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Teams, Career Horizon Diversity, and Tacit Collusion

Jonathan Glover Eunhee Kim

Columbia Business School City University of Hong Kong

PRELIMINARY

Draft: July 2019

Abstract

We study optimal team design for multiple agents with possibly heterogeneous expected career

horizons (i.e., discount factors). A principal assigns four agents to two teams, creating either

homogenous teams by assigning the two agents with shorter horizons to one team and the two

agents with longer horizons to a second team or heterogeneous (diverse) teams by mixing

horizons within each team. The optimal team composition and explicit contract specified by the

principal set the stage for implicit contracting between agents—to motivate desirable implicit

contracting (mutual monitoring) and to prevent undesirable implicit contracting (collusion).

When all agents have relatively short horizons, the contract is designed to motivate mutual

monitoring that prevents free-riding, and homogenous and diverse teams perform equally. When

all agents have relatively long horizons, collusion is always a pressing concern, and

heterogeneous teams are optimal. For intermediate horizons, team assignment affects whether or

not the collusion problem arises, which can lead to either diverse or homogenous teams being

optimal. We view our model as providing a new theory that helps explain both observed

diversity in team assignments and diversity in pay-for-performance sensitivity.

Keywords: team diversity, team composition, mutual monitoring, tacit collusion, pay-for-

performance sensitivity

E-mail addresses: jg3463@gsb.columbia.edu (J. Glover) and eunhee.kim@cityu.edu.hk (E. Kim).

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1.   Introduction

Demographic diversity, including gender, race, age or tenure, is an important consideration in

team design (Jackson, May, and Whitney, 1995; Lazear, 1999). Although management and

organizational behavior research has shown that team composition significantly influences team

performance, the evidence on whether demographically diverse teams outperform homogeneous

teams is mixed.1

Age or tenure (expected horizon) diversity in the workplace has become more common than

similarity (Jackson, May, and Whitney, 1995; Kunze, Boehm, and Bruch, 2011). Diversity can

create opportunities for learning. Also, team members with different career horizons are less

likely to face emotional conflict (Pelled, Eisenhardt, and Xin, 1999). However, diversity can also

make communication within teams less effective and interactions more challenging (Zenger and

Lawrence, 1989; Lazear, 1999; Timmerman, 2000). Age discrimination climates in workgroups

and generational differences in work attitudes are also important considerations (Ely 2004;

Kunze, Boehm, and Bruch, 2011).

We focus on a benefit of horizon diversity that has not yet been studied—the role of diversity

in facilitating desirable implicit incentives (mutual monitoring) and preventing undesirable

implicit incentives (collusion) that team members provide to each other. We study optimal team

design for multiple agents with possibly heterogeneous expected career horizons (i.e., discount

factors). A principal assigns four agents to two teams, creating either homogenous teams by

assigning the two agents with shorter horizons to one team and the two agents with longer

horizons to a second team or heterogeneous (diverse) teams by mixing horizons within each

team. The optimal team composition and explicit contract specified by the principal set the stage

for implicit contracting between agents. When all agents have relatively short horizons, the

contract is designed to motivate mutual monitoring that prevents free-riding, and both teams

perform equally. When the agents have relatively long horizons, collusion becomes a pressing

concern, and diverse teams are optimal. In this case, the optimal contract treats the agents

                                                                                                                         1  For evidence on the impact of age and racial diversity on the performance of basketball and baseball teams, see Timmerman (2000). For evidence on the effectiveness of bonus plans at retail bank branches as a function of age and tenure, see Ely (2004). For evidence on the performance of retail stores with age diversity vs. similarity, see Leonard, Levine, and Joshi (2004). For evidence on project teams of engineers and their effectiveness on team communication when co-workers have diverse vs. similar ages or tenures, see Zenger and Lawrence (1989). For surveys on the effectiveness of team diversity, see Milliken and Martins (1996) and Reiter-Palmon, Wigert, and de Vreede (2012).  

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asymmetrically, providing lower-powered mutual monitoring incentives to the agent with the

longest horizon and higher-powered collusion-proof incentives to the agent with the shortest

horizon. We view our model as providing a new theory that helps explain both observed

diversity in team assignments and diversity in pay-for-performance sensitivity.

In our model, once teams are formed and the principal specifies the explicit contract, each

agent chooses either to work or shirk in each period. Our model has the following additional

features. First, team production exhibits a productive substitutability, which is intended to

capture teams of agents with interchangeable inputs (agents with similar expertise) to focus the

model on the role of horizon rather than productive diversity. Second, due to the proximity of

team members, they observe each other’s actions and can potentially use implicit contracts to

motivate each other via mutual monitoring. By mutual monitoring, we mean the agents police

each other by threatening to punish free-riding with shirking in all future periods. Mutual

monitoring allows the principal to take advantage of the agents’ mutual observation of each

other’s actions.

However, mutual observation of efforts also creates the potential for the agents to collude on

other actions the agents prefer to (work, work). Regardless of whether mutual monitoring and/or

collusion incentives exist within a team, such implicit incentives must be self-enforcing for both

agents: from the agents’ perspective, the agreed upon plan must be both Pareto optimal and

constitute a subgame perfect equilibrium. We allow for collusion in a tractable way by confining

attention to stationary strategies, allowing the team members to use an agreed upon correlation

device to assign probabilities to (work, work), (work, shirk), (shirk, work), and (shirk, shirk). In

our main analysis, we consider regret-free collusion: agents must find the collusion Pareto

dominating relative to joint working at every point in time (including just after the correlation

device has assigned the agents to play a pure strategy). We also consider an alternative notion of

collusion, ex ante collusion, in the extension section: agents most only find the collusion Pareto

dominating ex ante (before the correlation device is employed).

Because the agents are asymmetric, we allow the principal to treat them asymmetrically by

offering them different contracts. Thus, the optimal (explicit) contract the principal offers to the

agents must generate (implicit) mutual monitoring incentives for both agents but eliminate

(implicit) collusion incentives for only one of the agents in the team. Mutual monitoring and

collusion both rely on a grim trigger strategy—if the agents observe a deviation from the agreed

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upon play, they revert to the lowest payoff stage game (one-shot) equilibrium in all future

periods.

When all agents have short horizons (low discount factors), mutual monitoring is the only

implicit incentive, and the optimal contract is symmetric. In this case, homogenous and diverse

teams are equivalent from an incentive standpoint. However, as agents’ discount factors increase,

a collusion problem may arise under homogenous teams but not under diverse teams: when one

agent has a relatively short expected career horizon and another agent has a sufficiently long

horizon, the agent with the short horizon is simply unwilling to collude. This feature is a first

(and somewhat straightforward) benefit of team diversity.

When all agents have long horizons (high discount factors), the collusion problem arises

across all teams. Given the binding collusion constraint, the optimal contract treats the agents

asymmetrically: one team member receives a higher-powered collusion-proof wage with the

other team member receives a lower-powered mutual-monitoring wage. It is the heterogeneity in

discount factors within a team that strengthens the efficiency of asymmetric contracting, which is

a second benefit of team diversity.

Specifically, the principal targets the agent with the lower discount factor (the shorter

expected career horizon) in upsetting collusion. Heterogeneous assignment reduces the mutual

monitoring wages, since the mutual monitoring wage is now offered only to high discount factor

(patient) agents. The impact on the collusion constraints is less straightforward. Compared to a

team of short-horizon players, a heterogeneous team increases the collusion-proof wage because

a long-horizon player is willing to pay a greater bribe (in terms of effort) to the short-horizon

player than another short-horizon player would in order to induce collusion. For the team of

long-horizon players, switching to a heterogeneous team decreases the collusion-proof wage.

The net impact of these factors favors diverse assignment.

Under ex-ante collusion, the results are similar to regret-free collusion when all agents have

relatively long or relatively short horizons. However, for intermediate discount factors, either

heterogeneous or homogenous assignment can be optimal. Here, the result is not driven by the

level of collusion-proof wages per se. Instead, the optimal assignment is driven by whether that

assignment results in collusion being a pressing concern (a binding constraint).

From the perspective of the theory of the firm as team production (Alchian and Demsetz,

1972), our model highlights the role of diverse assignment and asymmetric contracts as an

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optimal means of creating a common interest in non-shirking. Our focus is on the link between

team design (and, more broadly, organizational design) and implicit contracting between agents

via long-term relationships.2

Our theory also develops a dependency of pay-for-performance sensitivity (and disparity in

that sensitivity) that depends on both the expected career horizons of the agents and the diversity

of those horizons. In particular, a high level of pay-for-performance disparity arises only when

the agents all have relatively long expected horizons and, as a result, are offered asymmetric

contracts that provide only the shorter-horizon agent with high-powered collusion-proof

incentives. Bushman, Dai, and Zhang (2016) find that the duration of a team’s tenure mitigates

undesirable consequences of estimated suboptimal pay-for-performance disparity. Our theory

can be viewed as providing an alternative interpretation of their results: long expected horizons

create a demand for such pay-for-performance disparity because of the threat of collusion.

There are large theoretical and empirical literatures that explore the role of individual

incentives provided by explicit contracts. However, much less attention has been given to the

role of mutual monitoring and the role of implicit incentives that team members provide to each

other. The handful of articles that have studied mutual monitoring and implicit incentives (e.g.,

Arya, Fellingham, and Glover, 1997; Che and Yoo, 2001; Glover and Xue, 2019; and Glover and

Kim, 2019)3 have confined attention to ex ante identical agents.4 In this article, we expand this

theory to include agents that are heterogeneous in their expected horizons and to allow the

principal to offer the agents asymmetric contracts.

A related literature shows that the principal can be better off by offering asymmetric

contracts to symmetric agents (e.g., Demski and Sappington, 1984, Ishiguro, 2004, and

Baldenius, Glover, and Xue, 2016). Ishiguro (2004) shows that discriminatory wage schemes for

symmetric agents can be optimal to prevent collusion under relative performance evaluation.

                                                                                                                         2 Slivinski (2002) studies the interaction between organizational form (for-profit and not-for-profit) and free-riding problems. 3  On the empirical front,  Guay, Kepler, and Tsui (2019) find evidence that heterogeneous executives (with different tasks/expertise) are often provided with the same cash bonuses, which they interpret as being designed to encourage mutual monitoring. Li (2019) provides evidence that the total compensation (including equity incentives) of executives is more consistent with mutual monitoring than individual (Nash) incentives. Duchin, Goldberg, and Sosyura (2017) find a similar result for divisional managers.  4  Glover and Kim (2019) consider productively diverse teams; however, all agents in their model have the same expected career horizon.  

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However, in his model, the nature of the collusion problem is different from ours because agents

do not observe each other’s effort choices, so there is no role for mutual monitoring.

Gibbons and Murphy (1992) also argue that agents near the end of their careers will receive

higher-powered explicit incentives. Their model is of individual implicit incentives whereas our

model is of team implicit incentives (mutual monitoring and collusion), with our model yielding

predictions that do not arise in theirs based on the distribution of career horizons within

organizations.

Our article is also related to studies on job design problems (e.g., Holmstrom and Milgrom,

1991; Itoh, 1992; Hemmer, 1995). The key insights from these static models is the dependence

of optimal team assignment on technological parameters such as performance signals, production

costs, or productive synergies.5 In a dynamic setting with unverifiable performance measures,

Mukherjee and Vasconcelos (2011) show that a team assignment that resolves the multitasking

problem requires larger bonuses (paid out less often), which increases the principal’s incentive to

renege on the promised bonus. Based on a repeated version of Itoh (1991), Ishihara (2017) shows

that the optimal organizational structure is either specialization in which agents focus on their

own tasks or teamwork in which agents within a team help each other. These studies are of

relational contracts between a principal and agents, whereas our focus is on implicit contracts

between agents and the impact of those implicit contracts on team composition.

There is also related research on team composition (or task assignment) for heterogeneous

agents. In Meyer (1994), a principal optimally pairs a junior manager with a senior manager to

enhance the principal’s learning about the junior’s type because the senior’s type is already

known. Lazear (1999) highlights gains from skill complementarity/knowledge and costs from

inefficient communication that arise in multi-cultural teams. Kaya and Vereshchagina (2014)

study endogenous team composition and analyze how the cost of preventing free-riding problems

affects team assignment depending on the organizational form (partnerships vs. corporations).

Amaldoss and Staelin (2010) study strategic alliances and show how alliance structures (same-

function alliances vs. cross-function alliances) affect individual firms’ investment behaviors.

None of these articles develop a role for implicit incentives that agents provide to each other.

Glover and Kim (2019), which study productive diversity vs. specialization, is an exception.

                                                                                                                         5  Che and Yoo (2001) discuss a job design interpretation of their findings. However, agents are identical in their model, so they are silent about team composition.  

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Productive diversity does not create a demand for asymmetric contracts, whereas horizon

diversity (our focus) does.

The remainder of the article is organized as follows. Section 2 provides a motivating

example. Section 3 describes the team assignment model. Section 4 studies expected career

horizon (discount factor) diversity with regret-free collusion. Section 5 studies how our results

change with the alternative notion of collusion. Section 6 concludes.

2.   Motivating Example

Consider a firm with four agents: two of type h and two of type l. The types (h or l) are

observable and characterize the agent’s expected career horizon summarized by their discount

factors: the two of agent hs have a discount factor 𝛿" and the two agent ls have a discount factor

𝛿#, where 𝛿" > 𝛿#. The discount factor can be interpreted as the probability an agent will survive

to the next period, so can be thought of as capturing the agent’s expected career horizon. To keep

things simple (and avoid the issue of turnover), we study infinitely-lived agents whose discount

factors capture the agent-specific time value of money.6 The principal permanently groups the

agents into two teams for repeated team production. The principal can group the same types h

and h into one team and l and l into the other team (homogeneous teams) or mix different types h

and l into each team (diverse teams). For convenience, we refer to the principal as “she” and the

agent as “he.” Each task is independent and has an outcome of 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 = 8 or 𝑓𝑎𝑖𝑙𝑢𝑟𝑒 = 0

depending on the agents’ efforts. Each agent’s effort is either 0 at no personal cost or 1 at cost of

𝑐 = 1. Within a team, each agent’s effort contributes to production symmetrically. The

probability of 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 is given as follows.

effort (1,1) (1,0) (0,0)

probability 0.9 0.75 0.51

There is an assumed productive substitutability: each agent’s marginal productivity is greater

when the other agent is shirking.

                                                                                                                         6  If we were to model turnover explicitly, we would also need to assume that there is a sufficient supply of each type of agent waiting in the wings who can observe the previous play: if one of the agents within a team exits, a waiting agent with the same discount factor would replace the existing agent.  

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In addition to deciding on team composition, the principal designs an explicit contract for

each team to maximize her lifetime expected payoff. We confine attention to stationary incentive

contracts (the same periodic wages are provided in all periods). In this stationary environment,

maximizing the principal’s lifetime expected payoff is equivalent to maximizing her per-period

payoff.

Assume that agents (within a team) can observe each other’s effort choices. The mutual

observation of efforts and repeated production generate implicit incentives within a team. The

role of the explicit contract the principal offers to the agents is to elicit desirable mutual

monitoring incentives while preventing undesirable collusion incentives. The agents’ incentives

to sustain (work, work) are created by a punishment—playing the worst Nash equilibrium of

their one-shot (stage) game—that is triggered upon free-riding. Also, to prevent collusion, the

agents must find the (work, work) equilibrium more appealing than any collusive strategy.

Implicit contracting between agents (within a team) is sustained only if both agents agree to

do so (it is Pareto optimal for the agents) and constitutes a subgame perfect equilibrium in their

subgame. The principal can treat the agents (within a team) asymmetrically by offering them

each different contracts. These asymmetric explicit contracts may lead to asymmetric implicit

contracts between the agents. To make the problem tractable, we also assume the agents will

play stationary strategies. We allow for correlated randomization in which agents assign the

same probabilities over (work, work), (work, shirk), (shirk, work), and (shirk, shirk) in each

period (unless a punishment is triggered). We will show the binding collusion constraint will

assign positive probabilities only to (work, shirk) and (shirk, work).

An agent’s probabilities of working and shirking as part of collusion determine his per-period

payoff. As such, when an agent strictly prefers collusion, then he is willing to work more often

(by increasing his probability of working) as a bribe so that his teammate is willing to collude. It

turns out that the optimal correlated randomized collusion (the binding collusion constraint) is

asymmetric under both assignments. Because the probabilities the agents play depend on the

contracts offered to them, the principal must find the optimal contract that is immune to any

feasible correlated randomization.

Let 𝑤:; > 0 denote the optimal stationary bonus paid to agent 𝑖, 𝑖 ∈ {ℎ, 𝑙} when 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 is

realized under team composition 𝑘 ∈ {𝑠, 𝑑}, where 𝑠 denotes a homogeneous (similar) team and

𝑑 denotes a diverse team. It is optimal to pay zero if the project fails. We will show that it is

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efficient to offer the asymmetric collusion-proof wage (if the collusion problem arises) to one

agent in each team. Targeting a single agent in each team to combat collusion suffices to restore

the mutual monitoring incentive of the untargeted agent. In this example, the collusion problem

arises in both compositions if 𝛿" > 𝛿# > 0.79. Let 𝑥 = C D EC(F)C D EC(G)

= 2.6. The lowest cost mutual-

monitoring wage for agent 𝑖 ∈ {ℎ, 𝑙} is 𝑤:;∗ =G

F.GKG

GLMN(OEG) across all teams, and the lowest cost

collusion-proof wage for agent 𝑖 ∈ {ℎ, 𝑙} is 𝑤P;∗∗ =G

F.GKMNQ(OEG)

GLMN(OEG) under homogeneous teams and

𝑤R#∗∗ =G

F.GKMSMT(OEG)GLMS(OEG)

under diverse teams.7 Paying the collusion-proof wage to the targeted agent

ensures that there is no way for the untargeted teammate to bribe the targeted agent (by working

more often).

Suppose 𝛿" = 0.9 and 𝛿# = 0.8. In each composition, we have the following optimal wages:

Homogeneous teams:

𝑤P"∗ = 2.73  for one of the agent hs and 𝑤P"∗∗ = 3.54  for the other agent h,  

𝑤P#∗ = 2.92  for one of the agent ls and 𝑤P#∗∗ = 2.99  for the other agent l.  

Diverse teams:

𝑤R"∗ = 2.73  for both agent hs and 𝑤R#∗∗ = 3.15  for both agent ls.

Under homogeneous assignment, the collusion-proof wage for agent ℎ is greater than for

agent 𝑙 (3.54 > 2.99), whereas the mutual-monitoring wage for agent ℎ is smaller than for agent

𝑙 (2.73 < 2.92). Agent ℎ has a relatively strong implicit incentive for both collusion and mutual

monitoring. By contrast, agent 𝑙 has relatively weak implicit incentives. Under diverse

assignment, the collusion-proof wage for agent 𝑙 (targeted agent) is smaller than for agent ℎ but

greater than for agent 𝑙 under homogeneous assignment (2.99 < 3.15 < 3.54). The reason that

𝑤P#∗∗ < 𝑤R#∗∗ is that agent ℎ is willing to work more often (as a bribe) to entice his teammate

agent 𝑙 to collude. As a result, the collusion-proof wage for agent 𝑙 is greater when his teammate

is agent ℎ than when his teammate is agent 𝑙. The binding collusion constraint has the targeted

agent 𝑙 working with probability 0.365 when his untargeted teammate is agent ℎ but working

                                                                                                                         7  Glover and Kim (2019) show that, under productive substitutability, the one-shot stage game equilibrium (used as a punishment to foster mutual monitoring) depends on the discount factors. For small discount factors, the stage game equilibrium is either (work, shirk) or (shirk, work); for high discount factors, the stage game equilibrium is (shirk, shirk). We will show that, in our model, the unique stage game equilibrium is (shirk, shirk) if the collusion problem arises (i.e., for high discount factors).  

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with probability 0.393 when his untargeted teammate is agent 𝑙. This is a disadvantage of diverse

assignment.

However, by targeting agent 𝑙, the principal can ignore agent ℎ’s strong incentive for

collusion. When the two of agent ℎs work together as a team, due to their strong implicit

incentive for collusion, the collusion-proof wage for either agent ℎ is greater than for agent 𝑙:

𝑤R#∗∗ < 𝑤P"∗∗. Therefore, the targeting approach allows the principal to exploit agent 𝑙’s weak

incentive for collusion and to restore agent ℎ’s strong incentive for mutual monitoring. These are

the benefits of diverse assignment.

Taken together, the principal’s expected per-period net payoff for each composition is:

homogeneous: 0.9× 8 − 2.73 − 3.54 + 0.9× 8 − 2.92 − 2.99 = 3.438

diverse: 2×0.9× 8 − 2.73 − 3.15 = 3.816.

Therefore, the principal is better off under diverse team assignment.

This example highlights a latent aspect of team diversity. When the team production is

subject to collusion problems, team diversity in terms of discount factors (career horizons) is

optimal because the principal assigns one easy-to-target agent to each team, which relaxes both

the collusion and mutual monitoring constraints across teams. This assignment is efficient

because asymmetric targeting not only exploits a weak collusion incentive of the agent with the

low discount factor but also restores a strong mutual monitoring incentive of the agent with the

high discount factor. If instead all agents have sufficiently low discount factors, the collusion

constraints do not bind, and it suffices to provide all agents with incentives for mutual

monitoring. In this case, the cost of providing incentives is not affected by team assignment. As

we will show toward the end of this paper, for intermediate discount factors, the results will

depend on the notion of collusion (regret-free as in this example vs. ex ante).

3.   Model

A principal assigns four agents to perform two tasks in each period. Each task requires two

agents who each makes a binary effort decision 𝑒 ∈ 0,1 at personal cost 𝑐𝑒, where 𝑒 = 1

denotes work and 𝑒 = 0 denotes shirk. The effort choices are not observed by the principal. The

agents have publicly observable types, ℎ or 𝑙, and there are two agents of each type. The type

characterizes the agent’s discount factor: type ℎ agent has a discount factor 𝛿", and type 𝑙 agent

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has a discount factor 𝛿#, where 𝛿" > 𝛿#. For agents who are approaching a voluntary retirement

and/or agents who plan to sit out the labor market earlier than others for other reasons, his 𝛿 will

be smaller. Thus, 𝛿 captures the probability of continuing in the work relationship, which affects

the agents’ implicit incentives.8 Let 𝛿[ > 0 denote the principal’s discount factor.

There are two possible team assignments: two of agent ℎ perform one task together and two

of agent 𝑙 perform the other task, which we call homogenous teams, or two sets of agent ℎ and 𝑙

perform each task, which we call diverse teams. If type 𝑖, 𝑗 ∈ {ℎ, 𝑙} are matched to perform a task

as a team with unobservable effort (𝑒;, 𝑒]), then the task generates a success 𝑆 > 0  with

probability 𝑓 𝑒;, 𝑒] ∈ (0,1) or a failure 𝐹 = 0  with probability 1 − 𝑓 𝑒;, 𝑒] . 𝑓 𝑒;, 𝑒] is

increasing in the agents’ efforts. The production technology for each task is independent and

identical. Within a team, each agent’s effort contributes to production symmetrically (𝑓 0,1 =

𝑓 1,0  for  all  𝑖, 𝑗). Because the agents’ contributions are symmetric within a team, for notational

convenience, we use 𝑓 𝑒;; to denote the production function. We assume that team

production exhibits a productive substitutability.

𝑓 2 − 𝑓 1 < 𝑓 1 − 𝑓 0 .

That is, each agent’s marginal productivity is greater when his teammate is shirking than when

he is working.

Due to their close work interactions, each agent can observe the effort choice of the other

agent within the team. However, communication from the agents to the principal about their

observations of each other’s actions is blocked.9 We focus on implicit side contracting between

agents instead of explicit side contracting considered in Itoh (1993). The agents’ effort strategies

map any possible history into current effort decisions. Without loss of generality, we restrict

attention to grim trigger strategies for the agents. We also assume the principal’s decision on

team composition is made at the start of the relationship and permanent once determined.

                                                                                                                         8  Heterogeneity in 𝛿 within the workforce is prevalent. Age diversity is one example. Even if they are the same age, employees may have different plans for career breaks or leaves. In this regard, the heterogeneity in 𝛿 can also capture distinct probabilities of continuing work relationship that depend on gender. For example, the total years in the workforce are, on average, 29 years for women and 38 years for men (Garnick, 2016). Women take more career breaks than men do (Pew Research Center survey, 2013). 9  We rule out message games between the principal and agents in order to highlight the role of implicit incentives for effort between agents. See related discussions in Arya, Fellingham, Glover (1997), Che and Yoo (2001), Kvaloy and Olsen (2006), and Baldenius, Glover, and Xue (2016).  

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For analytical tractability, we assume stationary wage contracts that have wages depending

only on current period performance and that are applied to all subsequent periods once

determined at the beginning of the work relationship. However, we allow for asymmetric

contracting. Let 𝑤:; ≥ 0  and  𝑣:; ≥ 0 denote the principal’s payments to agent 𝑖 ∈ {ℎ, 𝑙} in team

𝑘 ∈ {𝑠, 𝑑} contingent on performance 𝑆 and 𝐹, respectively, where 𝑠 and 𝑑 represent

homogeneous and diverse teams. The non-negativity constraint can be interpreted as capturing

the agents’ limited liability and is the source of the contracting friction, along with the

unobservability of their actions by the principal. To highlight the role of team incentives in team

composition, we assume the principal always wants to elicit 𝑒 = 1 from both agents in each

period. All parties are risk neutral, and each agent’s reservation utility is normalized to zero.

The principal’s objective is to maximize her lifetime expected payoff by solving the

assignment and contracting problems: at the beginning of the relationship, she 1) (permanently)

assigns agents to teams and 2) designs a stationary wage contract to induce each agents to work

(e = 1) as a Pareto-undominated subgame-perfect equilibrium. For each composition, the wage

contracts are said to be optimal if the principal elicits (work, work) at the minimum cost. With

optimal contracts, a team composition is said to be optimal if the principal’s expected payoff

under that team composition is the highest among all other compositions. The optimal team

composition with the optimal contract maximizes the principal’s lifetime expected payoff. In this

stationary environment, this is equivalent to maximizing the principal’s per-period payoff. Thus,

we do not use 𝛿[ throughout the article and instead focus on the principal’s per-period payoff.

The principal either assigns the same types for each task, (ℎ, ℎ) and (𝑙, 𝑙), or mixes the types,

(ℎ, 𝑙).

Benchmarks: First-best and One-shot Games Consider the first-best solution with no

moral hazard. Because homogeneous teams dominate diverse teams in terms of productivity

without any frictions, this leads to a positive assortative assignment: ℎ and ℎ for one task and 𝑙

and 𝑙 for the other. To see this, suppose that agents’ efforts are observable to the principal and

verifiable/contractible. Thus, each agent is paid 𝑐 for their effort 𝑒 = 1, and the principal’s

expected payoff (depending on team composition) is:

𝑓 2 + 𝑓 2 𝑆 − 4𝑐.

13  

Under moral hazard in the one-shot game, the principal designs a contract that ensures (work,

work) is a stage game Nash equilibrium, which yields her an expected payoff of:

𝑓 2 + 𝑓 2 𝑆 − 4𝑐

𝑓 2 − 𝑓(1).

The principal’s payoff remains the same regardless of team assignment. So, static incentives

make the two assignments identical from an incentive standpoint.

4.   Diversity in Career Horizons: heterogeneity in 𝜹

In this section, we investigate diversity in expected career horizons between agents within a

team. We first derive the mutual monitoring incentive compatible wages, initially ignoring the

threat of collusion.

Mutual Monitoring Recall that agents sustain the (work,work) equilibrium using the stage

game equilibrium as a punishment upon any deviation. When team production exhibits a

productive substitutability, the stage game equilibrium that serves as the punishment depends on

the discount factor. See Glover and Kim (2019) for the case of agents who have identical

discount factors. In particular, if the production function exhibits a weak substitutability and the

discount factor is not too low, then both agents choosing shirk is the unique equilibrium of their

one-shot game and is used as the punishment. If the discount factor is sufficiently low, there are

two stage-game equilibria: one agent chooses shirk while the other chooses work and vice versa.

Thus, the mutual monitoring incentive compatibility constraints depend on the discount factor.

Because 𝛿" > 𝛿#, it is possible that the stage game equilibrium differs across teams. For implicit

contracting between agents to be feasible, we assume that 𝛿" and 𝛿# are either low enough or

high enough so that the stage game equilibrium used as a punishment is well-defined.10

                                                                                                                         10  If (shirk, shirk) is used as the punishment under a team of agent hs, whereas (work, shirk) and (shirk, work) are used under a team of agent ls, then a diverse team has no pure strategy stage game equilibrium the agents can use as a punishment. The excluded case results in a mixed strategy stage game equilibrium (and no equilibrium in pure strategies). Including this case would complicate our analysis but not affect our results qualitatively.    

14  

Assumption.

A.1 Either 𝛿# < 𝛿" ≤ 𝛿l or 𝛿" > 𝛿# ≥ 𝛿l, where 𝛿l = DC G EC D EC FC G EC F

∈ (0,1). This ensures

that 𝛿" and 𝛿# are either both sufficiently small that the stage game equilibria are (work, shirk)

and (shirk, work) or sufficiently large that the unique stage game equilibrium is (shirk, shirk).

For agent 𝑖 in team 𝑘, the mutual monitoring incentive compatible (M-IC) constraints are:

𝑓 2 𝑤:; − 𝑐 ≥ 1 − 𝛿; 𝑓 1  + 𝛿;𝑓 0 𝑤:; or (M-IC)

𝑓 2 𝑤:; − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤:;  + 𝛿; 𝑓 1 𝑤:; − 𝑐 .

Throughout the article, we normalize both sides of the constraints by multiplying by 1 − 𝛿; .

The left hand side represents the present value of the agent’s expected payoff from working in all

periods and the right hand side captures the agent’s payoff from deviating and being punished in

all future periods, either bilateral shirking (in the first constraint) or the deviating agent’s

working accompanied by the non-deviating agent’s shirking (in the second constraint). We

formulate the program for the principal’s contracting problem in Appendix A. The following

lemma presents the optimal mutual-monitoring wage as a function of the discount factors.

Lemma 1. (Mutual-Monitoring Wage) Let 𝛿l ∈ (0,1) denote the value of 𝛿; at which the

punishment equilibrium changes from (work, shirk) or (shirk, work) to (shirk, shirk). Suppose 𝛿"

and 𝛿# are both sufficiently low that the collusion constraints do not bind for either assignment.

Then, all agents are offered the mutual-monitoring wage:

i)   𝑤:;∗ =GEMN m

C D EC G if 𝛿# < 𝛿" ≤ 𝛿l,

ii)   𝑤:;∗ =m

GEMN C D EC G LMN C D EC F if 𝛿l ≤ 𝛿# < 𝛿".

In this case, the total expected payments are identical under homogenous and diverse teams.

To foster mutual monitoring, the principal must incorporate the (M-IC) constraints for both

agents when determining explicit contracts and pays the mutual-monitoring wages which depend

on 𝛿;. This makes the optimal contracts symmetric,11 and the total expected cost of providing

                                                                                                                         11  The only difference in the mutual-monitoring wages for different agents results from their discount factors.      

15  

incentives does not depend on the team composition. In this case, heterogeneous discount factors

do not play an important qualitative role.

Collusion Mutual observation of effort within a team also creates opportunities for unwanted

collusive behavior. In particular, under strategic substitutes, agents may find playing collusive

strategies other than working appealing. To make the problem tractable, we confine attention to

collusive strategies that are stationary but allow for an asymmetric treatment by introducing

correlated randomization. Moreover, if the productive substitutability is strong, then the

possibility of collusion can also upset the (shirk, shirk) stage-game equilibrium. To avoid this

possibility, we assume that the productive substitutability is a weak enough one that this does not

occur.

Assumption.

A.2. C G EC FC D EC G

< GL KD

. This ensures the collusion-proof wage does not upset the (shirk, shirk)

equilibrium.

Consider team  𝑘 consisting of agents 𝑖 and 𝑗, 𝑖, 𝑗 ∈ {ℎ, 𝑙}. The agents are assumed to have

access to a randomized device that will assign any probabilities to the action profiles 𝑒;, 𝑒] :=

1,1 , 1,0 , 0,1 , and 0,0 , where 𝑖 = 𝑗 if 𝑘 = 𝑠, and 𝑖 = ℎ and 𝑗 = 𝑙 if 𝑘 = 𝑑. Once assigned

to play a particular action profile by the randomization device, that action profile must be self-

enforcing. Under a productive substitutability, the agents may find it profitable to collude on

1,0 and 0,1 with positive probabilities but never (0,0) or (1,1). (We will prove this in Lemma

2.) Let 𝛽P ∈ 0,1 denote the probability that the first agent 𝑖 in homogenous teams works (𝑒; =

1) and the second agent 𝑗 shirks (𝑒] = 0), and the other way around with probability 1 − 𝛽P.

Similarly, let 𝛽R ∈ 0,1 denote the probability that agent ℎ works and agent 𝑙 shirks, and the

other way around with probability 1 − 𝛽R. Then, under collusion, the per-period expected payoff

of each agent conditional on the team composition 𝑘 is:

𝑈; 𝛽:;𝑤:; = 𝛽: 𝑓 1 𝑤:; − 𝑐 + 1 − 𝛽: 𝑓 1 𝑤:;    and

𝑈] 𝛽:;𝑤:] = 𝛽:𝑓 1 𝑤:

] + 1 − 𝛽: 𝑓 1 𝑤:] − 𝑐 .

16  

In order for collusion to arise, (i) the agents must find the collusion Pareto dominating

relative to (work, work) in any states, and (ii) the collusion must be self-enforcing. To upset

collusion, it is sufficient for the principal to violate (i) or (ii). We focus our attention on regret-

free collusion. That is, the two agents strictly prefer collusion regardless of whether they play

1,0 or 0,1 .12 Essentially, our focus on regret-free collusion requires that each agent’s payoff

(from any collusion) be bounded below from the payoff from the desirable action, (work, work),

at any history on the equilibrium path. Because our results qualitatively depend on the notion of

collusion we use, in Section 5, we also consider alternative notion: agents may collude as long as

such play is self-enforcing and it gives them a greater expected payoff than joint working.

To break condition (i), the contract must satisfy the following (Pareto) constraint:

𝑓 2 𝑤:; − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤:; − 𝑐 + 𝛿;𝑈; 𝛽:; 𝑤:;      for  all  𝛽:. (Pareto)

The left hand side of the (Pareto) constraint is the present value of the expected payoff from

working in the current and all future periods, whereas the right hand side represents the expected

payoff of the agent who is supposed to work in the current period under the proposed collusion

and adheres to the proposed collusion in all future periods.13 The optimal contract must be

collusion-proof for any feasible 𝛽:.

To break condition (ii), the contract must ensure that an agent who is supposed to work today

wants to deviate:

𝑓 0 𝑤:; ≥ 1 − 𝛿; 𝑓 1 𝑤:; − 𝑐 + 𝛿;𝑈 𝛽:;𝑤:; . (No Self-Enforcing Collusion)

The left hand side of the (No Self-Enforcing Collusion) constraint is the present value of the

expected payoff from deviating from the collusion to shirk and being punished by joint shirking

indefinitely, and the right hand side is the expected payoff under working today and adhering to

the collusion in all future periods. We will show in the proof of Proposition 1 that the optimal

(least costly) means of preventing collusion while maintaining the mutual monitoring incentive

for the principal is to use the (Pareto) constraint.

                                                                                                                         12  Most breakups of collusion (among bidders in auctions or companies in cartels) in practice often result from whistle-blowing or confessions by a dissatisfied player who eventually turns in his conspirators (McAfee and McMillan, 1992; Spagnolo, 2008; Levenstein and Suslow, 2011). Such potential obstacles can voluntarily break collusion even without any explicit (collusion-proof) contracts or regulatory agencies. We focus on regret-free collusion in order to highlight how an optimal contract—which is immune to regret-free collusion—interacts with a decision of optimal team composition. 13  Under the incentive contract in which one agent strictly prefers collusion and the other agent is indifferent, they may sustain collusion.  To break condition (i), the principal has to make the (Pareto) constraint strict by paying slightly more to the agent who is indifferent.

17  

Although there is no direct monetary transfer between agents within a team, the agents can

adjust their payoffs by fine-tuning 𝛽: which also depends on the wages offered to the agents.

Fine-tuning 𝛽: can be interpreted as a bribe: when agent 𝑖 (a briber) who works with probability

𝛽: wants to induce his teammate agent 𝑗 (a bribee) to collude, agent 𝑖 can do so by increasing his

working probability 𝛽:. We will show in the proof of Lemma 2 that the set of feasible collusion

is C D EC Gm

𝑤:], 1 − C D EC G

m𝑤:; and this set is not empty. If the contract is immune to the most

demanding (from the principal’s perspective) collusion, then it is immune to any feasible

collusion. The binding collusion constraint is when there are no more bribes between agents. For

the team of agent 𝑖 and agent 𝑗 under team 𝑘, the most demanding collusion is derived as

follows:

maxst

𝑓 1 𝑤:; − 𝑐 1 − 𝛿; − 𝛿;𝛽:𝑐, 𝑓 1 𝑤:] − 𝑐 1 − 𝛿] − 𝛿] 1 − 𝛽: 𝑐   , (1)

where the expressions in (1) are the two agents’ expected payoffs when they are supposed to

work in the current period under the proposed collusion and adhere to the proposed collusion in

all future periods (i.e., the right hand side of (Pareto) constraint). As shown in (1), for a given

wage, agent 𝑖’s payoff is linearly decreasing in 𝛽:, whereas his teammate agent 𝑗’s payoff is

linearly increasing in 𝛽:. Thus, the most demanding collusion is when one of agent 𝑖 and 𝑗 works

with the smallest feasible probability (and his teammate works with the largest feasible

probability), or the other way around: the briber agent who works with the largest feasible

probability cannot bribe the bribee agent beyond the largest feasible probability.

We show in Lemma 2 that the least costly collusion-proof contract is asymmetric: the bribee

agent receives a high-powered collusion-proof wage and the briber agent receives a low-powered

mutual-monitoring wage. To illustrate, suppose agent 𝑗 is the targeted bribee: the expression (1)

obtains the maximum when agent 𝑗 (agent 𝑖) works with the smallest (the largest) feasible

probability, i.e., 𝛽: = 1 − C D EC Gm

𝑤:; . Rewriting the bribee’s (Pareto) constraint yields:

𝑓 2 − 𝑓 1 𝑤:] + 𝛿]𝑤:; ≥ 𝛿]𝑐.

By paying a high-powered wage to the bribee agent, it makes the bribee regret once it is his turn

to work. Although paying a high-powered wage to the briber agent 𝑖 also relaxes agent 𝑗’s

(Pareto) constraint, it is more efficient to directly increase the bribee’s wage because the bribee

18  

agent 𝑗 discounts such a bribe by his discount factor 𝛿]. The following lemma summarizes the

discussion.

Lemma 2. (Binding Collusion Constraint) For any contract that fosters mutual monitoring, if

collusion arises, it will have the agents playing (1,0) and (0,1) with positive probabilities but not

(0,0) and (1,1). Suppose that agent 𝑖 works with probabitliy 𝛽: and agent 𝑗 works with

probability 1 − 𝛽:. For any contract that is collusion-proof, we obtain the optimal correlated

randomization for team composition 𝑘 as follows.

i)   When agent 𝑖 is the briber and agent 𝑗 is the bribee, the optimal correlated

randomization sets 𝛽:∗ = 1 − C D EC Gm

𝑤:; and the minimum necessary collusion-proof

wage is 𝑤:]∗∗ = mMust

∗

C D EC G.

ii)   When agent 𝑗 is the briber and agent 𝑖 is the bribee, the optimal correlated

randomization sets 𝛽:∗ =C D EC G

m𝑤:] and the minimum necessary collusion-proof

wage is 𝑤:;∗∗ =mMN GEst

∗

C D EC G.

Recall that the principal need upset only one of the agents’ incentives to collude, so offers

one agent in each team the more costly collusion-proof wage and the other agent in that team the

mutual-monitoring wage. The principal may offer a symmetric collusion-proof wage to both

agents—in which case the optimal correlated randomization is also altered—however, we show

that asymmetric contracting is less costly than symmetric one. The intuition is because paying

symmetric collusion-proof wages does not make use of the agents’ (desirable) mutual-

monitoring incentives.

Observe that the minimum necessary collusion-proof wage is increasing in the bribee agent’s

discount factor. Recall that the mutual-monitoring wage for agent 𝑖 is decreasing in his discount

factor 𝛿;. Hence, under diverse teams, agent ℎ is offered the mutual-monitoring wage and agent 𝑙

is offered the collusion-proof wage, whereas under homogenous teams, one of agent ℎs and one

of agent 𝑙s are each offered the collusion-proof and mutual-monitoring wages. It is useful to

define 𝑥 ≡ C D EC(F)C D EC(G)

and rewrite the mutual-monitoring wage as 𝑤P;∗ =m

C D EC(G)G

GLMN(OEG). Due to

19  

weak substitutability assumption, we have 2 < 𝑥 < wL KD

. The following proposition states the

optimal contract for each composition.

Proposition 1. (Optimal Contract) Let 𝑥 ≡ C D EC FC D EC G

. Under homogenous assignment, the

collusion constraint binds for 𝛿; > 1/(𝑥 − 1). Under diverse assignment, the collusion

constraint binds for 𝛿" > 1/(𝑥 − 1) and 𝛿# > 𝛿 𝛿" , where 𝛿 𝛿" < 1/(𝑥 − 1) is a

decreasing function of 𝛿" and the expression of 𝛿 𝛿" is presented in the Appendix A. Suppose

𝛿" and 𝛿# are both sufficiently high that the collusion constraints bind for both team

assignments.

i)   Under homogeneous teams, the optimal contract arbitrarily selects one of the two

agents in each team to receive the collusion-proof wage, 𝑤P;∗∗ =m

C D EC GMNQ(OEG)

GLMN(OEG) ,

and the other agent receives the mutual-monitoring wage, 𝑤P;∗ =m

C D EC(G)G

GLMN(OEG).

ii)   Under diverse teams, the principal offers the collusion-proof wage,  𝑤R#∗∗ =m

C D EC GMSMT(OEG)GLMS(OEG)

, to agent 𝑙 and offers the mutual-monitoring wage, 𝑤R"∗ =

mC D EC(G)

GGLMS(OEG)

, to agent ℎ.

To understand the reason why the diverse teams have different thresholds for the agents’

discount factors, recall that agent 𝑙 has a relatively weaker incentive to collude than agent ℎ.

Thus, when agent ℎ is the briber and agent 𝑙 is the bribee, the briber agent ℎ must work more

often (as a bribe) to induce agent 𝑙 to collude. If agent 𝑙’s discount factor is low enough that

agent ℎ’s bribe cannot induce agent 𝑙 to collude, then the collusion constraint does not bind even

when 𝛿" > 1/(𝑥 − 1) (i.e., if 𝛿# ≤ 𝛿 𝛿" ). In this case, under homogenous assignment, the

collusion problem arises in the team of agent ℎs, whereas under diverse assignment, the

collusion problem does not arise. Figure 1 depicts a threshold for 𝛿; at which the collusion

constraint binds in each composition.

When the collusion constraints bind across all teams (𝛿" > 𝛿# > 1/(𝑥 − 1)),

heterogeneous discount factors serve an important role in combatting collusion more efficiently.

20  

Although the optimal contracts are asymmetric under both homogeneous and diverse assignment,

diverse assignment facilitates a more targeted (and less costly) approach. In each team, the

impatient agent 𝑙 is targeted to upset collusion. Agent ℎ would like to collude but finds agent 𝑙

unwilling. Moreover, agent ℎ has a lower mutual-monitoring wage than agent 𝑙. The asymmetric

collusion-proof wage for each agent in each composition is summarized in Table 1. Figure 2 in

Appendix B depicts the optimal contracts based on a numerical example: mutual-monitoring

wages are decreasing in the agents’ discount factors, whereas the collusion-proof wages are

increasing in the agents’ discount factors.

Proposition 1 also has implications for the disparity in pay-for-performance sensitivity within

teams. Without collusion, every agent is offered a mutual-monitoring wage. When all team

members have the same discount factors (homogenous teams), they have the same pay-for-

performance sensitivity and, thus, no disparity in pay-for-performance sensitivity arises under

homogenous teams.

However, when the agents within a team have different discount factors (diverse teams), each

agent’s mutual-monitoring wage depends on his discount factor, thereby creating a disparity

Homogenous assignment Diverse assignment

Figure 1. Binding Collusion Constraints

Figure 1 depicts the region in which the collusion constraint binds in each composition. If 𝛿" > 𝛿# >y1/(𝑥 − 1), the collusion constraints bind in both teams under both assignments (dark gray area). If 𝛿" > y1/(𝑥 − 1) > 𝛿(𝛿") ≥ 𝛿#, the collusion constraint binds only under the team of agent ℎs (light gray area). If 𝛿" > y1/(𝑥 − 1) > 𝛿# > 𝛿(𝛿"), the collusion constraint binds under the team of agent ℎs (light gray area) and under the two teams in diverse assignment (dark gray area).

21  

within teams: a long-horizon agent receives a lower mutual-monitoring wage than his short-

horizon teammate. As both agents in diverse teams become more patient (high discount factors),

their mutual-monitoring wages decrease and the disparity in pay-for-performance sensitivity

decreases too.

Once collusion is a pressing concern, the optimal asymmetric contracts create a disparity in

pay-for-performance sensitivity across all teams: one agent receives a higher-powered collusion-

proof wage and the other agent receives a lower-powered mutual-monitoring wage. In this case,

the qualitative nature of the disparity is the same in both homogeneous and diverse teams: the

disparity within teams is increasing in the agents’ discount factors. Figure 3 in Appendix B

depicts the disparity in pay-for-performance sensitivity based on a numerical example.

We now investigate the overall efficiency of each composition with respect to both 𝛿" and 𝛿#.

The following proposition presents the conditions under which the total wages under diverse

teams are strictly less than under homogenous teams and characterizes how the difference in the

total wages of homogeneous and diverse teams changes with respect to discount factors.

Agent ℎ Agent 𝑙

Homogeneous

mC D EC(G)

MSQ(OEG)

GLMS(OEG) (C), m

C D EC(G)MTQ(OEG)

GLMT(OEG) (C)

mC D EC(G)

GGLMS(OEG)

(M), mC D EC(G)

GGLMT(OEG)

(M)

Diverse m

C D EC(G)G

GLMS(OEG) (M), m

C D EC(G)MSMT(OEG)GLMS(OEG)

(C)

Table 1. Optimal Asymmetric Contracts

This table summarizes the optimal asymmetric contracts for homogeneous teams and diverse

teams when the collusion constraints bind, where C denotes the collusion-proof wage and M

denotes the mutual-monitoring wage. Under homogeneous teams in which the same type of

agents work together as a team, one agent ℎ (𝑙) receives the collusion-proof wage and the other

agent ℎ (𝑙) receives the mutual-monitoring wage. Under diverse teams in which agent ℎ and 𝑙

work together as a team, agent ℎ receives the mutual-monitoring wage and agent 𝑙 receives the

collusion-proof wage.

22  

Proposition 2. Let 𝑊: ≡ 2 𝑤:;;{",# denote the total wages under team 𝑘.    

i)   For 𝛿# < 𝛿" ≤ 1/(𝑥 − 1), 𝑊P = 𝑊R.

ii)   For 𝛿# ≤ 𝛿(𝛿") < 1/(𝑥 − 1) < 𝛿", the collusion constraint binds only under the

team of agent  ℎs. Under the diverse assignment, the collusion problem does not arise,

and 𝑊P −𝑊R =m

C D EC GMSQ OEG EGGLMS OEG

> 0.

iii)   For 𝛿 𝛿" < 𝛿# ≤ 1/(𝑥 − 1) < 𝛿", the collusion constraints bind under the team of

agent  ℎs and under the diverse assignment, and 𝑊P −𝑊R = 𝛿" − 2𝛿# +D

GLMT OEG−

GLMSEDMTGLMS(OEG)

> 0.

iv)   For 1/(𝑥 − 1) < 𝛿# < 𝛿", the collusion constraints bind across all teams, and

𝑊P −𝑊R =m(OEG)(MSEMT)C D EC G

GL MSEMT LMSMT(OEG)GLMS OEG GLMT OEG

> 0.

Whenever 1/(𝑥 − 1) < 𝛿", we have 𝑊P −𝑊R > 0. Provided that 𝑊P −𝑊R > 0, | }~E}�|MS

>

0 ≥ |(}~E}�)|MT

.

As shown previously in Lemma 1, when there is no collusion in both compositions, then the

total mutual-monitoring wages are the same and the team composition is irrelevant (with respect

to total wages). If the collusion constraint binds in composition 𝑘 only, then the total wages of

that composition are strictly greater than the other composition. This happens if 𝛿# ≤ 𝛿(𝛿") and

1/(𝑥 − 1) < 𝛿". In this case, the collusion constraint binds under the team of agent ℎs only,

and we have 𝑊P −𝑊R > 0. Even for 𝛿 𝛿" < 𝛿# ≤ 1/(𝑥 − 1) < 𝛿" (the collusion problem

arises both teams under diverse assignment but does not arise in the team of agent 𝑙s), we show

that 𝑊P −𝑊R > 0. The intuition is because exploiting agent 𝑙’s weaker incentive for collusion

(by assigning them across teams under diverse assignment) is better than combating one of agent

ℎs’ stronger incentive for collusion and utilizing two of agent 𝑙s’ weaker incentive for mutual

monitoring (under homogenous assignment).

Both compositions face collusion problems if 𝛿" and 𝛿# are both sufficiently high,

1/(𝑥 − 1) < 𝛿# < 𝛿". In this case, we still have 𝑊P −𝑊R > 0. Asymmetric contracts are

optimal across all teams however, career horizon diversity makes asymmetric contracting more

23  

efficient because the principal efficiently targets agent 𝑙s by assigning them across teams under

diverse assignment. Moreover, as 𝛿" increases, the incentive gap 𝑊P −𝑊R strictly increases,

whereas, as 𝛿# increases, the incentive gap 𝑊P −𝑊R weakly decreases.

To summarize, team diversity serves an important role when asymmetric contracts are

allowed. First, the principal always targets the two agent 𝑙s for collusion-proof wages under

diverse assignment, whereas the principal targets one agent ℎ and one agent 𝑙 under

homogeneous teams. Targeting agent 𝑙 reduces the payment for agent ℎ who is more tempted to

collude within a team. This is because the principal can ignore agent ℎ’s (Pareto) constraint as

the targeted agent will never agree on collusion. This allows the principal to enjoy a wage

reduction from agent ℎ who has a strong incentive for mutual monitoring without collusion.

Therefore, heterogeneous discount factors strengthen the team incentive efficiency when team

production is subject to collusion and asymmetric contracting is allowed. This result highlights a

latent advantage of fostering team diversity in career horizons.

5.   Extension

So far, we have studied collusion that is ex post regret-free. In this extension, we consider an

alternative notion of collusion: agents may collude as long as the expected payoff from collusion

gives them a greater ex ante (before the correlation device is employed) payoff than (work,

work). To break this ex ante collusion, the contract must satisfy the following (Pareto′)

constraints.

𝑓 2 𝑤:; − 𝑐 ≥ 𝛽: 𝑓 1 𝑤:; − 𝑐 + 1 − 𝛽: 𝑓 1 𝑤:;  or (Pareto′)

𝑓 2 𝑤:] − 𝑐 ≥ 𝛽:𝑓 1 𝑤:

] + 1 − 𝛽: 𝑓 1 𝑤:] − 𝑐    for  all  𝛽:.

The left-hand side of the (Pareto′) constraint is the expected payoff from working and the right-

hand side represents the expected payoff under the proposed collusion. Observe that neither side

of (Pareto′) constraints depends on the agents’ discount factors. Contrary to the regret-free

collusion case, the total collusion-proof wages are the same across all teams given that the

collusion constraint binds. However, as in the regret-free collusion case, when the collusion

constraints bind depends on the team assignment.

Proposition 3. (Ex ante Collusion) Suppose that agent 𝑖 works with probability 𝛽: and agent 𝑗

works with probability 1 − 𝛽: in team 𝑘. For any contract that is collusion-proof, the optimal

24  

correlated randomization under the ex ante collusion is the same as that under the regret-free

collusion: either 𝛽: =C D EC G

m𝑤:] or 𝛽: = 1 − C D EC G

m𝑤:; . The minimum necessary collusion-

proof wages satisfy 𝑤:;∗∗ + 𝑤:]∗∗ = m

C D EC G. In homogeneous teams, the collusion constraint

binds for 𝛿; > 1/(𝑥 − 1), whereas in diverse teams, it binds for 𝛿"𝛿# > 1/ 𝑥 − 1 D. Let 𝑊:

denote the total wages under team 𝑘.

i)   For 𝛿# < 𝛿" ≤ 1/(𝑥 − 1) or 𝛿" > 𝛿# ≥ 1/(𝑥 − 1), 𝑊P = 𝑊R.

ii)   For 𝛿# ≤ 1/𝛿" 𝑥 − 1 D and 1/(𝑥 − 1) < 𝛿", 𝑊P > 𝑊R.

iii)   For 1/𝛿" 𝑥 − 1 D < 𝛿# ≤ 1/(𝑥 − 1) < 𝛿" , 𝑊P < 𝑊R.

If both discount factors are sufficiently low (high) that collusion does not arise in any team

(arises for all teams), heterogeneous and homogeneous assignments are equivalent.

Heterogeneous assignment is optimal when 𝛿" is sufficiently high (𝛿" >G

OEG) and 𝛿# is

sufficiently low (𝛿"𝛿# ≤G

OEG Q). In this case, the collusion problem arises only for a team of hs,

so heterogeneous assignment eliminates the collusion problem and is optimal. However, for

slightly higher values of 𝛿# (𝛿# > 1/𝛿" 𝑥 − 1 D), the collusion problem arises in a team of hs

and in diverse teams, so homogenous assignment eliminates the collusion problem for one of the

teams and is optimal. With ex ante collusion, incentive efficiency is simply characterized by

whether the collusion constraints bind.

6.   Conclusion

In our model, an organization faces a team composition problem due to the various implicit

incentives via repeated work relationships. Without collusion concerns within teams (low

discount factors), team composition is irrelevant and both assignments perform equally well.

When the collusion problem arises for all teams (high discount factors), the principal uses

asymmetric contracting to combat collusion while eliciting mutual monitoring. When collusion

arises for some assignments but not others, the results depend on the notion of collusion we

adopt (regret-free vs. ex ante).

Our results have implications for pay-for-performance sensitivity for multiple agents.

Without collusion concerns, every agent is offered a mutual-monitoring wage. Thus, under

homogeneous assignment, all team members have identical pay-for-performance sensitivity,

25  

whereas under diverse assignment, a pay-for-performance sensitivity disparity arises within the

teams because of individual team members’ distinct discount factors. Once collusion is a

pressing concern, a pay-for-performance sensitivity disparity arises even in homogenous teams

because the optimal contract treats the agents asymmetrically. This asymmetric treatment

increases the pay-for-performance disparity under diverse teams. The broader point is that both

the quantitative and qualitative natures of pay-for-performance disparity depend on the

distribution of career horizons within organizations.

As an extension, one might consider teams of three or more agents, each with their own

(observable) discount factors. Suppose there are three heterogeneous agents in a team with the

most senior agent targeted to receive a collusion-proof wage. Are mutual monitoring incentives

sufficient for the remaining two agents? Whether the prospect of being punished by the most

senior agent is sufficient to deter collusion by the remaining agents seems likely to depend on the

productive contribution made by the senior agent. If there is a residual opportunity of collusion,

then the principal seems likely to target the next most senior agent to upset their collusion. This

suggests that, for 𝑁 ≥ 3 agents, as more senior agents receive collusion-proof wages, it will

gradually reduce junior agents’ incentives to collude and, thus, the collusion-proof wages needed

to prevent them from colluding with ever smaller coalitions.

Despite the early studies (e.g., Itoh, 1993; Arya, Fellingham, and Glover, 1997; and Che and

Yoo, 2001), the role of implicit team incentives within organizations has been relatively

underexplored. The repeated interactions among team members and resulting team incentives are

important in improving our understanding of the performance of various types of teams

including (but not limited to) top management teams, boards of directors, audit engagement

teams, product development teams, etc. In addition to team design and the explicit principal-

multi-agent contracts used to foster implicit incentives within the team that we considered in this

article, understanding the implications of implicit team incentives for job rotation policies (e.g.,

how often to rotate group members) and/or on the choice of performance measures (team vs.

individual, conservative vs. unbiased, aggregate vs. disaggregate, etc.) seem to be natural next

steps.

26  

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Appendix A.

For the mutual-monitoring wage, recall our definition of 𝑥 = C D EC(F)C D EC(G)

and rewrite the

expression as follows:

𝑐1 − 𝛿; 𝑓 2 − 𝑓 1 + 𝛿; 𝑓 2 − 𝑓 0

=𝑐

𝑓 2 − 𝑓(1)1

1 + 𝛿;(𝑥 − 1).

29  

Due to our assumption of weak productive substitutability, 𝑥 ∈ 2, wL KD

. We use both

expressions for the mutual-monitoring wage interchangeably in the appendix.

The Programs for heterogeneous discount factors.

In homogeneous teams, the same types of agents work together. To distinguish the agents in

each team of agent 𝑖s, 𝑖 ∈ {ℎ, 𝑙}, we denote by 𝑤P;�, 𝑛 = 1,2, the wage for 𝑛th agent in the team

of agent 𝑖s.

1)   Homogeneous teams

max�~N�

𝑓(2) 𝑆 − 𝑤P"G − 𝑤P"D + 𝑓 2 𝑆 − 𝑤P#G − 𝑤P#D

Subject to

𝑓 2 𝑤P;� − 𝑐 ≥ 0, for  𝑖 = ℎ  and  𝑙, and  𝑛 = 1  and  2, (IR)

𝑓 2 𝑤P;� − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤P;� − 𝑐 + 𝛿;𝑈; 𝛽P; 𝑤P;�    for  all  𝛽P, 𝑖 = ℎ  𝐚𝐧𝐝  𝑙, 𝑛 = 1 or 2, (Pareto)

𝑓 2 𝑤P;� − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤P;� + 𝛿;𝑓 0 𝑤P;�, for  𝑖 = ℎ  and  𝑙, and  𝑛 = 1  and  2, (M-IC)

𝑓 2 𝑤P;� − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤P;� + 𝛿;(𝑓 1 𝑤P;� − 𝑐), for  𝑖 = ℎ  and  𝑙, and  𝑛 = 1  and  2.

In diverse teams, agent ℎ and agent 𝑙 work together as a team, thus, their types fully

characterize their identities.

2)   Diverse teams

max��N2×𝑓 2 𝑆 − 𝑤R" − 𝑤R#

Subject to

𝑓 2 𝑤R; − 𝑐 ≥ 0, for  𝑖 = ℎ  and  𝑙, (IR)

𝑓 2 𝑤R; − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤R; − 𝑐 + 𝛿;𝑈; 𝛽R; 𝑤R;  for  all  𝛽R, 𝑖 = ℎ  𝐨𝐫  𝑙, (Pareto)

𝑓 2 𝑤R; − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤R; + 𝛿;𝑓 0 𝑤R; , for  𝑖 = ℎ  and  𝑙, (M-IC)

𝑓 2 𝑤R; − 𝑐 ≥ 1 − 𝛿; 𝑓 1 𝑤R; + 𝛿; 𝑓 1 𝑤R; − 𝑐 , for  𝑖 = ℎ  and  𝑙.

30  

Proof of Lemma 1.

As discussed in the main text, the agents sustain the (work, work) equilibrium with

punishments of either (work, shirk)/(shirk, work) or (shirk, shirk). In the case of (work, shirk) or

(shirk, work), the deviating agent plays work while the non-deviating agent plays shirk after the

deviation conditional on such play being self-enforcing. Then, the (M-IC) is:

𝑓 2 𝑤:; − 𝑐 ≥ 1 − 𝛿 𝑓 1 𝑤:; + 𝛿 𝑓 1 𝑤:; − 𝑐 .

The binding wage of the above constraint is 𝑤:; = 1 − 𝛿 mC D EC(G)

. To show that the

punishment is self-enforcing, plug 𝑤:; into:

1)   𝑓 1 𝑤:; − 𝑐 ≥ 𝑓 0 𝑤:; ⇔ 1 − 𝛿 C G EC FC D EC G

≥ 1 ⇔ 𝛿 ≤ DC G EC D EC FC G EC F

≡ 𝛿l,

2)   𝑓 1 𝑤:; ≥ 𝑓 2 𝑤:; − 𝑐 ⇔ 1 ≥ 1 − 𝛿 C D EC(G)C D EC(G)

, which is always true.

Thus, for 𝛿 ≤ 𝛿l, (work, shirk) (or (shirk, work)) is self-enforcing.

Similarly, for (shirk, shirk), the (M-IC) constraint is:

𝑓 2 𝑤:; − 𝑐 ≥ 1 − 𝛿 𝑓 1 𝑤:; + 𝛿𝑓 0 𝑤:; .

The binding wage of the above constraint is 𝑤:; =m

GEM C D EC G LM C D EC F. To see (shirk,

shirk) is self-enforcing, plug 𝑤:; into:

𝑓 0 𝑤:; ≥ 𝑓 1 𝑤:; − 𝑐 ⇔ 1 ≥ C G EC FGEM C D EC G LM C D EC F

= C G EC FC D EC G LM C G EC F

⇔ 1 ≥   G� Q �� �� � �� � LM

⇔ 𝛿 ≥ 𝛿l.

Therefore, for 𝛿 ≥ 𝛿l, (shirk, shirk) is self-enforcing.

Because the mutual-monitoring wages do not depend on the team composition, the total wages

for homogeneous teams are the same as the total wages for diverse teams.

Q.E.D.

Proof of Lemma 2.

31  

First, we show that, if collusion arises, agents are tempted to play correlated randomization

by putting positive probabilities only on (1,0) and (0,1) for a given contract that fosters mutual

monitoring. We then find the optimal correlated randomization for a given contract and the

minimum necessary collusion-proof wages.

Correlated randomization: Let 𝛼, 𝛽, 𝛾 and 1 − 𝛼 − 𝛽 − 𝛾, denote probabilities over

1,1 , 1,0 , 0,1 and (0,0), respectively. Let agent 𝑖 be the agent who works when 1,0 is

realized and agent 𝑗 be the agent who works when 0,1 is realized. Given wage 𝑤:; and 𝑤:], the

agents find 𝛼, 𝛽, 𝛾 to maximize

𝛼 𝑓 2 𝑤:; + 𝑤:] − 2𝑐 + (𝛽 + 𝛾) 𝑓 1 𝑤:; + 𝑤:

] − 𝑐 + 1 − 𝛼 −  𝛽 −  𝛾 𝑓 0 𝑤:; + 𝑤:] ,

which can be written as follows:

𝑓 0 𝑤:; + 𝑤:] + 𝛼 𝑓 2 − 𝑓 0 𝑤:; + 𝑤:

] − 2𝑐

+ 𝛽 + 𝛾 𝑓 1 − 𝑓 0 𝑤:; + 𝑤:] − 𝑐 .

Because the payoffs are linear in 𝛼 and 𝛽 + 𝛾, the maximum of the agents’ payoffs is obtained

either at 𝛼 = 1 if

𝑓 2 − 𝑓 0 𝑤:; + 𝑤:] − 2𝑐 ≥ 𝑓 1 − 𝑓 0 𝑤:; + 𝑤:

] − 𝑐,

and 𝑓 2 − 𝑓 0 𝑤:; + 𝑤:] − 2𝑐 ≥ 0,

or at 𝛽 + 𝛾 = 1 if

𝑓 2 − 𝑓 0 𝑤:; + 𝑤:] − 2𝑐 < 𝑓 1 − 𝑓 0 𝑤:; + 𝑤:

] − 𝑐,

and 𝑓 1 − 𝑓 0 𝑤:; + 𝑤:] − 𝑐 ≥ 0,

We first consider the case in which symmetric mutual-monitoring wages are offered

(characterized in Lemma 1). In the proof of Proposition 1, we will show that the collusion

constraints bind under homogeneous teams for 𝛿; ≥C D EC GC G EC F

= GOEG

and under diverse

teams for 𝛿"𝛿# >C D EC GC G EC F

D GLMS OEGGLMT(OEG)

, and that C D EC GC G EC F

> 𝛿l. Combined with Assumption

32  

A1, whenever collusion arises, we have 𝛿; > 𝛿l. In this case, the mutual-monitoring wage is

𝑤:;∗ =m

GEM C D EC G LMN C D EC F. It is immediate to see that 𝑓 2 − 𝑓 0 𝑤:;∗ + 𝑤:

]∗ −

2𝑐 > 0. Moreover, observe that

𝑓 2 − 𝑓 0 𝑤:;∗ + 𝑤:]∗ − 2𝑐 < 𝑓 1 − 𝑓 0 𝑤:;∗ + 𝑤:

]∗ − 𝑐

⇔ 𝑓 2 − 𝑓 1 𝑤:;∗ + 𝑤:]∗ < 𝑐

⇔

1𝑥 − 1 < 𝛿;    if  𝛿; =   𝛿],1

𝑥 − 1 D < 𝛿"𝛿#    if  𝛿; >   𝛿].

So, we have 𝛽 + 𝛾 = 1 when GOEG

< 𝛿; for homogeneous teams and when GOEG Q < 𝛿"𝛿# for

diverse teams. Therefore, under the optimal mutual monitoring wages, they would collude by

setting 𝛽 + 𝛾 = 1 for GOEG

< 𝛿; for homogeneous teams and 𝛿"𝛿# >G

OEG Q for diverse teams.

We now consider the contracts that foster mutual monitoring but allow for asymmetric

contracts. Because the mutual monitoring incentive must be maintained, 𝑤:; ≥ 𝑤:;∗ and 𝑤:] ≥

𝑤:]∗ are required. This ensures that 𝑓 2 − 𝑓 0 𝑤:; + 𝑤:

] − 2𝑐 > 0. As before, because the

agents’ payoffs are linear in 𝛼 and 𝛽 + 𝛾, there are two cases, either 𝛼 = 1 or 𝛽 + 𝛾 = 1. By the

same logic above, 𝛽 + 𝛾 = 1 if and only if

𝑓 2 − 𝑓 1 𝑤:; + 𝑤:] < 𝑐 ⇔ 𝑤:; + 𝑤:

] <𝑐

𝑓 2 − 𝑓 1 .

Therefore, as long as the contracts satisfy 𝑤:;∗ + 𝑤:]∗ ≤ 𝑤:; + 𝑤:

] < mC D EC G

, the agents put

positive probabilities only on 1,0 and   0,1 . In the proof of Proposition 1, we will show that

this condition is always satisfied for the optimal collusion-proof contracts.

Optimal correlated randomization and Collusion-Proof wages: We now find the optimal

correlated randomization for a given contract. Because 𝛽 + 𝛾 = 1 when collusion arises, we use

𝛾 = 1 − 𝛽. To distinguish the different team assignments, we use 𝛽: for team 𝑘 ∈ {𝑠, 𝑑}. We

first derive the set of feasible collusion. Given wages 𝑤:; and 𝑤:], for the agents to prefer

33  

correlated randomization, each agent’s expected payoff under correlated randomization must be

greater than or equal to their stage game payoff under joint working:

𝛽: 𝑓 1 𝑤:; − 𝑐 + (1 − 𝛽:)𝑓 1 𝑤:; ≥ 𝑓 2 𝑤:; − 𝑐

⇔ 𝛽: ≤ 1 −𝑓 2 − 𝑓 1

𝑐 𝑤:; .

That is, agent 𝑖 is willing to work up to 1 − C D EC Gm

𝑤:; as a bribe in order to ensure his

teammate agent 𝑗 is willing to collude. Agent 𝑖’s bribe must be self-enforcing: when agent 𝑖 is

supposed to work as part of collusion, he still wants to collude instead of deviating to shirk and

being punished by joint shirking forever:

(1 − 𝛿;) 𝑓 1 𝑤:; − 𝑐 + 𝛿; 𝛽: 𝑓 1 𝑤:; − 𝑐 + 1 − 𝛽: 𝑓 1 𝑤:; ≥ 𝑓 0 𝑤:;

⇔ 𝛽: ≤𝑓 1 − 𝑓 0

𝑐𝛿;𝑤:; −

1 − 𝛿;𝛿;

.

Thus, agent 𝑖’s self-enforcing bribe is to work up to 𝛽: = min 1 − C D EC Gm

𝑤:; ,C G EC F

mMN𝑤:; −

GEMNMN

. We show that 1 − C D EC Gm

𝑤:; <C G EC F

mMN𝑤:; −

GEMNMN

for any 𝛿; and any contract. Observe

that because the contract must provide the mutual-monitoring incentive, 𝑤:; ≥ 𝑤:;∗ (characterized

in Lemma 1).

1 −𝑓 2 − 𝑓 1

𝑐 𝑤:; <𝑓 1 − 𝑓 0

𝑐𝛿;𝑤:; −

1 − 𝛿;𝛿;

⇔𝑤:;

𝑐𝑓 1 − 𝑓 0

𝛿;+ 𝑓 2 − 𝑓 1 >

1𝛿;.

(2)

Because the left-hand side is increasing in 𝑤:; and we know that 𝑤:; ≥ 𝑤:;∗, if the above

inequality is satisfied at 𝑤:; = 𝑤:;∗, then the inequality is satisfied for any 𝑤:; ≥ 𝑤:;∗. Plug 𝑤:;∗

into the left-hand side, we have:

1𝑓 2 − 𝑓 1

11 + 𝛿; 𝑥 − 1

𝑓 1 − 𝑓 0𝛿;

+ 𝑓 2 − 𝑓 1 >1𝛿;⇔ 𝑥 − 1 1 − 𝛿; > 1 − 𝛿;,

which is always satisfied because 𝑥 − 1 > 1 due to substitutability.

Similarly, the agent 𝑗 is willing to work up to 1 − 𝛽: if:

34  

𝛽:𝑓 1 𝑤:] + (1 − 𝛽:) 𝑓 1 𝑤:

] − 𝑐 ≥ 𝑓 2 𝑤:] − 𝑐

⇔ 𝛽: ≥𝑓 2 − 𝑓 1

𝑐 𝑤:].

By the same logic, the briber agent 𝑗’s bribe is self-enforcing if

(1 − 𝛿]) 𝑓 1 𝑤:] − 𝑐 + 𝛿] 𝛽:𝑓 1 𝑤:

] + 1 − 𝛽: 𝑓 1 𝑤:] − 𝑐 ≥ 𝑓 0 𝑤:

]

⇔ 𝛽: ≥1𝛿]−𝑓 1 − 𝑓 0

𝑐𝛿]𝑤:].

Thus, agent 𝑗’s self-enforcing bribe is to work up to 1 − 𝛽:, where 𝛽: = max C D EC Gm

𝑤:], GMu−

C G EC FmMu

𝑤:] . As above, we show that C D EC G

m𝑤:] > G

Mu− C G EC F

mMu𝑤:] for any 𝛿] and any

contract 𝑤:] ≥ 𝑤:

]∗.

𝑓 2 − 𝑓 1𝑐 𝑤:

] >1𝛿]−𝑓 1 − 𝑓 0

𝑐𝛿]𝑤:] ⇔

𝑤:]

𝑐𝑓 1 − 𝑓 0

𝛿]+ 𝑓 2 − 𝑓 1 >

1𝛿],

which is same inequality as (2). Thus, by the same logic for agent 𝑖, we conclude that C D EC G

m𝑤:] > G

Mu− C G EC F

mMu𝑤:] for any 𝛿] and any contract 𝑤:

] ≥ 𝑤:]∗.

Let 𝑤: = (𝑤:; , 𝑤:]). Thus, for any contract 𝑤:, the set of feasible 𝛽: is:

𝑓 2 − 𝑓 1𝑐 𝑤:

], 1 −𝑓 2 − 𝑓 1

𝑐 𝑤:; ≡ 𝐵 𝑤: .

Observe that the set 𝐵(𝑤:) is non-empty if and only if 𝑤:; + 𝑤:] ≤ m

C D EC G. If a contract is

immune to the most demanding collusion among all feasible collusion, then the contract is

immune to any feasible collusion. The most demanding collusion is the one that maximizes the

expected payoff of either agent under collusion given that the collusion is feasible. Thus, for the

(Pareto) constraint, the principal must consider:

maxst∈� �t

𝑓 1 𝑤:; − 𝑐 1 − 𝛿; − 𝛿;𝛽:𝑐, 𝑓 1 𝑤:] − 𝑐 1 − 𝛿] − 𝛿] 1 − 𝛽: 𝑐   ≡ Π 𝛽:, 𝑤: .

35  

For any contract 𝑤:, the agents’ payoffs are linear in 𝛽:. Thus, Π(𝛽:;𝑤:) obtains the

maximum at the boundary. Because 𝑓 1 𝑤:] − 𝑐 1 − 𝛿] − 𝛿] 1 − 𝛽: 𝑐 is increasing in 𝛽:, one

potential maximum is when 𝛽: is the largest, i.e., 𝛽:∗ = 1 − C D EC Gm

𝑤:; . In this case, agent 𝑗 is

the bribee (who works less often) and agent 𝑖 is the briber. Because 𝑓 1 𝑤:; − 𝑐 1 − 𝛿; − 𝛿;𝛽:𝑐

is decreasing in 𝛽:, another potential maximum is when 𝛽: is the smallest, i.e., 𝛽:∗ =C D EC G

m𝑤:]. In this case, agent 𝑖 is the bribee and agent 𝑗 is the briber. In both cases, the (Pareto)

constraint of the bribee agent is the most demanding constraint. When agent 𝑗 is the bribee, we

target agent 𝑗:

𝑓 2 𝑤:] − 𝑐 ≥ 𝑓 1 𝑤:

] − 𝑐 1 − 𝛿] − 𝛿](1 − 𝛽:)𝑐 ⇔ 𝑤:] ≥

𝑐𝛽:𝛿]𝑓 2 − 𝑓 1 .

Thus, the minimum necessary collusion-proof wage for the bribee is 𝑤:]∗∗ = mst

∗MuC D EC G

. Similarly,

when agent 𝑖 is the bribee, we target agent 𝑖 and the minimum necessary collusion-proof wage is

𝑤:;∗∗ =m(GEst

∗)MNC D EC G

.

Q.E.D.

Proof of Proposition 1

The proof consists of two parts. The first part of the proof is to find the optimal collusion-

proof contract given the optimal correlated randomization we found in Lemma 2. For the optimal

collusion-proof contract, we also derive the threshold of the discount factor at which the

collusion constraint binds in each composition. We also show that the optimal contracts satisfy

𝑤:; + 𝑤:] < m

C D EC G (which is equivalent to saying that agents put positive probabilities only on

(1,0) and (0,1)). The second part of the proof is to show why the derived collusion-proof wage is

optimal to combat collusion.

Optimal Collusion-Proof Contract: Recall Π 𝛽:;𝑤: denotes the maximum payoff of

either agent in team 𝑘 under collusion 𝛽: given contract 𝑤: = 𝑤:; , 𝑤:] .

Π 𝛽:;𝑤: = maxst∈� �t

𝑓 1 𝑤:; − 𝑐 1 − 𝛿; − 𝛿;𝛽:𝑐, 𝑓 1 𝑤:] − 𝑐 1 − 𝛿] − 𝛿] 1 − 𝛽: 𝑐   .

36  

As shown in Lemma 2, Π(𝛽:;𝑤:) obtains the maximum at the boundary: either 𝛽: = 1 −C D EC G

m𝑤:; (i.e., agent 𝑖 is the briber and agent 𝑗 is the bribee) or 𝛽: =

C D EC Gm

𝑤:] (i.e., agent 𝑗

is the briber and agent 𝑖 is the bribee). The bribee agents’ payoffs are:

Π 𝛽:;𝑤: st{GEC D EC G

m �tN = 𝑓 1 𝑤:

] − 𝑐 1 − 𝛿] − 𝛿] 𝑓 2 − 𝑓 1 𝑤:; ≡ 𝑢:],

Π 𝛽:;𝑤: st{C D EC G

m �tN = 𝑓 1 𝑤:; − 𝑐 1 − 𝛿; − 𝛿; 𝑓 2 − 𝑓 1 𝑤:

] ≡ 𝑢:;.

We first consider homogeneous teams, 𝑘 = 𝑠. Because 𝑖 = 𝑗, we use 𝑖1 (who works with

probability 𝛽P) and 𝑖2 (who works with probability 1 − 𝛽P) to distinguish agents. Observe that:

𝑢P;G > 𝑢P;D ⇔ 𝑓 1 𝑤P;G − 𝑤P;D > −𝛿; 𝑓 2 − 𝑓 1 𝑤P;G − 𝑤P;D ⇔ 𝑤P;G > 𝑤P;D.

Therefore, there are three cases to consider: binding collusion at 1) 𝛽P = 1 − C D EC Gm

𝑤P;G   ⇔

𝑤P;G < 𝑤P;D , 2) 𝛽P =C D EC G

m𝑤P;D   ⇔ 𝑤P;G > 𝑤P;D , or 3) both 𝛽P =

C D EC Gm

𝑤P;D and 𝛽P = 1 −

C D EC Gm

𝑤P;G   ⇔ 𝑤P;G = 𝑤P;D . If the binding collusion is 𝛽P = 1 − C D EC Gm

𝑤P;G, the maximum

Π 𝛽P;𝑤P is agent 𝑖2’s payoff. We target agent 𝑖2. Plug 𝛽P = 1 − C D EC Gm

𝑤P;G into the (Pareto)

constraint of agent 𝑖2:

𝑓 2 𝑤P;D − 𝑐 ≥ 𝑓 1 𝑤P;D − 𝑐 1 − 𝛿; − 𝛿; 𝑓 2 − 𝑓 1 𝑤P;G

⇔ 𝑓 2 − 𝑓 1 𝑤P;D + 𝛿;𝑤P;G ≥ 𝑐𝛿;.

The minimum necessary collusion-proof contract and the binding collusion constraint is:

𝑤P;D = 𝑤P;∗∗ = 𝛿;𝑐

𝑓 2 − 𝑓 1 − 𝑤P;G , 𝑤P;G = 𝑤P;∗, 𝛽P∗ =𝑓 2 − 𝑓 1

𝑐 𝑤P;∗.

Because 𝑤P;∗ =m

C D EC(G)G

GLMN(OEG), the above can be written as:

𝑤P;∗∗ =𝑐

𝑓 2 − 𝑓(1)𝛿;D(𝑥 − 1)

1 + 𝛿;(𝑥 − 1), 𝑤P;∗ =

𝑐𝑓 2 − 𝑓(1)

11 + 𝛿;(𝑥 − 1)

, 𝛽P∗ =1

1 + 𝛿;(𝑥 − 1).

It is immediate to see that 𝑤P;∗ + 𝑤P;∗∗ =m

C D EC(G)GLMN

Q(OEG)GLMN(OEG)

< mC D EC(G)

because 𝛿; < 1.

37  

By symmetry, if the binding collusion is 𝛽P∗ =C D EC G

m𝑤P;D, then we obtain the same result

but agent 𝑖1 receives a collusion-proof wage, agent 𝑖2 receives a mutual monitoring wage, and

𝛽P∗ =MN(OEG)

GLMN(OEG). Lastly, if collusion constraints bind both at 𝛽P∗ =

C D EC Gm

𝑤P;D and 𝛽P∗ = 1 −

C D EC Gm

𝑤P;G (i.e., 𝑢PG = 𝑢PD ⇔ 𝑤P;G = 𝑤P;D), then we target either agent but both agents receive

symmetric collusion-proof wages. Without loss of generality, target agent 𝑖1 (targeting agent 𝑖2

generates the same result):

𝑓 2 𝑤P;G − 𝑐 ≥ 𝑓 1 𝑤P;G − 𝑐 1 − 𝛿; − 𝛿; 𝑓 2 − 𝑓 1 𝑤P;D

⇔ 𝑓 2 − 𝑓 1 𝑤P;G + 𝛿;𝑤P;D ≥ 𝑐𝛿;.

The minimum necessary collusion-proof contract and the binding collusion constraint is:

𝑤P;∗∗ =𝑐

𝑓 2 − 𝑓 1𝛿;

1 + 𝛿;, 𝛽P∗ = 1 −

𝛿;1 + 𝛿;

 .

Targeting agent 𝑖2 generates 𝛽P∗ =MN

GLMN. Observe that 2𝑤P;∗∗ =

mC D EC(G)

DMNGLMN

< mC D EC(G)

because

𝛿; < 1. It is straightforward to show that 𝑤P;∗∗ > 𝑤P;∗ if and only if 𝛿; >C D EC(G)C G EC(F)

= GOEG

regardless of whether 𝑤P;∗∗ =m

C D EC(G)MNQ(OEG)

GLMN(OEG) or 𝑤P;∗∗ =

mC D EC G

MNGLMN

. Recall from Lemma 2

that agents would collude (𝛽 + 𝛾 = 1) if 𝛿; >G

OEG. Because G

OEG< 1, we have G

OEG> G

OEG. The

reason why there is a gap between the threshold at which the collusion constraint binds ( GOEG

)

and the threshold above which agents prefer collusion ( GOEG

) is because any collusion that can

arise for 𝛿; ∈G

OEG, G

OEG for homogeneous teams is not regret-free: the bribee agent regrets

when it’s his turn to work. For instance, if agent 𝑖1 is the bribee and  𝛽P∗ =G

GLMN(OEG), agent 𝑖1

regrets if:

𝑓 2 𝑤P;∗ − 𝑐 > 1 − 𝛿; 𝑓 1 𝑤P;∗ − 𝑐 + 𝛿;𝑈 𝛽P; 𝑤:;∗ ⇔ 𝛿; < 1/ 𝑥 − 1.

38  

The same conclusion is made for 𝛽P∗ =MN(OEG)

GLMN(OEG) and 𝛽P∗ =

MNGLMN

(or 1 − MNGLMN

). Thus, any potential

collusion for 𝛿; ∈G

OEG, G

OEG does not meet the regret-free collusion requirement. As we will

see in the proof of Proposition 3, GOEG

for homogenous teams becomes the thresholds once we

consider ex ante collusion (that is not regret-free).

The following claim shows that the total wages under symmetric (collusion-proof) contracts

are always greater than the total wages under asymmetric contracts.

Claim 1. The minimum necessary symmetric (collusion-proof) wages are always greater than the

asymmetric wages.

Proof of Claim 1)

2𝑐

𝑓 2 − 𝑓 1𝛿;

1 + 𝛿;;{",#

>𝑐

𝑓 2 − 𝑓 11 + 𝛿;D 𝑥 − 11 + 𝛿; 𝑥 − 1;{",#

⇔1− 𝛿"1 + 𝛿"

𝛿"D 𝑥 − 1 − 11 + 𝛿" 𝑥 − 1

�F

>  1 − 𝛿#1 + 𝛿#

1 − 𝛿#D 𝑥 − 11 + 𝛿# 𝑥 − 1

�F

.

Given that the collusion constraints bind (𝛿; > 1/(𝑥 − 1)), the left hand side is always positive

whereas the right hand side is always negative, thus ensuring the above inequality for any 𝛿; >

1/(𝑥 − 1). ⎕

Thus, under homogenous teams, the optimal contract is paying one agent mC D EC(G)

MNQ(OEG)

GLMN(OEG) and

paying his teammate mC D EC(G)

GGLMN(OEG)

, and the collusion constraint binds if and only if 𝛿; >

1/(𝑥 − 1).

We now consider diverse teams. Recall that agent ℎ works with probability 𝛽R and agent 𝑙

works with probability 1 − 𝛽R. As before, there are three cases to consider: binding collusion at

1) 𝛽R = 1 − C D EC Gm

𝑤R", 2) 𝛽R =C D EC G

m𝑤R# , or 3) both 𝛽R =

C D EC Gm

𝑤R# and 𝛽R = 1 −

C D EC Gm

𝑤R". Contrary to homogenous teams, because agents have different discount factors,

case 3) is not equivalent to symmetric collusion-proof wages. Thus, additionally, we will

39  

consider symmetric collusion-proof wages. If the binding collusion is 𝛽R = 1 − C D EC Gm

𝑤R", we

target agent 𝑙:

𝑓 2 𝑤R# − 𝑐 ≥ 𝑓 1 𝑤R# − 𝑐 1 − 𝛿# − 𝛿# 𝑓 2 − 𝑓 1 𝑤R"

⇔ 𝑓 2 − 𝑓 1 𝑤R# + 𝛿#𝑤R" ≥ 𝑐𝛿#.

The minimum necessary collusion-proof contract and the binding collusion constraint is:

𝑤R#∗∗ =𝑐

𝑓 2 − 𝑓 1𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

,𝑤R"∗ =𝑐

𝑓 2 − 𝑓 11

1 + 𝛿" 𝑥 − 1, 𝛽R∗ =

𝛿" 𝑥 − 11 + 𝛿" 𝑥 − 1

.

Observe that 𝑤P#∗∗ + 𝑤P"∗ =m

C D EC(G)GLMSMT OEGGLMS OEG

< mC D EC(G)

because 𝛿# < 1. It is straightforward

to show that 𝑤P#∗∗ > 𝑤P#∗ if and only if 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

. From Lemma 2, agents in diverse

teams would collude if 𝛿"𝛿# >G

OEG Q. We will shortly show that any collusion that satisfies

𝛿"𝛿# >G

OEG Q but does not satisfy 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

is not regret-free for agent 𝑙.

Similarly, if the binding collusion is 𝛽R =C D EC G

m𝑤R# , we target agent ℎ, and the minimum

necessary collusion-proof contract and the binding collusion constraint is:

𝑤R"∗∗ =𝑐

𝑓 2 − 𝑓 1𝛿"𝛿# 𝑥 − 11 + 𝛿# 𝑥 − 1

,𝑤R#∗ =𝑐

𝑓 2 − 𝑓 11

1 + 𝛿# 𝑥 − 1, 𝛽R∗ =

11 + 𝛿# 𝑥 − 1

.

Observe that 𝑤P"∗∗ + 𝑤P#∗ =m

C D EC(G)GLMSMT OEGGLMT OEG

< mC D EC(G)

because 𝛿" < 1. It is straightforward

to show that 𝑤P"∗∗ > 𝑤P"∗ if and only if 𝛿"𝛿# >G

OEGGLMT(OEG)GLMS(OEG)

. From Lemma 2, the necessary

condition for collusion is 𝛿"𝛿# >G

OEG Q. Thus, GOEG

GLMT(OEG)GLMS(OEG)

≥ GOEG Q is required. We will shortly

show that any collusion that satisfies 𝛿"𝛿# >G

OEG Q but does not satisfy 𝛿"𝛿# >G

OEGGLMT(OEG)GLMS(OEG)

is

not regret-free for agent ℎ. When the binding constraints are both 𝛽R =C D EC G

m𝑤R# and 𝛽R =

1 − C D EC Gm

𝑤R", we target either agent ℎ or agent 𝑙, respectively. In this case, the minimum

necessary collusion-proof wages are the same as above.

40  

When symmetric contracts are offered, the most demanding collusion is either agent ℎ works

with probability 𝛽R <MS

MSLMT (in which case we target agent ℎ) or agent 𝑙 works with probability

1 − 𝛽R and 𝛽R >MS

MSLMT (in which case we target agent 𝑙). Use the (Pareto) constraint, when agent

ℎ is targeted, then:

𝑤R"∗∗ = 𝑤R#∗∗ =𝑐

𝑓 2 − 𝑓 1𝛿"

1 + 𝛿", 𝛽R∗ =

𝛿"1 + 𝛿"

.

When agent 𝑙 is targeted, then:

𝑤R"∗∗ = 𝑤R#∗∗ =𝑐

𝑓 2 − 𝑓 1𝛿#

1 + 𝛿#, 𝛽R∗ =  

11 + 𝛿#

.

Because mC D EC G

MSGLMS

> mC D EC G

MTGLMT

, it is efficient to target agent 𝑙 under symmetric contracts.

Regardless of which agent is targeted, 𝑤R"∗∗ + 𝑤R#∗∗ =m

C D EC GDMNGLMN

< mC D EC G

because 𝛿; < 1.

Moreover, 𝑤R#∗∗ > 𝑤R#∗ if and only if 𝛿# >G

OEG. Recall from Lemma 2 that agents in diverse

teams would collude (𝛽 + 𝛾 = 1) if 𝛿"𝛿# >G

OEG Q, which is satisfied for 𝛿" > 𝛿# >G

OEG.

The following claim shows that asymmetric contracting with targeting agent 𝑙 is always least

costly relative to any other collusion-proof contracts.

Claim 2. The least costly minimum necessary collusion-proof contract is asymmetric with

targeting agent 𝑙.

Proof of Claim 2)

Given that asymmetric wages are offered, the total wages when targeting agent ℎ are more

expensive than the total wages when targeting agent 𝑙:

2𝑐

𝑓 2 − 𝑓 11 + 𝛿"𝛿# 𝑥 − 11 + 𝛿# 𝑥 − 1

> 2𝑐

𝑓 2 − 𝑓 11 + 𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

,

41  

which is always satisfied because 𝛿# < 𝛿". Observe also that the total wages when targeting

agent 𝑙 but offering symmetric wages are more expensive than the total wages when targeting

agent 𝑙 and offering asymmetric wages:

4𝑐

𝑓 2 − 𝑓 1𝛿#

1 + 𝛿#> 2

𝑐𝑓 2 − 𝑓 1

1 + 𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

⇔ 𝛿"𝛿# >1

𝑥 − 1,

which is always satisfied because collusion constraint binds under the symmetric contract

whenever 𝛿" > 𝛿# >G

OEG. When targeting agent 𝑙 with asymmetric contracting, collusion

constraint binds whenever 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

> GOEG

. Thus, even in case GOEG

GLMS(OEG)GLMT(OEG)

>

𝛿"𝛿# >G

OEG (collusion constraint does not bind when targeting agent 𝑙 with asymmetric

contracting), the inequality is satisfied. ⎕

Thus, under diverse teams, the optimal collusion-proof contract is paying agent 𝑙 the collusion-

proof wage mC D EC(G)

MSMT(OEG)GLMS(OEG)

and paying agent ℎ the mutual monitoring wage

mC D EC(G)

GGLMS(OEG)

, and the collusion constraint binds if and only if 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

.

As we discussed in homogenous teams, there is a gap between GOEG

GLMS OEGGLMT(OEG)

and GOEG Q. We

first note that both thresholds can be written as a threshold for 𝛿#. First, 𝛿"𝛿# >G

OEG Q is

equivalent to 𝛿# >G

MS OEG Q. The condition, 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

, on the other hand, is written as

follows:

𝛿"𝛿# >1

𝑥 − 11 + 𝛿"(𝑥 − 1)1 + 𝛿#(𝑥 − 1)

⇔ 𝛿# >4𝑥 − 3 + 4/𝛿" − 1

2(𝑥 − 1) ≡ 𝛿 𝛿" .

To see if 𝛿 𝛿" is well-defined (i.e., 0 < 𝛿 𝛿" < 1):

4𝑥 − 3 + 4/𝛿" − 1 > 0 ⇔ 𝑥 + 1/𝛿" > 1,

which is always satisfied. Moreover,

4𝑥 − 3 + 4/𝛿" − 12(𝑥 − 1) < 1 ⇔ 𝛿" >

1𝑥 − 1 D,

42  

which is always satisfied because 𝛿" >GOEG

and GOEG

> GOEG Q for 𝑥 ∈ 2, wL K

D.

Notice also that 𝛿 𝛿" < 1/ 𝑥 − 1:

𝛿 𝛿" < 1/ 𝑥 − 1 ⇔ 𝛿" > 1/ 𝑥 − 1.

Now, we show that any collusion for 𝛿# ∈G

MS OEG Q , 𝛿 𝛿" under diverse teams is not regret-

free: the bribee agent 𝑙 regrets when it’s his turn to work. To see this, plug the mutual monitoring

wage and 𝛽R∗ into agent 𝑙’s (Pareto) constraint:

𝑓 2 𝑤R#∗ − 𝑐 > 1 − 𝛿# 𝑓 1 𝑤R#∗ − 𝑐 + 𝛿#𝑈 𝛽R;𝑤R#∗ ⇔ 𝛿"𝛿# <1

𝑥 − 11 + 𝛿" 𝑥 − 11 + 𝛿# 𝑥 − 1

.

Thus, any collusion with agent 𝑙 being the bribee for 𝛿# ∈G

MS OEG Q , 𝛿 𝛿" does not meet the

regret-free collusion requirement. Observe that, for the symmetric collusion-proof contract

mC D EC G

MTGLMT

with 𝛽R =G

GLMT, the binding collusion is equivalent to 𝛿# >

GOEG

. However, this

does not mean that there is no regret-free collusion for 𝛿 𝛿" < 𝛿# ≤G

OEG. This simply means

that the most demanding collusion cannot be GGLMT

because agents can play 𝛽R∗ =MS(OEG)

GLMS(OEG) to

maximize the sum of their payoffs while no one regrets. For the asymmetric collusion-proof

contract with targeting agent ℎ, we can similarly derive 𝛿 𝛿# = �OEwL�/MTEGD(OEG)

(from 𝛿"𝛿# >

GOEG

GLMT(OEG)GLMS(OEG)

) and show that collusion with agent ℎ being the bribee for 𝛿" ∈G

MT OEG Q , 𝛿 𝛿#

does not meet the regret-free collusion requirement. Because 𝛿# < 𝛿", we have 𝛿 𝛿# > 𝛿 𝛿" .

However, as in the symmetric contract case, this does not mean that, for 𝛿 𝛿" < 𝛿# < 𝛿" <

𝛿 𝛿# , there is no regret-free collusion. In this parameter region, the regret-free collusion cannot

be agent ℎ being the bribee and 𝛽R∗ =MS(OEG)

GLMS(OEG) is the regret-free collusion.

To summarize, collusion constraint binds under diverse teams for 𝛿" > 1/ 𝑥 − 1 and 𝛿# >

𝛿 𝛿" .

43  

Recall that 1 − 𝛿; >G

GLMN OEG is equivalent to 𝛿; < 𝛿l = DC G EC D EC(F)

C G EC(F) (characterized in

Lemma 1). Also, GGLMN OEG

< GD is equivalent to

𝛿; >1

𝑥 − 1 >12.

(3)

The last inequality of (3) is due to weak substitutability, which also ensures that 𝛿l < GD: 𝛿l <

1 − DGL K

< GD. When 𝛿; ≤ 1/ 𝑥 − 1 (the collusion constraint does not bind), the binding

constraints are the (M-IC) constraints and, thus, symmetric mutual-monitoring wages are

optimal. When 𝛿; > 1/ 𝑥 − 1,  (shirk, shirk) is a unique stage game equilibrium because GOEG

>

GOEG

> GD> 𝛿l. In this case, the (Pareto) constraint binds for homogeneous teams and, thus, it is

optimal to offer the collusion-proof wage to one of agent ℎs and one of agent 𝑙s in each team and

to pay the counterpart agent in each team the mutual-monitoring wage.

For diverse teams, when 𝛿" ≤ 1/ 𝑥 − 1, the binding constraints are the (M-IC) constraints

and, thus, symmetric mutual-monitoring wages are optimal. Notice that, even if agent ℎ wants to

collude (i.e., 𝛿" > 1/ 𝑥 − 1), they cannot collude if agent 𝑙 does not agree (i.e., 𝛿# ≤ 𝛿 𝛿" ). In

this case, symmetric mutual-monitoring wages are optimal. When 𝛿# > 𝛿 𝛿" , the binding

constraint is the (Pareto) constraint, and it is optimal to offer the collusion-proof wage to agent 𝑙s

and to pay agent ℎs the mutual-monitoring wage in each team because agent ℎ’s mutual-

monitoring wage is less costly than agent 𝑙’s.

(Pareto) constraint vs. (No Self-Enforcing Collusion) Constraint: We now show the

second part of the proof. Given that there is a potentially profitable collusion, the agents may

sustain such collusion using the unique stage game equilibrium, which is playing (shirk, shirk)

(because 𝛿; >GOEG

> GOEG

> 𝛿l for the collusion to arise). The constraint that upsets the agent’s

incentive to sustain such collusion for the agent who is supposed to work is:

1 − 𝛿; 𝑓 0 𝑤:; + 𝛿;𝑓 0 𝑤:; ≥ 1 − 𝛿; (𝑓 1 𝑤:; − 𝑐) + 𝛿;𝑈 𝛽:;𝑤:; for all 𝛽:. (4)

Observe that the right hand side of inequality (4) is the same as that of the (Pareto) constraint. As

in the (Pareto) constraint, if the contract is immune to the most demanding collusion, then it is

immune to any feasible collusion. Thus, the contract that satisfies the (No Self-enforcing)

44  

constraints for all 𝛽: is derived as in (1). Therefore, it is sufficient to check for 𝛽:∗ that we

derived for the (Pareto) constraint.

When agent 𝑖 is the briber and agent 𝑗 is the bribee, 𝛽:∗ = 1 − C D EC Gm

𝑤:; and (4) is written

as:

𝑤:; ≤𝑐 1 − 𝛿; 1 − 𝛽:∗

𝑓 1 − 𝑓 0 ⇔ 𝑤:; ≤𝑐

𝑓 1 − 𝑓 0 + 𝛿; 𝑓 2 − 𝑓 1≡ 𝑤:;

�.

Notice that 𝑤:;� < 𝑤:;∗ because of substitutability:

𝑐𝑓 1 − 𝑓 0 + 𝛿; 𝑓 2 − 𝑓 1

<𝑐

1 − 𝛿; 𝑓 2 − 𝑓 1 + 𝛿; 𝑓 2 − 𝑓 0⇔ 𝛿; < 1.

In other words, using the upper bound 𝑤:;� eliminates the briber agent’s mutual monitoring

incentive too.

For the bribee agent 𝑗, (4) is written as:

𝑤:] ≤

𝑐 1 − 𝛿]𝛽:∗

𝑓 1 − 𝑓 0 ⇔ 𝑤:] ≤

𝑐 1 − 𝛿]𝑓 1 − 𝑓 0 + 𝛿]

𝑓 2 − 𝑓 1𝑓 1 − 𝑓 0 𝑤:; .

(5)

There are two cases to consider: 𝑘 = 𝑠 or 𝑑. When 𝑘 = 𝑠, inequality (5) is equivalent to:

𝑤P] ≤

𝑐 1 − 𝛿]𝑓 1 − 𝑓 0 − 𝛿] 𝑓 2 − 𝑓 1

≡ 𝑤P]�.

Notice that 𝑤P]� = (1 − 𝛿])𝑤P;

�. Because 𝑤P;� < 𝑤P;∗ and 𝑖 = 𝑗 (homogenous team), we conclude

that 𝑤P]� < 𝑤P

]∗. Hence, using the upper bound 𝑤P]� eliminates the bribee agent’s mutual

monitoring incentive in homogenous teams.

When 𝑘 = 𝑑, inequality (5) is 𝑤R] ≤ m GEMu

C G EC F+ 𝛿]

C D EC GC G EC F

𝑤R; ≡ 𝑤R]�. Notice that 𝑤R

]�

depends on 𝑤R; . We will show that 𝑤R]�

��N∗ is always less than 𝑤R

]∗: paying the mutual monitoring

wage to the briber agent 𝑖 and paying 𝑤R]�

��N∗ eliminates the bribee agent’s mutual monitoring

incentive. We then show that, with the change in 𝑤R; to ensure 𝑤R]� ≥ 𝑤R

]∗ always leads to more

45  

expensive total wages relative to using the (Pareto) constraint of the bribee agent 𝑗. The

following claim shows 𝑤R]�

��N∗ < 𝑤R;

�.

Claim 3. 𝑤R]�

��N∗ < 𝑤R;

�.

Proof of Claim 3) Observe that

𝑤R]�

��N∗ < 𝑤R;

� ⇔ 𝛿; < 𝛿;𝛿] 𝑥 − 1 .

When 𝑖 = ℎ, 𝑗 = 𝑙, then the left hand side, 𝛿", is less than 1 but the right hand side, 𝛿"𝛿# 𝑥 − 1 ,

is greater than 1 because the collusion constraint binds for 𝛿"𝛿# >G

OEGGLMS(OEG)GLMT(OEG)

under the

diverse teams, thus, 𝛿"𝛿# 𝑥 − 1 > GLMS(OEG)GLMT(OEG)

> 1. Similarly, when 𝑖 = 𝑙, 𝑗 = ℎ, the collusion

constraint binds for 𝛿"𝛿# >G

OEGGLMT(OEG)GLMS(OEG)

, thus 𝛿#𝛿" 𝑥 − 1 > GLMT(OEG)GLMS(OEG)

. Notice that:

𝛿# <1 + 𝛿#(𝑥 − 1)1 + 𝛿"(𝑥 − 1)

⇔ 0 < 1 − 𝛿# + 𝛿# 𝑥 − 1 1 − 𝛿" ,

which is always true. Therefore, 𝛿# <GLMT(OEG)GLMS(OEG)

< 𝛿#𝛿" 𝑥 − 1 . ⎕

Because we know that 𝑤R;� < 𝑤R;∗, this implies that 𝑤R

]�

��N∗ < 𝑤R;∗. Recall that 𝑤R"∗ < 𝑤R#∗.

Therefore, for 𝑖 = ℎ, 𝑗 = 𝑙, we have 𝑤R#���S∗ < 𝑤R#∗. For 𝑖 = 𝑙, 𝑗 = ℎ, we directly show that

𝑤R"���T∗ < 𝑤R"∗ in the following claim.

Claim 4. 𝑤R"���T∗ < 𝑤R"∗.

Proof of Claim 4) Observe that 𝑤R"���T∗ < 𝑤R"∗ is equivalent to

⇔𝑐 1 − 𝛿"𝑓 1 − 𝑓 0 + 𝛿"

𝑓 2 − 𝑓 1𝑓 1 − 𝑓 0

𝑐𝑓 2 − 𝑓 1

11 + 𝛿# 𝑥 − 1

<𝑐

𝑓 2 − 𝑓 11

1 + 𝛿" 𝑥 − 1

⇔1− 𝛿"𝑥 − 1 +

𝛿"𝑥 − 1

11 + 𝛿# 𝑥 − 1

<1

1 + 𝛿" 𝑥 − 1

46  

⇔1+ 𝛿# 1 − 𝛿" 𝑥 − 1

1 + 𝛿# 𝑥 − 1<

𝑥 − 11 + 𝛿" 𝑥 − 1

⇔ 𝑥 − 1 D𝛿# 1 − 𝛿" 1 − 𝛿" + 𝑥 − 1 1 − 𝛿" 1 − 𝛿# > 1

⇔ 𝑥 − 1�G

1 − 𝛿" 𝑥 − 2 𝛿# + 1�G

+ 𝛿"D𝛿# 𝑥 − 1 D

�MS

> 1. (6)

Here, 𝑥 − 1 > 1 and 𝑥 − 2 𝛿# + 1 > 1 because of substitutability, 𝑥 > 2, and 𝛿"D𝛿# 𝑥 − 1 D >

𝛿" because the necessary condition for collusion (𝛽 + 𝛾 = 1) is 𝛿"𝛿# >G

OEG Q. Because 1 − 𝛿"

is multiplied by some constant greater than 1 and added by another constant greater than 𝛿", the

left-hand side of (6) is always greater than 1. ⎕

From Claim 3 and Claim 4, we conclude that 𝑤R]�

��N∗ eliminates the bribee agent 𝑗’s mutual

monitoring incentive.

Now, we show that using the (No Self-enforcing collusion) constraint of the bribee agent 𝑗

(i.e., paying 𝑤R]�) by adjusting 𝑤R;

� is always more expensive than the contract we find. Let 𝑤R;

��

denote that the wage paid to agent 𝑖 that ensures that 𝑤R]� ≥ 𝑤R

]∗. The minimum necessary

collusion-proof wage is to set 𝑤R]� = 𝑤R

]∗, which leads to:

𝑤R;�� = 𝑤R

]∗ −𝑐 1 − 𝛿]𝑓 1 − 𝑓 0

1𝛿]𝑓 1 − 𝑓 0𝑓 2 − 𝑓 1 .

The following claim shows that 𝑤R]� + 𝑤R;

�� > mC D EC(G)

GLMSMT(OEG)GLMS(OEG)

(total wages using the

(Pareto) constraint).

Claim 5. 𝑤R]� + 𝑤R;

�� > mC D EC(G)

GLMSMT(OEG)GLMS(OEG)

.

Proof of Claim 5) First, consider 𝑗 = ℎ, 𝑖 = 𝑙. Then, the inequality is written as:

𝑤R"� + 𝑤R#

�� >𝑐

𝑓 2 − 𝑓 11 + 𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

⇔ 𝑤R"∗ 1 +𝑥 − 1𝛿"

−𝑐

𝑓 2 − 𝑓 11 − 𝛿"𝛿"

>𝑐

𝑓 2 − 𝑓 11 + 𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

47  

⇔1

1 + 𝛿" 𝑥 − 11 +

𝑥 − 1𝛿"

−1 − 𝛿"𝛿"

>1 + 𝛿"𝛿# 𝑥 − 11 + 𝛿" 𝑥 − 1

(7)

⇔ 𝑥 − 1 1 − 𝛿" + 𝛿"D 1 − 𝛿# > 1 − 𝛿",

which is always satisfied because 𝑥 − 1 > 1 and 𝛿"D 1 − 𝛿# > 0. Thus, when agent ℎ is the

bribee, it is never optimal to use his (No Self-enforcing collusion) constraint.

Consider the other case (𝑗 = 𝑙, 𝑖 = ℎ), 𝑤R#� + 𝑤R"

�� > mC D EC(G)

GLMSMT(OEG)GLMS(OEG)

. Instead of directly

showing the inequality, observe that the left hand side of (7) is a decreasing function of the

bribee agent’s discount factor:

𝜕𝜕𝛿"

11 + 𝛿" 𝑥 − 1

1 +𝑥 − 1𝛿"

−1 − 𝛿"𝛿"

= −𝑥 − 2 1 + 𝑥 − 1 𝛿" 2 − 𝛿"

𝛿"D 1 + 𝛿" 𝑥 − 1D < 0.

Because 𝛿" > 𝛿#, we have 𝑤R#� + 𝑤R"

�� > 𝑤R"� + 𝑤R#

��.

Therefore, 𝑤R#� + 𝑤R"

�� > mC D EC(G)

GLMSMT(OEG)GLMS(OEG)

. ⎕

Hence, under diverse teams, using the bribee agent’s (No Self-enforcing collusion) constraint

either eliminates the bribee’s mutual monitoring incentive or generates more expensive total

wages.

To summarize, in both compositions, using the briber agent’s (No Self-enforcing collusion)

constraint eliminates the briber’s mutual monitoring incentive. In homogeneous teams, using the

bribee agent’s (No Self-enforcing collusion) also eliminates the bribee agent’s mutual

monitoring incentive. In diverse teams, using the bribee agent’s (No Self-enforcing collusion)

also eliminates the bribee agent’s mutual monitoring incentive (if the briber receives his mutual

monitoring wage) or generates more expensive total wages (if the briber’s wage is adjusted so

that it ensures the bribee agent’s mutual monitoring incentive).

The same conclusion is made for the opposite case in which agent 𝑖 is the bribee and agent 𝑗

is the briber (thus, 𝛽:∗ =C D EC G

m𝑤:]). Therefore, from the principal’s perspective, it is optimal to

use the (Pareto) constraint for both compositions.

Q.E.D.

48  

Proof of Proposition 2

For notational convenience, we normalize both the mutual-monitoring and collusion-proof

wages by the common term mC D EC G

, i.e., the normalized mutual-monitoring wage is 1 − 𝛿; or

GGLMN(OEG)

, where 𝑥 = C D EC(F)C D EC(G)

. The normalized collusion-proof wage is MNQ(OEG)

GLMN(OEG) for

homogenous teams and MSMT(OEG)GLMS(OEG)

for diverse teams. Recall Assumption A2 that 𝛿# < 𝛿" ≤ 𝛿l or

𝛿" > 𝛿# ≥ 𝛿l. The (normalized) total wages are:

𝑊P =

2 1 − 𝛿" + 2 1 − 𝛿#    if  𝛿# < 𝛿" < 𝛿l,2

1 + 𝛿" 𝑥 − 1+

21 + 𝛿# 𝑥 − 1

   if  𝛿l ≤ 𝛿# < 𝛿" ≤1𝑥 − 1

,

𝛿"D(𝑥 − 1)1 + 𝛿"(𝑥 − 1)

+1

1 + 𝛿" 𝑥 − 1+

21 + 𝛿# 𝑥 − 1

 if  𝛿l ≤ 𝛿# ≤1𝑥 − 1

< 𝛿",

𝛿"D(𝑥 − 1)1 + 𝛿"(𝑥 − 1)

+𝛿#D(𝑥 − 1)

1 + 𝛿#(𝑥 − 1)+

11 + 𝛿" 𝑥 − 1

+1

1 + 𝛿# 𝑥 − 1  if  

1𝑥 − 1

< 𝛿# < 𝛿",

𝑊R =

2 1 − 𝛿" + 2 1 − 𝛿#    if  𝛿# < 𝛿" < 𝛿l,2

1 + 𝛿" 𝑥 − 1+

21 + 𝛿# 𝑥 − 1

     if  𝛿l ≤ 𝛿# < 𝛿" ≤1𝑥 − 1

,

21 + 𝛿" 𝑥 − 1

+2

1 + 𝛿# 𝑥 − 1  if  𝛿l ≤ 𝛿# ≤ 𝛿 𝛿" <

1𝑥 − 1

< 𝛿",

and            2𝛿"𝛿#(𝑥 − 1)1 + 𝛿"(𝑥 − 1)

+2

1 + 𝛿" 𝑥 − 1                        if  𝛿 𝛿" < 𝛿# <

1𝑥 − 1

< 𝛿".                    

Part i) When 𝛿# < 𝛿" ≤GOEG

, then 𝑊P = 𝑊R due to Lemma 1.

Part ii) When 𝛿# ≤ 𝛿 𝛿" < GOEG

< 𝛿", the collusion problem arises under homogeneous

assignment (in the team of agent ℎs) but does not arise under diverse assignment, and 𝑊P −

𝑊R =MSQ OEG EGGLMS OEG

> 0.

Part iii) When 𝛿 𝛿" < 𝛿# <GOEG

< 𝛿", the collusion problem arises under the team of agent ℎs

and under diverse assignment, and 𝑊P −𝑊R = 𝛿" − 2𝛿# +D

GLMT OEG− GLMSEDMT

GLMS OEG, which can be

written as follows:

49  

𝑊P −𝑊R

=𝛿"D 𝑥 − 1 − 1 𝛿# 𝑥 − 1 + 𝛿"𝛿# 𝑥 − 1 − 1 D + 𝛿" 2 + 𝛿" 1 − 𝛿#D 𝑥 − 1 𝑥 − 1

1 + 𝛿" 𝑥 − 1 1 + 𝛿# 𝑥 − 1> 0.

The last inequality is always satisfied. To see this, note that GOEG

< 𝛿" which suggests that

𝛿"D 𝑥 − 1 − 1 > 0, and 𝛿# <GOEG

which suggests that 1 − 𝛿#D 𝑥 − 1 > 0. Therefore, 𝑊P −

𝑊R > 0.

Part iv) When GOEG

< 𝛿# < 𝛿", the collusion problem arises across all teams, and 𝑊P −𝑊R =

OEG (MSEMT) GL MSEMT LMSMT OEGGLMS OEG GLMT OEG

> 0.

Therefore, 𝑊P −𝑊R > 0 when GOEG

< 𝛿". We next observe that 𝑊P −𝑊R increases as 𝛿"

increases:

𝜕𝜕𝛿"

𝑊P −𝑊R

=

𝑥 − 1 1 + 𝛿"D 𝑥 − 1 + 2𝛿"1 + 𝛿" 𝑥 − 1

D > 0                            if  𝛿# ≤ 𝛿 𝛿"

𝛿" 𝑥 − 11 + 𝛿" 𝑥 − 1

+𝑥 − 1 (1 − 𝛿#) + 𝛿" − 𝛿#

1 + 𝛿" 𝑥 − 1D > 0    if  𝛿 𝛿" < 𝛿# <

1𝑥 − 1

𝑥 − 1 1 + 𝛿"D 𝑥 − 1 + 2 𝛿" − 𝛿#

1 + 𝛿" 𝑥 − 1D > 0          if  

1𝑥 − 1

< 𝛿#

We lastly observe that 𝑊P −𝑊R weakly decreases as 𝛿# increases:

𝜕𝜕𝛿#

𝑊P −𝑊R

=

                                                                         0                                                                                          if  𝛿# ≤ 𝛿 𝛿"

                                     −2𝛿" 𝑥 − 11 + 𝛿" 𝑥 − 1

−2 𝑥 − 1

1 + 𝛿# 𝑥 − 1D < 0                if  𝛿 𝛿" < 𝛿# <

1𝑥 − 1

−𝛿" 𝑥 − 1 − 11 + 𝛿" 𝑥 − 1

−𝑥

1 + 𝛿# 𝑥 − 1D < 0                    if  

1𝑥 − 1

< 𝛿#

Q.E.D.

50  

Proof of Proposition 3

As before, we have the same set of feasible collusion, 𝐵(𝑤:) for team 𝑘. Recall the ex ante

Pareto constraints:

𝑓 2 𝑤:; − 𝑐 ≥ 𝑈; 𝛽:;𝑤:;  or (Pareto′)

𝑓 2 𝑤:] − 𝑐 ≥ 𝑈] 𝛽:;𝑤:

]    for  all  𝛽:,

Consider a homogeneous team first. The principal must consider: maxs~∈�(�~)

{𝑓 1 𝑤P;G −

𝛽P𝑐, 𝑓 1 𝑤P;D − 1 − 𝛽P 𝑐}, which obtains the maximum at 𝛽P =C D EC G

m𝑤P;D and/or 𝛽P = 1 −

C D EC Gm

𝑤P;G. If 𝛽P =C D EC G

m𝑤P;D is the binding constraint, we target agent 𝑖1.

𝑓 2 𝑤P;G − 𝑐 ≥ 𝑓 1 𝑤P;G − 𝑓 2 − 𝑓 1 𝑤P;D ⇔ 𝑓 2 − 𝑓 1 𝑤P;G + 𝑤P;D ≥ 𝑐.

Thus, the minimum necessary collusion-proof wages must satisfy 𝑤P;G + 𝑤P;D =m

C D EC G. Recall

that the set of feasible collusion is non-empty when 𝑤P;G + 𝑤P;D ≤m

C D EC G. Thus, for any optimal

collusion-proof wages, the optimal correlated randomization is uniquely characterized as 𝛽P =C D EC G

m𝑤P;D = 1 − C D EC G

m𝑤P;G.

Because both 𝑤P;G and 𝑤P;D must ensure mutual monitoring incentive, 𝑤P;� ≥ 𝑤P;∗, 𝑛 = 1,2 is

required. Regardless of whether the contract is symmetric ( mD C D EC G

) or asymmetric (𝑤P;G =

mC D EC G

− 𝑤P;∗, 𝑤P;D = 𝑤P;∗, or any other combinations), the collusion constraint binds for 𝛿; >

GOEG

. Because

𝑓 2 − 𝑓 1 𝑤P;∗ + 𝑤P;∗ ≥ 𝑐 ⇔ 𝛿; ≤1

𝑥 − 1.

That is, given that the mutual-monitoring wages are offered, agents find collusion Pareto-

dominant for 𝛿; >G

OEG.

Now, consider diverse teams. Because maxs�∈�(��)

{𝑓 1 𝑤R" − 𝛽R𝑐, 𝑓 1 𝑤R# − 1 − 𝛽R 𝑐} does

not depend on the agents’ discount factors, we make the same conclusion. The maximum payoff

under feasible collusion is at 𝛽R =C D EC G

m𝑤P# and/or 𝛽R = 1 − C D EC G

m𝑤P". Contrary to the

51  

regret-free collusion, whether agent ℎ is targeted or not, the minimum necessary collusion-proof

wages satisfy 𝑤R" + 𝑤R# =m

C D EC G where 𝑤R; ≥ 𝑤R;∗, 𝑖 = ℎ, 𝑙. The collusion constraint binds for

𝛿"𝛿# >G

OEG Q. Because

𝑓 2 − 𝑓 1 𝑤P"∗ + 𝑤P#∗ ≥ 𝑐 ⇔ 𝛿"𝛿# ≤1

𝑥 − 1 D.

Whenever 𝛿" >G

OEG, we have G

MS OEG Q <G

OEG. Thus, for 𝛿# <

GMS OEG Q <

GOEG

< 𝛿", the collusion

constraint does not bind. For GMS OEG Q < 𝛿# <

GOEG

< 𝛿", the collusion constraint binds.

Let 𝑊: denote the sum of wags under team 𝑘 ∈ {𝑠, 𝑑}. Recall that mutual-monitoring wages

do not depend on team composition and that the sum of minimum necessary collusion-proof

wages is mC D EC G

. Therefore, if collusion does not arise (or arise) across all teams, we have 𝑊P =

𝑊R: for 𝛿# < 𝛿" ≤ 1/(𝑥 − 1) or 𝛿" > 𝛿# ≥ 1/(𝑥 − 1), we have 𝑊P = 𝑊R. For 𝛿# ≤

1/𝛿" 𝑥 − 1 D and 1/(𝑥 − 1) < 𝛿", collusion arises under the team of agent ℎs (homogenous

team), but does not arise under the team of agent 𝑙s and diverse teams. Thus,

𝑊P =𝑐

𝑓 2 − 𝑓 1 1 +2

1 + 𝛿#   𝑥 − 1

> 𝑊R =𝑐

𝑓 2 − 𝑓 12

1 + 𝛿"   𝑥 − 1+

21 + 𝛿#   𝑥 − 1

.

For 1/𝛿" 𝑥 − 1 D < 𝛿# ≤ 1/(𝑥 − 1) < 𝛿", collusion arises under the team of agent ℎs and

diverse teams, but does not arise under the team of agent 𝑙s. Thus,

𝑊P =𝑐

𝑓 2 − 𝑓 1 1 +2

1 + 𝛿#   𝑥 − 1< 𝑊R =

2𝑐𝑓 2 − 𝑓 1 .

Q.E.D.

52  

Appendix B.

   

 

𝑤  

𝛿"  

M for agent 𝑙

C for agent ℎ under homogenous teams

M for agent ℎ

1√𝑥 − 1

Figure 2. Optimal Contracts

Figure 2 depicts optimal contracts for the following parameter values: = 1, 𝑓(2) = 0.9, 𝑓(1) = 0.75, 𝑓(0) = 0.51, and 𝛿# = 𝛿" − 0.15. M and C denote the mutual monitoring wage and the collusion-proof wage, respectively. In this parameter region, y1/(𝑥 − 1) = 0.79. The downward sloping lines are the mutual-monitoring wages for agent ℎ (dashed line) and for agent 𝑙 (solid line), respectively. The upward slopping lines are the collusion-proof wages for agents ℎ (dashed line) and for agent 𝑙 (solid line). At point Δ = 0.91 at which the collusion constraint binds under diverse teams, 𝛿"𝛿# =

GOEG

GLMS(OEG)GLMT(OEG)

is satisfied. In the team of agent 𝑙s under homogeneous assignment, the mutual monitoring wage is the same as the mutual-monitoring wage for agent 𝑙 under diverse teams (downward solid line) and the collusion-proof wage is lower than that under diverse teams (the upward solid line).

C for agent 𝑙 under diverse teams

Δ

53  

 

0.6 0.7 0.8 0.9 1.0

0.5

1.0

1.5

𝛿"  

Figure 3. Disparity in Pay-for-Performance Sensitivities

Figure 3 depicts the disparity in PPS across teams for the following parameter values: 𝑐 = 1, 𝑓(2) =0.9, 𝑓(1) = 0.75, 𝑓(0) = 0.51, 𝛿# = 𝛿" − 0.15. In this parameter region, y1/(𝑥 − 1) = 0.79. The downward sloping solid line is the disparity in mutual-monitoring wages under diverse teams. The upward slopping lines are the disparity in mutual-monitoring and collusion-proof wages under diverse teams (solid line) and under homogeneous teams—team of agent ℎs (dashed line). At point Δ = 0.91 at which the collusion constraint binds under diverse teams, 𝛿"𝛿# =

GOEG

GLMS(OEG)GLMT(OEG)

is satisfied. In the team of agent 𝑙s under

homogeneous teams, the qualtitative nature of disparity is the same as the team of agent ℎs but the magnitude of disparity is lower than the dashed line.

Diverse teams

Team of agent ℎs

1√𝑥 − 1

Δ