CUBO A Mathematical Journal

Vol.11, N
o
¯ 02, (15–36). May 2009

Optimal Effort in Heterogeneous Agents Population

with Global and Local Interactions

Arianna Dal Forno and Ugo Merlone

Statistics and Applied Mathematics Department, University of Torino,

Corso Unione Sovietica 218/bis, Turin, I-10134, Italy

emails: dalforno@econ.unito.it, merlone@econ.unito.it

ABSTRACT

A game where agents interact in small teams is proposed; the interaction is examined

when the population consists of different types of agent and a reward mechanism devised

to increase competition is introduced. We prove that such a mechanism may expand

the set of Nash equilibria and, in particular, reduce the production level of some agents.

Finally, we extend our results to heterogeneous populations by means of agents based

modeling. This way we can study the dynamics of adjustment of agents response and

extend our results when considering local interaction and a egocentric knowledge of the

population composition.

RESUMEN

Se propone un juego en el cual agentes interactuan en un pequeño grupo, la interacción

es examinada cuando la población contituye diversos tipos de agentes y se introduce un

mecanismo de recompenza para aumentar la competición. Se demuestra que tal mecan-

ismo puede expandir el conjunto de equilibrio de Nash y, en particular, reduce el nivel



16 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

de produción de algunos agentes. Finalmente, extendemos nuestros resultados a pobla-

ciones heteregeneas mediante un modelo de agente artificial. De esta manera es posible

estudiar la dinámica de ajustamiento de respuesta de agentes y estender los resultados

considerando interacción local y un conocimiento egocentrico de la composición de la

población.

Key words and phrases: Bounded rationality, Mathematical Organization Theory, Public Goods,

Heterogeneous agents, Agent Based Simulation.

AMS Classification: 91A80,91B18,91A26,91D10.

1 Introduction

The problem of incentives and compensation is cardinal in modern economic literature; many pa-

pers address this problem considering moral hazard in agency relationships. Other papers address

the problem of optimal form of hierarchy in firms and give interesting results (for a survey, see e.g.

[6] and [10]).

The approach we use is different; we provide a model of firm which is very simple, and do not

consider hierarchy explicitly. The model of firm we introduce may be interpreted as an Interaction

Game [8] with agents playing bounded rationality strategies. This approach may fail to take into

account some relevant phenomena but allows us to shed light on aspects that are usually neglected

such as the interpretation of equilibria as corporate culture.

We consider a population of 2n agents randomly paired in teams
1
; each member supplies a

non-observable individual effort in order to produce a good. Members are rewarded according to

their joint production, yet each agent bears its own private cost in providing effort. In a first

analysis we limit our study to symmetric agents; when considering only rational agents this game

has one single Nash equilibrium. We are interested in outcomes which are less predictable than

Nash equilibria and, in particular, we consider a profile of strategies which is not only ideal from

the firm’s prospective but also maximizes the workers’ welfare. To achieve this particular profile

we introduce a class of agents which is able to commit to a coordinated effort even if they have

incentive to shirk; the effort provided by these agents is the maximum feasible one and any other

effort cannot be greater than this one. By contrast, we assume that some of our agents may not

be fully aware of the set of alternatives from which they have to choose, or may have not the skills

necessary to make whatever complicated calculations are needed to discover its optimal course of

action, or, finally, do not clearly perceive the action-consequence relationship especially when they

face uncertainty. In other words, we assume bounded rationality of some agents and consider their

effort fixed to some level determined exogenously.

1It must be noted that teams we consider differ from the ones considered in [7]



CUBO
11, 2 (2009)

Optimal Effort in Heterogeneous ... 17

Obviously, a population consisting of heterogeneous agents may affect behavior of rational

agents and, consequently, the equilibria of the game. The results we provide are interesting since

we find, for example, that the presence of low fixed effort individuals induces rational agents to

work harder. The relative composition of the population is used to study the equilibria with

heterogeneous agents and these equilibria are compared to the ones in homogeneous population.

In [1] it is argued that a thorough understanding of internal incentives is critical to develop a

viable theory of the firm. Furthermore, in the literature individual vs group incentives plans have

been compared. Some of the drawbacks of individual incentives are well known in the literature

(see for instance [12]). In this paper we do not approach these problems from the classical point

of view of incentive and contracts literature (for a first introduction see [6]), rather we propose a

simple ranking policy clearly showing some of these drawbacks. This way even a simple model, like

the one we propose, encompasses some of the crucial points in the comparison between individual

and group incentives plans. In particular, we rank agents according to their profit and, even if

this should increase competition between agents, it just expands the equilibrium set of the game

including only equilibria dominated by the Nash equilibrium. Furthermore, we show that this

policy, in some cases, reverses the effect of fixed low effort individuals on rational agents in terms

of optimal effort.

Performing simulation with artificial agents allows us to extend the theoretical results in

two more directions. First, we can consider heterogeneous populations and observe the best reply

dynamical adjustments when the population compositions vary. Second, we can consider the effects

of local interaction when the agents have no longer the knowledge about the global composition of

the population, but they can observe only the composition of their neighborhood in order to make

their decisions.

The paper is structured in the following way: in Section 2 we present the model of the game

and some results about equilibria in the symmetric case; in Section 3 we discuss how the proposed

game may be seen as a model of a firm; in Section 4 we introduce bounded rationality agents and

some results about the case with the population partitioned in different classes of agents. Section

5 defines the ranking policy and studies how this affects the equilibrium set and the performance

of rational agents. Section 6 provides a discussion of the results when the theoretical model

is extended by artificial simulation. Finally, in the last section we summarize our findings and

provide possible interesting directions in further research.

2 The model

A population of 2n agents is randomly partitioned, following a uniform distribution, in n couples

of two players; the partition is not revealed to the agents. As a consequence, no agent knows

who his/her mate is; this avoids some of the problems related to repeated games and simplify the

analysis. In this one-shot game each team is supposed to produce a good; the final production

is the only verifiable variable and depends on the vector of the non-observable efforts each agent



18 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

supplies: this is a joint production model (for a survey, see e.g. [6]). Agent i ∈ I = {1, 2, . . . , 2n}

receives a monetary payoff for the final production of its team and supplies an effort ei which

implies some cost to him/her; we assume ei ∈ E ⊆ R
+
, with E compact, convex and nonempty.

For sake of simplicity we consider only the utility of the payoff each agent receives: if agent i is

paired with agent j and the vector of their efforts is (ei, ej), their utility will be f (ei + ej); each

agent’s disutility of effort is c (e). Furthermore, we assume:

• f : R+ → R

• c : R+ → R

• f, c ∈ C2

• f (0) = c (0) = 0

• f ′ > 0, c′ > 0

• f ′ (0) > c′ (0)

• f ′′ < 0, c′′ > 0

When agents i and j are the members of the same team, their payoffs are respectively:







πi (ei, ej) = f (ei + ej) − c (ei)

πj (ei, ej ) = f (ei + ej ) − c (ej )
(1)

and they are not transferable. Since the partition is random, agent i’s profit is a random variable

Πi depending also on the effort its randomly paired mate exerts. The expected value of its profits

is

E [Πi (e,X)] =
1

2n − 1

n
∑

k=1

πi (ei, ek) , k 6= i

where X is the random variable determining the partition and e = (ei, e−i).

2.1 Equilibria of the game in some particular cases

Since for all players the set of strategies E is a compact, convex and nonempty subset of R
+
, and

the payoff functions are continuous and concave w.r.t. the strategies, the existence of at least one

equilibrium is guaranteed.

We remark that the production function is symmetrical and agents have the same cost function.

In this section we limit our study to the case in which the agents choose the same effort, i.e., we

consider symmetric equilibria. Therefore, the expected profit of each agent is actually (1). In



CUBO
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Optimal Effort in Heterogeneous ... 19

addition, if we assume that players can commit to an effort, they decide the optimal effort e
C

solving the following problem:

max
e

f (2e) − c (e) , e ∈ E ⊂ R+

where E is the set of feasible efforts. We call this profile of strategies
2

the coordination equilibrium
(

e
C

, e
C
)

.

Obviously, this is not a Nash equilibrium, in fact:

Proposition 1 The coordination equilibrium is Pareto optimal, but even if the two team-mates

decide to coordinate to this equilibrium, there is an incentive to shirk. Furthermore, this profile of

strategies maximizes the social welfare.

Proof: The payoff function is strictly concave so FOCs are sufficient to determine the interior

coordination equilibrium. The coordination equilibrium is therefore characterized by:

2f
′
(

2e
C
)

− c′
(

e
C
)

= 0. (2)

Let us consider the partial derivative of agent i
′
s profit with respect to its effort when the coordi-

nation equilibrium is played:

∂

∂ei

(

πi

(

ei, ej = e
C

))

|ei=eC
= f

′
(

2e
C
)

− c′
(

e
C

)

< 2f
′
(

2e
C
)

− c′
(

e
C

)

= 0.

This partial derivative is negative: πi decreases with respect to ei. Therefore, choosing a lower

effort close enough to e
C

, player i has a larger profit. The second part is obvious since this profile

of strategies is also the solution of

max
ei,ej

2f (ei + ej ) − c (ei) − c (ej ) , ei, ej ∈ E ⊂ R
+
.

�

To obtain this equilibrium we need something more, something that makes each player commit

to provide effort e
C
; for example, it could be achieved by signing an enforceable contract or

assuming repetition of this situation (for a deep analysis refer to [5]). We assume the existence of

such agents
3

and consider this profile of strategies for the following reasons. First of all, from the

social point of view it would be desirable to have this situation; then, it is an important yardstick

to which compare the other suboptimal equilibria; finally, in this situation the agents exert the

maximal effort and this is one of the goals of the firm.

If we do not assume the existence of such ideal agents, this game has a unique Nash equilibrium
(

e
N

, e
N

)

and it holds:

2Each symmetric profile of strategies can, at least theoretically, be coordinate between players; nevertheless, we
reserve this name to this particular one.

3With this assumption this profile of strategies constitutes an equilibrium in the sense that, these committed
agents have no incentive to deviate.



20 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

Theorem 1 The effort exerted in the Nash equilibrium is lower than the effort exerted in the

coordination equilibrium, i.e., eN < eC .

Proof: The Nash equilibrium can be found solving









max
ei

f (ei + ej ) − c (ei) ,

max
ej

f (ei + ej ) − c (ej ) ,
ei, ej ∈ E

and is characterized by the following FOC:

f
′
(

2e
N

)

= c
′
(

e
N

)

. (3)

By contradiction let e
C ≤ eN ; since f is concave and c is convex, it follows:







c
′
(

e
C

)

≤ c′
(

e
N

)

,

f
′
(

2e
C
)

≥ f
′
(

2e
N

)

,

by condition (2) on the coordination equilibrium and (3), this means







c
′
(

e
C

)

≤ c′
(

e
N

)

c
′
(

e
C

)

= 2f
′
(

2e
C
)

> f
′
(

2e
C
)

≥ f
′
(

2e
N

)

= c
′
(

e
N

)

,

clearly absurd.

�

Obviously, this game is a Public Goods Game [4].

3 The game as a model of firms

We propose this model to study the possible equilibria in a complex structure such as a firm; in

fact, we can think of these equilibria as corporate culture (see e.g. [5]). Corporate culture may

be defined as the basic assumptions and beliefs that are shared by the members of a group or

organization and that are used as norm. The organization’s problem is to identify a rule that

allows relatively efficient transactions to take place and devise some way to communicate that rule

to all current and potential trading partners.

We do not expect that all the equilibria of the game we consider may represent efficient

equilibria.

The organization has an interest in preserving and promoting a good reputation to allow for

future beneficial transactions; nevertheless, it is quite common to observe situations where there is



CUBO
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Optimal Effort in Heterogeneous ... 21

no interest in reputation and members’ main interest is free riding. This situation is widespread in

Italian public sector offices and often it is claimed that a higher level of competition could avoid such

misbehavior. The ranking procedure we introduce in Section 5 may be compared to some individual

merit compensation incentives used by organizations. Although individual incentive systems often

lead to improved performance, these programs may, at times, lead to employees competing with

one another, with undesirable results. Finally, it must be observed that the particular ranking

policy we consider may be unrealistic since the principal usually cannot observe private costs of

its agents; nevertheless there are always costs to take into account and our model can be easily

modified to describe situations where the agents are provided with a fixed resource and have to

allocate it efficiently. Furthermore, very often there is a social mechanism which tends to reward

people who obtain just a higher profit than the others’, without necessarily maximizing it.

4 Some results in heterogeneous populations

Even if this model is very simple we assume that not all agents may:

• be fully aware of the set of alternatives from which they have to choose

• have the skill necessary to make whatever complicated calculations are needed to discover

their optimal course of action

• do clearly perceive the action-consequence relationship especially when facing uncertainty.

For an analysis of some other motives that may conflict with the rational man paradigm the reader

may refer to [11].

We are interested in considering simple “bounded rationality” strategies and how they affect

the equilibria of the game with heterogeneous agents. In particular, we consider agents who

stubbornly provide a fixed effort regardless of their results. The rationales of such a behavior may

be different: for example, an agent, given the difficulties to find an optimal effort, may resolve to

providing a fixed effort which it consider appropriate for the situation.

Consider then three classes of agents depending on their behaviors:

• bounded rationality agents playing a fixed effort ē ∈
[

0, e
C
]

• committed agents who play the optimal effort knowing the fact that all the agents of this

class play the same coordinated effort

• rational agents who play the optimal effort and do not commit to any coordinated effort.

In the following, we assume that the type of agent is private information while the composition

of the population is common knowledge. Let e (ē) be the optimal effort an agent, either committed



22 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

or rational, exerts when its team-mate is a bounded rationality agent playing effort ē. Then e (ē)

is the solution of the following problem:

max
e

f (e + ē) − c (e) .

Proposition 2 Consider a team consisting of a rational (committed) agent and a fixed effort agent.

The higher is the fixed effort provided, the lower is the optimal effort of the rational (committed)

agent:

ē1 < ē2 =⇒ e (ē1) > e (ē2) .

Furthermore, a rational (committed) agent will always provide an effort lower than eC .

Proof: By contradiction let e (ē1) ≤ e (ē2); this means e (ē1) + ē1 < e (ē2) + ē2. Since f
′

is

decreasing:

c
′
(e (ē1)) = f

′
(e (ē1) + ē1) > f

′
(e (ē2) + ē2) = c

′
(e (ē2))

absurd since e (ē1) ≤ e (ē2) =⇒ c
′
(e (ē1)) < c

′
(e (ē2)).

For the second part we know by the first part of this proposition that e (ē) will be maxi-

mum when the fixed effort is null, so it is sufficient to prove that when ē = 0 then e (0) < e
C

;

e (0) is characterized by the FOC f
′
(e (0)) = c

′
(e (0)). By contradiction assume e (0) ≥ eC , by

monotonicity of f
′
and c

′
it follows:







2f
′
(

e
C

)

> f
′
(

e
C

)

≥ f ′ (e (0))

c
′
(

e
C

)

≤ c′ (e (0)) ,

clearly absurd.

�

This result states the fact that a rational agent will exert higher effort when paired with

somebody exerting a lower effort. This is interesting in terms of free riding: it may appear that

the rational agent’s behavior incentives free riding, but it must be recalled that rational agents are

interested only in maximizing their profit. In Section 5 we introduce an incentive to competition

and the results will be different. In the following, we consider the equilibria resulting as the

composition of population varies.

4.1 Fixed effort agents vs committed agents

Let us consider the profit maximizing game; the population of 2n agents is partitioned in two

subsets:

1. m fixed effort ē agents



CUBO
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Optimal Effort in Heterogeneous ... 23

2. (2n − m) committed agents who coordinate on the best effort knowing the fact that there

are m agents providing the fixed effort.

A single committed agent does not know which kind of mate it will be paired with so it has

to solve the following problem:

max
e

2n − m − 1

2n − 1
(f (2e) − c (e)) +

m

2n − 1
(f (e + ē) − c (e)) . (4)

Let e
∗

be the optimal solution effort to problem (4), it holds:

Theorem 2 As the number of committed agents increases to 2n the optimal effort provided by the

committed agents increases to the coordination equilibrium effort eC . As the effort ē provided by

the fixed effort agents increases, the optimal effort provided by the committed agents decreases.

Proof: Let us consider FOC of problem (4):

F (e
∗
, m, ē) =

= 2 (2n − m − 1) f ′ (2e∗) + mf ′ (e∗ + ē) − (2n − 1) c′ (e∗) = 0.

By implicit function theorem it is possible to write:

∂e
∗

∂m
= −

∂F/∂m

∂F/∂e∗
=

= −
−2f ′ (2e∗) + f ′ (e∗ + ē)

4 (2n − m − 1) f ′′ (2e∗) + mf ′′ (e∗ + ē) − (2n − 1) c′′ (e∗)
.

By assumptions on concavity/convexity of f and c the denominator is negative while, as it concerns

the numerator, consider the following strictly concave functions

g1 (e) = f (e + ē) − c (e) ,

g2 (e) = f (2e) − c (e) .

Since by Proposition 2 we have e (ē) < e
C
, then ∀e such that e (ē) < e < eC it is:







g
′
1 (e) < 0,

g
′
2 (e) > 0,

because e (ē) is the optimal point of g1 (e) and e
C

is the optimal point of g2 (e). It follows:

−g′2 (e
∗
) + g

′
1 (e

∗
) = −2f ′ (2e∗) + f ′ (e∗ + ē) < 0.

This way if e (ē) < e
C

it follows that ∂e
∗
/∂m < 0 and therefore the optimal effort to problem (4)

is decreasing with respect to m. In the limit case, all agents are committed, and obviously, the



24 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

optimal effort will be e
C
. Finally, let us consider:

∂e
∗

∂ē
= −

∂F/∂ē

∂F/∂e∗
=

= −
mf

′′
(e

∗
+ ē)

4 (2n − m − 1) f ′′ (2e∗) + mf ′′ (e∗ + ē) − (2n − 1) c′′ (e∗)
< 0.

and the second part of the theorem follows.

�

Figure 1 summarizes the results stated in Proposition 2 and Theorem 2.

Figure 1: Fixed vs Committed

As the number of committed players approaches the whole population, their effort approx-

imates e
C

and, the higher is the effort provided by fixed effort agents, the lower is the optimal

effort supplied by committed players. This result may be explained since when the population

tends to consist of almost only committed agents, the probability for a single committed agent to

be paired with a different kind of agent will be low. Furthermore, when few committed agents face

a population of fixed effort agents, they will expect a low probability of facing another committed

agent. As it concerns the second part it may be interpreted analogously to Proposition 2.

4.2 Rational agents vs committed agents

Let us consider rational agents and recall that while their goal is to maximize their profit, they are

not able to coordinate as committed agents do. It is obvious that if the population consists only

of rational agents they will play the Nash equilibrium. In this section we study the equilibria in a

mixed population composed of both committed and rational agents.



CUBO
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Optimal Effort in Heterogeneous ... 25

Consider:

1. m rational agents playing the effort ei , i = 1, . . . , m

2. (2n − m) committed agents who play the best effort eco knowing the fact that there are m

rational agents.

A single committed agent does not know which kind of mate it will be paired with so it has

to solve the following problem:

max
eco

2n − m − 1

2n − 1
(f (2eco) − c (eco)) +

m

2n − 1
(f (eco + ej) − c (eco)) (5)

j = 1, . . . , m

while a rational agent i will solve:

max
ei

2n − m

2n − 1
(f (ei + eco) − c (ei)) +

1

2n − 1

∑

j 6=i

(f (ei + ej ) − c (ei)) . (6)

Proposition 3 In a mixed population composed of both committed and rational agents, the effort

provided by the committed is not inferior to the effort provided by the rational agents.

Proof: Let us consider FOC of problem (5) and (6):

{

F (m, eco) = (2n − m − 1) [2f
′
(2eco) − c

′
(eco)] + m [f

′
(eco + ej ) − c

′
(eco)] = 0,

G (m, ei) = (2n − m) [f
′
(ei + eco) − c

′
(ei)] +

∑

j 6=i [f
′
(ei + ej ) − c

′
(ei)] = 0.

All rational agents solve the same problem, therefore we consider symmetrical equilibria: ei = er,

i = 1, . . . , m

{

F (m, eco) = (2n − m − 1) [2f
′
(2eco) − c

′
(eco)] + m [f

′
(eco + er) − c

′
(eco)] = 0,

G (m, er) = (2n − m) [f
′
(er + eco) − c

′
(er)] + (m − 1) [f

′
(2er) − c

′
(er)] = 0.

(7)

By contradiction let er > eco, it follows 2er > er + eco > 2eco.

Now consider f
′
(eco + er) − c

′
(eco):

• if f ′ (eco + er) − c
′
(eco) ≥ 0, we have:

er + eco > 2eco ⇒ f
′
(2eco) > f

′
(er + eco)

and it follows

2f
′
(2eco) − c

′
(eco) > f

′
(2eco) − c

′
(eco) > 0

absurd since it contradicts the first FOC in (7).



26 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

• if f ′ (eco + er) − c
′
(eco) < 0, we have:

er > eco ⇒ f
′
(er + eco) − c

′
(er) < f

′
(er + eco) − c

′
(eco) < 0

and it is

2er > er + eco ⇒ f
′
(2er) − c

′
(er) < f

′
(er + eco) − c

′
(er) .

Putting together we obtain:

f
′
(2er) − c

′
(er) < f

′
(er + eco) − c

′
(er) < f

′
(er + eco) − c

′
(eco) < 0

absurd since it contradicts the second FOC in (7).

It follows er ≤ eco.

�

This may be easily interpreted since rational agents deviate and shirk in order to maximize

their profit.

Theorem 3 As the number of committed agents increases to 2n, the optimal effort provided by

the rational agents decreases to the best reply effort to the coordination equilibrium e
(

e
C

)

and the

optimal effort provided by the committed agents increases to the coordination equilibrium effort eC .

Proof: Let us consider conditions (7), by implicit function theorem it is possible to write:

∂er

∂m
= −

∂G/∂m

∂G/∂er
= −

−f ′ (er + eco) + f
′
(2er)

(2n − m) f ′′ (er + eco) + 2 (m − 1) f ′′ (2eco) − (2n − 1) c′′ (er)
.

By assumptions on concavity/convexity of f and c the denominator is negative, and, as it concerns

the numerator, since er < eco and f is a marginal decreasing function, it is positive. Therefore,

∂er/∂m > 0. Obviously, when a single rational agent faces only committed agents it will play the

best reply effort to the coordination equilibrium e
(

e
C
)

.

As it concerns the second part of the statement, it is obvious that when a single committed

agent faces only rational agents it will play the Nash effort e
N

and, vice versa, if the population

consists only of committed agents, they all will play the coordinated effort e
C
. Furthermore,

committed agent’s effort eco depends on m, the number of rational agents, both directly and

indirectly, via the optimal effort of rational agents:

eco = g (m, er (m)) .

We consider the infinitesimal variation in eco as m increases:

deco

dm
=

∂g

∂m
+

∂g

∂er

der

dm
.

By Theorem (2) ∂g/∂m is negative; we just proved that der/dm > 0, and by second part of

Theorem (2) also ∂g/∂er < 0. It follows the second part of the thesis.



CUBO
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Optimal Effort in Heterogeneous ... 27

�

Figure 2 summarizes the results stated in Proposition 3 and Theorem 3.

Figure 2: Rational vs Committed

When few rational agents face many committed agents, they expect that the latter will tend

to exert an effort close to the committed one and, as a consequence, they will tend to play the best

reply to the expected effort. Vice versa, when few committed agents face many rational agents,

their effort will be lower since the probability to be paired with agents maximizing their own profit

will be higher.

4.3 Fixed effort agents vs rational agents

Consider:

1. m fixed effort ē agents;

2. (2n − m) rational agents playing the effort ei , i = 1, . . . , 2n − m.

A single rational agent i will solve:

max
ei

m

2n − 1
(f (ei + ē) − c (ei)) +

1

2n − 1

∑

j 6=i

(f (ei + ej ) − c (ei)) . (8)

Theorem 4 Assume ē > eN (ē < eN ), as the number of fixed effort agents increases to 2n, the

optimal effort provided by the rational agents decreases (increases) to e (ē).



28 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

Proof: When a single rational agent faces only fixed effort agents it will play e (ē). By proposition

(2) it is ē > e
N ⇒ e (ē) < e

(

e
N

)

= e
N

and, obviously, e (ē) ≤ er ≤ e
N

.

Let us consider FOC of problem (8) when we consider symmetrical equilibria: ei = er, i =

1, . . . , 2n − m :

H (m, er) = m [f
′
(er + ē) − c

′
(er)] + (2n − m − 1) [f

′
(2er) − c

′
(er)] = 0. (9)

By implicit function theorem it is possible to write:

∂er

∂m
= −

∂H/∂m

∂H/∂er
= −

f
′
(er + ē) − f

′
(2er)

m [f ′′ (er + ē) − c′′ (er)] + (2n − m − 1) [2f ′′ (2er) − c′′ (er)]
.

And it follows ∂er/∂m < 0.

Similarly, the case in which ē < e
N

can be proved.

�

Figure 3 summarizes the results stated in Theorem 4.

Figure 3: Rational vs Fixed

This may be explained taking into account the relative proportion of the population and the

fact that the rational agents maximize their profit.

5 A ranking policy

The firm may consider to introduce some mechanism to increase productivity, for example, reward-

ing employees who maximize their profit. Nevertheless, such mechanisms must be devised carefully



CUBO
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Optimal Effort in Heterogeneous ... 29

since a known problem of individual incentives is that they may lead employees competing with

one another (see for instance [12]). In particular, we introduce a ranking policy and will show what

may be some of the undesirable results for the agents. Consider the following individual incentive

plan: all agents are ranked according to their payoff and normalized in the sense that the agent

with the higher payoff will get 1 and the one with the lower will get 0; should all the agents have

the same payoff, every agent will get 1. Formally:

ri =

















πi − min
j∈I

{πj}

max
j∈I

{πj} − min
j∈I

{πj}
if max

j∈I
{πj} 6= min

j∈I
{πj}

1 if max
j∈I

{πj} = min
j∈I

{πj}

(10)

This policy may recall the ”Employee of the Year” prize even if in a sketched way. Obviously,

it must be noted that usually firms give this kind of prize observing agents’ efforts and that in

our model effort is not observable and costs are private. Nevertheless, this ranking policy may be

viewed as a social policy where the best individual is the one with higher personal profit, and this

kind of pressure may not be ignored since here we do not assume any social norm. In this section

we will assume that agents want to maximize rank and examine how this ranking policy can affect

the equilibrium set.

It is obvious that any profile of strategies where all players exert the same effort gives the

maximum outcome. Nevertheless, some may prefer to avoid tie results and this may give incentive

to deviation. This can be explained in different ways: for example, if an agent is particularly

competitive it may deviate to be the only winner, while, if some others expect such a deviation,

they could anticipate it and behave consequently. In particular:

Theorem 5 Consider the team game defined in Section 2; if we introduce the ranking policy (10)

the set of Nash equilibria expands to the set (e∗) where e∗ = (e∗, e∗, . . . , e∗) ∈ R2n, e∗ ∈
[

0, e
N

]

.

Proof: Let all the players exert the same effort ê, then player i will exert an effort ei such that






f (ei + ê) − c (ei) ≥ f (ei + ê) − c (ê) ,

f (ei + ê) − c (ei) ≥ f (2ê) − c (ê) .

The first condition guarantees that player i’s profit will be not lower than its mate’s j, while the

second condition means that player i’s profit is not lower than the profits of the members of the

other teams. It should be noted that the first condition is equivalent to ei ≤ ê.

Let us consider a profile of strategies e
∗

= (e
∗
, e

∗
, . . . , e

∗
), where e

∗ ∈
[

0, e
N

]

. If player i exerts

effort ei = e
∗

his payoff is the same as the other players’ and its rank will be 1. Furthermore,

player i has no incentive to exert a lower effort since if we consider its profit, the partial derivative

with respect to its effort is positive when ei = e
∗
:

∂

∂ei
(πi (ei, ej = e

∗
))

|ei=e∗
= f

′
(2e

∗
) − c′ (e∗) > f ′

(

2e
N

)

− c′
(

e
N

)

= 0.



30 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

The second pure partial derivative is negative and this means that:

∂

∂ei
(πi (ei, e

∗
)) = f

′
(ei + e

∗
) − c′ (ei) > 0 ∀ei ∈ [0, e

∗
]

so there is no incentive to deviate.

Finally, let us consider a profile of strategies e
∗

= (e
∗
, e

∗
, . . . , e

∗
), where e

∗
> e

N
.

Player i has incentive to shirk since, considering its profit, the partial derivative with respect

to its effort is positive when ei = e
∗
:

∂

∂ei
(πi (ei, ej = e

∗
))

|ei=e∗
= f

′
(2e

∗
) − c′ (e∗) < f ′

(

2e
N

)

− c′
(

e
N

)

= 0

�

The set of Nash equilibria is expanded by the ranking policy; this adds some further problems

in predicting which equilibrium will be selected. One way to overcome this could be to assume

that agents have a lexicographic utility and among different equilibria would prefer the one with

higher profit.

Finally, even in very simple cases we show how this policy may not increase the productivity

of agents. Consider the situation in which a rational agent is employed in a firm where a corporate

culture consisting in providing the same effort ē is established. Without the incentive policy its

optimal effort will be e (ē); this effort will be higher than ē if and only if ē < e
N

and, in particular,

will be the same as the other agents’ if and only if ē = e
N

.

Now, introducing the incentive policy, it holds:

Theorem 6 Consider 2n − 1 fixed effort ē agents, and a single rational agent, then the incentive

policy never incentives the rational agent to provide an effort larger than ē.

Proof:The rational agent will solve:

max
e

f (e + ē) − c (e)

and take into account also its partner profit.

Let e
∗

:= arg max [f (e + ē) − c (e)]; three cases are given:

1. e
∗

< ē: in this case the rational agent will provide this effort since [f (e + ē) − c (e)] <

[f (2ē) − c (ē)] and will be the only one to get payoff 1;

2. e
∗

= ē: this case is trivial since all agents will get 1;

3. e
∗

> ē: in this case should the rational agent provide this effort its partner would free ride

and would be the only one to get 1.



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Optimal Effort in Heterogeneous ... 31

Clearly there is no incentive to provide efforts higher than ē.

�

This simple result is very important because it shows, in this simple model, some of the

“undesirable results” [12]) of the individual incentive. In particular, if the fixed level ē is low, the

effort of a new rational employee is bounded by this individual incentive plan.

6 Simulation Results

In the following we propose a simulation approach in order to extend the theoretical results we

derived in previous sections. The agent based simulations we performed were implemented on a

customized version of the platform described in [2]. The platform has been modified simply in-

cluding the behavioral classes we described in the theoretical analysis. In particular, we considered

the same functional form which was used in human subject experiments in [3], that is, f (e) = 5
√

e

and c (e) = e
2
; when agents i and j are in the same team, their payoff is respectively







πi (ei, ej) = 5
√

ei + ej − e
2
i ,

πj (ei, ej ) = 5
√

ej + ei − e
2
j .

(11)

It is worth noting that, considering the one-shot game, the Nash equilibrium is e
N

=
3

√

25/32 ≃

.92100787466 while the coordination equilibrium is e
C

=
3
√

25/2 ≃ 1.4620088691.

We are interested in extending the results about optimal effort when considering a mixed

population of 2n agents which is partitioned as follows:

1. rational agents 1, 2, . . . , m1 who maximize their individual profit;

2. m2 fixed effort ē agents;

3. (2n − m1 − m2) committed agents who coordinate on the best effort.

For all the agents the population composition is common knowledge. As a consequence an

equilibrium configuration can be found solving

















max
ei

2n−m1−m2
2n−1

5
√

ei + eco +
1

2n−1

∑

j ≤ m1
j 6= i

5
√

ei + ej +
m2

2n−1
5
√

ei + ē − e
2
i

(i = 1, 2, . . . , m1) ,

max
eco

2n−m1−m2−1
2n−1

5
√

2eco +
m1

2n−1
5
√

eco + er +
m2

2n−1
5
√

eco + ē − e
2
co.

As we assume that rational agents are symmetric an equilibrium configuration may be (e
∗
r , e

∗
co)

where e
∗
r indicates the optimal effort for the m1 rational agents and e

∗
co indicates the optimal



32 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

effort for the m2 committed agents. In the simulation, by the symmetry assumption, rational and

committed agents decide their efforts dynamically as follows:









e
t+1
r = arg max

er

2n−m1−m2
2n−1

5
√

er + etco +
m1−1

2n−1
5
√

er + etr +
m2

2n−1
5
√

er + ē − e
2
r,

e
t+1
co = arg max

eco

2n−m1−m2−1

2n−1
5
√

2eco +
m1

2n−1
5
√

eco + etr +
m2

2n−1
5
√

eco + ē − e
2
co.

(12)

Since in our case the best reply functions are given implicitly, in the simulation we use an

iterative method in order to solve the maximization problem and therefore the best reply functions

for each agent are numerically evaluated. In our simulations we compute the steady state for

different population compositions. Assuming 2n = 900 agents, we represent on the x−axis the

number of fixed agents and on the y−axis the number of rational agents; it follows that point

(x, y), where 0 ≤ x, y ≤ 900 and x + y ≤ 900, represents a population consisting of x fixed agents,

y rational agents and 900 − x − y committed agents. For each population composition we compute

the rational agent and committed agent steady effort assuming that all agents know the population

composition. The respective efforts
4

are illustrated in Figures 4 and 5.

Figure 4: Rational (left) and Committed (right) agent effort with global interaction and fixed
effort level ē = 0.1

It can be observed that the results we obtain by simulation are consistent with all the theoreti-

cal we obtained in previous sections. In fact, when considering the restrictions for zero fixed agents

in Figure 4, we obtain respectively rational and committed agents’ efforts; combining opportunely

these restrictions we obtain the results stated in Proposition 3 and Theorem 3, as depicted in

Figure 2. Observe that the same results hold for Figure 5 as we do not consider fixed effort agents.

Then, when considering the restrictions for zero rational agents in Figures 4 (right) and 5

(right), we obtain respectively the committed agents’ efforts in a 0.1 and 1.2 fixed effort agents

population; an opportune combination of these restrictions allows us to obtain the results stated

in Proposition 2 and Theorem 2, as summarized in Figure 1.

4Recall that, given the population composition, efforts are defined only for x + y ≤ 900.



CUBO
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Optimal Effort in Heterogeneous ... 33

Finally, when considering the restrictions for zero committed agents in Figures 4 (left) and 5

(left), i.e, y = 900 − x, x = 0, 100, 200, . . . , 900, we obtain respectively the rational agents’ efforts

in a 0.1 and 1.2 fixed effort agents population. Once again, with an appropriate combination of

these restrictions we obtain the results stated in Theorem 4 and summarized in Figure 3.

Figure 5: Rational (left) and Committed (right) agent effort with global interaction and fixed
effort level ē = 1.2

So far, in the simulations, we have assumed that the population composition was common

knowledge and that agents could interact globally, as in a complete graph (see [13]). In the

following, we analyze the consequence of a local interaction. We assume that agents can interact in

a Von Neumann neighborhood, and do not know the population composition. This has important

consequences in the dynamic process described in (12) as agents must have knowledge of their

opponents’ best reply in order to make a decision. In particular, each agent assumes that all the

agents in its neighborhood share its own information about neighborhood composition. We can

say that the agents have egocentric thought about neighborhood in the sense of Piaget [9].

Figure 6: An example of egocentric neighborhood



34 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

For example, in Figure 6 assume that a rational agent interacts with two fixed effort agents

and two committed agents. That is, its neighborhood consists of one rational, two committed

and two fixed effort agents. This agent does not have the complete composition of its neighbors’

neighborhood; in our model we assume that the agents consider all their neighbors having their

same neighborhood composition. That is, the local composition is assumed to be constant.

Figure 7: Rational (left) and Committed (right) agent effort with local interaction and fixed effort
level ē = 0.1

Figure 8: Rational (left) and Committed (right) agent effort with local interaction and fixed effort
level ē = 1.2.

In our simulations the agents are randomly located on a toroidal lattice; the results we obtain

with this assumption and local interaction are presented in Figures 7 and 8. Comparing these

results to those presented in figures 4 and 5, we can observe that the average efforts are quite

close to those in global interaction. So, global interaction behavior is a good approximation of

the average behavior we obtain when considering local interaction in Von Neumann neighborhood

with egocentric agents. Finally, it is interesting to observe that when agents are located according

to their behavioral class instead of being randomly located, the results are quite similar to those

obtained when considering homogeneous populations .



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Optimal Effort in Heterogeneous ... 35

7 Conclusions and further research

In this paper we described a game which tries to model some of the transactions taking place in

a firm. We discussed some possible equilibria of the game in the simpler cases and observed how

the game may be compared to a version of the Prisoner’s Dilemma. The results we found and the

stability of equilibria are interpreted in terms of corporate culture.

We considered also some “bounded rationality” agents and the impact their presence has on

the ideal cases equilibria.

Then, we discussed a ranking incentive devised to increase competition between agents; we

showed how this ranking policy affects the performance of the subjects and found the set of

equilibria of the modified game. Since the modified game has a continuum of equilibria we suggested

one possible way to refine them. The results we found are interesting in terms of free riding and

shirking and may help to shed light on how the different composition of employees may result in

different equilibria.

Furthermore, we showed in the simple model of firm how an individual incentive plan may

result in limiting the optimal effort of some agents.

Finally, by means of agent based simulation two more directions were investigated. First, we

could consider heterogeneous populations with coexistence of all the kinds of behavior we studied

in the theoretical analysis. This way, we could overcome the problem of agents having the best

reply functions in implicit form by a numerical iterative method. The comparison of the results,

when assuming that population composition was common knowledge across the agents, exhibited

consistency with the theoretical results. Second, we assumed that the agents could interact only

in their Von Neumann neighborhood, without knowing the global population composition. Since

for practical reasons each agent must know the best reply of its opponents, we modeled a sort

of egocentric thought about the neighbors’ neighborhood composition. That is, we assumed that

each agent had an egocentric perspective in a sense close to Piaget about the composition of its

neighbors’ neighborhood. In this case the local interaction results were quite similar to the global

ones, and therefore coherent with the theoretical analysis.

In further research it would be interesting to analyze which equilibria may be selected in a

dynamic setting of the game. It would be also interesting to consider also some more sophisticated

models of bounded rationality.

Received: March 14, 2008. Revised: May 9, 2008.



36 Arianna Dal Forno and Ugo Merlone CUBO
11, 2 (2009)

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