Strategic Interaction and Aggregate Incentives
Mohamed Belhaj† and Frédéric Deroı̈an‡
†
‡
Centrale Marseille (Aix-Marseille School of Economics), CNRS and EHESS
Aix-Marseille University (Aix-Marseille School of Economics), CNRS and EHESS
——————————————————————————————————
Abstract
We consider a model of interdependent efforts, with linear interaction and
lower bound on effort. Our setting encompasses asymmetric interaction and
heterogenous agents’ characteristics. We examine the impact of a rise of crosseffects on aggregate efforts. We show that the sign of the comparative static
effects is related to a condition of balancedness of the interaction. Moreover,
we point out that asymmetry and heterogenous characteristics are sources of
non-monotonic variation of aggregate efforts.
Keywords: Strategic Interaction, Social Network, Aggregate Efforts, Asymmetric Interaction, Heterogenous Characteristics
JEL: C72, D85
——————————————————————————————————
∗
We would like to thank participants at the 15th Coalition Theory Network Conference and at
seminars in GREQAM. We are extremely grateful to the editor and two anonymous referees, who
substantially contributed to improve the quality of the paper.
E-mail addresses: [email protected], [email protected].
1
1
Introduction
The importance of economic and social networks has been recently emphasized in a wide range of
economic contexts, including job search, partnerships between firms, free trade agreements, social
influence, crime economics, etc.1 Networked interdependencies between individual actions concern
many applications, such as crime economics (Calvó-Armengol and Zénou [2004]), local public goods
(Bramoulé and Kranton [2007]), equilibrium consumptions in pure exchange economies with positional
goods (Ghiglino and Goyal [2010]), pricing with local network externalities (Bloch and Quérou [2011]),
risk taking under informal risk sharing (Belhaj and Deroı̈an [2011]). A standard comparative statics,
often relevant for policy consideration, consists in raising cross-effects. For instance, this can fit
with an increase of the level of synergy between individual actions (either by increasing the level of
complementarity or by reducing the level of substitutability) and/or a reduction of activity cost. A
central concern is how the sum of individual efforts varies with the rise of cross-effects. For example,
a policy maker may be interested in decreasing the level of criminality, or in increasing the provision
of public good or the investment in a new technology, etc.
Ballester et al. (2006) discuss this issue in the context of linear and symmetric interaction, lower
bound on effort, and homogenous individual characteristics. They show that, when the intensity of
interaction is sufficiently low, raising cross-effects in a way which preserves symmetry generates an
increase of the sum of individual efforts. Exploiting the fact that games with symmetric interaction
admit a potential function, Bramoullé et al. (2011) complement this result under large intensity
of interaction. However, in real world, interactions are in general both nonlinear and asymmetric,
and agents have heterogenous characteristics. If approximating nonlinearity by linear interaction is
sometimes, at least locally, acceptable, there is no general way to reduce asymmetric interactions to
symmetric ones. Moreover, individual characteristics may differ across agents; for instance, individual
costs of effort can vary strongly from one individual to another.
This paper analyzes the impact of a rise of cross-effects on aggregate efforts in presence of asymmetric interaction and heterogenous individual characteristics. Our main contribution is to relate the
variation of aggregate efforts to a condition of balancedness of the interaction. Precisely, we select any
initial equilibrium of the game, irrespective of the possible existence of other equilibria, and whether
the equilibrium includes corner agents or not (by corner agent, we mean an agent who exerts no
effort).2 Then, we introduce a perturbation which raises cross-effects; in particular, the perturbed
system may possess multiple equilibria. We are then able to compare the aggregate efforts of the
initial equilibrium to that of all equilibria of the perturbed system satisfying that corner agents in
the initial equilibrium stay corner. We give a condition of balancedness of interaction, condition (C1)
1
Some recent books present different applications of the role of networks in economic activity- see
Goyal (2009), Jackson (2008), or Rauch (2007).
2
Even in this linear context, multiplicity is a matter. Indeed, efforts are bounded from below, and
corner equilibria can emerge.
2
thereafter, under which aggregate efforts are enhanced after the perturbation. This condition states
that there exists a nonnegative solution to the transposed initial system with homogenous constant.
When condition (C1) is violated, there exist in general perturbations which both raise cross-effects
and lead to a decrease of aggregate efforts, and we build such a perturbation.
We then illustrate our results on specific models of linear interaction. For games with complementarities, condition (C1) always holds, and thus comparative statics are monotonic. For games
with shifted complementarities, the condition of balancedness is useful for large levels of interaction.
For games with substitutabilities, we give an original condition which guarantees that the comparative statics are monotonic, and we present two polar examples illustrating that both asymmetric
interaction and heterogenous characteristics can generate non-monotonic statics.
This paper is organized as follows. Section 2 describes the model and introduces the definition of
a rise of cross-effects. Section 3 studies how raising cross-effects affects aggregate efforts. Section 4
concludes. The last section is an appendix collecting all proofs.
2
A model of linear interaction
We consider a collection N = {1, · · · , n} of agents. Agent i plays some uni-dimensional action given
by the nonnegative real number xi ∈ R+ . We denote 1 as the column-vector of ones and we let
G be the set of n-square real matrices with positive diagonals. Letters with upper-script T denote
transposes. Consider a matrix Γ ∈ G and a vector A ∈ Rn , we define system (1) as follows:

P
P

 γii xi + γij xj = ai if ai − γij xj > 0
j6=i
j6=i
P

x
=
0
if
a
−
γij xj ≤ 0
 i
i
(1)
j6=i
The matrix Γ represents the matrix of interaction. When γij < 0 (resp. γij > 0), agent j’s effort
is a strategic complement (resp. substitute) to agent i’s effort. The set G contains the economically
important class of symmetric matrices (i.e. γij = γji ), among which it allows for mixed effects,
meaning that some bilateral interactions are complements, others substitutes. The set G also includes
asymmetric matrices.
We assume that system (1) characterizes pure strategy Nash equilibria of some underlying game,
i.e. system (1) is necessary and sufficient for Nash. This system selects (pure) Nash equilibria in
many economic contexts, like synergistic efforts with linear quadratic utilities (Ballester et al. [2006]),
local public goods (Bramoulé and Kranton [2007]), pure exchange economies with positional goods
(Ghiglino and Goyal [2010]). Indeed, since we assume γii > 0, all utility functions are strictly concave
in own-effort and the system (1) represents the first order conditions of a utility maximization problem.
In general, the constant ai represents either some individual return or some cost to effort. Note that
our setting allows idiosyncratic constant ai , possibly negative.3
3
See Belhaj and Deroı̈an (2010) for a model with possibly negative constant. A negative constant
3
The following definitions are useful. Let X be an equilibrium associated with system (1). An
interior agent i is such that xi > 0, and a corner agent i is such that xi = 0. A knife-edge agent
P
γij xj . We define respectively I(X), C(X), K(X) as the sets
i is a corner agent satisfying ai =
j6=i
of interior, corner, and knife-edge agents associated with X. We let ΓI(X) denote the matrix of
interaction restricted to interior agents in X. Conform to Ballester et al. (2006), raising cross-effects
is formally defined as follows:
Definition (Raising cross-effects). A perturbation Θ = [θij ] raises cross-effects with respect
to Γ ∈ G if θij ≤ 0 for all i, j, and Γ + Θ ∈ G, that is, γii + θii > 0 for all i.
Everything else equal, when θij < 0, the influence exerted by agent j on agent i’s effort is shifted
upward; that is, either the initial level of complementarity is enhanced, or the initial substitutability
level is decreased, or the initial substitutability becomes a complementarity. Similarly, everything else
equal, when θii < 0, agent i’s sensitiveness to others’ efforts is increased. For instance, this can fit
with an increase of the level of synergy between individual actions (either by increasing the level of
complementarity or decreasing the level of substitutability) and/or reduction of activity cost.4 When,
following a perturbation which raises cross-effects, aggregate efforts are enhanced, we shall say that
the comparative statics is monotonic. When raising cross-effects leads to a decrease of aggregate
efforts, we shall say that the comparative statics is non-monotonic.
In this paper, we examine the impact of a perturbation on efforts, not on payoffs. In general,
understanding the impact of a rise of cross-effects on payoffs is an open issue. However, in linearquadratic setting with complementarities (as in Ballester et al. [2006]), since each payoff is positively
proportional to the square of equilibrium effort, raising cross-effect increases both aggregate efforts
and aggregate payoffs. In the local public good game of Bramoullé and Kranton (2007), which is a
game with strategic substitutes, the variation of aggregate payoffs is negatively proportional to the
variation of aggregate efforts.
3
Do larger cross-effects enhance aggregate efforts?
In this section, we explore how aggregate efforts respond to a rise of cross-effects. Our main finding is
that a condition of balancedness of interaction is key to the analysis. We also present circumstances
in which comparative statics are monotonic, and we identify some mechanisms which can generate
non-monotonic statics.
We start by presenting the condition which is crucial to the analysis. Let X be a solution to
system (1):
ai entails that, in absence of interaction, agent i would find optimal to exert no effort.
4
In terms of primitive utilities, raising cross-effects consists in increasing cross-derivatives of utilities
and/or decreasing second derivative of utilities with respect to own action.
4
Condition (C1). There is a nonnegative solution Z to the system (ΓI(X) )T Z = 1I(X) .
Condition (C1) may be interpreted as a condition of balanced interaction among interior agents of the
equilibrium. Condition (C1) means that there is a nonnegative weight vector Z such that (ΓI(X) )T Z
is a constant vector, that is, all the rows in ΓI(X) can be combined to get a balanced interaction
vector (proportional to 1I(X) ). This means that there is a virtual agent, set up as a weighted sum of
agents, whose received externalities are balanced so that she is affected by the same intensities types
of externalities from all agents.
Note that, if the matrix ΓI(X) is not invertible, there may not be a unique solution Z to
(ΓI(X) )T Z = 1I(X) . More, if ΓI(X) contains a line with homogenous elements, condition (C1) holds.
Note also that condition (C1) does not require that the inverse of the transposed matrix of interaction
is nonnegative, it only requires its row sums to be nonnegative.
Our main finding is summarized in a theorem, which compares any initial equilibrium to all
possible equilibria of a perturbed game satisfying that initial corner agents stay corner:
Theorem 1 (i). Suppose that condition (C1) holds. Then, given any perturbation raising crosseffects, for any equilibrium X of the original system and any equilibrium X̃ of the perturbed system
such that corner agents in the initial equilibrium stay corner in the perturbed equilibrium (i.e., C(X) ⊂
n
n
P
P
C(X̃)), the aggregate efforts increase (weakly), i.e.
x̃i ≥
xi .
i=1
i=1
(ii). Conversely, suppose that condition (C1) does not hold. Let X be a solution to system (1).
If X contains no knife-edge agent (i.e. K(X) = ∅), there exists a (local) perturbation raising crosseffects and an equilibrium X̃ of the perturbed system, such that initial corner agents stay corner and
n
n
P
P
x̃i <
xi .
i=1
i=1
When X contains no strict corner agent (i.e., C(X) \ K(X) = ∅), Theorem 1 reads as follows:
consider a game with interaction matrix Γ such that there is a positive solution X > 0 (satisfying
ΓX = A). Then, departing from X, any perturbation raising cross-effects will enhance aggregate
efforts in any equilibrium of the new game with interaction matrix Γ + Θ if and only if condition (C1)
holds; for the if part, the weaker condition X ≥ 0 applies.
Theorem 1 is compatible with the existence of multiple equilibria regarding both initial and
perturbed systems. Part (i) in Theorem 1 states that, starting from any equilibrium associated with
any constant A, to guarantee that an increase of cross-effects enhances aggregate efforts, the solution
of the transposed sub-system of interior agents with homogenous constant should be nonnegative.
More precisely, for a given perturbation, condition (C1) is sufficient to obtain a monotonic statics over
the set equilibria whose corner agents encapsulates initial corner agents.
Part (ii) of Theorem 1 considers the case where condition (C1) does not apply. Provided that
the initial equilibrium does not contain knife-edge agents, there always exists a perturbation raising
cross-effects and leading to a decrease of aggregate efforts; to prove the existence of non-monotonic
statics, we build a perturbation Θ in some appropriate direction, its magnitude being sufficiently
5
small to guarantee that initial corner agents stay corner. We illustrate that the condition K(X) = ∅
is necessary by presenting an example where part (ii) does not hold because we have K(X) 6= ∅.

1
2
0



T

Take Γ = 
 0 1 0 , A = (3 1 1) . This is a game with substitutabilities, with one unique
1 0 1
equilibrium X = (1 1 0)T , and x1 + x2 + x3 = 2. Agent 2 does not interact with other agents,
thus she is interior (as a2 > 0). Agent 3 is a knife-edge agent, thus I(X) = {1, 2}. Moreover,
(ΓTI(X) )−1 1 = (1 − 1)T . Thus, condition (C1) is not satisfied. Yet, for any perturbation raising
cross-effects, we have x̃1 + x̃2 + x̃3 ≥ 2.5
We discuss the relationships between Theorem 1 and the existing literature. Many games can be
framed in our paper. We explore games with complementarities, games with shifted complementarities
(see Ballester et al. [2006], and Ballester and Calvó-Armengol [2010]), games with substitutabilities
(Bramoullé and Kranton [2007]).
Games with shifted complementarities. Consider the following utility function:
Ui (X; Γ) = ai xi −
γii 2 X
x −
γij xi xj
2 i
(2)
j6=i
with γii > 0, γij ∈ R. When ai > 0, this quantity can be interpreted as the gross marginal return of
effort (net of any synergistic effect), γii as the intensity of the (quadratic) cost of effort, and γij as
the level of interaction between neighbors. A perturbation raising cross-effects can lower the cost of
effort (γii ) and/or decrease the coefficients γij (i.e., either increasing the level of complementarity or
reducing the level of substitutability). A solution to utility maximization problem with utilities as in
equation (2) satisfies system (1).
Ballester et al. (2006) consider ai = αi , γii = −σ for all i and γij = −σij . Their setting imposes
both α > 0 and σ < min{σ, 0}, where σ = min{σij |i 6= j}. Under these conditions, they obtain
a game with shifted complementarities with homogenous substitutability shift, that is, games of the
form Γ = Γ0 + φ11T , where Γ0 corresponds to a game with complementarities and φ ≥ 0. Under
the condition that the spectral radius of Γ0 is less than one (condition (CBCZ ) thereafter), there is a
unique equilibrium and raising cross-effects always induces an increase of aggregate efforts. Regarding
the statics, established under the crucial hypotheses of the symmetric interaction and homogenous
constant, it has to be stressed that their proof still works when the initial equilibrium is interior,
5
Technically, consider the perturbed interaction matrix Γ̃ = Γ + Θ, with Θ ≤ 0, and a solution
X̃ to the perturbed system. First, we get x̃2 =
1−θ21 x̃1 −θ23 x̃3
,
1+θ22
thus x̃2 ≥ 1. Second, we have
x̃1 + x̃3 ≥ 1. Indeed, consider agent 3’s incentives: (1 + θ31 )x̃1 + θ32 x̃2 + (1 + θ33 )x̃3 ≥ 1, with equality
if (1 + θ31 )x̃1 + θ32 x̃2 ≤ 1. Three cases can arise. Either x̃3 = 0 (in this case, the perturbation satisfies
that initial corner agents stay corner) and then x̃1 ≥
x̃3 ≥
1−θ32 x̃2
1+θ33
1−θ32 x̃2
1+θ31
entailing x̃1 ≥ 1; or x̃1 = 0 and then
entailing x̃3 ≥ 1; or both x̃3 > 0, x̃1 > 0 and thus x̃1 + x̃3 = 1−θ32 x̃2 −θ31 x̃1 −θ33 x̃3 ≥ 1.
6
irrespective of condition (CBCZ ).
To understand how Theorem 1 generalizes the statics, two remarks are in order. First, condition
(CBCZ ) implies condition (C1). Indeed, by condition (CBCZ ), system (1) admits a unique solution,
and this solution is nonnegative. Since any matrix admits the same spectral radius as its transpose,
the transposed system also satisfies condition (CBCZ ), and thus has a nonnegative solution. As a
direct implication, Theorem 1 extends the analysis of Ballester et al. (2006) to the following enlarged
contexts. If the initial equilibrium satisfies condition (CBCZ ), raising cross-effects always induces an
increase of aggregate efforts irrespective of symmetry of interaction, of the homogeneity of the diagonal
of the matrix of interaction, of the sign and homogeneity of the constant, and whether the perturbed
equilibrium is interior or not.
Second, and perhaps more importantly, Theorem 1 extends the analysis to situations in which
condition (CBCZ ) does not hold, i.e. to cases where the level of interaction is such that balancedness is an issue; hence it applies typically to the region for which uniqueness of equilibrium is not
guaranteed. One important message is that, to know if raising cross-effects generates necessarily an
increase of aggregate efforts, the relevant information is given by condition (C1) applied to the system
of interaction between interior agents of the initial equilibrium. To illustrate, we provide an example
where condition (CBCZ ) does not hold but comparative 
statics are monotone
 because of condition
3
1 −1


(C1).6 Consider A = 1 and the interaction matrix Γ = 
−1
1
0 
 which has shifted com
−1 −1 1
plementarities: each diagonal is strictly greater than the other positive entries in the same row. This
game has a unique equilibrium (1 2 4)T . It is easy to see that (ΓT )−1 1 > 0 and thus condition (C1)
holds and Theorem 1 applies. However, condition (CBCZ ) can never hold, whatever the shift we apply
(even if the shift is not homogenous, as below):

3 − u1
1 − u1

T
0

Γ = Γ + u1 =  −1 − u2 1 − u2
−1 − u3
−1 − u3
−1 − u1
−u2
1 − u3



u1
u1
u1
 
 +  u2
 
u3
u2

u2 

u3
u3
with 1 ≤ u1 ≤ 3, 0 ≤ u2 ≤ 1, 0 ≤ u3 ≤ 1 (inequalities are set up to induce that Γ0 is a matrix of
complementarities). Here, the matrix with complementarities Γ0 can never satisfy condition (CBCZ )
because det(Γ0 ) = 2 − 4u1 − 4u2 − 6u3 ≤ −2 < 0.
A special case is games with complementarities, where γij ≤ 0 for all i, j 6= i and γii > 0, A ∈ Rn .
If interaction is sufficiently low (the spectral radius of the matrix of interaction is less than one - see
Ballester and Calvó-Armengol [2010, corollary 1, p. 14]), such games admit a unique equilibrium.
This condition also guarantees that the transposed system meets condition (C1). Thus the following
result obtains.
Corollary 1 If a system of type (1) with complementarities has a nonnegative inverse Γ−1 ≥ 0 (thus
it has a unique equilibrium), then raising cross effects increases aggregate efforts.
6
We would like to thank an anonymous referee for suggesting this example.
7
Games with strategic substitutes (Γ ≥ 0). Consider the local public good game with utilities
written as follows:
Ui (X; Γ) = b(γii xi +
X
γij xj ) − κi xi
(3)
j6=i
where b(.) is strictly increasing and strictly concave on R+ , and b0 (+∞) < κi < b0 (0) for all i. When
κi > 0, this quantity may represent the constant marginal cost of own action. Fix ai = b0−1 (κi ).
Bramoullé and Kranton (2007) and Bramoullé et al. (2011) consider the following particular case:
define the matrix G = [gij ] such that gii = 0, gij ∈ {0, 1}, and gij = gji (G represents the adjacency
matrix of a non directed network), and set γii = 1, γij = δgij with δ ≤ 1, κi = κ > 0. Bramoullé
et al. (2011, Proposition 6) show that, under symmetric interaction and homogenous (and positive)
constant, starting from any initial equilibrium of the game, raising cross-effect, in such a way that
initial corner agents stay corner in the new equilibrium, generates an increase of aggregate efforts.
An equilibrium related to utilities as in equation (3) satisfies system (1). In particular, the setting
of Bramoullé and Kranton is such that interior agents satisfy ΓI XI = 1. Then, by symmetry we have
ΓTI XI = 1, and Theorem 1 applies. Theorem 1 therefore complements the results found in Bramoullé
et al. (2011). Indeed, the utility represented in equation (3) covers asymmetric bilateral influences,
and both heterogenous own influences (γii ) and costs of effort. In this enlarged setting, condition (C1)
is crucial to understand aggregate efforts.
From Theorem 1 we derive some conditions under which games with strategic substitutes admit
monotonic comparative statics. We consider the following condition. Let X be a solution to system
(1), with Γ ≥ 0:
Condition (C2). The matrix ΓI(X) is such that, for all i ∈ I(X),
P
j∈I(X)\{i}
γji
γjj
≤ 1.
We note that condition (C2) is given with weak inequalities. We obtain:
Proposition 1 Suppose that there are only substitutabilities (Γ ≥ 0). Consider an equilibrium X
associated with system (1). If condition (C2) holds, any perturbation raising cross-effects and such
that initial corner agents stay corner will enhance aggregate efforts.
It is worth mentioning that if the matrix Γ satisfies condition (C2), then, for any subset S of agents,
condition (C2) is also verified by the sub-matrix ΓS . This means that, in games with substitutes, the
condition that the matrix Γ satisfies condition (C2) is sufficient for monotonic statics whatever the
initial solution to system (1).
We turn to non-monotonic comparative statics. We illustrate below by means of examples that two
forms of heterogeneity can generate non-monotonic comparative statics. The first type of heterogeneity
is related to the asymmetry of bilateral interactions. For instance in local public good games, one
agent may provide more externality to another agent than the level of externality that she receives
8
from her. The second type of heterogeneity pertains to constant A. This corresponds for instance to
heterogenous individual costs of effort.
Example 1. 
Consider
1 0

where we set Γ = 
 .6 1
.6 0
0−1
if b (κi ) = 1 (that is, A
the 
3-agent local public goods game with utilities given by equation (3),
0

0 
. An equilibrium of this local public good game satisfies system (1)
1
= 1). In this system of interaction, agent 1 influences the behavior of
agents 2 and 3, and not vice-versa; further, agents 2 and 3 do not interact with each other. Then
n
P
X = Γ−1 1 = (1 .4 .4)T is a positive solution of the game, and
xi = 1.8. However, (−.2 1 1)T is
i=1
the unique solution to the transposed system with homogenous constant. Since it contains a negative
component, we know by Theorem 1 that there always exists a perturbation generating a non-monotonic
comparative statics. For instance, we select the perturbation Θ such that θ11 = −.05, and θij = 0
otherwise. Then X̃ ' (1.05 .36 .36)T is a solution of the perturbed system (Γ + Θ), with aggregate
n
n
P
P
efforts
x̃i ' 1.78 <
xi .7 The intuition is easily grasped. Agent 1 has strong influence on others,
i=1
i=1
but it not influenced by anyone. The perturbation has a direct effect on that agent, pushing agent 1 to
exert higher effort. This generates a large decrease of others’ efforts, in such a way that the resulting
net effect on aggregate efforts is negative. Since neither agent 2 nor agent 3 affect agent 1’s behavior,
the system is stabilized at this step.
Example 2. Consider
the 3-agent
local public good game with utilities given by equation (3),


1 .6 .6


T

where we set Γ = 
 .6 1 0 , and A = (1.25 1 1) . An equilibrium of this local public good
.6 0 1
game satisfies system 1. In this example, Γ is symmetric, and with an homogenous diagonal. However,
A is not homogenous. Then X = Γ−1 A ' (.17 .89 .89)T is a solution to the game, and thus
n
P
xi ' 1.96. Further, the unique solution to the transpoed system with homogenous constant is
i=1
T −1
(Γ )
1 = (−.71 1.42 1.42)T , and thus condition (C1) does not hold. As in example 1, we select
the perturbation Θ such that θ11 = −.05, and θij = 0 otherwise. This perturbation leads to a new
n
n
P
P
equilibrium X̃ ' (.21 .86 .86)T , which yields
x̃i ' 1.95 <
xi . Intuitively, the same type of
i=1
i=1
mechanism as in example 1 operates. Following the perturbation, consider the sequence of play in
which agent 1 computes optimal effort first, then agents 2 and 3 simultaneously, then again agent
1, etc. As first reaction to the perturbation, agent 1 increases effort, then agents 2 and 3 decrease
effort in such a way that the sum of both variations is of larger magnitude than the initial increase of
agent 1’s effort. This mechanism propagates through the entire sequence of play leading to the new
equilibrium, and therefore the static is non-monotonic. In that example, the heterogeneity of constant
A makes the initial equilibrium interior.
7
Even if numerical values are approximated, the actual values satisfy the comparison provided.
This is as well in both examples 1 and 2.
9
Remark. The existence of non-monotonic comparative statics is not related to the asymptotic
stability8 of the system of interaction. For instance, in Example 1, increasing cross-effects does not
always result in an increase of aggregate efforts, while Γ is a stable matrix.9
4
Conclusion
This paper has explored the impact of a rise of cross-effects on aggregate efforts in games with
piece-wise linear best-replies where strategies are bounded below, in a setting including possibly
heterogenous constant and asymmetric interactions. Such comparative statics, which can correspond
to a variation of synergies or activity costs, are relevant for policy intervention. Essentially, this
paper has shown that the response of aggregate efforts is related to a condition of balancedness of
interaction. This condition is particularly useful to guarantee monotonic statics under large level
of interaction. In particular, when interactions are not balanced, raising cross-effects may generate
a decrease of aggregate efforts. Both the asymmetry of bilateral interactions and heterogeneity of
individual characteristics are possible sources for such non-monotonic statics.
It would be interesting to deeper our understanding of perturbations which increase or decrease
aggregate efforts. Furthermore, the study of the impact of perturbations on aggregate payoffs is, in
general, an open issue.
5
Appendix: proofs
The following lemma is adapted from Farkas’s lemma.
Lemma 1 Let Q be an k × k matrix. Then, there exists a nonnegative solution Y to QT Y = 1 if and
n
P
only if, for all Y ∈ Rk such that QY ≥ 0, we have
yi ≥ 0.
i=1
Proof of Theorem 1.
(i). We consider an initial equilibrium X associated with system (1) with a set of interior agents
I and a set of corner agents C, including possibly knife-edge agents (there could be other equilibria).
The matrix ΓI describes the interaction pattern between interior agents in equilibrium X. We will
T
prove that if there exists a nonnegative solution Z to ΓI Z = 1I , then every perturbation Θ, which
raises cross-effects, induces an increase of aggregate efforts as soon as corner agents in the initial
equilibrium stay corner.
8
See Weibull (1995) for an introduction of the concept of asymptotic stability. In a word, a Nash
equilibrium is asymptotically stable if, following any sufficiently small perturbation, a naı̈ve bestresponse dynamics goes back to the original equilibrium.
9
A linear system is asymptotically stable when the real parts of all its eigenvalues are positive.
10
For any profile say T ∈ Rn , we define TI the sub-profile of T restricted to interior agents (if I = N ,
we have TI = T ). By system (1) we have
ΓI XI = AI
(4)
Now, let X̃ be any equilibrium of the perturbed game with interaction matrix Γ̃ = Γ + Θ, where
Θ ≤ 0, and where corner agents in X stay corner in X̃ (i.e., C(X) ⊂ C(X̃)). Or equivalently, X̃ solves
the linear complementarity problem
X̃ ≥ 0
(5)
Γ̃X̃ ≥ A
(6)
with equality for all interior agents. In particular, since corner agents in X stay corner in X̃, we have
Γ̃I X̃I ≥ AI
(7)
Expression (7) is equivalent to ΓI X̃I + ΘI X̃I ≥ AI . By (5) and the fact that Θ ≤ 0 (and thus ΘI ≤ 0),
we have ΘI X̃I ≤ 0. This implies:
ΓI X̃I ≥ AI
(8)
Combining equation (8) with equation (4), we obtain that ΓI (X̃I − XI ) ≥ 0. We can then apply
lemma 1 with Q = ΓI and Y = X̃I − XI and we are done.
(ii). We consider an initial solution X to system (1) with a set of k interior agents I, and without
any knife-edge agent (K = ∅). Suppose that there exists a solution Y to (ΓI )T Y = 1I with a negative
component. We will prove that there always exists a perturbation Θ which both raises cross-effects
and strictly reduces aggregate efforts.
By lemma 1, there exists a profile Y = (y1 · · · yk )T such that ΓI Y ≥ 0 while
k
P
yj < 0. We will
i=1
show that there exists a matrix Θ ≤ 0, and vector X̃ solution of the game with interaction matrix
n
n
P
P
(Γ+Θ, A), such that
x̃i <
xi . To proceed, we select a perturbation which only affects interaction
i=1
i=1
between interior agents, and which is related to the profile Y as follows. Consider a real number > 0
satisfying the following two conditions:
xi + yi > 0 , for all i ∈ I
ai <
X
(γij + θij )(xj + yj ) , for all i ∈ C
j6=i
11
(9)
(10)
Such a positive number exists since there is no knife-edge agent (indeed, condition (10) is verified
for = 0 with strict inequality). Define the n × n matrix Θ = [θij ] such that for all i, j ∈ I,
θij =
−[ΓI Y ]i
k
k
P
P
xi +
yi
i=1
(that is, the sub-matrix ΘI has uniform lines), while θij = 0 if at least one agent
i=1
among i or j is not interior. Basically, condition (9) guarantees that Θ ≤ 0. By construction, we have
ΘI (XI + · Y ) = − · ΓI Y
(11)
Define X̃ such that x̃i = 0 if xi = 0 and X̃I = XI + · Y (equivalently −Y = XI − X̃I ). The equation
(11) writes therefore
ΘI X̃I = ΓI (XI − X̃I )
(12)
(ΓI + ΘI )X̃I = AI
(13)
That is, X̃ satisfies
Hence, X̃ is a solution of system (1) with interaction matrix Γ + Θ. Indeed, by condition (9), initial
interior agents stay interior and by condition (10), initial corner agents stay (strict) corner. Now,
k
k
k
n
n
P
P
P
P
P
since X̃I − XI = · Y , the inequality
yi < 0 implies
x̃i <
xi , and thus
x̃i <
xi . i=1
i=1
i=1
i=1
i=1
Proof of Proposition 1. Consider an equilibrium X with I(X) = {1, 2, · · · , k} without loss of
generality. Suppose that the matrix ΓI(X) satisfies condition (C2). We have to check that condition
(C1) is satisfied.
Let D be a vector of size k with di =
1
γii ,
and let H be a k × k matrix with, for all i, j 6= i, hii = 0
γji
γii .
Define function f such that, for any X ∈ Rn , f (X) = D − HX. Since ΓI(X) satisfies
1 condition (C2), f is a continuous function from 0, γ111 × · · · × 0, γkk
to itself; indeed, on the one
and hij =
hand, if, for all j 6= i, we have xj ≥ 0, clearly xi ≤ γ1ii ; on the other hand, if xj = γ1jj for all j 6= i,
P γji
then xi ≥ 0 if γ1ii − γ1ii
γjj ≥ 0, which is exactly condition (C2). We conclude that function f
j6=i
admits a fixed point (by Brouwer Fixed Point Theorem), that is, there is a nonnegative solution to
X = D − HX (since the null vector is not a solution, the solution contains at least one positive
component). To finish, we note that X = D − HX if and only if ΓT X = 1. References
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