Strategic interaction and aggregate incentives

Journal of Mathematical Economics 49 (2013) 183–188

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Strategic interaction and aggregate incentives✩ Mohamed Belhaj a , Frédéric Deroïan b,∗ a

Ecole Centrale Marseille (Aix-Marseille School of Economics), CNRS and EHESS, France

b

Aix-Marseille University (Aix-Marseille School of Economics), CNRS and EHESS, France

article

info

Article history: Received 4 March 2011 Received in revised form 2 October 2012 Accepted 17 February 2013 Available online 27 February 2013

abstract We consider a model of interdependent efforts, with linear interaction and lower bound on effort. Our setting encompasses asymmetric interaction and heterogeneous agents’ characteristics. We examine the impact of a rise of cross-effects 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 heterogeneous characteristics are sources of non-monotonic variation of aggregate efforts. © 2013 Elsevier B.V. All rights reserved.

Keywords: Strategic interaction Social network Aggregate efforts Asymmetric interaction Heterogeneous characteristics

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 (Bramoullé 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 Deroïan, 2011). A standard comparative statics, often relevant for policy consideration, consists of 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

✩ 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. ∗ Correspondence to: GREQAM - Centre de la Vieille Charité - 2 rue de la Charité, 13002 Marseille, France. Tel.: +33 491140262. E-mail addresses: [email protected] (M. Belhaj), [email protected] (F. Deroïan). 1 Some recent books present different applications of the role of networks in

economic activity — see Goyal (2009), Jackson (2008), or Rauch (2007). 0304-4068/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmateco.2013.02.003

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 homogeneous 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 a large intensity of interaction. However, in the real world, interactions are in general both nonlinear and asymmetric, and agents have heterogeneous 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 heterogeneous 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

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no effort).2 Then, we introduce a perturbation which raises crosseffects; 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) 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 homogeneous 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 heterogeneous 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:

  γii xi + γij xj = ai    j̸=i

if ai −

   xi = 0

if ai −



γij xj > 0

j̸=i



γij xj ≤ 0.

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 i is a corner agent satisfying ai = j̸=i γij xj . We define respectively I (X ), C (X ), K (X ) as the sets 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 crosseffects 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 a linear–quadratic 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.

(1)

j̸=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 (Bramoullé 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

2 Even in this linear context, multiplicity is a matter. Indeed, efforts are bounded from below, and corner equilibria can emerge. 3 See Belhaj and Deroïan (2010) for a model with possibly negative constant. A negative constant ai entails that, in absence of interaction, agent i would find it optimal to exert no effort.

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): 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 intensity types of externalities from all agents.

4 In terms of primitive utilities, raising cross-effects consists of increasing crossderivatives of utilities and/or decreasing the second derivative of utilities with respect to own action.

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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 ) . Moreover, if ΓI (X ) contains a line with homogeneous 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) is satisfied for an equilibrium X , solution to system (1). Then, given any perturbation raising cross-effects, and any equilibrium X˜ of the perturbed system such that corner agents in the initial equilibrium stay corner ˜ in the perturbed equilibrium (i.e., n C (X ) ⊂ Cn (X )), the aggregate ˜ efforts increase (weakly), i.e. x ≥ i=1 i i=1 xi . (ii). Conversely, suppose that Condition (C1) does not hold for an equilibrium X , solution to system (1). If X contains no knife-edge agent (i.e. K (X ) = ∅), there exists a (local) perturbation raising cross-effects and an equilibrium X˜ of the perturbed system, n n such that initial corner agents stay corner and i=1 x˜ i < i=1 xi . 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 Γ X = 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 subsystem of interior agents with homogeneous 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 encapsulate 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, 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 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 ) ̸=  ∅.  Take Γ

=

1 0 1

2 1 0

0 0 1

, A = (3 1 1)T . This is a game with

substitutabilities, with one unique 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, (ΓIT(X ) )−1 1 = (1 − 1)T . Thus, Condition (C1) is not satisfied. Yet, for any perturbation raising cross-effects, we have x˜ 1 + x˜ 2 + x˜ 3 ≥ 2.5

5 Technically, consider the perturbed interaction matrix Γ˜ = Γ + Θ , with Θ ≤ 0, 1−θ21 x˜ 1 −θ23 x˜ 3 and a solution X˜ to the perturbed system. First, we get x˜ 2 = , thus 1+θ 22

x˜ 2 ≥ 1. Second, we have x˜ 1 + x˜ 3 ≥ 1. Indeed, consider agent 3’s incentives: (1 + θ31 )˜x1 + θ32 x˜ 2 + (1 + θ33 )˜x3 ≥ 1, with equality if (1 + θ31 )˜x1 + θ32 x˜ 2 ≤ 1. Three cases can arise. Either x˜ 3 = 0 (in this case, the perturbation satisfies that 1−θ x˜ initial corner agents stay corner) and then x˜ 1 ≥ 1+θ32 2 entailing x˜ 1 ≥ 1; or 31 x˜ 1 = 0 and then x˜ 3 ≥

1−θ32 x˜ 2 1+θ33

entailing x˜ 3 ≥ 1; or both x˜ 3 > 0, x˜ 1 > 0 and

thus x˜ 1 + x˜ 3 = 1 − θ32 x˜ 2 − θ31 x˜ 1 − θ33 x˜ 3 ≥ 1.

185

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; Ballester and CalvóArmengol, 2010), games with substitutabilities (Bramoullé and Kranton, 2007). Games with shifted complementarities. Consider the following utility function: Ui (X ; Γ ) = ai xi −

γii 2

x2i −



γij xi xj

(2)

j̸=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 Eq. (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 ̸= j}. Under these conditions, they obtain a game with shifted complementarities with homogeneous substitutability shift, that is, games of the form Γ = Γ ′ + φ 11T , where Γ ′ corresponds to a game with complementarities and φ ≥ 0. Under the condition that the spectral radius of Γ ′ 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 homogeneous constant, it has to be stressed that their proof still works when the initial equilibrium is interior, 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 (C1).6 Consider A = 1 and   the interaction matrix Γ

3

=

−1 −1

1 1 −1

−1 0 1

which has shifted

complementarities: each diagonal is strictly greater than the other positive entries in the same row. This game has a unique equilibrium

6 We would like to thank an anonymous referee for suggesting this example.

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(1 2 4)T . It is easy to see that (Γ T )−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 homogeneous, as below):

 Γ = Γ + u1 = ′

T

 +

u1 u2 u3

u1 u2 u3

3 − u1 −1 − u 2 −1 − u 3

u1 u2 u3

1 − u1 1 − u2 −1 − u 3

−1 − u 1 −u 2 1 − u3





with 1 ≤ u1 ≤ 3, 0 ≤ u2 ≤ 1, 0 ≤ u3 ≤ 1 (inequalities are set up to induce that Γ ′ is a matrix of complementarities). Here, the matrix with complementarities Γ ′ can never satisfy condition (CBCZ ) because det(Γ ′ ) = 2 − 4u1 − 4u2 − 6u3 ≤ −2 < 0. A special case is games with complementarities, where γij ≤ 0 for all i, j ̸= 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 is obtained. 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. Games with strategic substitutes (Γ ≥ 0). Consider the local public good game with utilities written as follows:

 Ui (X ; Γ ) = b γii xi +

 

γij xj − κi xi

(3)

j̸=i

where b(·) is strictly increasing and strictly concave on R+ , and b′ (+∞) < κi < b′ (0) for all i. When κi > 0, this quantity may represent the constant marginal cost of own action. Fix ai = b′−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 = δ gij with δ ≤ 1, κi = κ > 0. Bramoullé et al. (2011, Proposition 6) show that, under symmetric interaction and homogeneous (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 Eq. (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 ΓIT XI = 1, and Theorem 1 applies. Theorem 1 therefore complements the results found in Bramoullé et al. (2011). Indeed, the utility represented in Eq. (3) covers asymmetric bilateral influences, and both heterogeneous 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 ),  γji j∈I (X )\{i} γ ≤ 1. jj

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 from her. The second type of heterogeneity pertains to constant A. This corresponds for instance to heterogeneous individual costs of effort. Example 1. Consider the 3-agent local public goods gamewith  utilities given by Eq. (3), where we set Γ =

1 0.6 0.6

0 1 0

0 0 1

. An

equilibrium of this local public good game satisfies system (1) if b′−1 (κi ) = 1 (that is, A = 1). In this system of interaction, agent 1 influences the behavior of agents 2 and 3, and not viceversa; further, agents 2 and 3 do not interact with each other. Then X Γ −1 1 = (1 0.4 0.4)T is a positive solution of the game, and = n T i=1 xi = 1.8. However, (−0.2 1 1) is the unique solution to the transposed system with homogeneous 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 = −0.05, and θij = 0 otherwise. Then X˜ ≃ (1.05 0.36 0.36)T is a solution of the perturbed system (Γ + Θ ), with aggregate efforts n  ˜ i ≃ 1.78 < ni=1 xi .7 The intuition is easily grasped. Agent 1 i=1 x has a strong influence on others, but it is not influenced by anyone. The perturbation has a direct effect on that agent, pushing agent 1 to exert more 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 Eq. (3), where we set Γ

=

1 0.6 0.6

0.6 1 0

0.6 0 1

,

and A = (1.25 1 1)T . An equilibrium of this local public good game satisfies system (1). In this example, Γ is symmetric, and with an homogeneous diagonal. However, A is not homogeneous. 1 Then X = Γ − A ≃ (0.17 0.89 0.89)T is a solution to the n game, and thus i=1 xi ≃ 1.96. Further, the unique solution to the transposed system with homogeneous constant is (Γ T )−1 1 = (−0.71 1.42 1.42)T , and thus Condition (C1) does not hold. As in Example 1, we select the perturbation Θ such that θ11 = −0.05, and θij = 0 otherwise. This perturbation leads to a new equilibrium

˜ i ≃ 1.95 < i=1 xi . X˜ ≃ (0.21 0.86 0.86)T , which yields i=1 x Intuitively, the same type of mechanism as in Example 1 operates. Following the perturbation, consider the sequence of play in n

n

7 Even if numerical values are approximated, the actual values satisfy the comparison provided. This is also in both Examples 1 and 2.

M. Belhaj, F. Deroïan / Journal of Mathematical Economics 49 (2013) 183–188

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. 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 heterogeneous 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 deepen 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. Appendix. Proofs

Now, let X˜ be any equilibrium of the perturbed game with interaction matrix Γ˜ = Γ + Θ , where Θ ≤ 0, and where corner agents in X stay corner in X˜ (i.e., C (X ) ⊂ C (X˜ )). Or equivalently, X˜ solves the linear complementarity problem X˜ ≥ 0

(5)

Γ˜ X˜ ≥ A

(6)

with equality for all interior agents. In particular, since corner agents in X stay corner in X˜ , we have

Γ˜ I X˜ I ≥ AI .

(7)

Expression (7) is equivalent to ΓI X˜ I + ΘI X˜ I ≥ AI . By (5) and the fact that Θ ≤ 0 (and thus ΘI ≤ 0), we have ΘI X˜ I ≤ 0. This implies:

ΓI X˜ I ≥ AI .

(8)

Combining Eq. (8) with Eq. (4), we obtain that ΓI (X˜ I − XI ) ≥ 0. We can then apply Lemma 1 with Q = ΓI and Y = X˜ I − 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 ki=1 yj < 0. We will show that there exists a matrix

Θ ≤ 0, and vector X˜ solution of the game with interaction matrix  n (Γ + Θ , A), such that ni=1 x˜ i < i=1 xi . To proceed, we select a perturbation which only affects interaction between interior agents, and which is related to the profile Y as follows. Consider a real number ϵ > 0 satisfying the following two conditions: xi + ϵ yi > 0,

for all i ∈ I  (γij + θij )(xj + ϵ yj ), ai <

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 Q T Y = 1 if and only if, for all Y ∈ Rk such that QY ≥ 0, n we have i=1 yi ≥ 0. 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 prove that if

 T

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. 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 .

187

(4)

(9) for all i ∈ C .

(10)

j̸=i

Such a positive number ϵ exists since there is no knife-edge agent (indeed, condition (10) is verified for ϵ = 0 with strict inequality). Define the n × n matrix Θ = [θij ] such that for all i, j ∈ I , θij = k

−ϵ[ΓI Y ]i k

i=1 xi +ϵ

i=1 yi

(that is, the sub-matrix ΘI has uniform

lines), while θij = 0 if at least one agent among i or j is not interior. Basically, condition (9) guarantees that Θ ≤ 0. By construction, we have

ΘI (XI + ϵ · Y ) = −ϵ · ΓI Y .

(11)

Define X˜ such that x˜ i = 0 if xi = 0 and X˜ I = XI + ϵ · Y (equivalently −ϵ Y = XI − X˜ I ). The Eq. (11) writes therefore

ΘI X˜ I = ΓI (XI − X˜ I ).

(12)

That is, X˜ satisfies 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 naïve best-response dynamics goes back to the original equilibrium. 9 A linear system is asymptotically stable when the real parts of all its eigenvalues are positive.

(ΓI + ΘI )X˜ I = AI .

(13)

Hence, X˜ 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, since

188

M. Belhaj, F. Deroïan / Journal of Mathematical Economics 49 (2013) 183–188

X˜ I − XI = ϵ · Y , the inequality

k

i=1 xi , and thus

˜i < i =1 x

n

n

k

i =1

i =1 x i .

yi < 0 implies

k

i=1

x˜ i <



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 = γ1 , and let H be a k × k matrix ii

γ

with, for all i, j ̸= i, hii = 0 and hij = γji . Define function f such ii that, for any Y ∈ Rk , f (Y ) = D − HY . Since  ΓI(X ) satisfiesCondition 

(C2), f is a continuous function from 0, γ1 × · · · × 0, γ1 to kk 11 itself; indeed, on the one hand, if, for all j ̸= i, we have yj ≥ 0, clearly yi ≤ γ1 ; on the other hand, if yj = γ1 for all j ̸= i, ii

then yi ≥ 0 if γ1 − γ1 ii ii

j=k

j=1 j̸=i

γji γjj

jj

≥ 0, which is exactly Condition

(C2). We conclude that function f admits a fixed point (by Brouwer Fixed Point Theorem), that is, there is a nonnegative solution to Z = D − HZ (since the null vector is not a solution, the solution contains at least one positive component). To finish, we note that Z = D − HZ if and only if (ΓI (X ) )T Z = 1I (X ) . 

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