than two players, it isshown this is no longer true by providing counterexamples. Part (ii) of Proposition 2.2 states that the equilibrium bid in the privateinformation case is lower than in the public information case for players with sufficiently high costs. For continuously distributedcosts, it implies that the inequality holds in an interval of sufficiently high costs near the upper end of the distribution's support.For a discrete distribution of costs, it implies that the inequality holds at least for players with the highest possible cost.

3. Numerical illustrations

The equation b(c)=T [c ;b(⋅)], with operator T [c ;σ] defined above, is a multi-dimensional nonlinear integral equation. Suchequations can only be solved numerically, and even then finding a solution becomes increasingly difficult and ultimatelyunfeasible for large N due to the so-called “curse of dimensionality” (the exponential growth in the number of summation termsneeded to evaluate a multi-dimensional integral). In this section, a numerical solution, b*(c), is presented for relatively smallvalues of N=2,3, and 4. For comparability to Fey (2008), the lower bound of the cost parameter is set to be c0=0.01.

idding functions for N=2 players. Left panels: The symmetric equilibrium bidding functions, b (c), for the uniform (empty squares) and linearly increasing) distributions of cost parameter c. The full information symmetric equilibrium input, b0(c), is shown by the solid curve. Right panels: The relativece between b*(c) and b (c). The two upper panels correspond to h(x)=x (g(y)=y); the two lower panels correspond to h(x)=x1/2 (g(y)=y2/2).

Fig. 2. Bidding functions for N=3 players. Left panels: The symmetric equilibrium bidding functions, b (c), for the uniform (empty squares) and linearly increasing(crosses) distributions of cost parameter c. The full information symmetric equilibrium input, b0(c), is shown by the solid curve. Right panels: The relativedifference between b*(c) and b0(c). The two upper panels correspond to h(x)=x (g(y)=y); the two lower panels correspond to h(x)=x1/2 (g(y)=y2 /2).

Fig. 3. Bidding functions for N=4 players. Left panels: The symmetric equilibrium bidding functions, b (c), for the uniform (empty squares) and linearly increasing(crosses) distributions of cost parameter c. The full information symmetric equilibrium input, b0(c), is shown by the solid curve. Right panels: The relativedifference between b*(c) and b0(c). The two upper panels correspond to h(x)=x (g(y)=y); the two lower panels correspond to h(x)=x1/2 (g(y)=y2 /2).

562 D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

563D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

A solution for two players, N=2, the uniform distribution of players' cost parameters, f(c)= I [c∈D] (here, I [⋅] is the indicatorfunction equal to one if the condition is satisfied and zero otherwise), and the Tullock (1980) CSF, with h(x)=x (g(y)=y), wasnumerically obtained by Fey (2008). He showed that the symmetric equilibrium bidding function b*(c) in this case is below the fullinformation symmetric equilibrium effort b0(c) for all values of c.

Here, the analysis of Fey (2008) is extended along three dimensions. First, in addition to N=2, I consider the values of NN2;second, in addition to the uniform distribution of cost parameters, I consider a non-uniform distribution f(c)=2(c−c0)I [c∈D];third, in addition to the Tullock (1980) CSF, I consider a CSF with h(x)=x1/2, or, in our notation, g(y)=y2/2.5 As I show, thesemodifications may lead to interesting qualitative changes in the equilibrium bidding function.

The integral equation b(c)=T [c ;b] is solved numerically using a standard iterative procedure. The procedure starts with a trialsolution b1(c)=b0(c) and follows the recursion bn+1(c)=T [c ;bn] until convergence is reached.6 For all the results reported here,functions were defined on a grid of 20 nodes and integration was replaced with summation.7

The results for N=2, 3, and 4 are presented in Figs. 1–3, respectively. In all three figures, the two upper panels correspond tothe CSF with h(x)=x (g(y)=y), and the two lower panels to the CSF with h(x)=x1/2 (g(y)=y2/2). The panels on the left presentthe numerically obtained symmetric equilibrium bidding functions, b*(c), for the uniform distribution of costs (empty squares)and the linearly increasing distribution of costs, f(c)=2(c−c0)I [c∈D] (crosses), as well as the full information symmetricequilibrium input functions, b0(c) (solid curves). As the three functions in each of the left panels are hard to visually distinguish,especially for large values of c, the right panels in all three figures present the relative differences, [b*(c)−b0(c)] /b0(c), betweenthe private information bidding functions and the full information equilibrium input for the same symmetric value of c.

The empty squares in the upper left panel in Fig. 1 reproduce the result of Fey (2008). As seen from Fig. 1, and predicted byProposition 2.2, b*(c) is everywhere below b0(c), although the relative difference is nonmonotonic. Also, equilibrium inputs for thelinear distribution of costs are lower than those for the uniform distribution in the range of relatively low costs, but the reverse istrue in the range of higher costs. The reason is that the linearly increasing distribution of costs describes the situation in whichthere are relatively few “skilled” (low cost) players, and a relatively high number of “unskilled” (high cost) players in thepopulation. Thus, skilled players can decrease their effort compared to the uniform distribution case as they have a lowerprobability of encountering another skilled player, whereas the unskilled increase effort because their chances of encounteringanother unskilled player are higher.

Comparing the upper and the lower panels in Fig. 1, it is evident that the relative difference between b* and b0 is smaller (inabsolute value) for h(x)=x1/2 (g(y)=y2/2). This result is interesting, as one of the interpretations of the exponent, r, in theexpression h(x)=xr (this class of CSFs is used extensively in the literature) is the “decisiveness” or “discriminatory power” of thecontest (Hirshleifer, 1995). Higher values of r correspond to a highermarginal impact of any player's effort on his or her probabilityof winning. As r decreases, contest uncertainty plays a higher role in the determination of winning. At the same time, with thechange of variables from efforts to inputs undertaken here, 1/r becomes the exponent in the “input cost function.” Thus, one caninterpret a transition from h(x)=xr to h(x)=xs (with sb r) as either an increase in the level of winning uncertainty, or, keeping thelevel of uncertainty constant, an increase in the convexity of the cost function for inputs. As seen from Fig. 1 (and Figs. 2 and 3 aswell), equilibrium bidding functions for inputs in contests with more uncertainty (more convex cost functions) are closer to thecorresponding symmetric complete information input levels.

Fig. 2 shows the results for N=3. The key qualitative difference as compared to the case of N=2 is that there emerges a rangeof values of cwhere b*(c) exceeds b0(c). This occurs for the linearly increasing distribution of costs in the Tullock (1980) CSF case,and for both the uniform and linearly increasing distributions for the CSF with h(x)=x1/2. Thus, the intuition that “asymmetricinformation makes effort more risky, which tends to decrease effort levels” (Fey, 2008; Hurley and Shogren, 1998a) does notgeneralize to contests of more than two players.

This effect becomes even stronger for N=4 (see Fig. 3), where a range of c with b*(c)Nb0(c) exists for both CSFs and bothdistributions. Also, the relative difference between b*(c) and b0(c) appears to be increasing with N.

Finally, as seen from Figs. 1 through 3, in the range of relatively high costs b* (c) is below b0(c), as predicted by Proposition 2.2.The higher a player's cost, the more likely it is that his or her opponents are relatively more skilled, hence the incentive for playersat the upper end of the cost distribution to lower the input as compared to the symmetric public information case.

4. Discussion and concluding remarks

Despite the long history of the literature on contests, and a qualitative similarity between contests and auctions, little is knownabout bidding functions for contests in the classic symmetric IPV-like formulation. Until now, Fey (2008) has been the only studyof such bidding functions, and it focuses on a restrictive special case of contest models. In this paper, I extend the analysis of Fey(2008) to cover a very general class of contest models. I prove the existence of, and analyze numerically, the symmetricequilibrium bidding function for contests in the IPV-like formulation and compare it to the corresponding complete informationcase.

It is evident from the analysis that, despite the complexity involved in their calculation, symmetric equilibrium biddingfunctions for contests are smooth and qualitatively very similar to bidding functions in symmetric contests with complete

5 The factor of 1/2 in g(y) is chosen for convenience; it does not affect the CSF.6 The bidding function was iterated until the condition maxc∈D|bn+1(c)−bn(c)|b10−4 was met.7 The results are robust to an increase in the number of nodes. For N=3, the calculation was repeated with 50 nodes and the results did not change.

564 D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

information and IPV auctions. The fact that equilibrium bidding functions in contests with incomplete information exist, aresmooth, and can be found, at least in principle, is important for many applications. For example, in R&D competition it is unlikelythat different firms have information about each other's costs, but it is plausible that the distribution of possible values of suchcosts is common knowledge. Our results imply that one can assume a smooth monotonic equilibrium bidding function for firmsand explore more complex events in this setting, such as entry and exit, or response of R&D investments to government policies.

It is shown that in symmetric two-player contests equilibrium bids under private information are always lower than thoseunder public information about players' costs. I found, however, that this result does not generalize to contests of more than twoplayers. Although at the upper end of the cost distribution the equilibrium bids are still lower in the private information case, thereis a range of costs for which the equilibrium bids under private information may exceed those under public information. Thus,generally, the relationship between equilibrium bids under the two information conditions is not universal and depends on thenumber of players, the CSF, and the distribution of costs.

The qualitative difference between the cases of two and more than two players is somewhat puzzling. To gain some intuitionfor this result, consider a contest with only two types of players — low-cost and high-cost. If a player is low-cost, he or she iscertain, in the complete information case, that all his or her opponents are also low-cost. In the incomplete information case,however, he or she knows that with a positive probability at least some of his or her opponents are high-cost. Thus, compared tothe symmetric complete information case, the expected opponent's cost is higher for low-cost players in the incompleteinformation case. The opposite is true for high-cost players, whose expected opponent's cost is lower in the incompleteinformation case as compared to the symmetric complete information case. Because in the incomplete information case playerseffectively participate in contests of heterogeneous players, albeit of unknown abilities, some intuition can be gained from theprevious research on contests of heterogeneous players under complete information. Nti (1999) studied such contests for the caseof N=2 players, whom he called the “favorite” (low-cost or high prize valuation) and the “underdog” (high-cost or low prizevaluation). Nti (1999) found that the favorite's effort increases in the underdog's valuation, while the underdog's effort decreasesin the favorite's valuation. In our setting, this corresponds to both the low-cost and the high-cost players decreasing their effort inthe incomplete information case as compared to the corresponding symmetric complete information contests. Thus, the result ofNti (1999) explains what happens for N=2. However, the result of Nti (1999) breaks down for NN2, as shown by Stein (2002),who studied contests of players with heterogeneous prize valuations under complete information with more than two players.Specifically, Stein (2002) shows that if a low-cost player increases his or her valuation, all high-cost players will decrease theireffort, but if a high-cost player decreases his or her valuation, low-cost players may or may not decrease their effort (the decreaseis guaranteed only for N=2).

This work can be further extended along several dimensions. First, the existence of equilibrium with heterogeneous players(characterized by either different “input production functions” hi(x) in the CSF, or different distributions of costs, fi(⋅), or both) isstill an open question. Second, the fixed point techniques used in this paper, and by Fey (2008), do not allow one to address theissue of equilibrium uniqueness. Third, development of numerical techniques to allow for calculation of equilibrium biddingfunctions for larger Nwould be useful in applications of contest bidding functions as building blocks in more complexmodels. Onepossible direction is to use Monte Carlo simulations to evaluate the multi-dimensional integral.

Acknowledgements

I am grateful to the Editor, Arye L. Hillman, and two anonymous referees for their useful comments.

Appendix A. Proof of Proposition 2.1

A.1. Preliminaries

To simplify notation, let t=(t2,…, tN), dF(t)≡∏ j≥2dF(tj), s(t)≡∑ j≥2σ(tj). Also, let Φ1[y ;σ] denote the operator Φ1 with allplayers j≥2 choosing the same bidding function σ(⋅).

Player 1's best response function b(c) is defined by the equation.

with.

b cð Þ = T c;σ½ � = 0; if c N cm σð Þ = Φ1 0;σ½ �= g ′ 0ð Þ;y such that Φ1 y;σ½ � = cg ′ yð Þ; if c≤ cm σð Þ;

�ð9Þ

Φ1 y;σ½ � = ∫dF tð Þ s tð Þy + s tð Þ½ �2 : ð10Þ

Here and below, integration over the (N-1)-dimensional cube DN−1 is assumed.Consider C(D)— a set of real continuous functions defined on D=[c0,c0+1]. With the sup norm, ∥ f∥=supt∈D| f(t)|, this set is a

Banach space. A set CE of nonnegative continuous functions σ :D→ [0,E], where E=h(1/c0) is the upper bound on inputs identifiedin Section 2, is a closed and convex subset of C(D). For simplicity, similar to Fey (2008), assume that function σ=0 is not part of CE,which implies that CE is not closed. This issue can be resolved by considering instead a set of functions σ :D→ [�,E], which is closed,and letting �→0.

565D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

Operator T [c ;σ] acts on functions in CE, and the resulting best response functions are also in CE. To prove the existence of anequilibrium one needs to show that operator T has a fixed point. Since CE is not compact with the sup norm, it will be shown,similar to Fey (2008), that T is a compact operator and the Shauder fixed point theorem will be used.

Theorem A.1. (Schauder). Let X be a nonempty, closed, bounded, convex subset of a Banach space and suppose S :X→X is a compactoperator. Then S has a fixed point.

A compact operator is defined as follows.

Definition A.1. Let X,Y be Banach spaces. An operator S :X→Y is compact if it is continuous andmaps bounded sets into relativelycompact sets (i.e. sets whose closure is compact).

Thus, to establish compactness of T one needs to show that T is continuous and its range is relatively compact. Continuitywill beproved directly. To show the relative compactness of the range, the following theorem is applied.

Theorem A.2 (Arzelà-Ascoli). A set of functions in C [a,b]with the sup norm is relatively compact if and only if it is uniformly boundedand equicontinuous on [a,b].

This theorem uses the following two definitions.

Definition A.2. A set of real-valued functionsF is uniformly bounded on the intervalD if there exists anMb∞ such that for all f∈Fand all x∈D | f(x)|≤M.

Definition A.3. A set of real-valued functions F is equicontinuous on the interval D if for any �N0 there exists a δN0 such that forall f∈F and all x,y∈D | x−y|bδ implies | f(x)− f(y)|b�.

As discussed above, the range of T is uniformly bounded by E. Thus, to show that the range of T is relatively compact it suffices toshow that it is equicontinuous.

A.2. Proof of equicontinuity of the range of T

It is sufficient to show that there is a constant A such that for all c,c′∈D |b(c)−b(c′)|≤A|c−c′|. Let c,c′∈D and assume, withoutloss of generality, that cNc′. If cm(σ)≥c0+1, see case (iii) below; otherwise, there are three possible cases.

(i) c,c′≥cm(σ), then b(c)=b(c′)=0, and the result holds trivially;(ii) cNcm(σ) and c′≤cm(σ), then b(c)=0 and b(c′)≥0, with Φ1[b(c′) ;σ]=c′g ′(b(c′)). This gives, using Eq. (10),

c−c′ N cm−c′ =1

g ′ 0ð Þ∫dF tð Þs tð Þ − 1

g ′ b c′ð Þð Þ∫dF tð Þs tð Þ

b c′ð Þ + s tð Þ½ �2

≥ 1g ′ b c′ð Þð Þ∫dF tð Þ 1

s tð Þ−s tð Þ

b c′ð Þ + s tð Þ½ �2" #

=b c′ð Þ

g ′ b c′� �� � ∫ dF tð Þs tð Þ

s tð Þ b c′ð Þ + s tð Þ½ �2 + ∫ dF tð Þs tð Þ b c′ð Þ + s tð Þ½ �

" #

≥b c′ð Þ c′

N−1ð ÞE +g ′ 0ð Þ

g ′ b c′ð Þð ÞcmNE

� �≥ b c′ð Þ c0

N−1ð ÞE = b c′ð Þ−b cð Þ½ � c0N−1ð ÞE :

, I used Eq. (10) and the fact that functions b(⋅) and σ(⋅) are bounded by E.

Here(iii) c≤cm(σ), c′bcm(σ), then b(c)bb(c′) and g′(b(c))≤g ′(b(c′)). This gives, using Eq. (10),

c−c′ =1

g ′ b cð Þð Þ∫dF tð Þs tð Þ

b cð Þ + s tð Þ½ �2 −1

g ′ b c′ð Þð Þ∫dF tð Þs tð Þ

b c′ð Þ + s tð Þ½ �2

≥ 1g ′ b c′ð Þð Þ∫dF tð Þ s tð Þ

b cð Þ + s tð Þ½ �2 −s tð Þ

b c′ð Þ + s tð Þ½ �2" #

=b c′ð Þ−b cð Þg ′ b c′ð Þð Þ ∫dF tð Þ s tð Þ

b cð Þ + s tð Þ½ � b c′ð Þ + s tð Þ½ �2 +s tð Þ

b cð Þ + s tð Þ½ �2 b c′ð Þ + s tð Þ½ �

" #

≥ b c′ð Þ−b cð Þ½ � c′

NE+

g ′ b cð Þð Þg ′ b c′

� �� � cNE

" #≥ b c′ð Þ−b cð Þ½ � c0

NE:

where

566 D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

above, I used Eq. (10) and the fact that functions b(⋅) and σ(⋅) are bounded by E.

AsThus, in cases (ii) and (iii) inequalities of the form b(c′)−b(c)≤A(c−c′) are obtained, and, since both sides of the inequalities

are positive, they also hold for absolute values, and equicontinuity of b(⋅) follows.

A.3. Proof of continuity of T

Next, it is shown that operator T is continuous, i.e. for any sequence of functions σn(t) converging to σ(t) the correspondingsequence of best response functions bn(c) converges to b(c). Convergence in a space of functionswith the sup norm is equivalent touniform convergence, therefore one needs to show that bn(c) converges to b(c) uniformly on D.

Consider a sequence σn(t)→σ(t) and fix a point c∈D. One needs to show that for any �N0 there exists an N (independent of c)such that for all nNN |b(c)−bn(c)|b�. There are three possible cases to consider. (i) cNcm(σ), then there exists a large enough Nsuch that for all nNN cNcm(σn), i.e., |b(c)−bn(c)|=0.

(ii) cbcm(σ), then there exists a large enough N such that for all nNN cbcm(σn), i.e., bn(c)N0. Then b(c) and bn(c) satisfy Φ1[b(c) ;σ]=cg′(b(c)) and Φ1[bn(c) ;σn]=cg′(bn(c)), respectively. Assume, without loss of generality, that bn(c)bb(c) (the oppositecase can be considered similarly; if bn(c)=b(c), choose another n or another point c∈D). Then g′(bn(c))≥g′(b(c)), and Eq. (9)implies

∫ dF tð Þsn tð Þbn cð Þ + sn tð Þ½ �2 ≤∫ dF tð Þs tð Þ

b cð Þ + s tð Þ½ �2 :

Here, sn(t)≡∑ j≥2σn(tj). Letting B=[bn(c)+sn(t)]2[b(c)+s(t)]2 and following the same chain of algebraic transformations asin Fey (2008), obtain

b cð Þ−bn cð Þ½ �∫dF tð Þ s tð Þ b cð Þ + bn cð Þ½ � + 2s tð Þsn tð ÞB

≤b2 cð Þ∫dF tð Þ s tð Þ−sn tð ÞB

+ ∫dF tð Þ s tð Þsn tð Þ sn tð Þ−s tð Þ½ �B

:

Note that the left-hand side of this inequality is positive, therefore the absolute value of both sides can be taken. Further, usingthe triangle inequality and following the same arguments as in Fey (2008), obtain that |b(c)−bn(c)|bAsupt∈D|sn(t)−s(t)|, with Aindependent of c.

(iii) c=cm(σ), then b(c)=0 and necessarily bn(c)≥b(c). Going through the same steps as in case (ii), with b(c) replaced byzero, obtain the result.

The conclusion is that bn(c) converges uniformly to b(c), therefore, operator T is continuous, and by the Shrauder fixed pointtheorem it has a fixed point.

Appendix B. Proof of Proposition 2.2

The equilibrium bidding function b*(c) satisfies the equation

∫DN−1 ∏j≥2

dF tj� �" # ∑j≥2 b

�tj

� �b� cð Þ + ∑j≥2 b

� tj� �h i2 = cg ′ b� cð Þ� �

: ð11Þ

For brevity, denote the integral in Eq. (11) as ∫dF(t). The left-hand side can be transformed as

∫dF tð ÞN−1ð Þb� cð Þ + ∑j≥2 b� tj

� �−b� cð Þ

� �Nb� cð Þ + ∑j≥2 b� tj

� �−b� cð Þ

� �h i2 =N−1

N2b� cð Þ∫dF tð Þ1 + 1

N−1a tð Þ

1 + 1Na tð Þ

h i2 ; ð12Þ

a(t)=∑ j≥2[b*(tj)/b*(c)−1]. Combining Eq. (12) with Eq. (8), further obtain

∫dF tð Þ1 + 1

N−1a tð Þ

1 + 1Na tð Þ

h i2 =b�cð Þg ′ b

�cð Þ� �

b0 cð Þg ′ b0 cð Þð Þ =G b

�cð Þ� �

G b0 cð Þð Þ : ð13Þ

Here, G(y)≡yg′(y) is strictly increasing in y under our assumptions.

567D. Ryvkin / European Journal of Political Economy 26 (2010) 558–567

To prove part (i) of the proposition, consider the expression under the integral in Eq. (13). The difference between thedenominator and the numerator is

1 +a tð ÞN

� �2−1− a tð Þ

N−1=

a tð Þ2N2 +

N−2ð Þa tð ÞN N−1ð Þ : ð14Þ

This proves that forN=2 the left-hand side of Eq. (13) is less than one, which implies b*(c)bb0(c). The counterexamples shownnumerically in Section 3 for N=3 and 4 prove that the property does not hold for NN2.

For part (ii), let Pc be the maximal possible cost. Clearly, a(t)N0 for c = Pc for almost all t, therefore, it follows from (14) thatb� Pcð Þbb0 Pcð Þ. By continuity, this implies the existence of a neighborhood of Pc where this is also true.

References

Aidt, T., Hillman, A., 2008. Enduring rents. European Journal of Political Economy 24, 545–553.Amegashie, J., Kutsoati, E., 2007. (Non)intervention in intra-state conflicts. European Journal of Political Economy 23, 754–767.Anderson, S., Goeree, J., Holt, C., 1998. Rent seeking with bounded rationality: an analysis of the all-pay auction. Journal of Political Economy 106, 828–853

Reprinted in: Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008. 40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 225–250.Baye, M., Kovenock, D., de Vries, C., 1996. The all-pay auction with complete information. Economic Theory, 8, pp. 291–305. Reprinted in: Congleton, D., Hillman, A.,

Konrad, K. (Eds.), 2008. 40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 209–223.Berentsen, A., Bruegger, E., Loertscher, S., 2008. On cheating, doping and whistleblowing. European Journal of Political Economy 24, 415–436.Chang, Y.-M., Potter, J., Sanders, S., 2007. War and peace: third-party intervention in conflict. European Journal of Political Economy 23, 954–974.Cohen, C., Sela, A., 2008. Allocation of prizes in asymmetric all-pay auctions. European Journal of Political Economy 24, 123–132.Congleton, R., Hillman, A., Konrad, K., 2008. 40 Years of Research on Rent Seeking, volumes 1 and 2. Springer, Berlin.Corchón, L., 2007. The theory of contests: a survey. Review of Economic Design 11, 69–100.Cornes, R., Hartley, R., 2005. Asymmetric contests with general technologies. Economic Theory 26, 923–946.Fey, M., 2008. Rent-seeking contests with incomplete information. Public Choice 135, 225–236.Garfinkel, M., Skaperdas, S., 2007. Economics of conflict: an overview. In: Sandler, T., Hartley, K. (Eds.), Handbook of Defense Economics, vol. 2. North-Holland/

Elsevier, Amsterdam, pp. 649–709.Harstad, R., 1995. Privately informed seekers of an uncertain rent. Public Choice 83, 81–93.Hillman, A., Riley, J., 1989. Politically contestable rents and transfers. Economics and Politics 1, 17–39 Reprinted in: Congleton, D., Hillman, A., Konrad, K. (Eds.),

2008. 40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 185–207.Hirshleifer, J., 1995. Anarchy and its breakdown. Journal of Political Economy 103, 26–52.Hurley, T., Shogren, J., 1998a. Asymmetric information contests. European Journal of Political Economy 14, 645–665.Hurley, T., Shogren, J., 1998b. Effort levels in a Cournot–Nash contest with asymmetric information. Journal of Public Economics 69, 195–210.Klemperer, P., 2004. Auctions: Theory and Practice. Princeton University Press, Princeton, N.J.Kräkel, M., 2007. Doping and cheating in contest-like situations. European Journal of Political Economy 23, 988–1006.Krueger, A., 1974. The political economy of the rent-seeking society. American Economic Review, 64, pp. 291–303. Reprinted in: Congleton, D., Hillman, A., Konrad,

K. (Eds.), 2008. 40 Years of Research on Rent Seeking, volume 2, Springer, Berlin, pp. 151–163.Lazear, E., 1999. Personnel economics: past lessons and future directions. Journal of Labor Economics 17, 199–236.Leigh, A., 2008. Estimating the impact of gubernatorial partisanship on policy settings and economic outcomes: a regression discontinuity approach. European

Journal of Political Economy 24, 256–268.Lockard, A., Tullock, G. (Eds.), 2001. Efficient Rent Seeking: Chronicle of an Intellectual Quagmire. Kluwer Academic Publishers, Boston.Malueg, D., Yates, A., 2004. Rent seeking with private values. Public Choice, 119, pp. 161–178. Reprinted in Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008.

40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 503–520.Malueg, D., Yates, A., 2006. Equilibria in rent-seeking contests with homogeneous success functions. Economic Theory 27, 719–727.Moldovanu, B., Sela, A., 2001. The optimal allocation of prizes in contests. American Economic Review 91, 542–558 Reprinted in: Congleton, D., Hillman, A., Konrad,

K. (Eds.), 2008. 40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 615–631.Noussair, C., Silver, J., 2006. Behavior in all-pay auctions with incomplete information. Games and Economic Behavior 55, 189–206.Nti, K., 1999. Rent seeking with asymmetric valuations. Public Choice, 98, pp. 415–430. Reprinted in Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008. 40 Years of

Research on Rent Seeking, volume 1, Springer, Berlin, pp. 149–164.Pecorino, P., 2010. By-product lobbying with rival public goods. European Journal of Political Economy 26, 114–124.Pogrebna, G., 2008. Learning the type of the opponent in imperfectly discriminating contests with asymmetric informationAvailable at SSRN: http://ssrn.com/

abstract=12593292008.Schoonbeek, L., Winkel, B., 2006. Activity and inactivity in a rent-seeking contest with private information. Public Choice 127, 123–132.Skaperdas, S., 1996. Contest success functions. Economic Theory 7, 283–290 Reprinted in: Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008. 40 Years of Research

on Rent Seeking, volume 1, Springer, Berlin, pp. 263–270.Stein, W., 2002. Asymmetric rent seeking with more than two contestants. Public Choice 113, 325–336.Sui, Y., 2009. Rent-seeking contests with private values and resale. Public Choice 138, 409–422.Szidarovszky, F., Okuguchi, K., 1997. On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games and Economic Behavior 18, 135–140

Reprinted in: Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008. 40 Years of Research on Rent Seeking, volume 1, Springer, Berlin, pp. 271–276.Szymanski, S., 2003. The economic design of sporting contests: a review. Journal of Economic Literature 41, 1137–1187.Taylor, C., 1995. Digging for golden carrots: an analysis of research tournaments. American Economic Review 85, 872–890.Tullock, G., 1980. Efficient rent seeking. In: Buchanan, J., Tollison, R., Tullock, G. (Eds.), Toward a Theory of Rent-Seeking Society. Texas A&M University Press,

College Station, pp. 97–112.Wärneryd, K., 2003. Information in conflicts. Journal of Economic Theory 110, 121–136 Reprinted in: Congleton, D., Hillman, A., Konrad, K. (Eds.), 2008. 40 Years of

Research on Rent Seeking, volume 1, Springer, Berlin, pp. 487–502.