Optimal sorting in group contests

with complementarities

Philip Brookins John P. Lightle Dmitry Ryvkin∗

This version: September 3, 2014

Abstract

Contests between groups of workers are often used to create incentives in organiza-

tions. Managers can sort workers into groups in various ways in order to maximize

total output. We explore how the optimal sorting of workers by ability in such

environments depends on the degree of effort complementarity within groups. For

moderately steep costs of effort, we find that the optimal sorting is balanced (i.e.,

minimizing the variance in ability between groups) when complementarity is weak,

and unbalanced (i.e., maximizing the variance in ability) when complementarity is

strong. However, when the cost of effort is sufficiently steep, the optimal sorting

can be unbalanced for all levels of complementarity or even alternate between un-

balanced and balanced as the level of complementarity increases.

Keywords: group contest, complementarity, sorting, heterogeneity

JEL Classification: D72, C72, C02

∗Department of Economics, Florida State University, Tallahassee, FL 32306-2180, USA;dryvkin@fsu.edu

1

1 Introduction

In countless business settings, a worker or group of workers receives a reward based on

their performance relative to their peers. An economic contest, or tournament, is a

model of such a situation, where participants choose to expend resources, such as time or

effort, in order to increase their probability of being rewarded (Lazear and Rosen, 1981;

Lazear, 1995; Connelly et al., 2014). Contests are important tools that organizations use

to incentivize high productivity from workers when there is an indivisible reward (e.g.,

promotion); to reduce monitoring and measurement costs, or to filter out the risk of

common uncertainties (O’Keeffe, Viscusi and Zeckhauser, 1984).

Often, the performance of a firm depends on the combined input of a group of individu-

als. In this case, the firm’s management may wish to design a group contest where employ-

ees work together in teams, but compete against other similar groups, with a prize awarded

to the members of the winning group (Chen and Lim, 2013; Lim and Chen, 2014). For ex-

ample, consider a sales contest among branches of a chain store. To incentivize employee

effort and managerial oversight in each branch, the restaurant chain Dunkin’ Donuts of-

fered a reward to the best-run store within a region (O’Keeffe, Viscusi and Zeckhauser,

1984). Similarly, a Korean grocery store chain E-Mart Everyday used a sales competition

to increase its sales of U.S. beef.1

In this paper, we are interested in the following question: When an organization uses

an incentive scheme involving a group contest, how should it sort workers of heteroge-

neous abilities into groups in order to maximize the total output of all the participating

groups? Consider, for example, an architectural firm employing multiple designers, busi-

ness developers, and construction administrators, all with varying abilities. If the firm

has several ongoing projects, each of which requires one of each type of employee, the firm

will form teams and may use a contest to incentivize high effort. In order to maximize

total output (measured, for example, as a combination of quality, cost effectiveness, time-

liness of project completion and client satisfaction), should the firm put the best designer,

developer, and administrator together in the same team, or create a balance of highly

skilled and less skilled employees in each team?2

The principal result of this paper is in showing that the answer to this question de-

pends, sometimes in nontrivial and counter-intuitive ways, on the level of complementarity

between the efforts of the team members in the production process, as well as on the shape

1http://www.agweb.com/article/sales_competition_boosts_u.s._beef_at_korean_grocery_chain_NAA_News_Release/2Presumably, the firm does not want any of the projects to fail; therefore, the minimal necessary skill

level is still assumed even for the lowest-skill employees, otherwise they would not be employed by thefirm in the first place.

2

of the workers’ effort cost function. For illustration, suppose there are four workers enu-

merated 1 through 4 in the descending order of ability, and assume that the production

process involves groups of two workers. In a group contest with balanced sorting, group

(1,4) would compete with group (2,3); whereas, in the case of unbalanced sorting group

(1,2) would compete with group (3,4). Consider first the extreme case when within-group

efforts are perfect substitutes. Because of free-riding, competition between the groups

will be determined primarily by the effort of the best worker in each group.3 Thus, the

balanced contest will effectively reduce to a contest between workers 1 and 2, whereas

the unbalanced contest will be a contest between workers 1 and 3. Given that the av-

erage ability is higher in the former contest, it appears that the balanced sorting should

be preferred by the management in this case. In the opposite extreme case of perfect

complementarity of efforts within groups, the equilibrium effort will be determined by

a contest between the lowest ability workers in each group, i.e., between workers 3 and

4 in the case of balanced sorting and workers between workers 2 and 4 in the case of

unbalanced sorting. Here, the average ability is higher in the latter contest; therefore, it

appears that the unbalanced sorting should be preferred by the management.

Group production processes are characterized by different levels of complementarity

between workers. For example, an airport security checkpoint operates in a manner close

to perfect complementarity, while waiters in a restaurant or facilitators at a children’s

summer camp are close to perfect substitutes. Given the different effects of sorting on

aggregate contest output in the two extreme cases discussed in the previous paragraph,

there must be a cut-off level of complementarity at which the optimal sorting of workers

in a group contest switches from balanced to unbalanced.

We model a group contest with complementarities using a lottery (Tullock, 1980)

group contest success function (CSF) with a constant elasticity of substitution (CES)

aggregation of within-group efforts and workers with heterogeneous convex costs of effort.

We consider the combined output of all groups in equilibrium as a function of the level

of within-group effort complementarity. Surprisingly, we find that the way in which

complementarity interacts with optimal sorting is more complex than the simple intuition

above suggests, and depends on the shape of the workers’ cost function of effort. For

example, for certain parameters, there are intermediate levels of complementarity, such

as a Cobb-Douglas aggregation function, where balanced sorting is optimal, even though

unbalanced sorting is optimal at either extreme. In order to explore the effect of sorting

on output, similar to Ryvkin (2011), we use the quadratic approximation to the true

3When costs of effort are linear, group effort is determined only by the effort of the best worker (Baik,2008). For convex costs of effort, the marginal costs of effort within groups are equalized, therefore thefree-riding is not as extreme.

3

equilibrium efforts and develop an expansion of output in the moments of the distribution

of abilities. Within the quadratic approximation, we describe all possible cases for how

optimal sorting depends on within-group effort complementarity, and provide an example

of each case.

The problem of optimal sorting of heterogeneous players in a group contest with

perfect substitutes has been explored by Ryvkin (2011), who showed that the optimal

sorting is balanced as long as the players’ effort function is not too steep. Being an

important benchmark case, the perfect substitutes technology is not the most realistic

in applications. Indeed, within-group complementarities, or synergies, are one of the

key reasons group production exists in the first place (Alchian and Demsetz, 1972). In

this paper we extend the analysis of Ryvkin (2011) to arbitrary levels of within-group

complementarity.

The theoretical literature on group contests goes back to Katz, Nitzan and Rosenberg

(1990) and Nitzan (1991) who first considered symmetric group contests with a lottery

CSF, perfectly substitutable within-group effort, and linear effort costs. In a similar set-

ting, Baik (2008) considers the case of heterogeneous prize valuations and shows that

only the highest-valuation player in each group exerts positive effort in equilibrium.

Other within-group aggregation functions have also been analyzed. Lee (2012) considers

the weak-link (perfect complements) technology, while Chowdhury, Lee and Sheremeta

(2013) study the “best-shot” technology in which a group’s output is determined by the

maximal effort. The same aggregation functions have also been analyzed in an alter-

native perfectly discriminating contest (all-pay auction) setting in which the group pro-

ducing the highest output wins with certainty (e.g., Baik, Kim and Na, 2001; Topolyan,

2014; Chowdhury, Lee and Topolyan, 2013; Barbieri, Malueg and Topolyan, 2013). A

group contest involving groups with different aggregation technologies (one weak-link

and the other – best-shot) is analyzed by Chowdhury and Topolyan (2013). Finally,

Kolmar and Rommeswinkel (2013) use the CES aggregation function and allow for dif-

ferent complementarity levels in different groups and within-group player heterogeneity

with linear effort costs.

The rest of the paper is organized as follows. In Section 2, we formulate the general

model and prove the existence and uniqueness of equilibrium. In Section 3, we use the

quadratic approximation to calculate the equilibrium output and analyze the problem of

optimal sorting. In Section 4, we provide numerical illustrations of the various possible

scenarios, identified in Section 3, of how optimal sorting may change with the level of

complementarity and the shape of effort cost functions. Section 5 provides a discussion

and concluding remarks.

4

2 The model

Consider a contest between n groups, indexed by i = 1, . . . , n, of m risk-neutral players

each. The players in group i are indexed by ij = i1, . . . , im. All players in all groups

simultaneously and independently choose effort levels eij ≥ 0. The cost of effort eij to

player ij is cijg(eij), where cij > 0 is the player’s cost parameter and g is a strictly

increasing function. Parameters cij (i = 1, . . . , n; j = 1, . . . ,m) are common knowledge.

The output of group i is given by the constant elasticity of substitution (CES) aggre-

gation function Ei =(∑m

j=1 eρij

)1/ρ, with ρ ≤ 1. The probability of group i winning the

contest is pi = Ei/∑n

k=1Ek, with pi = 1/n if∑n

k=1Ek = 0. If group i wins the contest,

each of its players i1, . . . , im receives a prize normalized to one; otherwise, all players

receive zero, which is also all players’ outside option payoff.

The expected payoff of player ij is, thus, equal to

πij =Ei∑k Ek

− cijg(eij). (1)

Let cmin = mini,j cij and cmax = maxi,j cij denote, respectively, the lowest and highest cost

parameters among all players. Since each player can guarantee herself a payoff of zero

by choosing zero effort, the region of efforts (emax,∞), with cming(emax) = 1, is strictly

dominated for all players. Thus, emax is an upper bound on effort levels that can be chosen

in equilibrium. We assume that a finite emax exists and impose the following assumptions

on function g(·) on the interval [0, emax].

Assumption 1 (i) g(·) is continuously differentiable on [0, emax] and thrice continuously

differentiable on (0, emax];

(ii) g(0) = 0;

(iii) g′(e) > 0 and g′′(e) ≥ 0 on (0, emax];

(iv) cmaxn2

n−1max{m1/ρ,m}emaxg

′(0) < 1;

(v) g′(0) = 0 and g′′(e) > 0 on (0, emax] if ρ = 1.

Assumptions 1(i)-(iii) are standard. Assumptions 1(iv)-(v) ensure, as formulated in

Proposition 1 below, that a unique Nash equilibrium in pure strategies with all players

exerting positive efforts exists. To this end, for ρ = 1, Assumption 1(v) requires that the

marginal cost of effort at zero effort be zero and the cost of effort be strictly convex. When

ρ < 1, the strict convexity of g(·) and zero marginal costs at zero are no longer necessary,

but marginal costs at zero should not be too large, with an upper bound provided by

Assumption 1(iv).

5

Pure strategy equilibrium effort levels e∗ij satisfy the system of Kuhn-Tucker conditions

for maximization of πij with respect to eij:

(∑

l eρil)

(1−ρ)/ρ eρ−1ij

∑k =i (

∑l e

ρkl)

1/ρ[∑k (∑

l eρkl)

1/ρ]2 ≤ cijg

′(eij), i = 1, . . . , n; j = 1, . . . ,m. (2)

Equations (2) hold with equality for e∗ij > 0. It is easy to see that the left-hand side of (2) is

decreasing in eij; therefore, πij is a concave function of eij and the Kuhn-Tucker conditions

are necessary and sufficient for maximization. The following proposition establishes the

existence and uniqueness of equilibrium.

Proposition 1 In the contest game defined above, with Assumptions 1(i)-(v) satisfied,

there exists a unique pure strategy Nash equilibrium in which all players’ efforts e∗ij > 0

solve the system of first-order conditions (2) with equality.

All proofs are provided in the Appendix. Proposition 1 is a generalization of a similar

proposition in Ryvkin (2011) for ρ = 1. The proof relies on a reduction of the group contest

to an effective contest among n individuals and then uses the result of Cornes and Hartley

(2005).

3 The effects of sorting on output

Assume that there is a principal whose objective is maximization of total equilibrium

output in the contest,

E =n∑

i=1

(m∑j=1

e∗ρij

)1/ρ

. (3)

In this paper, we focus on the effects of sorting of players into groups by their ability.

Thus, we assume that the contest structure and the set of nm players with cost param-

eters (cij)i=1,...,n;j=1,...,m are fixed, and the only manipulation available to the principal is

the assignment of players to the groups. Different such assignments, or sortings, may

potentially lead to different levels of equilibrium output E, and the goal of this section is

to identify the optimal sorting.

The system of first-order conditions (2) has no closed-form solution, except for some

special cases (such as, for example, for cost functions g(e) = et with t > 1). Even in those

cases when a closed-form solution exists, equilibrium output E is a complicated function

of cost parameters cij, which is not helpful in the identification of optimal sortings. For

any given parameterization, one can potentially find the optimal sorting by going through

6

all possible sortings. This approach is not constructive, however, because, first, it becomes

very inefficient as n and/or m grow (the number of possible sortings is(nmn

)), and, second,

it does not provide any general insights into the properties of optimal sortings.

In this section, we use an alternative approach that addresses both issues raised

above. It is computationally efficient and provides a clear-cut criterion of optimality

for sortings. Importantly, the results do not rely on the availability of a closed-form

solution for equilibrium efforts. Such universality comes at a cost: the solution is obtained

approximately, in the form of a second-order expansion of aggregate output in the sample

moments of the distribution of cost parameters cij. The approximation relies on the weak

heterogeneity assumption for cost parameters, i.e., on the assumption that parameters

cij are not very different, in relative terms, from some average value c. As we show with

numerical illustrations, the approximation has a very high accuracy for small to moderate

levels of heterogeneity.

Without loss of generality, introduce the average cost parameter c = (nm)−1∑

i,j cij

and define relative abilities (or simply abilities) aij by writing cij = c(1 − aij). By con-

struction, aij < 1 and∑

i,j aij = 0. A player with a higher (lower) aij has a lower (higher)

cost of effort; moreover, a player with aij > 0 (aij < 0) has the cost parameter below

(above) the average level c. Let S2a = (nm)−1

∑i,j a

2ij denote the sample variance of abil-

ities. The weak heterogeneity assumption we make states that Sa ≪ 1, i.e., the spread in

relative abilities is small compared to unity.

Let e denote the equilibrium effort in the symmetric contest of nm players with cij = c

for all i, j. It is given by the solution to the equation

n− 1

n2me= cg′(e), (4)

which exists and is unique under Assumption 1(iv). In the contest with heterogeneous

players, we will write equilibrium efforts in the form e∗ij = e(1 + xij), where xij is the

relative effort of player ij as compared to the symmetric equilibrium level. Provided the

equilibrium described by Proposition 1 exists and is sufficiently smooth in parameters,

we expect that under the weak heterogeneity assumption relative efforts xij will be small

compared to unity, in the same sense as relative abilities aij are small. Thus, we will look

for relative efforts xij in the form of a Taylor expansion in the powers of aij. Due to the

symmetry of the contest, this will lead to a representation of aggregate output E in the

form of an expansion in the moments of aij. We restrict the expansion by the second

order, which, as we show, is the lowest order in which the effects of sorting on output can

be seen.

7

Let Ai =∑

j aij denote the aggregate ability of group i. Further, let S2A = n−1

∑iA

2i

denote the sample variance in ability across groups. The aggregate output in a contest

of homogeneous players is E = nm1/ρe. Similar to individual effort, we will write the

aggregate output for heterogeneous players in the form E = E(1 + Y ), where Y is small

compared to unity under the weak heterogeneity assumption. In the quadratic approxi-

mation, we present Y in the form Y = Y (1) + Y (2) + O(S3a), where Y (1) is the first-order

correction (linear in abilities), Y (2) is the second-order correction (quadratic in abilities),

and the error of the approximation O(S3a) is cubic in abilities and is, therefore, neglected.

As we show in the Appendix, Y (1) = 0, i.e., there is no first-order effect of sorting on

aggregate output. This result is standard and is due to the symmetric structure of the

contest and the mean-preserving nature of sorting. At the same time, Y (2) = 0, i.e., the

main effect of sorting is given by the second-order correction to aggregate output. Due

to the symmetric structure of the contest, the second-order correction has a particular

symmetric form described in Proposition 2 below.

It is convenient to introduce the following dimensionless coefficients:

k2 =g′′(e)e

g′(e), k3 =

g′′′(e)e

g′′(e). (5)

Coefficients k2 and k3 represent, respectively, the second- and third-order dimensionless

curvatures of the cost function g(·) at the symmetric equilibrium point. Coefficient k2 can

be interpreted as the elasticity of the marginal cost of effort, and k3 as the elasticity of the

second derivative of effort costs. Also, let ξ = 1− ρ denote the level of complementarity

of effort within groups, with ξ ≥ 0, and ξ = 0 corresponding to perfectly substitutable

effort.

Proposition 2 (a) In the contest defined above, in the quadratic approximation, the ag-

gregate equilibrium output is E = E(1 + Y (2)) + O(S3a), where Y (2) = λaS

2a + λAS

2A,

and

λa =2k2 + κ

2(k2 + 1)(k2 + ξ),

λA =Λξ

2m2(k2 + 1)[(n− 1)k2 + 1]2;

Λξ =[2(k2 + κ)((n− 1)k2 + 1) + κγ]γ

k2 + ξ+ (k2 − 1)(n− 1)2ξ, (6)

κ =ξ(1− k2)− k2k3

k2 + ξ, γ = (n− 1)ξ − 1.

(b) The optimal sorting of players into groups minimizes (maximizes) the variation in

8

ability across groups, in the quadratic approximation, if Λξ < 0 (Λξ > 0).

Part (a) of Proposition 2 shows that the second-order correction to aggregate output

is a linear combination of the sample variance in ability across all nm players, S2a, and the

sample variance in ability across groups, S2A. While S2

a is fixed for a given set of players, S2A

can be manipulated through sorting of players into groups. Specifically, S2A is minimized

(maximized) by the most balanced (unbalanced) sorting that makes aggregate abilities

across groups as equal (unequal) as possible. As stated in part (b) of the proposition, the

effect of such manipulations depends on the sign of coefficient Λξ.

Note that for ξ = 0 (i.e., for ρ = 1 when within-group efforts are perfect substitutes),

we have κ = −k3, γ = −1, and the expressions for λa and λA in (6) simplify to those in

Ryvkin (2011), with Λ0 = −[2− k3k2

+ 2(n− 1)(k2 − k3)]. For cost functions of effort that

are not too steep, i.e., k2 ≥ k3, we have Λ0 < 0 and hence λA < 0 and the optimal sorting

is balanced. For very steep effort cost functions such that Λ0 > 0, we have λA > 0 and

the optimal sorting is unbalanced.4

Consider now the opposite limit of a large ξ (i.e., a large negative ρ when within-group

efforts are strong complements). The following lemma describes the asymptotic behavior

of coefficient Λξ in this case.

Lemma 1

limξ→∞

Λξ ≡ Λ∞ = (n− 1)k2[(n− 1)(2k2 − k3) + 2]. (7)

Lemma 1 shows that Λξ converges to a constant for large ξ. Interestingly, Λ∞ may

be positive or negative. The following proposition describes what may happen for inter-

mediate values of ξ.

Proposition 3 In the quadratic approximation, the following is a complete list of possible

cases:

(a) Λ∞ ≥ 0, Λ0 ≤ 0, and there exists a unique ξ such that Λξ ≤ 0 (the balanced sorting is

optimal) for ξ ≤ ξ and Λξ ≥ 0 (the unbalanced sorting is optimal) otherwise.

(b) Λξ ≥ 0 (the unbalanced sorting is optimal) for all ξ ≥ 0.

(c) Λ∞ ≥ 0, Λ0 ≥ 0, and there exist ξ1 and ξ2 such that Λξ ≤ 0 (the balanced sorting is

optimal) for ξ ∈ [ξ1, ξ2] and Λξ ≥ 0 (the unbalanced sorting is optimal) otherwise.

(d) Λ∞ ≤ 0, Λ0 ≥ 0, and there exists a unique ξ such that Λξ ≥ 0 (the unbalanced sorting

is optimal) for ξ ≤ ξ and Λξ ≤ 0 (the balanced sorting is optimal) otherwise.

4In order for Λ0 to be positive it is necessary (although not sufficient) to have k3 > k2, i.e., the secondderivative of g(·) has to grow faster than the first derivative.

9

Cases (a) and (b) are the most straightforward extensions of the results of Ryvkin

(2011) for perfect substitutes to arbitrary levels of complementarity. In case (a), the

optimal sorting is balanced for perfect substitutes, and Λξ has a single crossing point ξ.

In this case, the optimal sorting continues to be balanced as long as ξ < ξ and becomes

unbalanced for ξ > ξ. In case (b), the optimal sorting is unbalanced for perfect substitutes

and remains unbalanced for all ξ as Λξ is always positive. It should be noted that case (b)

is somewhat less intuitive than case (a) and only arises for very steep effort cost functions

g(·).Cases (c) and (d) represent even less intuitive departures from case (a). In (c), the

unbalanced sorting is optimal for low and high levels of complementarity, but the balanced

sorting is optimal for intermediate levels of complementarity. In (d), the unbalanced

sorting is optimal for perfect substitutes but balanced sorting becomes optimal beyond a

crossing point ξ and continues to be optimal as ξ increases.

To get some intuition behind the behavior of Λξ in cases (c) and (d), it helps to

consider the extreme case of perfect complements with ξ → ∞. In this case, equilibrium

efforts in the group contest are determined by the lowest ability players in each group and

the equilibrium output is proportional to the aggregate effort in the individual contest

of these n players (Lee, 2012). For illustration, suppose there are two groups of two

players, and the four players are ordered by their ability from 1 (highest ability) to 4

(lowest ability). The balanced sorting would match players (1,4) against players (2,3),

while the unbalanced sorting would match players (1,2) against players (3,4). Thus, with

perfectly complementary efforts, equilibrium for the unbalanced sorting would correspond

to a contest between players 2 and 4, while for the balanced sorting it would be a contest

between players 3 and 4. Comparing the two contests, the effort of the underdog (player

4) is lower while the effort of the favorite (player 2 as compared to player 3) is higher

in the former contest than in the latter.5 The overall effect of sorting on the aggregate

effort, therefore, depends on which of the two effects dominates. When the effort cost

function is not too steep, the increase in the favorite’s effort is larger than the reduction in

the underdog’s effort, hence the unbalanced sorting is optimal. However, when the effort

cost function is very steep, the effect is reversed because it is prohibitively expensive for

the favorite, whose effort is higher, to increase effort by a lot. For such very steep cost

functions we have Λ∞ < 0 and hence 2k2 < k3, which implies Λ0 > 0, i.e., the unbalanced

sorting is optimal for perfect substitutes (case (d)).

Case (c) arises as an intermediate step between (b) and (d). The effort cost function is

5For linear costs of effort, this result has been obtained by Nti (1999). For an arbitrary convex costof effort, the result is easily obtained by differentiating first-order conditions.

10

sufficiently steep for the unbalanced sorting to be optimal for perfect substitutes (Λ0 > 0),

but it is not sufficiently steep to reverse the effect for perfect complements (Λ∞ > 0).

Consider again the contest involving players 1 through 4 from the paragraph above. When

ξ = 0, marginal effort costs within groups are equalized and free-riding is the strongest.

Because the cost function of effort is very steep, the equilibrium can be approximated

by that of a contest between n highest-ability individuals in each group, i.e., between

players 1 and 2 for the balanced sorting and players 1 and 3 for the unbalanced sorting.

Similar to the discussion in the previous paragraph, player 1’s effort is higher and the

underdog’s effort is lower in the latter contest; therefore, the overall effect is ambiguous.

The fact that the unbalanced sorting dominates for ξ = 0 implies the increase in player

1’s effort is larger than the reduction in the underdog’s effort. For intermediate levels

of complementarity, as ξ increases, equilibrium efforts of players within a group become

closer to each other and are determined by a mixture of the cost parameters of the low-

and high-ability players in the group. Somewhere along the way, Λξ may decrease below

zero and the balanced sorting may become optimal for a range of ξ where it is beneficial

to have more balanced average abilities. Note that a similar reduction in Λξ may occur

in case (b) but it is not strong enough to reach zero.

In the following section, we present numerical examples of the four cases identified in

Proposition 3.

4 Examples

For all examples in this section, we consider contests between n = 3 groups of m = 2

players each. The players’ relative abilities are τ1 = 5d/6, τ2 = d/2, τ3 = d/6, τ4 = −d/6,

τ5 = −d/2 and τ6 = −5d/6, where d is the heterogeneity parameter. For the weak

heterogeneity approximation to apply, parameter d is supposed to be “small” compared

to unity. In the examples below, we consider the values of d up to 0.5 and show that the

quadratic approximation for aggregate output works very well for d at least as high as 0.3,

which corresponds to about 50% spread in abilities across players. The balanced sorting of

players into groups corresponds to the assignments of abilities a11 = τ1, a12 = τ6, a21 = τ2,

a22 = τ5, a31 = τ3 and a32 = τ4, while the unbalanced sorting of players corresponds to

the assignment a11 = τ1, a12 = τ2, a21 = τ3, a22 = τ4, a31 = τ5 and a32 = τ6. The examples

differ by the shape of the effort cost function g(·).

Example 1 g(e) = et, with t > 1.

11

0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

Λξ

ξ

0.0 0.1 0.2 0.3 0.4 0.50.00

0.01

0.02

0.03

Y(2)

ξ=2 balanced unbalanced numerical

d0.0 0.1 0.2 0.3 0.4 0.5

0.00

0.05

0.10

0.15

0.20

Y(2)

ξ=0 balanced unbalanced numerical

d

Figure 1: g(e) = e1.2, c = 1. In the left and middle panels, the solid and dashed curvesshow the quadratic approximation, cf. Proposition 2; the squares show the results ofhigh-precision numerical solutions. The right panel shows Λξ, Eq. (6).

One of the simplest examples is that of a power function, which gives rise to case (i) in

Proposition 3. Equation (6) gives

Λξ = −2(n− 1)(t− 1)− t+ (n− 1)(t− 1)[(n− 1)t+ 2]ξ, (8)

i.e., Λ0 < 0 and Λξ increases linearly in ξ. There is a unique cutoff point,

ξ =2(n− 1)(t− 1) + t

(n− 1)(t− 1)[(n− 1)t+ 2], (9)

such that Λξ < 0 (> 0) for ξ < ξ (ξ > ξ). For such effort cost functions, the optimal sorting

is always balanced in the case of perfect substitutes (ξ = 0), and becomes unbalanced

as the level of complementarity increases. A straightforward calculation shows that the

cutoff ξ, Eq. (9), decreases in t.

For illustrations, we use the power function with t = 1.2 and c = 1. Equation

(4) for the symmetric equilibrium effort level gives e = 0.138, which implies k2 = 0.2

and k3 = −0.8. The left and middle panels in Figure 1 show the second-order relative

correction to aggregate output, Y (2), as a function of heterogeneity parameter d, for, re-

spectively, ξ = 0 (perfect substitutes) and ξ = 2. The solid and dashed curves show the

quadratic approximation computed using Proposition 2; the squares show the results of

high-precision numerical solutions. The solid curves correspond to the balanced sorting,

while the dashed curves to unbalanced sorting. As predicted, the balanced sorting pro-

duces a higher aggregate output than the unbalanced sorting for perfect substitutes, but

the reverse is true for the complementarity level ξ = 2, which is higher than the threshold

value ξ = 1.136 (cf. Eq. (9)). The right panel in Figure 1 shows Λξ as a function of ξ. It

starts at Λ0 = −10 and increases monotonically towards Λ∞ = 1.76 with a single root.

12

0 2 4 6 8 100

2

4

6

8

Λξ

ξ0.0 0.1 0.2 0.3 0.4 0.5

0.000

0.002

0.004

0.006

Y(2)

ξ=2 balanced unbalanced numerical

d0.0 0.1 0.2 0.3 0.4 0.5

0.000

0.002

0.004

0.006

Y(2)

ξ=0 balanced unbalanced numerical

d

Figure 2: g(e) = e∫ 1

1−et−1 exp(t)dt, c = 0.03. In the left and middle panels, the solid and

dashed curves show the quadratic approximation, cf. Proposition 2; the squares show theresults of high-precision numerical solutions. The right panel shows Λξ, Eq. (6).

Example 2 g(e) = e∫ 1

1−et−1 exp(t)dt, c = 0.03.

This example produces case (ii) of Proposition 3, with Λξ > 0 for all ξ ≥ 0. Solving Eq. (4)

numerically, we obtain e = 0.698, k2 = 2.123, k3 = 3.530, Λ0 = 5.294 and Λ∞ = 14.559.

Similar to Example 1, the left and middle panels in Figure 2 show Y (2) as a function

of d for, respectively, ξ = 0 and ξ = 2. As predicted, the unbalanced sorting produces

a higher aggregate output than the balanced sorting in both cases. The right panel in

Figure 2 shows Λξ as a function of ξ. Interestingly, although Λξ is positive for all ξ ≥ 0, it

is nonmonotonic. As shown in the following example, such a nonmonotonicity may lead

to Λξ becoming negative in an interval of intermediate complementarity levels.

Example 3 g(e) = 1− (1− e2)1/2, c = 0.105.

This example produces case (iii) of Proposition 3, with Λξ alternating between positive

and negative values as ξ increases. Solving Eq. (4), we obtain e = 0.798, k2 = 2.757,

k3 = 5.271, Λ0 = 9.968 and Λ∞ = 13.707. The lower right panel in Figure 3 shows Λξ

as a function of ξ. It is nonmonotonic and crosses zero at ξ1 = 1.102 and ξ2 = 5.014.

The remaining three panels show Y (2) as a function of d for the balanced and unbalanced

sortings for ξ = 0 (upper left), ξ = 2 (upper right) and ξ = 6 (lower left). As predicted,

the unbalanced sorting produces a higher output for ξ = 0 and ξ = 6, which are both

outside the interval [ξ1, ξ2]. For ξ = 2, which is inside the interval, the balanced sorting

leads to a higher output.

Example 4 g(e) = 1− (1− e2)1/2, c = 0.07.

This example produces case (iv) of Proposition 3, with Λξ starting at Λ0 > 0 and

converging to Λ∞ < 0 with a single crossing point ξ. Solving Eq. (4), we obtain e = 0.876,

13

0 2 4 6 8 10-2

0

2

4

6

8

10

Λξ

ξ0.0 0.1 0.2 0.3 0.4 0.5

0.000

0.001

0.002

0.003

0.004

Y(2)

ξ=6 balanced unbalanced numerical

d

0.0 0.1 0.2 0.3 0.4 0.50.000

0.001

0.002

0.003

0.004

Y(2)

ξ=2 balanced unbalanced numerical

d0.0 0.1 0.2 0.3 0.4 0.5

0.000

0.001

0.002

0.003

0.004

Y(2)

d

ξ=0 balanced unbalanced numerical

Figure 3: g(e) = 1 − (1 − e2)1/2, c = 0.105. In the upper left, upper right and lower leftpanels, the solid and dashed curves show the quadratic approximation, cf. Proposition 2;the squares show the results of high-precision numerical solutions. The lower right panelshows Λξ, Eq. (6).

14

0 2 4 6 8 10-30

-20

-10

0

10

20

Λξ

ξ0.0 0.1 0.2 0.3 0.4 0.5

-0.0005

0.0000

0.0005

0.0010

Y(2)ξ=2

balanced unbalanced numerical

d0.0 0.1 0.2 0.3 0.4 0.5

-0.002

-0.001

0.000

Y(2)

ξ=0 balanced unbalanced numerical

d

Figure 4: g(e) = 1 − (1 − e2)1/2, c = 0.07. In the left and middle panels, the solid anddashed curves show the quadratic approximation, cf. Proposition 2; the squares show theresults of high-precision numerical solutions. The right panel shows Λξ, Eq. (6).

k2 = 4.286, k3 = 9.859, Λ0 = 22.590, Λ∞ = −4.907 and ξ = 0.783. The left and middle

panels in Figure 4 show Y (2) as a function of d for, respectively, ξ = 0 and ξ = 2. As

predicted, the unbalanced sorting produces a higher aggregate output than the balanced

sorting for ξ = 0, and the reverse is true for ξ = 2. The right panel in Figure 2 shows Λξ

as a function of ξ.

5 Discussion and conclusions

A group contest is a natural way for a manager to increase productivity in organizations

that use team production, particularly when effort is difficult to monitor but the final

products of the teams are easy to compare. Giving a bonus to the best-performing team

will incentivize workers to give more effort than what is required to simply keep their

job, but exactly how much more effort they give depends, among other factors, on how

the manager chooses to sort workers into teams. In this paper, we focus on the question

of how a manager should sort a set of workers with heterogeneous abilities into teams

in order to maximize the combined output of the teams in an organization using team

contests. Unlike previous work on this subject, we allow for synergies among the members

of each team, so that individual efforts combine with some degree of complementarity to

produce team output.

The primary result of our paper is Proposition 2 which provides a criterion for how

a manager should sort her employees into teams to maximize output. We show that ag-

gregate output can be manipulated by sorting through the sample variance of aggregate

ability across groups. We find that a manager should either seek to minimize or maxi-

mize this variance, i.e., create balanced or unbalanced groups, depending on the sign of

a criterion Λξ that depends on the degree of effort complementarity in team production

15

and the properties of the workers’ effort cost function.

Our results, therefore, have a direct managerial application to sorting in team contests.

One could imagine certain production processes, such as airline security teams, where the

success of the team depends critically on the effort given by each member of the group.

Individual efforts complement each other and free-riding is not a viable option in such a

situation. In other production processes, such as tomato picking on a farm or working

a cash register in a supermarket, effort is much more substitutable. Interestingly, the

precise extent to which the efforts complement each other can change the optimal sorting

in non-trivial ways depending on the shape of the workers’ effort cost function. When

the costs of effort are not too steep, we find that balanced groups are optimal with less

complementarity, but a cutoff level of complementarity exists beyond which unbalanced

groups become optimal. Thus, for example, if a supermarket organizes a contest between

shifts of cashiers, the optimal sorting of cashiers into shifts is balanced; at the same time,

if an architectural firm uses a contest between teams working on similar projects, the

optimal sorting of workers into the teams is unbalanced. While these may, to an extent,

be considered expected results, the situation becomes much less intuitive when the costs

of effort are very steep. We find that in this case unbalanced groups may be optimal for

all levels of complementarity. It is also possible that unbalanced groups are optimal for

low and high complementarity levels, while balanced groups are optimal for intermediate

complementarity levels. Finally, it may be that unbalanced groups are optimal for low

complementarity and balanced groups for high complementarity, with a single cutoff. We

provide numerical examples of cost functions and parameterizations to illustrate each of

these cases.

Admittedly, effort cost functions are not directly observable in the field, and serve

merely as a building block of a model of decision making by employees. So, what does a

more or less steep effort cost function actually represent? One interpretation is that the

steepness of an effort cost function represents how easy it is for the worker to increase

effort, if necessary. In the numerical examples, we observe that the counter-intuitive

behavior of Λξ takes place for cost functions that have a threshold value of effort such that

the marginal cost of effort can become infinite. It is close to such threshold effort levels

that higher-order derivatives of effort costs increase faster than lower-order derivatives,

which is a necessary condition for a reversal in the behavior of Λξ. In applications, this

corresponds to workers choosing effort close to their natural effort capacity. For example,

a medical resident or an associate at a law firm working 80 hours a week, or an airline pilot,

or a truck driver may be operating very close to their effort capacity. In these industries,

it is very costly, if not impossible, for workers to increase their effort; therefore, their cost

16

of effort may be considered very steep.

We calculate the aggregate equilibrium output in the group contest in the form of an

expansion in the sample moments of the distribution of abilities. By cutting the expansion

at the second order, we thus find the quadratic approximation to the equilibrium output.

There are several advantages to using this approach. First, it allows us to evaluate

equilibrium output for a wide class of models, including those for which a closed-form

characterization of equilibrium is not available. Second, it provides a representation for

aggregate output in the form which is naturally amenable to solving the optimal sorting

problem. Indeed, the second-order moment expansion contains a term proportional to the

sample variance in abilities across groups, which the manager can manipulate by making

groups more or less balanced in terms of their aggregate ability.

An obvious drawback of the quadratic approximation approach is that the results it

provides are imprecise, and their accuracy relies on the weak heterogeneity assumption.

However, our numerical examples show that the quadratic approximation works remark-

ably well, as compared to the precise numerical solutions, for a range of heterogeneity up

to at least 50% variation in abilities across employees. In practice, the weak heterogeneity

assumption means that employees’ abilities are not too different from the firm’s average.

This assumption is justified in most situations because of the natural labor market sorting

of employees that takes place between firms. Any given firm simply would not hire an

employee whose ability is far below the firm’s average, and any given employee would be

unlikely to join a firm if her ability is far above the firm’s average. For example, a top-

notch law firm in New York City would only hire the best law school graduates from top

universities, while a law firm in a small town would hire graduates from a state or local

law school. A similar stratification of talent across universities takes place in academia.

Our results show that understanding the theory of group contests is perhaps a deeper

and more complex problem than it was previously believed to be. This paper represents a

framework upon which both further theoretical studies and new empirical studies of group

contests can be undertaken. An astute manager may suspect that certain behavioral

phenomena interact with the purely monetary incentives of workers to compete in a

group contest. The predictions of this paper can be used as a benchmark against which

to compare data from actual group contests, in order to better understand how a manager

can optimally assign her workers into teams.

References

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17

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A Proof of Proposition 1

The result of Proposition 1 has already been established for ρ = 1 (Ryvkin, 2011); there-

fore, for the remainder of the proof we assume that ρ < 1. First, we show that all players

in all groups are active in equilibrium, i.e., equilibrium efforts are positive. Clearly, all

players in all groups choosing zero effort is not an equilibrium because any player ij can

guarantee herself a payoff arbitrarily close to one by deviating to an infinitesimally small

positive effort. Thus, there is at least one group k in which at least one player is active.

19

Suppose now that in some other group i = k all players choose zero effort and thereby

guarantee themselves zero payoff. The payoff of player ij from a deviation to eij > 0 is,

in that case,eij

eij +∑

k =iEk

− cijg(eij),

which has the derivative with respect to eij at eij = 0 equal to

1∑k =i Ek

− cijg′(0) ≥ 1

(n− 1)m1/ρemax

− cmaxg′(0) > 0.

Here, the first inequality follows by replacing the effort levels of all players in groups

k = i by the upper bound emax and replacing cij with the upper bound cmax. The second

inequality follows from Assumption 1(iv). Thus, player ij’s marginal payoff at zero effort

is positive, hence she has an incentive to deviate to an infinitesimally small positive effort.

We conclude that there must be at least one active player in each group in equilibrium.

Suppose player il is active in group i and consider some other player ij = il. The left-hand

side of inequality (2) is infinite at eij = 0, i.e., player ij’s best response has to be positive

and thus all players’ equilibrium efforts are positive.

Given that all players are active, the system of first-order conditions (2) holds with

equality in equilibrium. The system then implies that for any two players ij and il in

group i we have cijg′(eij)e

1−ρij = cilg

′(eil)e1−ρil , i.e., each player’s effort can be written as

a function of the effort of player i1. Defining function h(e) = g′(e)e1−ρ and the ratio of

cost parameters rij = ci1/cij, obtain in equilibrium,

eij = µ(ei1, rij) ≡ h−1 (rijh(ei1)) . (10)

Note that under our assumptions function h is strictly increasing; therefore, the inverse of

h exists and is also strictly increasing, hence µ is strictly increasing in ei1 and µ(0, rij) = 0.

The aggregate output of group i can, therefore, be written as a function of the effort

of player i1:

Ei =

(eρi1 +

∑j>1

µ(ei1, rij)ρ

)1/ρ

.

Clearly, Ei is strictly increasing in ei1, and a strictly increasing inverse function can be

defined, ei1 = ν(Ei, ci), with ν(0, ci) = 0. Here, ci = (ci1, . . . , cim) is the vector of cost

parameters of players in group i.

The system of nm equations (2) then can be reduced to a system of n equations for

20

the unknowns Ei, i = 1, . . . , n:

∑k =iEk

(∑

k Ek)2 = ci1g

′(ν(Ei, ci))

(ν(Ei, ci)

ρ

ν(Ei, ci)ρ +∑

j>1 µ(ν(Ei, ci), rij)ρ

)(1−ρ)/ρ

. (11)

This system of first-order conditions is equivalent to the one arising in a lottery contest of

n individuals choosing effort levels Ei with heterogeneous marginal cost functions given

by the right-hand side of (11). The existence and uniqueness of equilibrium in such con-

tests has been established by Szidarovszky and Okuguchi (1997) and Cornes and Hartley

(2005) for the case when the cost functions are convex. Thus, in order to complete the

proof we need to show that the right-hand side of Eq. (11) is increasing in Ei.

Since νEi> 0, we only need to consider the derivative of the right-hand side of Eq. (11)

with respect to ν. Let µij ≡ µ(ν(Ei, ci), rij) and µ′ij ≡ µν(ν(Ei, ci), rij). Omitting the

factor ci1 in the front and the arguments of ν for brevity, obtain the derivative in the form

g′′(ν)

(νρ

νρ +∑

j>1 µρij

) 1−ρρ

+ g′(ν)1− ρ

ρ

(νρ

νρ +∑

j>1 µρij

) 1−2ρρ

1(νρ +

∑j>1 µ

ρij

)2×

[ρνρ−1

(νρ +

∑j>1

µρij

)− νρ

(ρνρ−1 + ρ

∑j>1

µρ−1ij µ′

ij

)]

=

(νρ

νρ +∑

j>1 µ(ν, rij)ρ

) 1−2ρρ

1(νρ +

∑j>1 µ

ρij

)2×

[g′′(ν)νρ

(νρ +

∑j>1

µρij

)+ (1− ρ)g′(ν)νρ−1

∑j>1

µρ−1ij (µij − νµ′

ij)

].

The sign of this expression is determined by the sign of the term in the square brackets,

which can be simplified as

g′′(ν)ν2ρ + νρ−1∑j>1

µρ−1ij [g′′(ν)νµij + (1− ρ)g′(ν)(µij − νµ′

ij)].

Thus, it is sufficient to show that for any player ij, with j > 1, we have

g′′(ν)νµij + (1− ρ)g′(ν)(µij − νµ′ij) ≥ 0. (12)

21

Recall that h(µij) = rijh(ν), which gives

µ′ij =

rijh′(ν)

h′(µij)=

h(µij)h′(ν)

h(ν)h′(µij).

The left-hand side of (12) then becomes

g′′(ν)νµij + (1− ρ)g′(ν)

(µij −

νh(µij)h′(ν)

h(ν)h′(µij)

)= µijg

′(ν)

[g′′(ν)ν

g′(ν)+ (1− ρ)

(1− νh′(ν)/h(ν)

µijh′(µij)/h(µij)

)]. (13)

From the definition h(ν) = g′(ν)ν1−ρ, obtain

νh′(ν)

h(ν)=

ν[g′′(ν)ν1−ρ + (1− ρ)g′(ν)ν−ρ]

g′(ν)ν1−ρ= 1− ρ+

νg′′(ν)

g′(ν).

The term in the square brackets in Eq. (13) then becomes

g′′(ν)ν

g′(ν)+ (1− ρ)

(1− 1− ρ+ νg′′(ν)/g′(ν)

1− ρ+ µijg′′(µij)/g′(µij)

)≥ g′′(ν)ν

g′(ν)+ (1− ρ)

(1− 1− ρ+ νg′′(ν)/g′(ν)

1− ρ

)= 0.

Thus, the effective marginal cost functions in the system of Eqs. (11) are increasing

in Ei, and hence the cost functions are convex, which implies that the equilibrium exists

and is unique.

22

B Proof of Proposition 2

We start off by obtaining the second-order Taylor expansion for aggregate equilibrium

output E. Plugging the representation e∗ij = e(1 + xij) into Eq. (3),

E =∑i

(∑j

(e(1 + xij))ρ

)1/ρ

= e∑i

(∑j

(1 + xij)ρ

)1/ρ

= e∑i

[∑j

(1 + ρxij +

ρ(ρ− 1)

2x2ij

)]1/ρ+O(S3

a)

= e∑i

[m+ ρ

∑j

xij +ρ(ρ− 1)

2

∑j

x2ij

]1/ρ+O(S3

a)

= em1/ρ∑i

[1 +

1

ρ

(ρ

m

∑j

xij +ρ(ρ− 1)

2m

∑j

x2ij

)

+1− ρ

2ρ2

(ρ

m

∑j

xij +ρ(ρ− 1)

2m

∑j

x2ij

)2+O(S3

a)

= em1/ρn

1 + 1

nm

∑i

∑j

xij −1− ρ

2nm

∑i

∑j

x2ij +

1− ρ

2nm2

∑i

(∑j

xij

)2+O(S3

a).

Recall that in the symmetric equilibrium with homogeneous players E = em1/ρn. It

is convenient to introduce the notation Xi =∑

j xij and X =∑

i

∑j xij. Equilibrium

output E then becomes

E = E

[1 +

1

nmX − 1− ρ

2nm

∑i

∑j

x2ij +

1− ρ

2nm2

∑i

X2i

]+O(S3

a).

Let xij = x(1)ij + x

(2)ij +O(S3

a), where x(q)ij denotes the qth order correction in the approxi-

mation of equilibrium effort. Similar notation will be used for aggregate effort corrections

X and Xi. In the quadratic approximation, output E is

E = E

[1 +

1

nm(X(1) +X(2))− 1− ρ

2nm

∑i

∑j

(x(1)ij )

2 +1− ρ

2nm2

∑i

(X(1)i )2

]+O(S3

a). (14)

Note that we dropped the second-order corrections in xij and Xi because when squared

they will turn into fourth-order terms. As seen from Eq. (14), in order to calculate

the aggregate output in the quadratic approximation we only need to find the first-order

23

individual and group effort corrections x(1)ij and X

(1)i and the second-order aggregate effort

correction X(2).

The rest of the proof is based on the second-order Taylor expansion of the first-

order conditions (2) (holding with equality) in small variables aij and xij. Plugging the

representations cij = c(1− aij) and eij = e(1 + xij) into (2), obtain

[∑

l(1 + xil)ρ](1−ρ)/ρ(1 + xij)

ρ−1∑

k =i[∑

l(1 + xkl)ρ]1/ρ

[∑

k[∑

l(1 + xkl)ρ]1/ρ]2= cem(1− aij)g

′(e(1+xij)). (15)

The numerator, N , of the fraction in the left-hand side of (15) has the following

second-order expansion:

N = n− 1 +1

m

∑k =i

∑l

xkl −1− ρ

2m

∑k =i

∑l

x2kl +

1− ρ

2m2

∑k =i

(∑l

xkl

)2

− (n− 1)(1− ρ)xij −1− ρ

mxij

∑k =i

∑l

xkl +(n− 1)(1− ρ)(2− ρ)

2x2ij

+(n− 1)(1− ρ)

m

∑l

xil +1− ρ

m2

(∑l

xil

)∑k =i

∑l

xkl −(n− 1)(1− ρ)2

mxij

∑l

xil

− (n− 1)(1− ρ)2

2m

∑l

x2il +

(n− 1)(1− ρ)(1− 2ρ)

2m2

(∑l

xil

)2

+O(S3a).

The denominator, D, of the fraction in the left-hand side of (15) has the following

second-order expansion:

D = n2 +1

m2

(∑k

∑l

xkl

)2

+2n

m

∑k

∑l

xkl −n(1− ρ)

m

∑k

∑l

x2kl

+n(1− ρ)

m

∑k

(∑l

xkl

)2

+O(S3a).

The right-hand side, R, of (15) has the following second-order expansion:

R = cem

[g′ + g′′exij +

g′′′e2

2x2ij − g′aij − g′′eaijxij

]+O(S3

a).

Here, and in what follows, the derivatives of g are evaluated at the symmetric equilibrium

effort e.

24

Multiplying the denominator and the right-hand side, obtain

DR = cem

n2g′ + n2g′′exij +n2g′′′e2

2x2ij − n2g′aij − n2g′′eaijxij +

g′

m2

(∑k

∑l

xkl

)2

+2ng′

m

∑k

∑l

xkl +2ng′′e

mxij

∑k

∑l

xkl −2ng′

maij∑k

∑l

xkl −n(1− ρ)g′

m

∑k

∑l

x2kl

+n(1− ρ)g′

m

∑k

(∑l

xkl

)2+O(S3

a).

From Eqs. (4) and (5), obtain

cemn2g′ = n− 1, ce2mn2g′′ = (n− 1)k2, ce3mn2g′′′ = (n− 1)k2k3.

Equation N = DR then becomes

n− 1 +1

m

∑k =i

∑l

xkl −1− ρ

2m

∑k =i

∑l

x2kl +

1− ρ

2m2

∑k =i

(∑l

xkl

)2

− (n− 1)(1− ρ)xij −1− ρ

mxij

∑k =i

∑l

xkl +(n− 1)(1− ρ)(2− ρ)

2x2ij

+(n− 1)(1− ρ)

m

∑l

xil +1− ρ

m2

(∑l

xil

)∑k =i

∑l

xkl −(n− 1)(1− ρ)2

mxij

∑l

xil

− (n− 1)(1− ρ)2

2m

∑l

x2il +

(n− 1)(1− ρ)(1− 2ρ)

2m2

(∑l

xil

)2

= n− 1 + (n− 1)k2xij +(n− 1)k2k3

2x2ij − (n− 1)aij − (n− 1)k2aijxij

+n− 1

n2m2

(∑k

∑l

xkl

)2

+2(n− 1)

nm

∑k

∑l

xkl +2(n− 1)k2

nmxij

∑k

∑l

xkl

− 2(n− 1)

nmaij∑k

∑l

xkl −(n− 1)(1− ρ)

nm

∑k

∑l

x2kl

+(n− 1)(1− ρ)

nm

∑k

(∑l

xkl

)2

+O(S3a). (16)

In the linear approximation, keeping only the first-order terms on both sides of (16),

25

obtain

1

m

∑k =i

∑l

x(1)kl − (n− 1)(1− ρ)x

(1)ij −+

(n− 1)(1− ρ)

m

∑l

x(1)il (17)

= (n− 1)k2x(1)ij − (n− 1)aij +

2(n− 1)

nm

∑k

∑l

x(1)kl .

Using the the notation X(1)i =

∑j x

(1)ij and X(1) =

∑i X

(1)i , Eq. (17) becomes

(n− 1)k2x(1)ij − (n− 1)aij +

2(n− 1)

nmX(1) (18)

=1

m

(X(1) −X

(1)i

)− (n− 1)(1− ρ)x

(1)ij +

(n− 1)(1− ρ)

mX

(1)i .

Summing up this equation over all players j = 1, . . . ,m in group i, obtain

(n− 1)k2X(1)i − (n− 1)Ai +

2(n− 1)

nX(1) = X(1) −X

(1)i . (19)

Summing up (19) over all groups i = 1, . . . , n, obtain

(n− 1)k2X(1) − (n− 1)

∑i

Ai + 2(n− 1)X(1) = (n− 1)X(1),

which gives X(1) = (1 + k2)−1∑

i Ai = 0 because∑

i Ai = 0 by construction. Thus, the

linear correction to aggregate effort is equal to zero. This result is standard and is due

to the symmetric structure of the contest and the mean-preserving parameterization of

heterogeneity.

Using the result X(1) = 0, obtain from Eq. (19)

X(1)i =

(n− 1)Ai

(n− 1)k2 + 1. (20)

Plugging this back into Eq. (18), obtain

x(1)ij = αaij + βAi; α =

1

k2 + 1− ρ, β =

(n− 1)[(n− 1)(1− ρ)− 1]

m(k2 + 1− ρ)[(n− 1)k2 + 1]. (21)

Thus, the first-order correction to individual equilibrium effort is a linear combination of

the player’s own ability and the aggregate ability of her group. For ρ = 1, i.e., in the case

of perfectly substitutable effort, we have β < 0, indicating free-riding. As ρ decreases, β

becomes positive and free-riding is replaced by within-group coordination.

26

Next, we find X(2). Equating the second-order terms in (16) and summing up over i

and j, obtain

(n− 1)X(2) − (1− ρ)(n− 1)

2

∑k,l

(x(1)kl )

2 +(1− ρ)(n− 1)

2m

∑k

(X(1)k )2

− (n− 1)(1− ρ)X(2) − 1− ρ

m(X(1))2 +

1− ρ

m

∑k

(X(1)k )2

+(n− 1)(1− ρ)(2− ρ)

2

∑k,l

(x(1)kl )

2 + (n− 1)(1− ρ))X(2)

+1− ρ

m(X(1))2 − 1− ρ

m

∑k

(X(1)k )2 − (1− ρ)2(n− 1)

m

∑k

(X(1)k )2

− (1− ρ)2(n− 1)

m

∑k,l

(x(1)kl )

2 +(n− 1)(1− ρ)(1− 2ρ)

2m

∑k

(X(1)k )2

= (n− 1)k2X(2) +

(n− 1)k2k32

∑k,l

(x(1)kl )

2 − (n− 1)k2∑k,l

aklx(1)kl

+n− 1

nm(X(1))2 + 2(n− 1)X(2) +

2(n− 1)k2nm

(X(1))2 − 2(n− 1)

nmX(1)

∑i

Ai

− (n− 1)(1− ρ)∑k,l

(x(1)kl )

2 +(n− 1)(1− ρ)

m

∑k

(X(1)k )2.

Using that X(1) = 0 and solving the equation above for X(2), obtain

X(2) =1

1 + k2

[(1− ρ− k2k3

2

)∑k,l

(x(1)kl )

2 − 1− ρ

m

∑k

(X(1)k )2 + k2

∑k,l

aklx(1)kl

]. (22)

Plugging this into (14), obtain the expression for aggregate output,

E = E

[1 +

1

nm

((1− ρ− k2k3

2

1 + k2− 1− ρ

2

)∑k,l

(x(1)kl )

2

+

(1− ρ

2m− 1− ρ

m(1 + k2)

)∑k

(X(1)k )2 +

k21 + k2

∑k,l

aklx(1)kl

)]+O(S3

a).

Introducing the second-order correction to output in the form E = E(1 + Y (2)) +O(S3a),

27

obtain

Y (2) =1

nm

[(1− ρ)(1− k2)− k2k3

2(1 + k2)

∑k,l

(x(1)kl )

2 − (1− ρ)(1− k2)

2m(1 + k2)

∑k

(X(1)k )2

+k2

1 + k2

∑k,l

aklx(1)kl

]. (23)

Recall that x(1)kl = αakl+βAk, with coefficients α and β given by Eq. (21). This gives,∑

k,l

(x(1)kl )

2 = α2∑k,l

a2kl + (2αβ + β2m)∑k

A2k = α2mnS2

a + (2αβ + β2m)nS2A,

∑k,l

aklx(1)kl = α2

∑k,l

a2kl + β∑k

A2k = αmnS2

a + βnS2A.

Similarly, Eq. (20) gives

∑k

(X(1)k )2 =

(n− 1)2

[(n− 1)k2 + 1)]2

∑k

A2k =

(n− 1)2nS2A

[(n− 1)k2 + 1)]2.

Combining these expressions and introducing the notation ξ = 1− ρ, obtain

Y (2) =ξ(1− k2)− k2k3

2(1 + k2)

(α2S2

a +

(2αβ

m+ β2

)S2A

)− ξ(1− k2)(n− 1)2

2m2(1 + k2)[(n− 1)k2 + 1)]2S2A

+k2

1 + k2

(αS2

a +β

mS2A

)=

α[2k2 + α(ξ(1− k2)− k2k3)]

2(1 + k2)S2a +

Λξ

2m2(1 + k2)[(n− 1)k2 + 1)]2S2A, (24)

where

Λξ = m2[(n− 1)k2 + 1)]2[(ξ(1− k2)− k2k3)

(2αβ

m+ β2

)+

2k2β

m

]− ξ(1− k2)(n− 1)2.

Introducing κ = α(ξ(1− k2)− k2k3), immediately obtain the coefficient λa on S2a in (24)

as in (6). For Λξ, simplify further:

Λξ = m[(n− 1)k2 + 1)]2β

(2κ+

κmβ

α+ 2k2

)− ξ(1− k2)(n− 1)2.

28

Using the definitions of α and β in (21), obtain

Λξ =2(k2 + κ)((n− 1)k2 + 1) + κ((n− 1)ξ − 1)

k2 + ξ[(n− 1)ξ − 1]− ξ(1− k2)(n− 1)2,

which gives the coefficient λA on S2A in (6).

C Proof of Lemma 1

Transform Λξ as follows:

Λξ =1

k2 + ξ

[(2(k2 + κ)((n− 1)k2 + 1) + κγ)γ + (k2 − 1)(n− 1)2ξ(k2 + ξ)

]=

1

k2 + ξ

[(2

(k2 +

ξ(1− k2)− k2k3k2 + ξ

)((n− 1)k2 + 1)

+ξ(1− k2)− k2k3

k2 + ξ((n− 1)ξ − 1)

)((n− 1)ξ − 1) + (k2 − 1)(n− 1)2ξ(k2 + ξ)

]=

1

(k2 + ξ)2[2(k2

2 + ξ − k2k3)((n− 1)k2 + 1)((n− 1)ξ − 1)

+(ξ(1− k2)− k2k3)((n− 1)ξ − 1)2 + (k2 − 1)(n− 1)2ξ(k2 + ξ)2].

At first glance, the expression in square brackets is a cubic polynomial in ξ. However, the

coefficient on ξ3 in that polynomial is

(1− k2)(n− 1)2 + (k2 − 1)(n− 1)2 = 0,

i.e., it is, in fact, a quadratic polynomial in ξ. Given that the denominator is also quadratic

in ξ, the limit limξ→∞ Λξ is finite and determined by the coefficient on ξ2 in the numerator.

That coefficient is

Λ∞ = 2((n− 1)k2 + 1)(n− 1)− 2(1− k2)(n− 1)− k2k3(n− 1)2 + 2(k2 − 1)(n− 1)2k2

= (n− 1)[2(n− 1)k2 + 2− 2 + 2k2 − k2k3(n− 1) + 2(k2 − 1)k2(n− 1)]

= (n− 1)k2[(n− 1)(2k2 − k3) + 2].

D Proof of Proposition 3

As shown in the proof of Lemma 1 above, the sign of Λξ is determined by the sign of a

quadratic polynomial in ξ that may have zero, one or two roots on [0,∞). Note that if

29

Λ∞ < 0, it necessarily implies that Λ0 > 0. Thus, it is impossible to have both Λ∞ < 0

and Λ0 < 0. The other three cases (both Λ∞ and Λ0 are positive, Λ∞ > 0 and Λ0 < 0, or

Λ∞ < 0 and Λ0 > 0) are possible. The case when both Λ∞ and Λ0 are positive has two

sub-cases: case (b) in the proposition, in which Λξ is positive for all ξ, and case (c) in

the proposition, in which Λξ is negative in some range [ξ1, ξ2]. The cases when Λ∞ > 0,

Λ0 < 0 and Λ∞ < 0, Λ0 > 0 correspond to cases (a) and (d) in the proposition, in both

of which there is exactly one point ξ where Λξ crosses zero.

30