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Social Preferences and the Distribution

of Rewards

Raphaël Soubeyran,

Nicolas Quérou

&

Mamadou Gueye

CEE-M Working Paper 2022-06

Social Preferences and the Distribution of Rewards∗

Raphael Soubeyran†Nicolas Qu´erou‡Mamadou Gueye§

This version: June 2022

Abstract

Motivated by the potential tension between coordination, which may require dis-

criminating between identical agents, and social comparisons, which may call for small

pay diﬀerentials, we analyze the optimal reward scheme in an organization involving

agents with social preferences whose tasks are complementary. Although a tension ex-

ists between the eﬀects of inequality aversion and altruism, there is always more reward

inequality when agents are inequality-averse and altruistic than when they are purely

self-interested. We then highlight how our results diﬀer when agents are not altruistic

but rather inequality-averse a la Fehr and Schmidt (1999).

JEL classiﬁcation: D91, D86, D62

Keywords: incentives, coordination, principal, agents, social comparisons.

∗This article builds on two preliminary contributions, respectively, Gueye, Qu´erou, and Soubeyran (2021)

and Soubeyran (2021). We thank Frederic Deroian, Brice Magdalou and Antoine Soubeyran for helpful

discussions. This project beneﬁted from the ﬁnancial support of the Chair Energy and Prosperity. N. Qu´erou

acknowledges the ANR project GREEN-Econ (Grant ANR-16-CE03-0005).

†CEE-M, Univ. Montpellier, CNRS, INRAe, Institut Agro, Montpellier, France. E-mail address:

raphael.soubeyran@inrae.fr

‡CEE-M, Univ. Montpellier, CNRS, INRAe, Institut Agro, Montpellier, France. E-mail address: nico-

las.querou@umontpellier.fr

§PSL - LEDa - Univ. Paris Dauphine, Paris, France. E-mail address: mamadou.gueye@dauphine.psl.eu

1

1 Introduction

While social comparisons do aﬀect workers’ performance, well-being and pay,1little is known

about how organizations should account for these features. Should inequality aversion yield

a decrease in reward inequalities within organizations? Should it be associated with lower

monetary incentives? More generally, how should social comparisons aﬀect the distribution of

rewards within organizations?

In order to address these questions, we analyze the implications of social preferences for

the optimal design of reward schemes. Speciﬁcally, when workers’ decisions in an organiza-

tion exhibit complementarity eﬀects, there is a potentially important tension between social

comparisons and incentives. Indeed, a given worker may fear the risk that the other workers

shirk, making her own eﬀort useless. One solution is to oﬀer a suﬃciently high reward to a

worker, so that exerting eﬀort is beneﬁcial to her even when the other workers shirk. This

in turn removes the risk (for others) that this worker does not perform her task. Using this

argument in a contracting setting with externalities,2diﬀerent contributions highlight that the

need to ensure agents’ coordination implies that the optimal reward scheme is discriminatory,

meaning that workers obtain diﬀerent rewards even if they are identical (Segal, 2003; Winter,

2004). When workers are averse to inequalities, unequal rewards are likely to negatively aﬀect

them and, as such, to weaken the power of incentives. Yet, when workers are altruistic this

may not be a problem, since they are willing to work to increase production in order to make

other workers better oﬀ. As such, a tension seems to exist between the likely eﬀects of aversion

to inequalities and altruism.

We explore the implications of such tension and introduce an organization setting with

multiple agents who exhibit social preferences and whose tasks are complementary. We ﬁrst

consider the case where symmetric agents are inequality averse and altruistic, in that an

agent’s social component of utility always increases when the other agents’ payoﬀs increase.

We then consider the case where agents are inequality averse a la Fehr and Schmidt (1999), a

prominent form of social preferences which is such that an agent’s social component of utility

may decrease when other agents’ payoﬀs increase. Speciﬁcally, each agent then negatively

values the diﬀerence between her material payoﬀ and that of any other agent: her utility thus

decreases as another agent’s payoﬀ increases provided she obtains a smaller payoﬀ than this

given agent.

Our ﬁrst main result addresses how inequality aversion with altruism (IAwA) aﬀects the

distribution of rewards. The optimal reward scheme (the least cost scheme inducing all agents’

eﬀort provision as the unique equilibrium of the induced game) is such that the agents obtain

lower rewards, and inequality in the reward distribution is higher, when the agents are inequal-

1See Bandiera et al. (2005); Card et al. (2012); Cohn et al. (2014); Breza et al. (2017); Dube et al. (2019)

2The seminal paper is Segal (1999).

2

ity averse and altruistic rather than purely selﬁsh. This result is far from straightforward, and

the intuition goes as follows. If the agents exhibit no social preferences, as more agents ex-

ert eﬀort, smaller rewards are required to induce the remaining agents to exert eﬀort. The

optimal reward exhibits a divide and conquer property: agents are ranked, and each agent is

indiﬀerent between exerting eﬀort and shirking when the higher ranked agents exert eﬀort and

the lower ranked agents shirk. Thus agents are discriminated as identical agents get diﬀerent

rewards. Social preferences may result in more or less inequality in the reward distribution.

Indeed, social preferences have both an extensive margin and an intensive margin eﬀect on

inequalities.

At the extensive margin, each additional agent who decides to exert eﬀort generates a

positive externality for the agent just ranked above her. Since the agents are altruistic and

thus value the positive externalities they generate for others, the principal can decrease each

agent’s reward. Therefore, the extensive margin eﬀect tends to increase inequality in the

reward distribution. However, along the intensive margin, each additional agent who decides

to exert eﬀort generates positive externalities for all the higher ranked agents, who already

beneﬁt from the positive externalities they generate for each other. Since the agents are averse

to inequality, an additional agent’s marginal contribution to the social component of utility

is lower as more agents exert eﬀort. As more agents exert eﬀort, the principal must increase

the rewards allocated to the remaining agents to induce them to also exert eﬀort. Thus the

intensive margin eﬀect tends to decrease inequality in the reward distribution. Moreover,

the extensive margin eﬀect applies to the agent whose material payoﬀ is the lowest, while the

intensive margin eﬀect applies to the other agents who exert eﬀort and who get higher rewards.

Since the agents are averse to inequality, the extensive margin eﬀect is always stronger than the

intensive margin eﬀect. A ﬁnal implication is then that inequality in the reward distribution

is higher when the agents are inequality averse and altruistic rather than purely selﬁsh.

Our second main result addresses how inequality aversion a la Fehr and Schmidt (1999)

aﬀects the distribution of rewards. We show that this type of inequality aversion (IneqA)

should be associated with larger rewards: compared to the case where agents are selﬁsh,

agents exhibiting inequality aversion should be all provided with larger rewards. We then

show that inequality is unambiguously lower compared to a situation where the agents are

not inequality averse. But provided the agents are already inequality averse, an increase in

inequality aversion may actually result in an increase in inequality.3We further highlight that

disadvantageous inequality aversion is of ﬁrst-order importance compared to advantageous

inequality aversion. Finally, the distribution of rewards may also be non monotonic, and the

most inequality-averse agents may lie at both ends of the reward distribution.

The mechanism at work can here be described as follows. When designing the optimal

3Montero (2007) also shows (in a very diﬀerent setting) that inequality aversion may increase inequality.

3

reward scheme, the principal takes into account that oﬀering a reward to an agent has both

direct and indirect eﬀects on the other agents’ decisions. The direct eﬀect is due to social

comparisons, while the indirect eﬀect is due to the complementarity eﬀect in the agents’ eﬀorts.

While the indirect eﬀect is clearly positive, the direct eﬀect is ambiguous, as an agent’s payoﬀ

may increase or decrease when her eﬀort increases. This follows from the characterization

of the optimal reward scheme, which again exhibits the divide and conquer property. This

implies that, in this case, an agent’s material payoﬀ equals the sum of the opportunity cost

and of the increase in disutility due to the inequality resulting from this agent’s decision to

exert eﬀort. As in the case of IAwA social preferences, the logic for the connection between the

unique implementation requirement and the divide and conquer property is fairly consistent

with the existing literature (Segal, 2003; Winter, 2004; Halac et al., 2020).

Our third main result concerns the role of coordination. So far the principal is assumed to

explicitly account for the existence of a coordination problem (due to the potential existence

of multiple equilibria) when designing the reward scheme: the optimal reward scheme must

induce her desired outcome as a unique equilibrium of the induced game. We come back to the

same problem and instead assume that the principal can costlessly select her most preferred

equilibrium outcome. As such, the principal now oﬀers the least-cost reward scheme that

implements her desired outcome as one (of possibly many) equilibrium outcome of the induced

game. This has a notable eﬀect on the analysis, as several qualitative conclusions are reversed.

For IAwA-type of social preferences, while symmetric agents always obtain identical rewards

unlike in the previous case, there is not much diﬀerence regarding the other main results:

for instance, the level of rewards is lower compared to the case where the agents exhibit no

social preferences. By contrast, for IneqA-type of social preferences, all important results are

reversed: inequality aversion results in a decrease in the individual rewards, and advantageous

inequality aversion is then of ﬁrst order importance compared to disadvantageous inequality

aversion. The fact that eﬀort provision by all agents be only one of several equilibria allows

the principal to reduce the overall cost of the contractual scheme by relaxing the agents’

incentive constraints: this actually implies that advantageous inequality aversion parameters

then become most relevant. Moreover, the reward distribution becomes very diﬀerent from

the one derived when the principal has to explicitly solve for the coordination problem.

Overall, our results highlight that agents’ altruism is beneﬁcial to the principal and results

in lower agents’ rewards, while agents’ inequality aversion is detrimental to the principal.

When the agents are altruistic, our results suggest that inequality aversion has less eﬀect than

altruism, a result that is consistent with empirical evidence (Gueye et al., 2020). We also ﬁnd

that optimal rewards are necessarily higher when agents are (IneqA) inequality averse rather

than purely selﬁsh. This is also an important result, which contradicts a conjecture made by

Cohn et al. (2014) that inequality aversion should cause a reduction in pay inequality, and

that it should be associated with smaller monetary incentives. Another important implication

4

of our analysis is that in any case, social preferences may yield more inequality.

Literature. This paper is related to two diﬀerent strands in the literature. The ﬁrst one

relates to an empirical literature on the potential eﬀect of social comparisons on workers’

well-being and performance in organizations.4Recent empirical studies highlight that social

comparisons do aﬀect workers’ pay and job satisfaction (Card et al., 2012), workers’ perfor-

mance (Bandiera et al., 2005; Cohn et al., 2014), output and attendance (Breza et al., 2017),

and even decisions to quit (Dube et al., 2019). While social comparisons seem to strongly

aﬀect workers well-being and performance, little is known about how organizations should

account for these features. This is the main goal of this paper which, to our knowledge, is

the ﬁrst to analyze the interplay between the problem of coordinating agents’ decisions and

social preferences. We provide theoretical results, and as such general conclusions, about the

speciﬁc characteristics of the optimal reward scheme when multiple agents exhibiting social

preferences interact within an organization.

The present contribution also relates to the literature on behavioral contract theory.5

Speciﬁcally, this study relates to the literature focusing on optimal contracting with mul-

tiple inequality-averse agents (Demougin et al., 2006; Rey-Biel, 2008).6Very few papers con-

sider a principal - multiple agent relationship when agents are inequality averse.7A part of

this literature provides results related to team-based incentives, as for instance Bartling and

Von Siemens (2010, 2011), Rey-Biel (2008), or Itoh (2004). The general focus of these studies

is on how the principal tailors agents’ incentives to account for agents’ preferences by oﬀering

more equitable contracts or team-based incentive schemes. All these papers diﬀer notably

from the present contribution in terms of the setting considered and the research questions.

Moreover, all these contributions focus on two-agent settings and none of them is suited to

analyze the (possibly non monotonic) distribution of rewards. Finally, these contributions do

not tackle the problem where a principal uses rewards to explicitly induce coordination among

agents.8

Structure of the paper. The remainder of the contribution is organized as follows. Section

2 introduces the model. In Section 3 we characterize the optimal reward scheme that induces

eﬀort provision and coordination when the agents exhibit IAwA-type of social preferences. In

Section 4 we study the case where they exhibit IneqA-type of social preferences. In Section 5

4Recent laboratory experimental evidence clearly rejects the assumption that individuals care only about

their material payoﬀs (Camerer, 2003).

5See Koszegi (2014), DellaVigna (2009), and Rabin (1998) for extended reviews.

6Englmaier and Wambach (2010) mainly consider the eﬀect of inequality aversion on contract design in a

single principal - single agent setting when the agent cares for the principal’s material payoﬀs.

7Goel and Thakor (2006) analyze the case where agents envy each other: The focus is on contracts inducing

surplus sharing in the case of homogeneous agents. G¨urtler and G¨urtler (2012) analyze the eﬀect of inequality

aversion on individuals’ behavior in a quite general setting. Yet, the focus is on the externalities resulting

from such preferences in an homogeneous population setting.

8Dhillon and Herzog-Stein (2009) analyze the coordination problem in a very diﬀerent setting where agents

are status-seeking.

5

we analyze the diﬀerences that emerge when the principal can costlessly solve the coordination

problem.

2 The Model

A principal oﬀers individual bilateral contracts to several agents in an environment character-

ized by positive externalities between the agents. First, the principal oﬀers a publicly observ-

able reward scheme to a set of agents. Second, the agents observe the principal’s proposition

and simultaneously decide whether to exert eﬀort or shirk.

The vector of agents’ decisions is e= (e1, ..., en)∈ {0,1}n, where ei= 1 means that agent

ichooses to exert eﬀort and ei= 0 means that that agent decides to shirk. In the absence

of monetary incentives, agent i’s payoﬀ is denoted bi(e). Any agents’ eﬀort eigenerates

positive externalities for the other agents and they are complementary (e.g. eﬀort cost-reducing

externalities): bis strictly increasing in each of its arguments and supermodular. When an

agent (say, agent i) is the only one to exert eﬀort, her payoﬀ is normalized to zero, that is

bi(e) = 0 if Pj6=iej= 0 and ei= 1. An agent who decides to shirk receives an outside option

c, that is bi(e) = cif ei= 0.

The principal aims at inducing all agents to exert eﬀort at least cost. To reach this goal,

the principal oﬀers a reward scheme v= (v1, v2, ..., vn)∈Rnto agents i= 1,2, ..., n, to

induce them to exert eﬀort. An agent’s reward is conditional on this agent exerting eﬀort:

agent iobtains reward vifrom the principal if he exerts eﬀorts and 0 otherwise. The reward

scheme vis designed such agent i∈Nreceives a unique oﬀer υi, meaning the principal can

use individualized rewards. We will denote yyi=athe vector y= (y1, ..., yn)∈Rnwhich ith

component equals a. We will denote y−ithe vector ywhich ith component is removed. For

instance, if n= 3 we have π−1= (π2, π3). Then yk= (y, ..., y) denotes the k-dimensional

vector such that each component equals y∈R. Finally, t= (z,y) denotes the vector which

ﬁrst components correspond to those of vector zand last components correspond to those of

vector y.

Agent i’s material payoﬀ is:

πi(e, vi) = eivi+bi(e).(1)

The agents exhibit social preferences and they award relative weight to their own payoﬀ

(a “self-interested” motive) and to the distribution of all agents’ payoﬀs (a social motive).

Formally, the social utility of agent iis:

Ui(e,v) = ui(πi(e, vi)) + θWi(π(e,v)),(2)

6

where π(e, v) denotes the vector of the agents’ payoﬀs, uiis the selﬁsh component of utility and

Widenotes the agent’s social component of utility. Parameter θ≥0 measures the magnitude

of the social component of an agent’s utility. When θ= 0, the agents are purely self-interested

and when θ > 0 the agents exhibit social preferences. The individual utility function uiis

strictly increasing.

The principal seeks to implement full eﬀort (the outcome where all agents exert eﬀort) in

the following way. A reward system vimplements full eﬀort as a unique Nash equilibrium

if e=1nis the unique Nash equilibrium of the game induced by v.9The set of schemes

that implement full eﬀort as a unique Nash equilibrium is open because vtakes on continuous

values. Thus, we deﬁne the optimal reward scheme as the least cost scheme such that, for

any > 0, increasing viby for any agent i∈Nimplements full eﬀort as a unique Nash

equilibrium.10 More precisely, we characterize the reward vector v∗that solves the following

optimization problem:

Minv∈<nX

j∈N

vj(3)

s.t. for all > 0, full eﬀort (e=1n) is a Nash equilibrium when the rewards are increased by

:

Ui(1n,v+n)≥Ui(1n

ei=0,v+n),(NE)

for all i∈N, and there is no other Nash equilibrium, that is, for all e6=1n,∃i∈Nsuch that

ei=a,a∈ {0,1}and:

Ui(eei=1−a,v+n)> Ui(e,v+n).(UC)

The set of constraints (NE) ensures that each agent has incentives to exert eﬀort when all

other agents also exert eﬀorts, and the set of constraints (UC) ensures that, for each of the

other outcomes, at least one agent has incentives to deviate.

3 Inequality Aversion with Altruism

This section concerns the implications of IAwA-type of social preferences on the distribution

of rewards. We ﬁrst set general assumptions on the social component of the utility function,

then we study speciﬁc reward schemes (which exhibit the divide and conquer property), and

ﬁnally provide our main result on optimal rewards and inequality.

Les us introduce our assumptions on the social component of the utility function.

9In Section 5 we investigate the case where the principal can costlessly select her most preferred equilibrium

outcome. Thus, we characterize the least-cost contract that ensures that the outcome where all agents exert

eﬀort be one (of possibly many) equilibrium (partial implementation), as is done in Segal (1999) for instance.

10This solution concept corresponds to that of Winter (2004) and Halac et al. (2020). Halac et al. (2021)

use a similar concept in a Bayesian game.

7

Assumption C-SupM (Inequality aversion): For any i∈Nthe social component Wi

is concave and supermodular.

Assumption A (altruism): For any i∈Nthe social component Wiis strictly increasing

in πjfor any j∈N.

Assumption NDC (no double counting): For any i∈Nthe social component Widoes

not depend on πi. Formally Wi=Wi(π−i(e,v−i)) where π−i(e,v−i) denotes the vector of

the agents’ payoﬀs except agent iand v−idenotes the vector of rewards of all agents except

agent i.

Assumption S (symmetry): The agents are symmetric, that is ui≡u,bi≡band Wi≡W

and then Ui≡Ufor all i.

Notice that Assumption S implies that the level of social component Wiis the same for any

permutation of the material payoﬀ of the other agents: in this sense, others are anonymous.

Assumption C-SupM models inequality aversion in the following natural way. Concavity

guarantees that the social component of the utility function increases when transfering (part

of) a reward from one agent to another poorer agent (Atkinson, 1970). Supermodularity

enables to consider general inequality aversion preferences where the social component of

utility is non-separable across the agents’ rewards (Meyer and Mookherjee, 1987) and it implies

that the social component of utility increases more with an agent’s reward if the rewards of

other agents are high.

Assumptions C-SupM and A are satisﬁed by well known social welfare functions, such as

the the constant elasticity of substitution (CES) social welfare function, or the quasi-maximin

social welfare function, and assumptions NDC and S can be applied to these functions.11

To illustrate our results and provide intuition, we will use the following example assuming

a utilitarian social welfare function and linear externalities in the agents’ eﬀorts:

Example [Utilitarianism]: Let ube a concave function, Withe utilitarian social welfare

function Wi(π−i) = Pk6=iu(πk), and the externalities as modeled by function bi(e) = eiPj6=iejw+

(1 −ei)cwhere w > 0. Let n= 3 so W1(π2, π3) = u(π2) + u(π3),W2(π1, π3) = u(π1) + u(π3),

W3(π1, π2) = u(π1)+u(π3), while b1(e) = e1(e2+e3)w+(1−e1)c,b2(e) = e2(e1+e3)w+(1−e2)c,

and b3(e) = e3(e1+e2)w+ (1 −e3)c.

11The CES social welfare function is such that Wi(π−i) = hPk6=iu(πk)s−1

sis

s−1, where s∈]0,+∞[ and

s6= 1. The quasi-maximin social welfare function, introduced in Charness and Rabin (2002), is such that

Wi(π−i) = ηmin{π1, ..., πi−1, πi+1 , ..., πn}+ (1 −η)Pj∈Nπjwhere η∈[0,1].

8

The main results of this section highlight that IAwA-type of social preferences result in

lower agents’ rewards and in lower inequality in the distribution of rewards compared to purely

selﬁsh preferences.

3.1 Ranking and Reward Scheme

In this section, we analyze the principal’s problem.

3.1.1 Ranking Agents and Reward Distribution

The following class of reward schemes will prove to be useful in the analysis:

Deﬁnition 1: A reward scheme vis a ranking scheme for agent iif the agents are ranked

in a given order, and agent iis indiﬀerent between exerting eﬀort or shirking when the higher

ranked agents also exert eﬀorts and the lower ranked agents shirk.

To save on notations, the payoﬀ that agent iobtains in this situation will be denoted ˜πi.

It is such that:

˜πi≡πi(1i,0n−i) (4)

Coming back to our example and assuming that the agents are ranked from 1 to 3, we have

˜π1=v1, ˜π2=v2+wand ˜π3=v3+ 2w.

Formally, Deﬁnition 1 means that, if the agents are ranked from 1 to n(without loss of

generality), and vis a ranking scheme for agent i, we must have:

ui(c)−ui(˜πi) = θhWi(π−i((1i,0n−i),v−i)) −Wi(π−i((1i−1,0n−i+1),v−i))i(5)

This condition corresponds to the case where agent iexerts eﬀort, while the higher ranked

agents also exert eﬀorts and the other agents shirk. It states that the marginal beneﬁt from

shirking (related to the selﬁsh component of utility) is equal to the marginal beneﬁt from

exerting eﬀort (related to the social component of utility).

In our example, we have the following condition for the highest ranked agent:

u1(c)−u1(˜π1) = 0 (6)

The highest ranked agent must be indiﬀerent between exerting eﬀort or shirking, when no other

agent exerts eﬀort. Hence, his reward has to be equal to his opportunity cost, ˜π1=v1=c.

9

Now consider the indiﬀerence condition for agent 2:

u2(c)−u2(˜π2) = θ

u1(˜π1+w)−u1(˜π1)

| {z }

B21

(7)

This condition states that this agent’s marginal beneﬁt from shirking (related to the selﬁsh

component of utility) must equal the social marginal beneﬁts that he provides to higher ranked

agent, namely agent 1.

Finally, consider the third agent’s indiﬀerence condition:

u3(c)−u3(˜π3) = θ

u1(˜π1+ 2w)−u1(˜π1+w)

| {z }

B31

+u2(˜π2+w)−u2(˜π2)

| {z }

B32

(8)

This condition states that this agent’s marginal beneﬁt from shirking (related to the selﬁsh

component of utility) must equal the sum of the social marginal beneﬁts that this agent

provides to the higher ranked agents (agents 1 and 2).

Using these conditions, we can show that the agents’ payoﬀs are lower than their oppor-

tunity cost:

Proposition 1 [Payoﬀs under IAwA]: If vis a ranking scheme for agent iand all higher

ranked agents exert eﬀorts while the lower ranked agents shirk, then agent i’s material payoﬀ

is lower than the opportunity cost, ˜πi≤c.

The intuition of this result is as follows. When an agent exerts eﬀort, he generates positive

externalities to the other agents who also exert eﬀorts, which in turn increases the level of his

social component of utility. It is thus not necessary to compensate for the full opportunity

cost to induce this agent to exert eﬀort. Inspecting conditions (6), (7), (8) highlights that this

holds in the case of our example.

We now show a much less intuitive result:

Proposition 2 [Distribution of Payoﬀs under IAwA]: If vis a ranking scheme for the

ﬁrst iagents according to a common ranking (1to nwithout loss of generality), then ˜πk−1≥˜πk

for all 2≤k≤i.

In this Proposition, we compare the payoﬀs of two subsequent agents in two diﬀerent

situations. For each agent, we consider the situation where they exert eﬀort together with the

higher ranked agents while the other agents shirk. We show that the highest ranked agent

among the two subsequent agents obtains a higher material payoﬀ than the other one. The two

10

payoﬀs are equal when the agents exhibit no social preferences (θ= 0) and they are diﬀerent

when they do exhibit such preferences (θ > 0). We explain the intuition in both cases below.

If the agents exhibit no social preferences (θ= 0), agent 1 must be indiﬀerent between

exerting eﬀort and shirking when no other agent exerts eﬀort. In this case, the agents do not

beneﬁt from positive externalities. Agent 2 must be indiﬀerent between exerting eﬀort and

shirking when agent 1 also exerts eﬀort. Agent 2 thus beneﬁts from a positive externality

and has greater incentives to exert eﬀort. Agent 2 thus obtains a reward that is equal to the

diﬀerence between agent 1’s reward and the positive externality generated by this agent. This

reasoning holds for any two subsequent agents.

Next consider the case where the agents exhibit social preferences (θ > 0). In our example

one can notice that ˜π1=c > ˜π2(by Proposition 1). Looking at the diﬀerence between the

indiﬀerence conditions for agents 3 and 2 (conditions (8) and (7)) we have:

u2(˜π2)−u3(˜π3) = θ× {B31 −B21

| {z }

(−)

+B32

|{z}

(+) }(9)

The sign of this diﬀerence is a priori ambiguous. The ﬁrst term between brackets on the

right hand side is the diﬀerence between the social marginal beneﬁt agent 1 provides to agent

3 and to agent 2. This intensive margin eﬀect is negative because of concavity of u. The

second term on the right hand side is the marginal social beneﬁt agent 3 provides agent 2

with. This extensive margin eﬀect is positive.

We can rewrite the diﬀerence as follows:

u(˜π2)−u(˜π3) = θ

B31

|{z}

(+) −

u1(˜π1+w)−u1(˜π1)

| {z }

B21

+u2(˜π2+w)−u2(˜π2)

| {z }

B32

| {z }

(+)

(10)

The ﬁrst term (B31) denotes the marginal social beneﬁt that agent 3 provides agent 1 with:

it is positive. The second term (B32 −B21) denotes the diﬀerence between the marginal social

beneﬁt that agent 3 provides to agent 2 and the marginal social beneﬁt that agent 2 provides

to agent 1. Since ˜π1>˜π2while the agents are symmetric and uis concave, this diﬀerence is

positive, and we can conclude.

The condition above proves that Proposition 2 holds in the utilitarian case with three

agents, concave utility uand linear externalities. The extensive margin eﬀect applies to the

agent with the lowest material payoﬀ, while the intensive margin eﬀect applies to the agents

who also exert eﬀort and whose material payoﬀs are higher. The concavity of Wimplies that

the extensive margin eﬀect is larger than the intensive margin eﬀect.

Proposition 2 also holds for more general settings than the one of our utilitarianism exam-

11

ple: the symmetry assumption together with the concavity of Wand supermodularity of W

and ballow to prove the result in the general case.

3.2 Optimal Reward Scheme and Inequality

We ﬁrst prove a preliminary result that will be used to analyze the optimal scheme:

Lemma 1: If e

vis such that each agent k≤iprefers to exert eﬀort when the higher ranked

agents (according to a common ranking, 1to n, without loss of generality) also exert eﬀort

and the remaining agents shirk, then each agent kalso prefers to exert eﬀort when jagents

(k≤j≤i) exert eﬀort.

The intuition of this result is simple if the agents exhibit no social preferences (θ= 0). As

more agents exert eﬀort, the agents who exert eﬀort beneﬁt from higher positive externalities

and thus they are more likely to exert eﬀort.

If the agents exhibit social preferences (θ > 0), the result is less straightforward. The

monotonicity result of Proposition 2 is needed to prove the result. To provide some intuition,

consider the case where externalities are linear and homogeneous and w > 0 denotes the

externality generated by each agent exerting eﬀort for any other agents who also exert eﬀort.

If e

vis a ranking scheme for the ﬁrst iagents according to a common ranking, then their

rewards are characterized by condition (5). Consider the situation where the ﬁrst iagents

exert eﬀort. If either agent ior agent kchooses to shirk, the material payoﬀ of each agent

ranked higher than iis reduced by w. Moreover, if agent ishirks, agent k’s payoﬀ is reduced

by wand if agent kshirks, agent i’s payoﬀ is reduced by w. Given that vk> viand Wis

concave, the social component of agent k’s utility decreases more (when agent kdecides to

shirk) than the social component of agent i’s utility (when agent idecides to shirk). Thus,

agent khas lower incentives to deviate than agent i. Since agent iis indiﬀerent between

exerting eﬀort and shirking, agent kstrictly prefers to exert eﬀort.

The following class of reward schemes will be useful to characterize the optimal reward

scheme:

Deﬁnition 2: A reward scheme vis a global ranking scheme if it is a ranking scheme for

all the agents according to a common ranking.

Notice that this scheme is unique for a given ranking. Each agent is indiﬀerent between

exerting eﬀort and shirking when the agents ranked before him also exert eﬀort, and the

remaining agents shirk. This deﬁnition is used to provide necessary conditions for a reward

scheme to be optimal:

12

Theorem 1 [Inequality under IAwA]: Any optimal reward scheme is a global ranking

scheme. Moreover, individual rewards are lower and the degree of inequality in the reward

distribution is higher when the agents exhibit IAwA-type of social preferences rather than purely

self-interested ones.

The ﬁrst part of the Theorem builds on the strategic complementarity property embedded

in the model due to the increasing externalities (in the sense of Segal 2003) and the supermod-

ularity property of the social component of agents’ utility. It states that the optimal scheme is

a divide-and-conquer scheme. The ﬁrst part of the Theorem thus shows the robustness of the

divide-and-conquer property when agents exhibit IAwA-type of social preferences. Notice that

the global ranking scheme is unique, up to a reordering of the (ex-ante identical) agents. This

result together with Propositions 1 and 2 yield the second part of the Theorem. It connects

the behavioral features of the model with the structure of the optimal scheme. If the agents

exhibit no social preferences, when more agents exert eﬀorts, an agent has higher incentives

to exert eﬀort since he beneﬁts from higher positive externalities. If the agents exhibit social

preferences, they derive utility from generating positive externalities to other agents: as such

the principal can provide them with lower rewards, and she can even more reduce the rewards

for the lower ranked agents. What is most surprising here, as discussed in Section 3.1.1, is

that this result holds even if the agents are both inequality averse and altruistic.

To prove suﬃciency, we need an additional assumption:

Assumption SOSC (second order social concerns): For any two reward vectors vand

v0, we have |u(πi(e, vi)) −u(πi(e, v0

i))|> θ |W(π(e,v)) −W(π(e,v0))|.

This assumption implies that social concerns are of second-order importance compared to

material payoﬀs (it has been also used in Cabrales and Calvo-Armengol 2008). Using this

assumption, we obtain the following result:

Theorem 2 [Suﬃciency]: If assumption SOSC holds, any global ranking scheme is an

optimal reward scheme.

Theorem 2 provides suﬃciency conditions. Namely, if the social component is of second-

order importance compared to the private component of the utility function, then full eﬀort

is a unique Nash equilibrium under any global ranking scheme. Suppose assumption SOSC

does not hold, and the agents are very inequality averse. An agent may prefer to shirk when

the higher ranked agents exert eﬀort and the remaining agents shirk if the agents’ rewards

are increased by a positive amount. Indeed, an increase in the agents’ rewards simultaneously

increases the marginal eﬀect of exerting eﬀort on the private component of utility and decreases

the marginal eﬀect of exerting eﬀort on the social component of utility. Thus, full eﬀort is

the unique Nash Equilibrium under a global ranking scheme if the latter eﬀect is larger than

13

the former for each agent when the higher ranked agents exert eﬀort while the lower ranked

agents shirk.

3.3 Heterogeneity

Up to now, we have focused on the case of identical agents and homogenous externalities.

One may wonder whether the main results of the paper hold in a situation where the agents

exhibit heterogeneous preferences and under heterogeneous externalities.

Let us analyze the role of heterogeneity using a simple form of altruistic preferences.

Assume that the agents exhibit heterogeneous preferences for the other agents’ payoﬀs and

that the social component of an agent’s utility is a weighted form of the other agents’ payoﬀs:

Wi(e) = X

j6=i

γijπj,(11)

where γij ≥0 denotes the intensity of the preference of agent ifor agent j’s payoﬀ.

Also assume that externalities are heterogeneous: when agents iand jexert eﬀort, agent

jgenerates positive externalities for agent iwhose related beneﬁts are denoted wij ≥0:

bi(e) = X

j6=i

wijejei+ (1 −ei)c. (12)

To keep things simple, also assume that the private component of utility is such that

ui(πi)≡πi. Using all the assumptions above, agent i’s utility can be written as:

Ui(e,v) = πi+θX

j6=i

γijπj,(13)

where

πi=ei

vi+X

j6=i

ejwij

+ (1 −ei)c. (14)

We can derive the optimal reward scheme using similar proofs as in Bernstein and Winter

(2012). We use their notations and denote ijthe agent characterized by rank j. If each agent

ijwith j < k exerts eﬀort, then agent ikalso prefers to exert eﬀort if:

vik+X

ij<ik

wikij+θX

ij<ik

γikijwijik> c. (15)

Thus, the optimal reward scheme is such that the agents are ranked from i1to inand the

14

optimal reward of agent ikis:

v∗

ik=c−X

ij<ik

(wikij+θγikijwijik),(16)

Hence, for a given ranking, the agents still obtain lower rewards when they exhibit hetero-

geneous altruistic preferences (θ > 0) than when they do not (θ= 0). The optimal ranking

depends on the virtual popularity tournament described in section A in Bernstein and Winter

(2012). In the present model, agent jbeats agent kif:

(1 −θγjk)wkj <(1 −θγkj )wj k (17)

This result is interesting since it implies that altruistic preferences aﬀect the agents’ optimal

ranking. Indeed, one may conclude that ijbeats ikif altruism is not accounted for (wjk > wkj )

while ikactually beats ijwhen altruism is accounted for ((1 −θγkj )wj k <(1 −θγj k)wkj ). This

occurs for instance when agent ijderives beneﬁts from high externalities generated by agent

ikand agent ikstrongly values agent ij’s payoﬀ.

How the introduction of altruistic preferences (θ > 0) aﬀects inequality is not straightfor-

ward. Let us consider the spread of the rewards distribution:

max

k∈Nvik−min

k∈Nvik=X

ij<il

(wilij+θγilijwijil),(18)

where ilis characterized by mink∈Nvik=vil(notice that we have l > 1). We have the following

conclusion:

Proposition 3 [Reward Distribution under Heterogeneity]: The spread of the reward

distribution is higher when the agents exhibit altruistic heterogeneous preferences than when

they exhibit no social preferences.

Thus, in this sense, altruistic preferences result in more inequality when one introduces

heterogeneity in the model. It is however important to notice that the results of this section are

limited to the case where utility is linear in the agents’ actions and externalities are additive.

They cannot be easily extended to the more general case of non linear preferences (that

encompasses inequality aversion) or of non linear externalities considered in the remainder of

the section.

4 Inequality Aversion without Altruism

Up to now, we have assumed that the agents’ social component of utility is always increasing

when the other agents’ payoﬀs increase. In this section, we depart from this assumption and

15

study a situation where the agents are assumed to be inequality averse a la Fehr and Schmidt

(1999). Speciﬁcally, the social component of agent i’s utility function is:

Wi(π) = −αi

n−1X

k6=i

max{πk(e, vk)−πi(e, vi),0}− βi

n−1X

k6=i

max{πi(e, vi)−πk(e, vk),0},(19)

where 0 ≤βi<1 and βi≤αi. The ﬁrst term on the right hand side of this equality

corresponds to disadvantageous inequality, while the second term on the right hand of the

equality corresponds to advantageous inequality. This speciﬁcation of preferences to model

aversion to inequalities has been widely used in the behavioral economics literature due to its

simplicity and to the consistency of theoretical ﬁndings with experimental evidence in diﬀerent

game situations.

This function departs from the assumptions considered in Section 3. It does not satisfy

Assumption A (altruism), since the social component of agent i’s utility can decrease when

the payoﬀ of another agent (say, j) increases. Indeed, assume there are only two agents (1 and

2) and that agent 1 is disadvantaged (π1< π2), then the social component of utility becomes

W1(π1, π2) = −α1(π2−π1) and thus it decreases when the second agent’s payoﬀ increases. It

does not satisfy Assumption NDC (no double counting) since it depends on the diﬀerences

between the agent’s payoﬀs and those of the other agents. However, it does satisfy Assumption

C-SupM since it is both concave and supermodular.

For simplicity, the externalities are assumed to be linear and homogenous:

bi(e) = X

j6=i

wejei+ (1 −ei)c, (20)

where w > 0.

We also assume that the private component of utility is linear and that θ= 1, so agent i’s

utility writes as follows:

Ui(e,v) = ei

vi+X

j6=i

wej

+ (1 −ei)c−αi

n−1X

k6=i

max{πk(e, vk)−πi(e, vi),0}

−βi

n−1X

k6=i

max{πi(e, vi)−πk(e, vk),0}.(21)

As in the previous section, we will characterize the least-cost contract that implements

eﬀort provision from all agents as a unique Nash equilibrium of the induced game.12

12In Section 5 we analyze the case where the principal can costlessly select her most preferred equilibrium

outcome. Thus, we characterize the least-cost contract that ensures that the outcome where all agents exert

eﬀorts be one (of possibly many) equilibrium (partial implementation), as in Segal 1999 for instance.

16

4.1 A simple example

Before developing the generic results and characterizations, we ﬁrst provide the intuition

underlying the characterization of the optimal reward scheme implementing full eﬀort as a

unique Nash equilibrium of the induced game (which will be provided in Proposition 5). We

do this by using a simple example. Suppose there are two agents, 1 and 2, and that the

principal’s reward scheme is a global ranking scheme, that is, agent 1 is indiﬀerent between

exerting eﬀort and shirking when agent 2 shirks and agent 2 is indiﬀerent between exerting

eﬀort and shirking when agent 1 exerts eﬀort. A restatement of the global ranking scheme

property given the agents’ inequality-averse preferences is that the following properties hold

(assuming v1≥v2):

1. The advantageous inequality generated by agent 1 exerting eﬀort when agent 2shirks

equals agent 1’s selﬁsh payoﬀ diﬀerence between exerting eﬀorts and shirking. Formally:

β1(v1−c) = v1−c⇐⇒ v1=c

where the equivalence statement follows from β1<1.

2. The net disadvantageous inequality generated by agent 2 exerting eﬀorts when agent

1exerts eﬀorts equals agent 2’s selﬁsh payoﬀ diﬀerence between exerting eﬀorts and

shirking. Formally,

α2(v1−v2)

| {z }

disadvantageous inequality when making eﬀorts −α2(v1−c)

| {z }

disadvantageous inequality under shirk

=

v2+w

| {z }

selﬁsh payoﬀ when exerting eﬀorts −c

|{z}

selﬁsh payoﬀ under shirk

Using v1=cwe notice that v2=c−w

1+α2.

From these simple calculations, we can understand why the agents must obtain a higher reward

when they are inequality averse. The counter-intuitive nature of this result lies in the following

reasoning: when agent 2 exerts eﬀorts, she actually increases disadvantageous inequality,

whereas commonplace intuition suggests that exerting eﬀorts should reduce it (inequality of

eﬀort is reduced). This property emerges because agent 2 is getting a lower reward than agent

1in spite of generating positive externalities that increases agent 1’s selﬁsh payoﬀs, that is, in

spite of her eﬀort contributions. Hence, each agent must receive higher monetary incentives

to exert eﬀorts, rather than downward reward compression.

17

4.2 The optimal reward scheme

The ﬁrst step to characterize the optimal reward scheme is to show that it is characterized

by the global ranking scheme property. The set of reward schemes exhibiting this property is

obtained by ranking agents in an arbitrary fashion, and by providing each agent with a reward

that would induce her to exert eﬀort assuming that all the higher ranked agents also exert

eﬀort and all lower ranked agents shirk. Intuitively, lower ranked agents are induced to exert

eﬀort by the others’ choice to do so and can be oﬀered smaller rewards.

So we ﬁrst consider an arbitrary ranking of the set of agents, and we provide a ﬁrst result:

Proposition 4 A reward scheme is a global ranking scheme if and only if it is an optimal

reward scheme.

This characterization result provides an important feature of the overall problem. Indeed,

the principal’s optimization problem then boils down to ﬁnding the optimal ranking and the

optimal scheme for that ranking. We now proceed in two steps. First, ﬁxing the ranking,

we characterize the optimal scheme. Let πj

idenote agent i’s material payoﬀ when the ﬁrst

jagents exert eﬀort while the remaining n−jagents shirk. Let Dj

i=αi

n−1Pk6=imax{πj

k−

πj

i,0}+βi

n−1Pk6=imax{πj

k−πj

i,0}denote agent i’s total disutility resulting from inequity.

Finally, introducing ∆Di

i=Di

i−Di−1

iwe obtain the following result:

Proposition 5 [Reward scheme under IneqA]: If the agents are averse to inequality, the

optimal reward scheme that implements full eﬀort as a unique Nash equilibrium of the induced

game is such that:13

v∗

i=c−(i−1)w+ ∆Di

i,(22)

where

∆Di

i=(i−1)αi

n−1+(i−1)αi−(n−i)βi

w, (23)

for all 1≤i≤n.

This result characterizes the additional material payoﬀ that the agents obtain because they

exhibit aversion to inequality. It highlights that the stronger the aversion to inequality (i.e.

the larger αior βi), the larger the agent’s reward. This static comparative result follows from

straightforward computations using the fact that disadvantageous inequality and advantageous

inequality increase when agent iexert eﬀort rather than when she shirks (see the discussion

based on a simple example in Section 4.1). When an agent is more averse to inequality, these

increases in inequality have to be compensated by an increase in the agent’s reward. The

Proposition also yields an interesting implication, which can be summarized as follows:

13The unique Nash equilibrium induced by these schemes is strict.

18

Lemma 2: For any agent i, the reward scheme satisﬁes πi

i−c= ∆Di

i≥0and this implies

that any agent obtains a larger payoﬀ if she is averse to inequality rather than selﬁsh.

This conclusion follows directly from Proposition 5, yet an initial intuition would suggest

that inequality aversion may cause a decrease in pay inequalities within organizations, which

in turn may be associated with smaller monetary rewards (see Cohn et al. 2014). In our

model, an intuition suggests that the principal, in order to minimize the rewards, might have

incentives to make the material payoﬀs smaller than the opportunity cost. However, this is

not possible due to the uniqueness constraint.

To understand the mechanism further, the main arguments are as follows. In order to

induce the highest ranked agent to exert eﬀort while all the other agents shirk, the principal

has to provide him with a reward that is at least equal to the opportunity cost, that is π1

1≥c.

For the other agents, it is actually not possible that the result holds for the ﬁrst i−1 agents

and not for agent i. Indeed, if the ﬁrst i−1 agents obtain a reward that is larger than cwhen

the higher ranked agents exert eﬀort, their payoﬀ increases when more agents exert eﬀort.

Hence, when the agents who are ranked higher than agent iexert eﬀort, if agent ireceives a

reward that is lower than the opportunity cost, she obtains the lowest payoﬀ, whatever her

decision. In this case, only disadvantageous inequality aversion plays a role and agent iis

more disadvantaged when she exerts eﬀort rather than when she shirks. We conclude that

agent i’s payoﬀ must satisfy πi

i≥c. Hence, agent i’s reward is v∗

i=c−(i−1)w+ ∆Di

i, where

∆Di

i≥0.

It is also interesting to notice that disadvantageous inequality aversion is of ﬁrst-order

importance compared to advantageous inequality aversion. Indeed, using a ﬁrst order approx-

imation of agent i’s reward around (αi, βi) = (0,0), we have vi∼c−(i−1)w+(i−1)αiwwhich

depends on αibut not on βi. This is due to the fact that, when the higher ranked agents exert

eﬀort, an agent is only advantaged compared to the lower ranked agents who do not exert ef-

fort and who get the opportunity cost c. The disutility that is due to advantageous inequality

aversion is thus proportional to the increase in the agent’s material payoﬀ when she decides

to exert eﬀort instead of shirking (still when only the higher ranked agents do exert eﬀort).

The divide and conquer nature of the optimal reward schemes awards a lot of importance to

higher ranked agents, and as such to disadvantageous inequality-aversion parameters.

The optimal scheme is a global ranking scheme with the optimal ranking, that is, the

ranking that minimizes the principal’s aggregate cost of providing incentives to exert eﬀort.

As such, it is important to notice that Proposition 5 does not provide insights on the agents’

optimal ranking.

19

4.3 Payoﬀs and inequality

The characterization provided in Proposition 5 can be used to provide some insights about

the eﬀects driven by inequality aversion compared to the case where the agents do not exhibit

social preferences. We obtain the following result:

Proposition 6 [Eﬀect of inequality aversion under IneqA]: Under the optimal reward

scheme, we have the following conclusions:

(i) An agent’s material payoﬀ is larger when he is inequality averse rather than purely selﬁsh

(i.e. when βi≥0and αi≥0instead of αi=βi= 0).

Assuming that the agents are symmetric (αk=αand βk=βfor all k), we also have that:

(ii) The magnitude of the diﬀerence between any two subsequent agents’ material payoﬀs,

|πi−πi+1|, is lower when the agents are averse to inequality rather than purely selﬁsh (i.e.

when β≥0and α≥0instead of α=β= 0)

These properties mostly follow from Proposition 5.14 For instance, regarding point (i), the

optimal reward scheme is such that disadvantageous inequality and advantageous inequality

increase when agent iexerts eﬀort rather than when she shirks. These increases in inequality

levels have to be compensated by an increase in the agent’s reward when she is averse to

inequality.

We can also use the characterization of the optimal reward scheme to highlight some non-

intuitive eﬀects of social comparisons on inequality levels. Here we just assume the simplest

case where αk=αand βk=βfor any k∈N, and we obtain:

Proposition 7 [Eﬀect on inequality under IneqA]: Under the optimal reward scheme,

we have the following conclusions:

(i) A marginal increase in disadvantageous inequality aversion may lead to an increase in

inequality at the bottom of the reward distribution. Formally, ∂|πi−πi+1 |

∂α ≥0if and only if i≥2

and α≥rn−1−(n−i)β

i(i−1) .

(ii) A marginal increase in advantageous inequality aversion may lead to an increase in in-

equality at the bottom of the reward distribution. Formally, ∂|πi−πi+1|

∂β ≥0if and only if i=n−1

or β∈1−ri(i−1)

(n−i)(n−i−1) (1 + α),1.

These eﬀects directly result from the characterization of optimal rewards provided in Proposi-

tion 5. The two cases highlight that an increase in the intensity of aversion to inequalities may

actually result in higher inequality through the impact of both types of inequality aversion

14The optimal ranking is generically unique: there might be knife-edge cases where two rankings are optimal

(for instance, when agents are fully symmetric).

20

on the agents’ rewards. Yet, these results also highlight that such eﬀect diﬀers depending on

whether the focus is on disadvantageous inequalities or on advantageous inequalities.

4.4 Optimal ranking

As a ﬁnal step, notice that we cannot provide a full characterization of the optimal ranking

as heterogeneity is bi-dimensional, each agent ibeing characterized by a pair of inequality-

aversion parameters, (αi, βi). However, we can characterize the optimal ranking when the

agents are identical with respect to one dimension and heterogeneous with respect to the

other one.

Let us ﬁrst consider the case where the agents have diﬀerent disadvantageous inequality-

aversion parameters:

Proposition 8 [Disadvantageous inequality under IneqA]: Assume βj=βfor all j. (i)

If the agents are weakly averse to disadvantageous inequality (αj<1for all j), then the opti-

mal ranking is such that an agent’s rank decreases as her disadvantageous inequality parameter

is lower (α1≥α2≥... ≥αn). (ii) If the agents are strongly averse to disadvantageous in-

equality (αj>1for all j), then the optimal ranking is such that the agents’ rank is a U-shaped

function of the disadvantageous inequality parameters (∃1< k < n such that α1≥... ≥αk

and αk≤... ≤αn).

If the agents value disadvantageous inequality more than their own monetary payoﬀ (case

(i)), the most averse agents lie at the top of the reward distribution. However, if the agents

value their own payoﬀ more than disadvantageous inequality (case (ii)), then the optimal

ranking is non monotonic and the most inequality-averse agents lie at both ends of the reward

distribution.

To get some intuition on this result, it is suﬃcient to focus on the disutility resulting from

disadvantageous inequality. Assume that agent iis ranked at position j. Her disutility result-

ing from disadvantageous inequality is αi∆Aj

j=αi

n−1hw−(πj

j−c)ij. A marginal increase in

αileads to a marginal disutility level ∆Aj

j+αi

∂Aj

j

∂αi. The ﬁrst term increases, while the second

term decreases, when the rank jincreases. When the degree of aversion to disadvantageous

inequality is suﬃciently low (αi<1), the ﬁrst term dominates, and then the principal has

incentives to ﬁrst rank the most inequality-averse agents. When the degree of aversion to

disadvantageous inequality is suﬃciently large (αi>1), the second term is suﬃciently strong

and the optimal ranking is non monotonic. In other words, the agents’ ranking is character-

ized by the magnitude of the disutility resulting from disadvantageous inequalities. An agent’s

disadvantageous inequality-aversion parameter has both a direct eﬀect on this disutility and

21

an indirect eﬀect resulting from its impact on this agent’s optimal reward. The direct eﬀect is

positive, while the indirect eﬀect is negative, and the net eﬀect depends on the fundamentals.

Proposition 8 only provides a partial characterization of the case where agents are ho-

mogeneous with respect to the advantageous inequality parameters. The following example

highlights that, when αj<1 is satisﬁed only for some agents j∈N, the optimal ranking

will depend on the relative values of agents’ disadvantageous inequality parameters together

with the absolute value of the advantageous inequality parameter. Thus, no fairly generic

conclusion may be expected in this case.

Let us consider the three-agent case. It is easily checked that the highest ranked agent’s

disadvantageous inequality parameter must satisfy α1≥α2. Indeed, the optimal ranking

must be such that the principal saves cost by relying on this ranking rather than switching

the position of the ﬁrst two highest ranked agents. Using Proposition 5 this implies that

α1

2+α1−β≥α2

2+α2−βmust be satisﬁed, which in turn is equivalent to α1≥α2. Now, moving on

to the comparison between the second highest ranked agent and the last agent in the ranking,

the optimal ranking must be such that the principal saves cost by relying on this ranking

rather than switching the position of these two agents. Using Proposition 5 this is equivalent

to α2

2 + α2−β+2α3

2+2α3≤α3

2 + α3−β+2α2

2+2α2

or

(1 −β)α3

(1 + α3)(2 + α3−β)≤(1 −β)α2

(1 + α2)(2 + α2−β)(24)

Since β < 1 is satisﬁed we only need to focus on function Φ(α) = α

(1+α)(2+α−β)and we obtain

Φ0(α) = 2−β−α2

(1+α)2(2+α−β)2so the sign of Φ0(α) is that of function g(α)=2−β−α2, which is

concave in α, increases up to α= 1 −β

2then decreases thereafter. This implies that, as

g(α) = 0 if and only if α=√2−β > 1, that Φ increases up to α=√2−βand decreases

thereafter. Thus inequality (24) holds when α3≤α2≤√2−βor √2−β < α2< α3holds.

Yet, these are not the only cases where this inequality holds. When α2<1<√2−β≤α3

holds then the conclusion depends on the relative values of α2and α3.

We now characterize the optimal ranking when the agents only diﬀer in terms of their

aversion to advantageous inequality:

Proposition 9 [Advantageous inequality under IneqA]: Assume αj=αfor al l j. The

optimal ranking is such that the rank is a U-shaped function of the advantageous inequality

parameters (∃1< k < n such that β1≥... ≥βkand βk≤... ≤βn).

22

The optimal ranking of heterogeneous agents who are averse to advantageous inequality is

non monotonic: The most inequality-averse agents lie at both ends of the reward distribution.

When agents are heterogeneous in their advantageous inequality-aversion parameters only, the

agents’ ranking is characterized by the magnitude of the disutility resulting from advantageous

inequalities. An agent’s disadvantageous inequality-aversion parameter has both a direct eﬀect

on this disutility and an indirect eﬀect resulting from its impact on this agent’s optimal reward.

As in the previous case, there is a trade-oﬀ between direct and indirect eﬀects, but this time

the trade-oﬀ has always bite: there is a U-shape relationship.

5 (How) does coordination matter?

In the previous sections, we have highlighted several qualitative properties of the optimal

reward scheme that solves any potential coordination problem. This raises the question about

whether the coordination problem does actually matter. Does the existence of such a problem

drastically aﬀect the characterization of the optimal reward scheme, or does it have little eﬀect

on it? To answer this question, we now solve for the optimal partial implementation reward

scheme, that is, the least-cost scheme inducing all agents to exert eﬀort as one of the Nash

equilibria of the induced game. We look for the solution to the principal problem that can

now be stated as follows:

Minv∈<nX

j∈N

vj(25)

s.t. full eﬀort (x=1n) is a Nash equilibrium:

Ui(1n,v)≥Ui(1n

xi=0,v),(26)

for all i∈N.

The set of constraints (26) ensures that each agent has an incentive to exert eﬀorts when

all other agents also exert eﬀorts. In order to minimize the cost of the scheme, the principal

will necessarily choose a reward scheme such that the constraints (NE) are binding, or:

ui(c)−ui(πi(1n)) = θhWi(π−i(1n,v−i)) −Wi(π−i((1i−1,0,1n−i),v−i))i,(27)

for all i∈N. This condition states that for each agent the marginal beneﬁt from shirking

(related to the private component of utility) must equal the marginal beneﬁt of exerting eﬀort

(related to the social component of utility) when all the other agents exert eﬀorts.

In each case (IAwA and IneqA) the assumptions made in the previous sections are main-

tained. Thus, for IAwA-type of preferences we keep the general functional form and assume

23

symmetric agents, while we keep the more speciﬁc form of utility a la Fehr and Schmidt (1999)

for IneqA-type of preferences and consider asymmetric agents.

5.1 The role of coordination under IAwA-type of social preferences

The properties of the optimal scheme under assumptions A, NDC, C-SupM and S can be

deduced from inspecting condition (27). Indeed, using symmetry, agent i’s optimal reward is

characterized by:

v∗

i=u−1u(c)−b(1n)−θhW(π−i(1n,v∗

−i)) −W(π−i((1i−1,0,1n−i),v∗

−i))i.(28)

This leads us to the following conclusion:

Proposition 10 [Partial implementation under IAwA]: The optimal partial implementa-

tion reward scheme is such that the agents obtain a lower reward when they exhibit IAwA-type

of social preferences (θ > 0) than when they exhibit no social preferences (θ= 0). Moreover,

all the (symmetric) agents obtain the same reward.

Hence, as in the case of unique implementation analyzed in Section 3, the agents obtain

lower rewards when they have IAwA-type social preferences rather than selﬁsh ones. However,

unlike the case of unique implementation, there is no inequality here whenever the (symmetric)

agents exhibit social preferences or are purely self-interested. Notice that here, there is no

need to rank the agents as in the case of unique implementation.

An interesting implication of IAwA-type of social preferences is that it aﬀects the diﬀerence

between the rewards agents obtain in the unique and in the partial implementation cases.

When agents are purely self-interested (θ= 0), the lowest ranked agent gets the same reward

in the unique implementation case and in the partial implementation case. On the other

hand, when agents exhibit IAwA-type of social preferences (θ > 0), the lowest ranked agent

gets a lower reward in the partial implementation case compared to the unique implementation

case. Indeed, the highest ranked agents get a lower reward in the partial implementation case

(compared to the unique implementation case). Thus, the marginal beneﬁt (related to the

social component of utility) that the lowest ranked agent obtains when he decides to exert eﬀort

instead of shirking is higher in the partial implementation case, because he generates positive

externalities to the other agents who obtain lower rewards than in the unique implementation

case.

5.2 The role of coordination under IneqA-type of social preferences

We now turn to analyze the role of coordination when the agents exhibit IneqA-type of social

preferences. We obtain the following ﬁrst result:

24

Proposition 11 [Partial Implementation under IneqA]: If the agents exhibit IneqA-type

of social preferences, the optimal partial implementation reward scheme is such that:

v∗

i=c−(n−1)w−Jiw, (29)

where

J1=β1

1−β1

(30)

and

Ji=βi+αi+βi

n−1Pi−1

j=1 Jj

1−βi+i−1

n−1(αi+βi),(31)

for all 2≤i≤nand given the ranking where higher ranked agents obtain higher rewards.

The main arguments of the proof of this result go as follows. First, the outcome where all

agents exert eﬀort is a Nash equilibrium of the induced eﬀort choice game if and only if, for

any i∈N, the following incentive constraint is satisﬁed:

(1−βi) [vi+ (n−1)w−c]+βiw−αi+βi

n−1X

k6=i

[max{vk−vi,0} − max{vk+ (n−2)w−c, 0}]≥0

Second, it must necessarily hold that vk+ (n−2)w−c≤0 for all k∈N: otherwise, the

reward scheme considered would not be least cost. This property enables us to simplify the

incentive constraint: the highest ranked agent’s reward has to satisfy the following incentive

constraint

(1 −β1) [v1+ (n−1)w−c] + β1w≥0

To induce the lowest feasible payment, this condition must be satisﬁed as an equality. Then,

for the agent whose reward is the ith largest one, the following constraint must be satisﬁed:

(1 −βi) [vi+ (n−1)w−c] + βiw−αi+βi

n−1X

k<i

[vk−vi]≥0

Again, to induce the lowest payment it must be satisﬁed as an equality. Solving the resulting

system of (n−1) equalities as functions of (v2, ..., vn), we obtain the desired expressions.

Proposition 11 allows to conclude that the coordination problem does matter as it deeply

aﬀects the characterization of the optimal reward scheme. Indeed, using Lemma 2 and Propo-

sition 11, we conclude that optimal unique and partial implementation reward schemes dras-

tically diﬀer. Under partial implementation, inequality aversion negatively aﬀect monetary

incentives: compared to the case of self-interested agents, the reward scheme provides all

agents with lower rewards. This conclusion is entirely reversed when dealing with unique im-

plementation: all agents are provided with higher rewards (compared to the case where they

25

do not exhibit social preferences). One must be cautious about the rankings corresponding to

the unique and partial implementation cases: in general, these rankings will diﬀer.

It is also interesting to notice that aversion to advantageous inequality is of ﬁrst-order

importance compared to disadvantageous inequality aversion. Indeed, using a ﬁrst-order ap-

proximation of agent i’s reward around (αi, βi) = (0,0), we have vi∼c−(n−1)w−βiwwhich

depends on βibut not on αi. This qualitative result is also entirely reversed when dealing

with unique implementation: disadvantageous inequality aversion is of ﬁrst-order importance

in this case.

Intuitively, the fact that full eﬀort be only one of several Nash equilibria allows the prin-

cipal to reduce the overall cost of the reward scheme by relaxing the incentive constraints

for all agents: this actually implies that advantageous inequality-aversion parameters then

become the most relevant parameters. This has to be contrasted with the case of optimal

unique implementation, where the divide and conquer structure of the scheme tends to award

more importance to higher ranked agents, and thus to aversion to disadvantageous inequalities.

We now rely on this characterization to highlight another notable eﬀect of the coordina-

tion problem: namely, the induced diﬀerences in terms of the agents’ optimal ranking. Since

there is a fundamental diﬀerence with the case of optimal unique implementation in terms of

the characterization of the optimal reward scheme, the same type of qualitative diﬀerences

is expected for the next results. In order to understand the eﬀect of each fundamental, we

proceed with the analysis in several steps. First, we assume that agents share the same degree

of aversion to advantageous inequalities. We then consider the case where they share the same

degree of aversion to disadvantageous inequalities. Finally, we provide some partial insights

about the general case. We have the ﬁrst following result:

Proposition 12 [Disadvantageous inequality under IneqA]: Assume βj=βfor all

j. Then aversion to disadvantageous inequality has no eﬀect on the agents’ ranking: vi=v

∀i∈N.

This important feature is perfectly illustrated by Proposition 12: Aversion to advantageous

inequalities has a ﬁrst-order eﬀect on the characterization of the optimal ranking induced by

the partial implementation reward scheme. We now analyze the polar case where the agents’

degree of aversion of disadvantageous inequalities is the same. We obtain the following results:

Proposition 13 [Advantageous inequality under IneqA]: Assume αj=αfor all j.

The optimal ranking is increasing in the magnitude of aversion to advantageous inequality:

β1≤... ≤βn.

This case highlights the asymmetric eﬀect of inequality aversion. Aversion to advanta-

geous inequality has a ﬁrst-order eﬀect on the optimal ranking, as it ﬂattens the distribution

26

of payments when agents are homogeneous with respect to this fundamental. Aversion to dis-

advantageous inequality does not have a similar type of eﬀect: when agents are homogeneous

with respect to this fundamental, the optimal ranking is characterized by increasing degrees

of aversion to advantageous inequality for lower-ranked agents.

We now move on to the general case, and we speciﬁcally highlight the potential non-

monotonicity of the optimal ranking. We obtain:

Proposition 14 [General case]: The optimal ranking is characterized by the fol lowing

conditions: β1≤β2and, for any i≥3:

(βi−βi−1)+ Pi−2

l=1 Il

n−1[(αi+βi)−(αi−1+βi−1)−βi−1αi+βiαi−1]+ i−2

n−1[βiαi−1−βi−1αi]≥0

(32)

As such, a suﬃcient condition for a ranking to be optimal is that it satisﬁes βi

βi−1>αi

αi−1≥1

for any i≥3, which also implies that this ranking satisﬁes αi≥αi−1and βi> βi−1for any

i≥3. When these conditions are not satisﬁed, there are cases where the optimal ranking

satisﬁes αi≤αi−1for some i∈.

The fact that it is not possible to obtain a closed-form characterization was fairly expected

as heterogeneity is again bi-dimensional. Nonetheless, Proposition 14 highlights a suﬃcient

condition ensuring that the optimal ranking satisﬁes some particular form of monotonicity.

When this condition is not satisﬁed, it is not possible to obtain a clear-cut conclusion and the

optimal ranking may be non-monotonic.

27

Appendix

Proof of Proposition 2:

Assume that vis a ranking scheme for the ﬁrst iagents. Using (5) for i= 1, we have

u(c)−u(v1) = θhW(cn−1)−W(cn−1)i= 0,(33)

and then v1=c.

Let ˆ

bi≡bi(1i,0n−i) for all i. Now we use induction to show that πk(1k,0n−k) = vk+ˆ

bk+1 ≤

πk−1(1k−1,0n−k+1) = vk−1+ˆ

bkfor all k≤i. Assume that this inequality holds for all k≤j+1

with j+ 2 ≤i. Using condition (5) for agents j+ 1 and j+ 2, we have:

u(vj+1 +ˆ

bj+1)−u(vj+2 +ˆ

bj+2)

=θhW(π−(j+2)((1j+2,0n−j−2),v−(j+2))) −W(π−(j+2) ((1j+1,0n−j−1),v−(j+2)))i

−θhW(π−(j+1)((1j+1,0n−j−1),v−(j+1))) −W(π−(j+1) ((1j,0n−j),v−(j+1)))i.(34)

It is suﬃcient to show that the right hand side term in condition (34) is positive. Let

y,∆,δ∈Rn−1. If δk≤∆kfor 1 ≤k≤n−1, concavity of Wimplies W(y+δ) + W(y−δ)≥

W(y+∆) + W(y−∆) (Rothschild and Stiglitz, 1970). Letting yk=vk+vk+1

2+ˆ

bj+ˆ

bj+2

2for

1≤k≤jand yk=cfor j+ 1 ≤k≤n−1, δk=vk−vk+1

2−ˆ

bj+2−ˆ

bj

2for 1 ≤k≤jand δk= 0

for j+ 1 ≤k≤n−1, and ∆k=vk−vk+1

2for 1 ≤k≤jand ∆k= 0 for j+ 1 ≤k≤n−1, we

must have:

W(v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−1) + W(v1+ˆ

bj, ..., vj+ˆ

bj,cn−j−1)

≥W(v1+ˆ

bj+ˆ

bj+2

2, ..., vj+ˆ

bj+ˆ

bj+2

2,cn−j−1)+W(v2+ˆ

bj+ˆ

bj+2

2, ..., vj+1+ˆ

bj+ˆ

bj+2

2,cn−j−1).

(35)

Since bis supermodular, we must have ˆ

bj+ˆ

bj+2

2≥ˆ

bj+1. Hence condition (36) implies:

W(v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−1) + W(v1+ˆ

bj, ..., vj+ˆ

bj,cn−j−1)

≥W(v1+ˆ

bj+1, ..., vj+ˆ

bj+1,cn−j−1) + W(v2+ˆ

bj+1, ..., vj+1 +ˆ

bj+1,cn−j−1).(36)

Since the agents are symmetric, we can permute the material payoﬀs without aﬀecting the

28

level of W. Hence, we obtain:

W(c, v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−2)−W(c, v2+ˆ

bj+1, ..., vj+1 +ˆ

bj+1,cn−j−2)

≥W(v1+ˆ

bj+1, ..., vj+ˆ

bj+1,cn−j−1)−W(v1+ˆ

bj, ..., vj+ˆ

bj,cn−j−1).(37)

Moreover, using supermodularity of Wand v1=c, we have

W(v1+ˆ

bj+1, v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−2) + W(c, v2+ˆ

bj+1, ..., vj+1 +ˆ

bj+1,cn−j−2)

≥W(c, v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−2) + W(v1+ˆ

bj+1, v2+ˆ

bj+1, ..., vj+1 +ˆ

bj+1,cn−j−2).

(38)

Combining (37) and (38), we ﬁnd:

W(v1+ˆ

bj+1, v2+ˆ

bj+2, ..., vj+1 +ˆ

bj+2,cn−j−2)−W(v1+ˆ

bj+1, v2+ˆ

bj+1, ..., vj+1 +ˆ

bj+1,cn−j−2)

≥W(v1+ˆ

bj+1, ..., vj+ˆ

bj+1,cn−j−1)−W(v1+ˆ

bj, ..., vj+ˆ

bj,cn−j−1).(39)

Since Wand bare non decreasing functions, condition (39) implies that the right hand side

in (34) is indeed positive.

Proof of Lemma 1: Assume that agent iprefers to exert eﬀort when the higher ranked

agents also exert eﬀort while the remaining agents shirk (assuming they are ranked from 1 to

n, without loss of generality). Hence:

u(vi+ˆ

bi) + θW (π−i((1i,0n−i),v−i)) ≥u(c) + θW (π−i((1i−1,0n−i+1 ),v−i)),(40)

where ˆ

bi=bi(1i,0n−i). Assume that agent k < i weakly prefers to shirk when each agent

j≤i, j 6=kexerts eﬀort while the remaining agents do not. Hence, we must have

u(c) + θW (π−k((1i

ek=0,0n−i+1),v−k)) ≥u(vk+ˆ

bi) + θW (π−k((1i,0n−i),v−k)) (41)

Using (40), we obtain:

u(vi+ˆ

bi) + θhW(π−i((1i,0n−i),v−i)) −W(π−i((1i−1,0n−i+1),v−i))i

≥u(vk+ˆ

bi) + θhW(π−k((1i,0n−i),v−k)) −W(π−k((1i

ek=0,0n−i+1),v−k))i(42)

29

which is equivalent to

θhW(π−i((1i,0n−i),v−i)) −W(π−i((1i−1,0n−i+1),v−i))i

−θhW(π−k((1i,0n−i),v−k)) −W(π−k((1i

ek=0,0n−i+1),v−k))i

≥u(vk+ˆ

bi)−u(vi+ˆ

bi).(43)

Using symmetry and writing the arguments more explicitly, we obtain:

θhW(v1+ˆ

bi, ..., vi−1+ˆ

bi,cn−i)−W(v1+ˆ

bi−1, ..., vi−1+ˆ

bi−1,cn−i)i

−θhW(v1+ˆ

bi, ..., vk−1+ˆ

bi, vi+ˆ

bi, vk+1 +ˆ

bi, ..., vi−1+ˆ

bi,cn−i)

−W(v1+ˆ

bi−1, ..., vk−1+ˆ

bi−1, vi+ˆ

bi−1, vk+1 +ˆ

bi−1, ..., vi−1+ˆ

bi−1,cn−i)i

≥u(vk+ˆ

bi)−u(vi+ˆ

bi).(44)

The second term in brackets in the left hand side term in (44) is identical to the ﬁrst term

in brackets except that vkis replaced by vi, which is smaller. Thus, concavity of Wimplies

that the left hand side term in (43) is negative. Hence:

u(vk+ˆ

bi)−u(vi+ˆ

bi)≤0 (45)

Thus, we must have vk≤vi, where k < i. This contradicts Proposition 2.

Proof of Theorem 1: Assume that vis an optimal reward scheme. Hence, if no agent

exerts eﬀort, one agent, say 1, must prefer to exert eﬀort (otherwise 0nis a Nash equilibrium).

Assume that each agent k≤iprefers to exert eﬀort when the agents ranked before this agent

(according to a common ranking, 1 to nwithout loss of generality) exert eﬀort while the

remaining agents do not. When the ﬁrst iagents exert eﬀort, none of these agents has an

incentive to deviate. Hence, another agent, say i+1, has an incentive to exert eﬀort, otherwise

(1i,0n−i) is a Nash equilibrium. Hence, the set of constraints (UC) can be reduced to:

u(vi+1 +ˆ

bi+1 +) + θW (π−(i+1)((1i+1,0n−i−1),v−(i+1) +n−1))

> u(c) + θW (π−(i+1)((1i,0n−i),v−(i+1) +n−1)),(46)

Given that the principal’s objective function is linear, the optimal reward vi+1 is charac-

terized either by the following condition:

u(vi+1+ˆ

bi+1)−u(c) = θhW(π−(i+1) ((1i,0n−i),v−(i+1))) −W(π−(i+1) ((1i+1,0n−i−1),v−(i+1)))i,

(47)

30

or by U(1n,v) = U(1n

ei+1=0,v), which is equivalent to

u(vi+1 +ˆ

bn)−u(c) = θhW(π−(i+1)(1n

ei+1=0,v−(i+1))) −W(π−(i+1) (1n,v−(i+1)))i.(48)

Let us denote v∗

i+1 the solution to (47) and v0

i+1 the solution to (48). We must have

v0

i+1 ≤v∗

i+1, </