GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 313

Multilayered Tournaments

Junichiro Ishida

August 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Multilayered Tournaments
Junichiro Ishida∗
August 6, 2013

Abstract
In many facets of life, we often face competition with a multilayered structure in which
different levels of competition take place simultaneously. In this paper, we propose a newclass of tournament models, called multilayered tournaments, to capture this type ofcompetitive environment. Among other things, we find that: (i) an increase in individual
incentives, holding the level of team incentives fixed, can lower total effort as it induces
inefficient allocation of effort; (ii) the optimal level of individual incentives depends on andis complementary to the level of team incentives. The analysis illuminates the essential
role of economic subgroups, such as firms, in achieving some degree of cooperation in an
inherently competitive environment, and provides an explanation for why high-powered
incentives are more common in market arrangements than within firms.

JEL Classification Number: D82; D86.
Keywords: Tournaments, Cooperation, Competition, Multi-task problem, Sabotage.
∗

Institute of Social and Economic Research, Osaka University. Email: jishida@iser.osaka-u.ac.jp

1

Introduction

In many facets of life, we often face competition with a multilayered structure in which withingroup internal competition and across-group external competition take place simultaneously.
A leading example of this situation is the competition among workers, where they compete
against each other within their respective firms for promotions while those firms in turncompete against each other for market shares. Similarly, politicians compete against eachother for candidacy within their political parties which in turn compete against each otherin elections. Professional athletes compete with teammates within their teams which inturn compete with each other in their respective leagues. Academic researchers compete
against each other for publication slots or research funds within their disciplines which inturn compete against each other for students or public recognition. Multilayered competition
is fairly common and ubiquitous in society: after all, human beings have engaged in various
types of competition within their respective communities, such as tribes, villages, states andcountries, for the pursuit of internal resources, while those communities have simultaneouslycompeted against each other for the pursuit of external resources.
Many of the instances raised above have a common simple structure which we characterize
by the following three features. First, the nature of competition is relative, at least to someextent, where more productive ones tend to capture more of the available resources.1 Second,even though intense internal competition undoubtedly underlies the strength of a team, teamproduction occupies an essential part of the production process in each of these cases, so that
it is essential to foster cooperation among teammates and let them help each other whennecessary. Finally and most importantly, agents compete simultaneously at two distinct
levels that are not clearly distinguishable in the sense that they cannot direct their resources(such as time and effort) exclusively to a particular level of competition.2 This last featureis the defining feature of multilayered competition which sets the current analysis apart
1

The nature of competition naturally becomes relative because the total amount of rewards is typicallybounded from above, in which case one’s gain necessarily becomes someone else’s loss at some point. It is wellrecognized that many forms of internal competition, such as promotion tournaments, have a relative aspect
because total prize, either pecuniary or nonpecuniary, is generally bounded from above. Likewise, many formsof external competition have this feature as well: for instance, the market share of a firm often depends on thefirm’s relative product quality level rather than the absolute level. A similar argument also applies to politicalcompetition in which the number of winners is exogenously fixed, both internally and externally.
2
For some of the examples raised above, the two levels of competition literally takes place simultaneously,
as in the case of the competition among workers, so that it is not feasible to make a clear distinction between
them: that is, it is in principle not feasible to allocate resources to one level of competition without affecting
the relative standing at the other level. In contrast, in political campaigns, the competition itself can besequential where the internal competition (primary) precedes the external competition. Even in this case,however, if what politicians need to do is to make investment in necessary skills and/or to frame basic policies,
it is hard to distinguish the two levels of competition.

1

from dynamic elimination tournaments (Rosen, 1990) where each round of competition is
independent from others with no strategic interactions.
In this paper, we propose a new class of tournament models which intend to capture these
aforementioned features of multilayered competition. We consider an environment in which
homogenous agents are divided into symmetric teams and compete with each other bothwithin and across teams simultaneously. To capture the possibility of within-group cooperation, the option of helping others a la Itoh (1991) is incorporated into this framework, so thateach agent must allocate his resources between improving his own productivity (own effort)
and helping teammates (helping effort). Although the analysis of multilayered competition
can easily get intractable due to its complicated hierarchical interactions between different
layers of competition, we devise a framework that is tractable enough to conduct various
comparative statics exercises. Viewing it as a model of markets and internal organizations,the present framework then illuminates the elusive link between these two primary allocation
mechanisms and provides answers to a set of questions concerning the relationship between
them, such as how incentive provision within firms should depend on the extent of market
competition, under what circumstances pro-competitive market policies are more desirable,
why high-powered incentives are more commonly found in markets than stronger than withinfirms, and above all, why firm organizations are so ubiquitous in the first place.
We obtain several results and implications. First, the analysis sheds some light on theessential role of economic subgroups, such as firms, in achieving the right balance between
competition and cooperation. It is in general impossible to induce cooperation which involves
all agents when they cannot be rewarded jointly. Even in this situation, however, it is still
possible to achieve some degree of cooperation by dividing agents into smaller subgroups andhaving them pit against one another. Our analysis shows that the possibility of cooperation is what makes multilayered competition both necessary and meaningful. A multilayered
tournament virtually has no bite when there is no possibility of helping others, as any feasible allocation can be achieved by a conventional single-layered tournament in this case.
Conversely, when the underlying production process calls for some degree of cooperation, amultilayered tournament provides a fairly effective way to induce helping effort even in an
inherently competitive environment where there are necessarily winners and losers.
Second, we show that an increase in individual incentives, holding the extent of teamincentives fixed, could induce inefficient substitution between the two types of effort andconsequently result in lower total effort. This result points to yet another route thoughwhich the adverse effect of individual incentives arises. While there is little doubt that
intense internal competition better motivates agents at the individual level, it also necessarily2impedes cooperation among them. We identify conditions under which an increase in own
effort induced by more intense internal competition is more than offset by a decrease inhelping effort, thereby leading to a reduction in total effort. We also relate this finding to
two widely cited results in the literature – sabotage in a tournament (Lazear, 1989) and themulti-task problem (Holmstrom and Milgrom, 1991) – and suggest an underlying connection
between them.
Third, the analysis illuminates the general superiority of team incentives, despite the factthat their effectiveness is generally limited by the notorious freerider problem. We show that
although the optimal level of individual incentives may be zero under some conditions, theoptimal level of team incentives is always bounded away from zero. The key observation
here is that individual incentives can induce own effort only at the expense of a decrease inhelping effort. In contrast, team incentives generally yield positive effects in both directions
as they are more inclusive and work at a higher level. We argue that this observation accords
well with Williamson’s view (1985) that high-powered incentives are more common in market
arrangements than within firms.
Forth, we provide a new perspective on how individual and team incentives should be
related to each other. We show that the optimal incentive scheme at one level of competition
cannot be determined independently of the extent of incentives provided for the other level.
We in particular show that the relationship between individual and team incentives is complementary, where a decrease in one should be accompanied by a decrease in the other. As
practical implications, the result suggests that: stronger individual incentives tend to work
well in environments with more intense market competition; pro-competitive policies are more
likely to be justified in environments where stronger individual incentives are provided withinfirms.
Finally, we also conduct various comparative statics exercises on team size. A larger
team obviously exacerbates the freerider problem, but could raise the incentive to help others
because it may be easier or less costly to help other teammates when more of them arearound. This is especially the case when helping effort takes the form of sharing information
with teammates (or contributing to a public good). In the presence of such a scale effect, if
the social planner can dictate the effort levels of each agent, it is always desirable to have aslarge a team as possible so as to fully exploit the scale effect, i.e., a grand coalition is always
socially optimal. This is not incentive compatible, however, as there is clearly no incentive
to help others when all of the agents are grouped into one team. The optimal team size must
balance these concerns, and we derive conditions under which a larger team size is warranted.
Related literature: The paper largely falls into a class of tournament models and extends
3this setting by incorporating team competition. While most tournament models, startingfrom Lazear and Rosen (1981), focus on competition among individuals, there is a handful ofworks which examine competition among teams (Drago, et al., 1996; Marino and Zabojnik,
2004; Ishida 2006; Gurtler, 2008).3 The major difference from these works is that they only
consider competition among teams with no regard to competition among individuals, where
the sharing rule within a team is exogenously given.4 The scope of the current analysis istotally different since our primary focus is on the interaction between the two distinct levelsof competition connected through two-dimensional effort choices.
One essential aspect of the model is the presence of team production in which each agent’s
contribution to total output cannot be measured accurately (Alchian and Demsetz, 1972;
Holmstrom, 1981). This naturally gives rise to the freerider problem, which substantiallydiminishes the effectiveness of team incentives. Although the freerider problem necessarily
entails efficiency loss, team production also has a bright side as it can encourage teammates
to cooperate with each other for the betterment of the team. Our setup with helping effort
is heavily indebted to Itoh (1991) who introduces the option of helping others to a standard
moral hazard problem.5 This paper follows this approach to capture a situation in which
mutual cooperation is at the core of the production process, although its extent is confinedwithin the boundary of a team. We then take a step further by integrating internal and external competition in order to illuminate the connection between them and derive implications
from it.
There are now numerous works which analyze incentive provision within firms in the filedof organizational economics; there are also many works on various market regulations andcompetition policies in the field of industrial organization. With few exceptions, however,these two strands of literature have developed rather independently from each other, partly
due to the technical difficulty of placing organizational and market interactions into a unified
3

For experimental studies on team tournaments, see Nalbantian and Schotter (1997) and Sutter and Strassmair (2009).
4
Technically, since our model builds on Tullock (1980), there is also some connection with works on rentseeking contests. The current analysis is especially related to works on “collective contests” (Nitzan, 1991;
Warneryd, 1998; Inderst et al., 2007; Munster, 2007; Nitzan and Ueda, 2011) although they differ in scopeand motivation to a great extent: the main focus of this strand of literature is on the equilibrium rate ofrent dissipation, i.e., the ratio of total contest expenditures to the total prize size. Along this line, Munster(2007) considers a situation where agents must make multi-dimensional effort choices as in ours. Aside fromthe differences in scope and motivation, the major difference is that in Munster, individual and collective
appropriative activities are totaly separated from each other, so that each agent can specifically direct his
resources to a particular level of competition.
5
Che and Yoo (2001) consider a relational contracting environment and show that there exists a cooperative
subgame perfect equilibrium which induces effort at a lower cost. In these works, absolute performance
evaluation is feasible for contracting. Ishida (2006) extends this idea and shows that this same mechanism
also works even when only relative performance measures are feasible for contracting.

4

framework. Still, some attempts have been made to integrate these two strands. One approach in this direction examines the effect of market competition on managerial behaviors
and incentives (Hart, 1983; Hermalin, 1992; Schmidt, 1997; and Raith, 2003). Yet anotherapproach looks at the effect of incentive provision on market outcomes through oligopolistic interactions (Fershtman and Judd, 1987). Also, see Morita (2012) for a recent attempt
to integrate human resource management and market competition. The paper intends tocontribute to this line of inquiry from a different perspective with particular focus on themultilayered nature of competition and to provide a framework which captures the elusive
link between markets and internal organizations in an intuitive and tractable manner.

2

Model

Environment: We consider a set I of N ≥ 4 homogeneous agents who are divided into msymmetric teams competing for prizes. We assume that N is some even (and relatively large)
integer. Throughout the analysis, the number of agents within each team is denoted by n ≥ 2
where N = mn. Note that the case with n = 1 represents the situation where each agent
forms his own team, so that the nature of competition is effectively single-layered with nowithin-group competition.6 For comparative statics exercises, we mainly focus on the range
N := [2, N2 ].
The structure of competition is multilayered in the sense that agents compete with eachother both at the individual level (the internal competition) and the aggregate level (theexternal competition) simultaneously. Throughout the analysis, we index agents by i =1, 2, ..., n and teams by j = 1, 2, ..., m, and use the following notations:• Ij : the set of agents in team j;• I−ij : the set of agents in team j other than agent i (or agent i’s “teammates”);• T : the set of all teams;• T−j : the set of teams other than team j.
Note that each agent is in principle identified by a pair (i, j); in what follows, however, weabbreviate the team index and refer to each agent as agent i for brevity.
6

Note also that the case with n = N where there exists only one team (a grand coalition) which involves
all the agents is strategically equivalent to the case with n = 1 if the reward structure is set optimally. As
we will see, however, the case with n = N is generally dominated by the case with n = 1 when the reward
structure is given arbitrarily.

5The current setting admits a number of possible interpretations, as mentioned in the introduction. Among those possibilities, we primarily view this as a model of markets and internalorganizations although no market competition which we observe in reality may correspondphysically to the external competition as specified here. Nonetheless, we here emphasize thefact that the current specification captures a situation where a team (or a firm) gains morewhen it does better than other competitors in a broad sense, which should be true in almost
any form of market competition. The external competition in the current setting should thus
be regarded as a “reduced-form” of market competition which can actually assume manydifferent forms at a more superficial level.
Effort choice: Each agent independently and simultaneously chooses his effort pair eij =(aij , bij ) ∈ R2+ , where aij is the level of own effort and bij is the effort level of helping effort.
The effort cost is given by
cij = C(aij , bij ).
A key departure from the previous literature is the addition of helping effort into a tournament framework. The necessity of helping each other arises largely from the fact that agentsare often heterogeneous in skills or sets of information they have: in that case, they can
improve the productive efficiency, e.g., by reallocating tasks to reap the benefit of skill complementarities and/or sharing critical knowledge that may be dispersedly distributed within
the team. Throughout the analysis, we simply assume that helping effort is allocated evenly
among the teammates and benefits each teammate equally (or alternatively focus on a classof cost functions under which
it is optimal to allocate helping effort evenly among the teamPmates).7 Define hij :=i′ ∈I−ij bi′ jn−1

as the amount of helping effort agent i receives from his

teammates. Agent i’s total effort, denoted as xij , is then given by
xij = F (aij , hij ),We assume that these functions satisfy the usual properties.
Assumption 1 (i) The cost function C is strictly increasing and weakly convex in each
respective argument. (ii) The production function F is strictly increasing and weakly concavein each respective argument.
For the analysis, it is often convenient to defineG(a, b) :=F1 (a, b)
F2 (a, b), H(a, b) :=.
C1 (a, b)
C2 (a, b)7
It is in principle possible to extend helping effort to agents in other teams. We rule out this possibility at
the outset, however, as there is no incentive to do so under any monotonic incentive scheme.

6Note that under the maintained assumption, G and H are strictly increasing in a and b,
respectively. Moreover, we impose the following assumptions .
Assumption 2 lima→0 G = ∞ and limb→0 H = ∞.
Assumption 3 G2 > 0 and H1 > 0.
Assumption 2 assures the existence of interior solutions. Assumption 3 indicates that thetwo types of effort are complementary to each other. We make this assumption to focus ourattention to the case where it is optimal to allocate resources between the two activities, rather
than to devote all resources to one activity. One can easily see that a sufficient condition forthem is that F12 ≥ 0 and C12 ≤ 0 with at least one holding with strict inequality.
Contest success function: Define rij as the probability that agent i wins the internal
competition, and qj as the probability that team j wins the external competition. We adopt
the (generalized) Tullock success function to determine the winner at each respective level ofP
competition. Letting zj := i∈Ij xij denote the total effort for team j, the contest successfunctions are specified as

rij = Pxαij

Pα , qj =i′ ∈Ij xi′ jzjα

α.
j ′ ∈T zj ′The Tullock specification is known to have some strategic foundations (Fullerton and McAfee,1999; Baye and Hoppe, 2003) as well as axiomatic justifications (Skaperdas, 1996; Clark andRiis, 1998). Important features of this specification are that: (i) taking all the others’ effort
choices as given, the winning probability is increasing in one’s effort level; (ii) taking one’s
effort choices as given, the winning probability is decreasing in the effort level of each of theother agents. We adopt this specification mostly for its greater tractability, among a plethoraof possibilities which can capture these features.
Prize allocation: We assume that the individual winner is identifiable only in the winningteam so that there are three possible contest outcomes: the winner in the winning team, thelosers in the winning team, and all the others in the losing teams.8 Given this information
structure, it suffices to consider a three-level reward schedule, denoted by (W, w, l), where
W is the reward for the winner in the winning team, w for the losers in the winning team,and l for all the agents in the losing teams. Define ∆I := W − w and ∆T := w − l. Since
8
In principle, we can also discriminate between the winner and the losers in a losing team. Here, we only
consider a three-level reward scheme to reduce the number of parameters and simplify the analysis. The
substance of our analysis is not affected even if we assume that the winner in a losing team earns more thanthe losers.

7

what matters for each agent is the reward dispersion, the incentive scheme can alternatively
written as (∆I , ∆T ): for expositional purposes, we refer to ∆I (∆T ) as the individual (team)∆I
denote the incentive ratio. The extent of the individual incentive
incentive. Let φ := ∆Tmeasures the intensity of the internal competition.
The total prize is fixed at R, which is entirely captured by the winning team and shared
among its members, possibly depending on the contest outcomes, so that R = W + (n − 1)w
and l = 0. We impose two assumptions on the set of feasible incentive schemes. First, there
is a liquidity constraint such that each reward must be nonnegative, i.e., W, w ≥ 0. Second,
we also require that an incentive scheme be monotonic in that W ≥ w ≥ l = 0.9 We use theprize size R as a measure of the intensity of the external competition.

3

Incentives in multilayered tournaments

3.1

Optimal effort choices

In this section, we characterize the optimal effort choices by taking the incentive scheme(∆I , ∆T ) and the team size n as given. Throughout the analysis, we seek for a symmetricequilibrium where all agents choose the same effort pair. When all the other agents choose(a, b), we can write
xij = F (aij , b), xi′ j = F (a,(n−2)b+bij), i′ ∈ I−ij .
n−1

The probability that agent i wins the internal competition conditional on his effort choice(aij , bij ) is then given by
rij = r(aij , bij ) =F (aij , b)αF (aij , b)α + (n − 1)F (a,(n−2)b+bij α
)
n−1

,

for n ≥ 2. Similarly, the probability that team j wins the external competition is given by
qj = q(aij , bij ) =(F (aij , b) + (n − 1)F (a,(F (aij , b) + (n − 1)F (a,(n−2)b+bij α))n−1(n−2)b+bij α)) + (m − 1)nα F (a, b)αn−1.

Taking (a, b) as given, the problem faced by each agent is formulated as follows:max

aij ,bij

qj rij ∆I + qj ∆T − C(aij , bij ).

9Needless to say, the optimal incentive scheme is indeed monotonic for a wide range of parameters. In thisframework, however, there may actually arise a case where it is optimal to inhibit the incentive to exert own
effort by offering a negative ∆I , if it feasible to do so. We rule out this possibility because a non-monotonic
incentive scheme may provide agents a detrimental incentive to “game the system,” e.g., by intentionallyhiding or destroying his own output. Moreover, a non-monotonic incentive scheme may not be optimal fromthe fairness point of view because it is highly demoralizing to punish the most productive agent.

8

The first-order conditions are obtained as∂rij∂qj(rij ∆I + ∆T ) + qj
∆ I = C1 ,
∂aij
∂aij
∂rij
∂qj
(rij ∆I + ∆T ) + qj
∆ I = C2 .
∂bij
∂bij
In a symmetric equilibrium, the first-order conditions are reduced to
αGPa = F (a, b),(1)

αHPb = F (a, b),

(2)

where
Pa :=

(N − 1)∆I + (N − n)∆T−∆I + (N − n)∆T
, Pb :=.
N2
N2

We assume that the second-order conditions are satisfied at the optimum: for instance, they
are satisfied if α is sufficiently small. See Appendix B for more detail on the first- andsecond-order conditions.
Let (a∗ (∆I , ∆T ), b∗ (∆I , ∆T )), or simply (a∗ , b∗ ), denote the solution to the pair of equations, (1) and (2), and x∗ = F (a∗ , b∗ ). The equilibrium effort pair (a∗ , b∗ ) is determinedlargely by Pa and Pb . It is important to note that an increase in the individual incentive ∆I
could diminish the incentive to help others as it unambiguously decreases Pb . The reason forthis is intuitive and fairly straightforward: more intense competition at the individual level
discourages agents to allocate resources to others, as there is more to lose by helping others.
This feature is the driving force of many results that follow.

3.2

On the role of multilayered tournaments

The current setup makes two key departures from standard tournament models that are
widely analyzed. First and foremost, competition is multilayered where the agents competewith each other simultaneously on the two different levels. Second, the effort-choice problem
in our setup is multi-dimensional in that each agent must allocate resources to two distinct
activities – own effort and helping effort – that may be imperfect substitutes for each other.
There is a close association between these two elements in that one ceases to have any value
when the other is absent. To illustrate what a multilayered tournament can give, we consider
two benchmark cases and contrast them with the current setup.

9A single-layered tournament: We first consider the case of single-layered competition
where each agent competes directly with all the others on the same level as usually assumed.10
In the current setup, the case with n = 1 (and also n = N ) corresponds to this standard
specification. It is evident that helping effort cannot be induced in single-layered tournaments
as it can only lower his chance of winning. Given this, (1) becomesαF (a∗ , 0)
(N − 1)(∆I + ∆T )
=
C1 ,
2
N
F1

for any given incentive scheme (∆I , ∆T ). Evidently, since ∆I and ∆T are perfect substitutes
in this case, each agent’s effort incentive depends only on ∆I + ∆T = W .
A multilayered tournament with no helping effort: As another benchmark, consider
the case where helping effort is not a viable option for the agents, either because it is prohibitively costly or because it carries no productive value, so that b∗ = 0. In this case, (1)becomesα(N − 1)∆I + (N − n)∆T
F (a∗ , 0)
=
C1 .
N2
F1

Note that an increase in ∆I is more effective in raising a∗ than an increase in ∆T due tothe freerider problem. This suggests that zero reward should be allocated to the external
competition in order to maximize the total effort.
Comparison: The two benchmark cases are instructive in illuminating the role of multilayered competition in general. To see this, we write the total effort as a function of n, denotedby x∗ (n), and compare these benchmarks in terms of the total effort induced.11 We establish
the following result which indicates some necessary conditions for multilayered tournaments
to have any value.
Proposition 1 Take (∆I , ∆T ) as given. If helping effort is not a viable option, x∗ (1) ≥ x∗ (n)for any n ∈ N .
Proof: To prove this, it suffices to show that(N − 1)∆I + (N − n)∆T
(N − 1)(∆I + ∆T )
≥
,
N2
N2
which holds with equality, for any n ∈ N , only if ∆T = 0.
10

There are no losers in the winning team in this case, and the winner simply captures W .
Define π(n) := vN x∗ (n) − ∆I − n∆T , where v is some constant, as total profit. Note that since the total
prize is weakly increasing in n, x∗ (1) ≥ x∗ (n) is a sufficient condition for π ∗ (1) ≥ π ∗ (n) for any n ≥ 2.
1110

In short, an integral ingredient of multilayered competition is the possibility or the necessity of helping others: the value of multilayered competition naturally diminishes when the
production process becomes more individualistic and requires less cooperation. The proposition states that multilayered competition has no bite when it is virtually impossible ormeaningless to help others (the case of purely individualistic production). The factor that is
especially critical is the imperfect substitutability of the two types of effort which gives riseto several interesting properties of multilayered tournaments as we will see shortly.

3.3

Efficient allocation

Properties of the efficient allocation are to a large degree independent of the shape of F .
Consider a social planner who can impose any effort pair (a, b) without any cost. The social
planner’s problem is then defined asmaxa,b

F (a, b) − C(a, b).

The first-order conditions with respect to a and b imply that
F1 = C1 , F2 = C2 .
The question of particular interest is whether and how this efficient allocation is implemented
via an appropriately designed incentive scheme. To do so, it is easy to observe that Pa = Pb ,which in turn implies ∆I = 0. Given this, the efficient allocation is implemented if ∆T is set
so as to satisfyα(N − n)∆T= F (a∗ (0, ∆T ), b∗ (0, ∆T )).
N2(3)

ef := n∆ef the total prize size
Let ∆ef
T denote the team incentive which satisfies (3), and R
T

required to achieve the (unconstrained) efficient allocation. We summarize this finding asfollows (the proof abbreviated).
Proposition 2 Suppose that the social planner can choose any R at no cost. The efficientef
allocation is then implemented by (0, ∆efT ) where ∆T solves (3).

3.4

Hidden costs of incentives and the multi-task interpretation of team

One important aspect of the model is that an increase in the individual incentive inhibits
cooperative behavior among teammates. Although intense internal competition can be beneficial up to some point, as it can provide stronger incentives for each agent and induce ahigher level of own effort, it is necessarily at the expense of a decrease in helping effort. This
could result in inefficient substitution of the two efforts and could even lower the total effortx∗ when the incentive ratio is highly unbalanced to begin with.
11

Proposition 3 Suppose that the two types of effort are complementary to each other, i.e.,
G2 > 0 and H1 > 0. Then, there exists some φ̄ ∈ [0, N − n) such that an increase in ∆I ,with ∆T fixed, reduces the total effort for φ ≥ φ̄ if there exists some finite number L(a) for
any given a > 0 such that
lim

b→0

F2 (F12 C2 − C12 F2 ) − F1 (F22 C2 − C22 F2 )< L(a).
F22 C2(4)

Proof: See Appendix A.
The proposition states that an increase in the individual incentive ∆I , with ∆T fixed,
may reduce the total effort supplied by each agent, thereby shedding light on yet anothermechanism through which the adverse effect of incentives arises. The condition we obtain is
not something that holds only under extreme conditions, as it is satisfied by many standard
specifications of F and C. To see this, when the cost is additively separable and linear ineach type of effort, (4) is reduced toFF1 F22 
12−< L(a),
lim
b→0 F2
F22
which is satisfied by many standard production functions. See section 5 for a more specificexample of the model.
When one side of the incentive scheme is exogenously fixed, the model yields implications
that are closely related to two well known results in the literature. On one hand, the adverse
effect of incentives is driven by the fact that an increase in ∆I induces inefficient substitution
between the two types of effort, where each agent shifts resources from more productive
helping effort to less productive own effort. In this sense, the result can be seen as yet
another manifestation of the multi-task problem (Holmstrom and Milgrom, 1991). On theother hand, this result also runs parallel to Lazear (1989) who points out that an increasein wage spread lowers total surplus (output net of effort cost), though not total effort, in atournament when agents have the option of sabotaging others. Here, an increase in ∆I leads
to a reduction in helping effort, which is qualitatively equivalent to an increase in sabotage,
when ∆T is held fixed for some exogenous reasons.
The result thus illuminates an inherent connection between the two widely cited rationales
for the adverse effect of incentives. The commonality of the two is that agents must allocate
resources into several different activities. To induce desirable behavior along several different
dimensions, the principal typically needs as many contracting measures. If there are notenough contractible variables to cover the whole set of activities, some of the incentives mayneed to be muted to achieve the right balance of efforts, which is precisely the point made by12Holmstrom and Milgrom (1991). In Lazear (1989), an increase in wage spread results in moresabotage (or less cooperation) because there is only one contractible measure, i.e., individual
ranking, while the agents must allocate resources into two distinct activities. The analysis
thus suggests a virtue of dividing agents into smaller subsets in a competitive environment
where performances can be measured only in relative terms, as it is instrumental in inducing
within-group cooperation by making an additional performance measure, i.e., team ranking,available for contracting.

4

Optimal provision of incentives

In this section, we assume that the total prize size is exogenously fixed at R ∈ (0, Ref )and examine how a change in the underlying environment, namely the incentive scheme(∆I , ∆T ) and the team size n, on each agent’s incentives and the aggregate outcome under
this restriction.

4.1

Effort-maximizing incentive scheme

The effort-maximizing incentive scheme is the solution to the following problem:
max F (a∗ , b∗ ),∆I ,∆Tsubject toR ≥ ∆I + n∆T .
Our primary interest is in how the extent of internal competition, measured by ∆I , is related
to the extent of external competition, measured by R, i.e., what fraction of the total prize
should be allocated to the individual winner.
Proposition 4 Suppose that F and C are homogenous functions. For any given R > 0, theeffort-maximizing incentive scheme can be written as
∆∗I =

φ∗
φ∗ + nR, ∆∗T =1R,
φ∗ + nwhere φ∗ ∈ [0, N − n) is some constant which is bounded from above.
Proof: See Appendix A.
A corollary of this result is that the effort-maximizing team incentive is always bounded
away from zero, while the effort-maximizing individual incentive can be zero in some cases.
13

To see under what conditions ∆∗I = 0 holds, note that when F and C are both homogeneous,the optimality condition can be written as
H( ab , 1)
Pa
=
.
Pb
G( ab , 1)
Under the maintained assumptions, there exists a unique ψ such that
1=

H(ψ, 1).
G(ψ, 1)This is the minimum effort ratio which can be implemented by a feasible incentive scheme:
for any ab > ψ, there is a unique incentive ratio which can implement the effort ratio. Thismeans that it is more likely to have ∆∗I = 0 if helping effort is relatively more productive
than own effort and ψ is higher.
Our analysis is instrumental in illuminating the elusive link between the two distinct levelsof competition: within-group (internal) competition and across-group (external) competition.
As emphasized, the most important example of this framework is the case where agentscompete for internal resources within organizations, such as firms, and for external resources
across organizations. Regarding this link, we emphasize two points that are of particularimportance.
General superiority of team incentives: As shown in Proposition 4, the optimal incentive
ratio can be zero but is bounded from above. There are actually two sides to this statement.
First, the optimal individual incentive might be zero, meaning that it is optimal to muteany within-group competition by treating all the teammates equally (low-powered individual
incentives) under some conditions. Second, the optimal team incentive is always positive andbounded from below, even though its effectiveness is severely limited by the freerider problem.
These two facts together point to the general superiority of team incentives in this type ofsetup, which is consistent with Williamson’s observation that that high-powered incentives
are more common in market arrangements than within firms (Williamson, 1985).
The reason why the team incentive is more effective in this context stems from the factthat it works at a higher level than the individual incentive and can hence yield a positive
impact on both types of effort. This draws clear contrast to the individual incentive which
can induce own effort only at the expense of a decrease in helping effort. Because of this,when the level of ∆T is small compared to ∆I , the agents choose to exert zero helping effort,
which is never optimal given that limb→0 H(a, b) = ∞. The same argument does not apply
to the case of the individual incentive because a positive team incentive, no matter how small
it is, can induce a positive level of own effort even if ∆I = 0.
14Complementarity between internal and external incentives: Our analysis also reveals
the importance of balancing the incentives at the two ends. Proposition 4 shows that given
an exogenous decrease in R, both ∆I and ∆T must be reduced proportionally, implying thatthe optimal level of incentives provided at one level of competition cannot be independent ofthat at the other level. We argue that the individual and team incentives are largely complementary to each other, so that a decrease in one should be accompanied by a decrease in
the other in order to maintain the right balance between competition and cooperation. As
more practical implications, the result suggests that: stronger individual incentives tend towork well in environments with more intense market competition; pro-competitive policies
are more likely to be justified in environments where stronger individual incentives are provided within firms. The model provides a critical policy insight that there exists no set ofcompetition policies that are universally effective because the extent of market regulations
should depend on how incentives are provided within firms, which could vary substantiallyacross industries as well as countries.

4.2

Effort-maximizing team size

It should be clear that the team size, or equivalently the number of teams, plays a criticalrole in almost every aspect of the model. We now turn our attention to the effect of theteam size n which is another key component of the incentive scheme.12 An increase in theteam size obviously exacerbates the freerider problem and yields a detrimental effect on effort
provision. Despite this, a larger team can also bring about a positive effect, as it may allow
the teammates to reap the benefit of the scale effect more effectively. The basic idea is thatwhen (i) the agents are heterogenous and complementary to each other, and (ii) the setof tasks they need to work on is highly diversified, it is intrinsically easier, or equivalently
more productive, to help each teammate by a small margin than to devote all of helping
effort intensively to one agent, or it is easier to find someone who an agent can help with his
expertise when there are more teammates to work with. Alternatively, the cost of helping
effort also decreases with n when it is a contribution to a public good, such as disclosure ofindispensable production knowledge, whose benefit is non-rival among the teammates.
To capture this idea and tradeoff, we add more structure to the cost function. Morespecifically, we now assume that the cost function is given bybij).
C(aij , bij ) = c(aij , β(n)That is, the production function is augmented with a new function β(n) which measures the12Here, we basically ignore the integer problem.

15

relative productivity of helping effort. For the subsequent analysis, we ignore the integer
problem and treat n as if it is a continuous variable for n ∈ N . We in general assume thatβ(n) is weakly increasing and weakly concave in n. Let x∗ (n) denote the equilibrium total
effort as a function of the team size. We are interested in how a change in the team size
affects each agent’s effort choices for n ∈ N .13
Due to the scale effect of helping effort, the socially optimal team size, when the social
planner can directly impose the effort levels, is n = N . This is not incentive compatible,
however, because there is clearly no incentive for any agent to exert helping effort when he isin a grand coalition which involves all the agents. Because of this inability to take advantageof the scale effect, the case with n = N is weakly dominated even by the least efficient casewhere each team consists of only one agent (n = 1).14 To induce a positive level of helping
effort, the team size should be bounded between n = 2 and n = N2 .
The optimal effort pair is now given by a∗ (∆I , ∆T , n), b∗ (∆I , ∆T , n). Taking (∆I , ∆T ) as
given, an increase in the team size raises the total effort if
F1

∂a∗
∂b∗
+ F2
> 0.
∂n
∂n

Let n∗ := argmaxn∈N x∗ (n) denote the effort-maximizing team size in the interval N . The
effort-maximizing team size depends largely on the magnitude of the scale effect, which is
captured by β ′ , and the incentive ratio φ. While it is intuitive and clear that a larger scaleeffect always favors a larger team size, the effect of φ is more somewhat complicated. We
state the following result concerning how the effort-maximizing team size is related to the
incentive scheme in effect.
Proposition 5 n∗ > 2 only ifNN ∆T=≥ β ′ (2).
(N − 1)∆I + (N − n)∆T
(N − 1)φ + N − n
Under the optimal incentive scheme (∆∗I , ∆∗T ) derived in Proposition 4, n∗ > 2 if and only if
β ′ (2) > 0.
Proof: See Appendix A.
13

There is of course a possibility that the effort-maximizing team size is one. This is more likely to be thecase if the two types of effort are more substitutive and/or the productivity of helping effort is very low. Given
the spirit of this problem, however, we do not consider this possibility and simply assume that x∗ (2) > x∗ (1).
14I=
To see this, note that b∗ (N ) = 0 for n = N , so that the total effort is given by α (N−1)∆
N2
F (a∗ (N),0) ∗γ−1
∗
∗
a
.
See
section
3.2
for
the
case
with
n
=
1.
Comparingthese,
it
is
clear
that
x
(1)
≥
x
(N
)
for
∗
F1 (a (N),0)
any given (∆I , ∆T ).

16

First, it is clear that the total effort is monotonically decreasing in n ∈ N in the absenceof the scale effect, i.e., when β ′ (n) = 0 for all n ∈ N . When β ′ (2) > 0, on the other hand,a larger team size may increase the total output. To see when a larger team size is moredesirable, observe that an increase in the team size renders the team incentive less effective
and hence yields an effect similar to a decrease in ∆T . Because of this, an increase in the teamsize tends to have a favorable effect on the total effort when the incentive ratio is relatively
low: a larger team is likely to be more desirable when the individual incentive ∆I is weakrelative to the team incentive ∆T . In contrast, when the individual incentive is relatively
strong (or the incentive ratio is relatively high), a small team is generally desirable, and it
may even be desirable to have no team at all.

5

An example: CES production function with linear cost

In this section, we provide a specific example of the model by employing more specific production and cost functions in order to obtain sharper predictions and implications. Specifically,
we assume that the production function takes the following CES form:
xij = F (aij , hij ) = a1−ρ
ij

2

+

 1
h1−ρ
1−ρ
ij
2

(5)

.

The key parameter is ρ ∈ [0, ∞) which measures the elasticity of substitution between own
effort and helping effort. Since we are interested in cases where helping effort is inherently
different from own effort and cannot be easily substituted, we assume that ρ > 1, i.e., thetwo types of effort are sufficiently complementary to each other. We also assume that thecost function is given by
C(aij , bij ) = aij +

bij
.
β

(6)

Under this specification, we obtain the following result.
Lemma 1 Under the production and cost functions specified in (5) and (6), the equilibrium
effort levels are given by1x∗ = αA(Paσ + (βPb )σ ) σ , a∗ =1
− 1−ρwhere σ := 1−ρρ and A := 2α(βPb )1+σ
αPa1+σ∗,b=,
Paσ + (βPb )σPaσ + (βPb )σ.

Proof: Given the production function (5), the equilibrium conditions are computed asρ1

αPa A(a∗ 1−ρ + b∗ 1−ρ ) 1−ρ a∗ −ρ = A(a∗ 1−ρ + b∗ 1−ρ ) 1−ρ ,
17

ρ

1

αβPb A(a∗ 1−ρ + b∗ 1−ρ ) 1−ρ b∗ −ρ = A(a∗ 1−ρ + b∗ 1−ρ ) 1−ρ .
With some computation, these can be written as
αPa = (a∗1−ρ + b∗1−ρ )a∗ρ = x∗ 1−ρ

αβPb = (a∗ 1−ρ + b∗ 1−ρ )b∗ ρ = x∗ 1−ρ

A

A

a∗ρ ,

b∗ ρ ,

from which we obtain the expressions in the lemma.
The lemma shows that the total effort is given by a very simple and tractable form whichallows us to conduct various comparative statics exercises and verify our preceding results.
Among them, we are particularly interested in the effect of an increase in the individual
incentive ∆I on the total effort x∗ . Note that in this setup, (4) can be written as
lim

F

b→0

12

F2

−

F1 F22  ρa−ρ (1 + ( ab )1−ρ ),=a1−ρ + b1−ρF22which is bounded from above for any given a > 0 under the maintained assumption of ρ > 1,
meaning that the condition in Proposition 3 is satisfied.
Proposition 6 There exists φ̄ ∈ [0, N − n) such that an increase in ∆I decreases x∗ for
φ > φ̄. If N − 1 > β σ , φ̄ > 0.
Proof: An increase in ∆I decreases x∗ if
1−σ
A σ(Pa + (βPb )σ ) σ ((N − 1)Paσ−1 − βnσ Pbσ−1 ) < 0.
N

(7)

The left-hand side represents the marginal increase in total effort per worker. It then follows
from this that an increase in ∆I reduces the total effort if(N − 1)Paσ−1 < β σ Pbσ−1 ,which is simplified top=1(N − 1)φ + N − n  N − 1  1−σ>.
−φ + N − nβnσNote that p ranges from one (at φ = 0) to infinity (at φ = N − n). There exists an interior
threshold φ̄ if N − 1 > β σ . Otherwise, the total effort is monotonically decreasing in ∆I .

18

We can also obtain sharper predictions on the effect of n on the total effort x∗ . To thisend, we now let the cost function (6) be given by
C(aij , bij ) = aij +bij,γ(n − 1)λwhere β = γ(n − 1)λ and λ ∈ [0, 1].15 It directly follows from Lemma 1 that the total effort
is increasing in n if−(n − 1)(Paσ−1 + β σ Pbσ−1 )∆T+ λ(βPb )σ > 0.
N2

(8)

The first term indicates the loss of incentives due to the freerider problem and is always
negative. The second term indicates the efficiency gain from the scale effect and is always
positive by definition. The effort-maximizing team size must balance this tradeoff.
The effort-maximizing team size is relatively straightforward to obtain when the scaleeffect of helping effort is absent. If λ = 0, i.e., helping effort exhibits no scale effect, thiscondition is reduced to−(Paσ−1 + β σ Pbσ−1 ) > 0,
which obviously never holds. This implies that the total effort is strictly decreasing in n for
n ∈ N due to the incentive to freeride. In contrast, when λ > 0, the total effort may increase
with the team size if
λ(γPb )σ > (n − 1)(Paσ−1 + γ σ Pbσ−1 )

∆T
.
N2

This condition is likely to be satisfied when the incentive ratio is small, so that the agents
allocate more resources to helping effort: as φ → 0, we haveλγ σ (N − n)
> 1 + γσ .
(n − 1)∆T

6

Conclusion

This paper proposes a tractable framework to analyze multilayered competition which weubiquitously observe. The framework allows us to analyze the interconnection between the
two levels of competition, internal (within-group) and external (across-group), and how they
are and should be related to each other. Among other things, we find that: (i) an increase
in individual incentives, holding the extent of team incentives fixed, can lower overall performances because it induces inefficient substitution of effort; (ii) the extent of individual
15
Under this specification, the scale effect is absent when λ = 0 while helping effort is a pure public goodwhose cost is independent of the number of beneficiaries when λ = 1.

19incentives should be complementary to that of team incentives. The latter finding suggeststhat when we devise an incentive scheme within a firm, the extent of market competition
that the firm faces is an important factor to be reckoned with. We also use our framework to
explain why high-powered incentives are more common in market arrangements than withinfirms.
As a final note, one limitation of the analysis is that we only consider symmetric cases
where agents are all identical and grouped into teams of the same size. One possible extension
along this line is therefore to introduce heterogeneity among agents and allow them to differ
in innate productivity. The extension adds another dimension to the problem and allows us
to ask how agents should be sorted into teams (positive or negative assortative matching).
The other possibility is to consider the case where teams vary in size and productivity so thatsome teams are larger than others. Although the analysis in either direction could becomemuch more complicated, as we rely on the imposed symmetric structure to make it tractable,
it is nonetheless of some interest to analyze these extended cases.

References
Alchian, A.A. and H. Demsetz, 1972, Produiction, Information Costs, and Economic Organization, American Economic Review, 62, 777-95.
Baye, M.R. and H.C. Hoppe, 2003, The Strategic Equivalence of Rent-Seeking, Innovation,
and Patent-Race Games, Games and Economic Behavior, 44, 217-26.
Clark, D.J. and C. Riis, 1998, Contest Success Functions: An Extension, Economic Theory,11, 201-4.
Drago, R., G.T. Garvey, and G.K. Turnbull, 1996, A Collective Tournament, Economics
Letters, 50, 223-7.
Fullerton, R.L. and R.P. McAfee, 1999, Auctioning Entry into Tournaments, Journal of
Political Economy, 107, 573-605.
Gurtler, O., 2008, On Sabotage in Collective Tournaments, Journal of Mathematical Economics, 44, 383-93.
Hart, O.D., 1983, The Market as an Incentive Mechanism, Bell Journal of Economics, 14,366-82.
Hermalin, B.E., 1992, The Effects of Competition on Executive Behavior, Rand Journal of
Economics, 23, 350-65.

20Holmstrom, B., 1982, Moral Hazrd in Teams, Bell Journal of Economics, 13, 324-40.
Holmstrom, B. and P. Milgrom, 1991, Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design, Journal of Law, Economics, and Organization,
7, 24-52.
Inderst, R., H.M. Muller, and K. Warneryd, 2007, Distributional Conflict in Organizations,
European Economic Review, 51, 385-402.
Ishida, J., 2006, Team Incentives under Relative Performance Evaluation, Joutnal of Economics and Management Strategy, 15, 187-206.
Itoh, H., 1991, Incentives to Help in Multi-Agent Situations, Econometrica, 59, 611-36.
Lazear, E., 1989, Pay Equality and Industrial Polictics, Journal of Political Economy, 97,
561-80.
Lazear, E. and S. Rosen, 1981, Rank Order Tournaments as Optimum Labor Contracts,
Journal of Political Economy, 89, 841-64.
Marino, A.M. and J. Zabojnik, 2004, Internal Competition for Corporate Resources andIncentives in Teams, Rand Journal of Economics, 35, 710-27.
Morita, H., 2012, Firm Dynamics and Labor Market Consequences, Journal of Law, Economics, and Organization, 28, 235-64.
Munster, J., 2007, Simultaneous Inter- and Intra-Group Conflicts, Economic Theory, 32,
333-52.
Nalbantian, H.R. and Schotter, A., 1997, Productivity under Group Incentives: An Experimental Study, American Economic Review, 87, 314-41.
Nitzan, S., 1991, Collective Rent Dissipation, Economic Journal, 101, 1522-34.
Nitzan, S. and K. Ueda, 2011, Prize Sharing in Collective Contests, European Economic
Review, 55, 678-87.
Raith, M., 2003, Competition, Risk, and Managerial Incentives, American Economic Review,93, 1425-36.
Rosen, S., 1986, Prizes and Incentives in Elimination Tournaments, American Economic
Review, 76, 701-15.
Schmidt, K.M., 1997, Managerial Incentives and Product Market Competition, Review of
Economic Studies, 64, 191-213.

21

Skaperdas, S., 1996, Contest Success Functions, Economic Theory, 7, 283-90.
Sutter, M. and C. Strassmair, 2009, Communication, Cooperation and Collusion in Team
Tournaments: An Experimental Study, Games and Economic Behavior, 66, 506-25.
Tullock, G., 1980, Efficient Rent Seeking. In J.M. Buchanan, R.D. Tollison, G. Tullock,(Eds.), Toward a Theory of the Rent-Seeking Society, College Station, TX: Texas A&M
University Press.
Warneryd, K., 1998, Distributional Conflict and Jurisdictional Organization, Journal of Public Economics, 69, 435-50.
Williamson, O., 1985, Economic Institutions of Capitalism, New York: Free Press.

Appendix
Proof of Proposition 3: Differentiating the first-order conditions with respect to ∆I
yields ∂a∗∂b∗  (N − 1)G
∂a∗
∂b∗
+ G2
=
F
+
F
,
+
Pa G1
1
2
∂∆I
∂∆I
N2
∂∆I
∂∆I

∂b∗ 
∂a∗
∂b∗
H
∂a∗
+ H2
+ F2
.
− 2 = F1
Pb H1
∂∆I
∂∆I
N
∂∆I
∂∆I
Solving these we obtain∂a∗(N − 1)(F2 − H2 Pb )G − (G2 Pa − F2 )H
= 2
,
∂∆I
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))(N − 1)(H1 Pb − F1 )G − (F1 − G1 Pa )H
∂b∗
.
= 2
∂∆I
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))It follows from these that∂a∗∂b∗
(N − 1)(F2 H1 − F1 H2 )GPb − (F1 G2 − F2 G1 )HPa
∂x∗
. (9)
= F1
+ F2
= 2
∂∆I
∂∆I
∂∆I
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))We first show that the denominator of (9) is always positive, i.e., (F1 −G1 Pa )(F2 −H2 Pb ) >(G2 Pa − F2 )(H1 Pb − F1 ). This can be written as(F2 H1 − F1 H2 )Pb + (F1 G2 − F2 G1 )Pa + (G1 H2 − G2 H1 )Pa Pb > 0,
which holds under the maintained assumption that G1 > 0, H2 > 0, G2 < 0 and H1 < 0.
Given this, it now suffices to show that the numerator is negative, i.e.,(F1 G2 − F2 G1 )HPa
> N − 1,
Pb →0 (F2 H1 − F1 H2 )GPb
lim

22

for any given Pa > 0. Since
H1 =F22 C2 − C22 F2 Pa
F12 C2 − C12 F2
H, H2 =
,
= ,
2
2
Pb
G
C2
C2

we have
lim

Pb →0

F2 2
11 F1
12 F1
)
− F2 F11 C1C−C)( C(F1 F12 C1C−C
2
2
2
1

1

F1 2
12 F2
2 F2
(F2 F12 C2C−C)− F1 F2 C2C−C
)( C
2
2
1
2
2

> N − 1,

which is further reduced toC1 F12F2 F11 C1+ F2FC111
F1 − C12 −
F12
lim
C2
Pb →0 C2 F12 − C12 − F1 F22+ F1FC222
F2
F22

F2 C11C1 F12C12F2 F11 C1
F1 C2 − C2 − F12 C2 + F1 C2= lim F12 C12 F1 F22 C2 F1 C22
Pb →0
+ F2
F2 − C2 −F22> N − 1, (10)Note that as Pb → 0, we have b → 0 and H → ∞ while a > 0 and G is bounded from above.
We first show that the numerator diverges to infinity as Pb → 0, i.e.,
C F
C12 F2 F11 C1 F2 C11 
1 12
−
−
lim
= ∞.
+
b→0 F1 C2C2
F1 C2
F12 C2
With some algebra, this condition can be written aslim C 2G

b→0

1

2

F1 C2

−

HF11
HC11 
+
= ∞,
GF1
F1

which holds true under the maintained assumptions. Given this, (10) holds if there exists
some finite number L(a) for any a > 0 such that
C12 F1 F22 C1 F1 C22
F2 (F12 C2 − C12 F2 ) − F1 (F22 C22 − F2 C2 C22 )
F12
−
−
=
lim
+
< L.
b→0
b→0 F2C2F2F22
F22 C2lim

Proof of Proposition 4: Under the restriction that R = ∆I + n∆T , the optimal incentive
scheme must satisfy
 ∂a∗
∂a∗
∂b∗ 
∂b∗
= F1
n F1
+ F2
+ F2
.
∂∆I
∂∆I
∂∆T
∂∆T

(11)

It is clear from this that ∆∗T > 0 under Assumption 3.
∗

∗

∂b
∂a
and ∂∆, we differentiate the first-order conditions with respect to ∆T :To obtain ∂∆
T
T


∂a∗
∂b∗  (N − n)G
∂a∗
∂b∗
Pa G1
+
+ G2
=
F
+
F
,
1
2
∂∆T
∂∆T
N2
∂∆T
∂∆T

∂b∗  (N − n)H
∂a∗
∂b∗
∂a∗
+ H2
=
F
+
F
.
+
Pb H1
1
2
∂∆T
∂∆T
N2
∂∆T
∂∆T
23

Solving these, we obtain(N − n)((G2 Pa − F2 )H + (F2 − H2 Pb )G)
∂a∗
= 2
,
∂∆T
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))(N − n)((H1 Pb − F1 )G + (F1 − G1 Pa )H)
∂b∗
.
= 2
∂∆T
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))It follows from these that
F1

∂a∗
∂b∗
(N − n)((F2 H1 − F1 H2 )GPb + (F1 G2 − F2 G1 )HPa )
.
+ F2
= 2
∂∆T
∂∆T
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))Combined with (9), we can rewrite (11) as(N − n)((F2 H1 − F1 H2 )GPb + (F1 G2 − F2 G1 )HPa
= n,
(N − 1)(F2 H1 − F1 H2 )GPb + (F2 G1 − F1 G2 )HPa
which is simplified to(n − 1)(F2 H1 − F1 H2 )G2 = (F1 G2 − F2 G1 )H 2 .
Given that F and C are homogenous functions, this condition can be rewritten asn − 1 = Ω( ab ) :=(F1 ( ab , 1)G2 ( ab , 1) − F2 ( ab , 1)G1 ( ab , 1))H( ab , 1)2.
(F2 ( ab , 1)H1 ( ab , 1) − F1 ( ab , 1)H2 ( ab , 1))G( ab , 1)2

Given that there exists an interior solution to satisfy this, we have a constant optimal effort
ratio ψ such that n − 1 = Ω(ψ) for any R. The optimal incentive scheme must solvePa(N − 1)φ + N − nH(ψ, 1)==,Pb−φ + N − nG(ψ, 1)
if there exists an interior solution.
Define ψ such that
1=

H(ψ, 1).
G(ψ, 1)

There exists an interior solution ifn − 1 < Ω(ψ).
If this condition is satisfied, there exists some φ∗ such that(N − 1)φ∗ + N − nH(ψ, 1)=.
∗−φ + N − nG(ψ, 1)
The resource constraint then implies thatR = φ∗ ∆T + n∆T .
24

Solving this yields the expressions in the proposition.
Proof of Proposition 5: Differentiating the first-order conditions with respect to n, we
obtain: ∂a∗∂b∗  ∆T
∂b∗
∂a∗
Pa G1
− 2 G = F1
+ G2
+ F2
,
∂n
∂n
N
∂n
∂n

 ∂a∗
∂b∗   ∆T
∂b∗
∂a∗
′
−
+ H2
+
F
.
−
β
P
H
=
F
Pb H1
2
1
b
n
∂n
∂n
N2
∂n
∂n
Solving these, we obtain(G2 Pa − F2 )(∆T − β ′ Pb N )H − (F2 − βH2 Pb )∆T G)
∂a∗
= 2
,
∂n
N ((F1 − G1 Pa )(F2 − βH2 Pb ) − (G2 Pa − F2 )(βH1 Pb − F1 ))∂b∗−(βH1 Pb − F1 )∆T G − (F1 − G1 Pa )(∆T − β ′ Pb N )H= 2.
∂n
N ((F1 − G1 Pa )(F2 − βH2 Pb ) − (G2 Pa − F2 )(βH1 Pb − F1 ))An increase in n increases the total effort if
F1

∂a∗
∂b∗
β(F2 H1 − F1 H2 )∆T GPb + (F1 G2 − F2 G1 )(∆T − β ′ Pb )HPa
+ F2
=− 2
> 0. (12)∂n∂n
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))It is evident from this that this condition holds only if∆T ≥ β ′ Pb N ⇔

N ∆T
≥ β′.
(N − 1)∆I + (N − n)∆T

Under the optimal contract (∆∗I , ∆∗T ), (13) becomesβ′(F1 G2 − F2 G1 )Pb> 0,
2
N ((F1 − G1 Pa )(F2 − H2 Pb ) − (G2 Pa − F2 )(H1 Pb − F1 ))(13)

which holds if and only if β ′ (2) > 0.

Appendix B: derivation of the first- and second-order conditions
B.1 The first-order conditions
The first-order conditions are given by∂qj∂rij(rij ∆I + ∆T ) + qj
∆ I = C1 ,
∂aij
∂aij
∂rij
∂qj
(rij ∆I + ∆T ) + qj
∆ I = C2 .
∂bij
∂bij
25

It is straightforward to computeαα(n − 1)xα−1
αxαij xα−1
∂rij
F2 ∂rij
∂rij
∂rij
ij x
= F1
= F1 α== −F2 α
,
,
∂aij
∂xij
(xij + (n − 1)xα )2 ∂bijn − 1 ∂x(xij + (n − 1)xα )2α(m − 1)zjα−1 z αα(m − 1)zjα−1 z α∂qj∂qj
F2 ∂qj∂qj= F1= F1 α
=
=
F
,
,
2 α
∂aij
∂xij
(zj + (m − 1)z α )2 ∂bijn − 1 ∂x
(zj + (m − 1)z α )2In a symmetric equilibrium (a∗ , b∗ ), we have∂rijα(n − 1) ∂rijα
∂qjα(N − n) ∂qjα(N − n)= F1,
= −F2 2 ∗ ,
= F1
,
= F2
,
2
∗
2
∗
∂aij
n x
∂bij
n x ∂aij
N x∂bij
N 2 x∗
It follows from these that the first-order conditions can be written asα(N − 1)∆I + (N − n)∆T−∆I + (N − n)∆T
F1 = C1 , α
F2 = C2 .
2∗N x
N 2 x∗

B2 The second-order conditions
The second-order conditions are given by∂qj ∂rij∂ 2 rij∂ 2 qj
(r
∆
+
∆
)
+
2
∆I < C11 ,
∆
+
q
ij
I
T
I
j
∂aij ∂aij
∂a2ij
∂a2ij
∂qj ∂rij∂ 2 qj∂ 2 rij
(r
∆
+
∆
)
+
2
∆I < C22 .
∆
+
q
ij
I
T
I
j
∂bij ∂bij
∂b2ij
∂b2ij
With some algebra, we obtainα α(n − 1)xα−1(α − 1)(xαij + (n − 1)xα ) − 2αxαij ∂ 2 rij
ij x
F
+
F
,
=
11
1
(xαij + (n − 1)xα )2(xαij + (n − 1)xα )xij
∂a2ij

αxαij xα−1
(α − 1)(xαij + (n − 1)xα ) − 2α(n − 1)xα ∂ 2 rij
=
−
,
F
+
F
22
2
(xαij + (n − 1)xα )2(n − 1)(xαij + (n − 1)xα )x∂b2ij
α(m − 1)zjα−1 z α (α − 1)(zjα + (m − 1)z α ) − 2αzjα ∂ 2 qj
=
F
+
F
,
11
1
(zjα + (m − 1)z α )2(zjα + (m − 1)z α )zj∂a2ij
α(m − 1)zjα−1 z α (α − 1)(zjα + (m − 1)z α ) − 2αzjα ∂ 2 qj
=
F
+
F
.
22
2
(zjα + (m − 1)z α )2(zjα + (m − 1)z α )zj∂b2ij
In a symmetric equilibrium (a∗ , b∗ ), we have
∂ 2 rij(α − 1)n − 2α  ∂rij  F11 (α − 1)n − 2α α(n − 1) 
+
F
+
F
=
,
=
11
1
n 2 x∗
nx∗∂aij F1
nx∗
∂a2ij
26

∂ 2 rij(α − 1)n − 2α(n − 1)  ∂rij  F22 (α − 1)n − 2α(n − 1) 
α 
+
F
+
F
=
,
=
−
22
2
n 2 x∗
n(n − 1)x∗
∂bij F2n(n − 1)x∗
∂b2ij
∂ 2 qj∂qj  F11 (α − 1)N − 2αn α(N − n) 
(α − 1)N − 2αn 
=
+
F
+
F
=
,
11
1
N 2 x∗
N nx∗∂aij F1N nx∗∂a2ij∂qj  F22 (α − 1)N − 2αn ∂ 2 qj(α − 1)N − 2αn α(N − n) 
+
F
+
F
=
,
=
22
2
N 2 x∗
N nx∗∂aij F2
N nx∗∂b2ij
where x∗ = F (a∗ , b∗ ). It follows from these that the second-order conditions can be writtenas(α − 1)N − 2αn  ∆Iα(N − n) 
+
∆
F
+
F
T
11
1
N 2 x∗N nx∗n
(n − 1)(N − n) α(n − 1)
(α − 1)n − 2α + 2(αF1 )2+F+F∆I < C11 ,111(N nx∗ )2
N nx∗
nx∗

α(N − n) 
(α − 1)N − 2αn  ∆I
+
∆
F
+
F
T
22
2
N 2 x∗
N nx∗
n

N
−
n
α
(α − 1)n − 2α(n − 1) − 2(αF2 )2
∆
−
F
+
F
∆I < C22 ,I
222(N nx∗ )2
N nx∗
n(n − 1)x∗

(14)

(15)

For any optimal solution (a∗ , b∗ ) where a∗ , b∗ > 0, the second-order conditions are satisfied if
α is sufficiently small.

27