Competing Activities in Social Networks

Appendix A: proofs

Bilateral symmetric interaction games²¹ have a potential function (Monderer and Shapley 1996). A potential function associated with the game satisfies that, for all i, for all xi,xi′∈[0,1] and all x−i∈[0,1]n−1, F(xi,x−i)−F(xi′,x−i)=vi(xi,x−i)−vi(xi′,x−i) (see Monderer and Shapley 1996). The following function F(.) is a potential function associated with the game:

The shape of the potential function gives information about uniqueness and stability. In particular, critical points of the potential function are critical points of the system of first-order conditions (FOCs). Concerning stability, the above potential having no minimum,²² all critical points are Nash equilibria. It contains either maxima or saddle points. Maxima correspond to strict (local) concavity of the potential. The maxima of the potential are thus asymptotically stable. In opposite, saddle points of the potential are unstable.

Since the function is defined on a convex compact, global concavity, which occurs when (In−γG) is positive definite, guarantees a unique maximum and its stability. In particular, the matrix (In−γG) is positive definite if and only if γ<1μ(G).²³ More generally, the local stability of any equilibrium obtains under a similar condition related to the intensity of interaction between interior agents:

Condition 3 (Local stability)Consider a network G. An equilibrium X, including possibly specialized agents, is (locally) asymptotically stable whenγμ(GI(X))<1.

Three useful properties of games with complementarities follow. We say that the SBRA {Xt}t>0 crosses a configuration Y when there is some t0 and some t′>t0 such that Xt0<Y and Xt′>Y. The following two simple properties, related to supermodular games, hold:

Property 1Consider a Nash equilibrium X and a configurationX0, with eitherX0≤XorX0≥X. Then, a SBRA, starting atX0, does not cross X.

Next property is related to the direction of move of a SBRA:

Property 2Start a SBRA atX0. IfX1≥X0(resp. X1≤X0), thenXt+1≥Xt(resp. Xt+1≤Xt) for allt>0.

Properties 1 and 2 are used to prove the next property, related to existence of equilibria:

Property 3If there are two distinct equilibriaX,Ywith neitherX≥YnorY≥X, there exists one third equilibrium Z withZ≥max(X,Y).

Proof of property 3. Let profile Z0=max(X,Y). By property 1, since X and Y are equilibria, a SBRA starting at X0=Z0 cannot decrease at first step. Using property 2, the SBRA increases at any step. Then, the process converges to a new equilibrium that covers both X,Y and their cover Z0. ∎

Properties 1 and 2 are also used to prove the following lemma, which is key to the analysis:

Lemma 1Consider the linear system(In−γG)X=K, whereki>0for all i. There is a solutionX≥0to the system (equivalently, the invert matrix(In−γG)−1exists and is non-negative) if and only ifγμ(G)<1. Moreover, ifX≥0, X≥K.

Proof of lemma 1.

If. If γμ(G)<1, the matrix (In−γG)−1 is non-negative and can be written (In−γG)−1=∑k=0∞(γG)k. It follows that X=∑k=0∞(γG)kK, and thus X≥K.

Only if. If γμ(G)≥1, the series ∑k=0∞(γG)kK diverges. Start a SBRA at X0=0. Then X1=K>0, meaning that the process increases for all at first step. Continuing the process, property 2 ensures that Xt increases at each step, and since the series diverges, the process goes to infinity. Now, assume that there is a solution X≥0 to the system. The series Xt crosses X, this contradicts property 1. ⃞

A direct and useful implication of lemma 1 reads as follows. Let G be non-negative and γμ(G)<1. If (In−γG)Z≥0, then Z≥0.

Proof of theorem 1.

(i) We prove that, for anyγ>0, region(E)contains a unique equilibrium, X∗.

•Step 1: a SBRA starting atX0=xe1goes upward.

Indeed, xi1=x0−γxedi+γ∑jgijxj0. Taking X0=xe1, we find X1=x01. Since m>0, we have X1>X0; and by property 2 we are done.

•Step 2: there is an equilibrium.

This is a basic result from supermodular games (see Milgrom and Roberts (1990) for instance). Indeed, property 2, combined with the fact that efforts are bounded above guarantees the existence of an equilibrium.

•Step 3: the equilibrium in region (E) is unique.

Second, consider two distinct equilibria X,Y. By property 3, we can assume X>Y without loss of generality. Since Y lies in region (E), we have Y>>0, and thus I(X)⊊I(Y).

Step 3.1: we have(I|I(Y)|−γGI(Y))(X−Y)I(Y)≤0.

For convenience, let di1(Y) (resp. di1(X)) denote the number of agent i’s neighbors that set effort to 1 in profile Y (resp. profile X). Clearly, di1(Y)≤di1(X) for all i. We first compute (I|I(Y)|−γGI(Y))YI(Y). For all i, we obtain

Then we compute (I|I(Y)|−γGI(Y))(X−Y)I(Y). Two cases can arise

• • Case 1: agent i∈I(X). We have

Moreover, as i∈I(X), we also have

Combining eqs [10] and [11], we obtain that

In total, from eqs [9] and [12], we find that for every agent i in I(X),

• • Case 2: agent i∈I(Y)∖I(X). We have

Thus, combining eqs [9] and [14], we get

Furthermore, the first-order condition imposes

From eq. [15] and inequality [16], we get that, for every agent i in I(Y)∖I(X),

and we are done.

Step 3.2: we haveγμ(GI(Y))<1.

Let Z=(Y−xe1)|I(Y)|. Note that Z>0 as Y lies in region (V). The linear system that Y solves can be written as:

where Di1(Y) is the sum of neighbors of agent i being at 1 in configuration Y. We are now ready to apply lemma 1, with ki=m+γ(1−xe)di1(G). Indeed, we have both m+γ(1−xe)di1(G)>0 for all i, and Z>0. Lemma 1 implies that γμ(GI(Y))<1.

Step 3.3: it holds thatX≤Y(a contradiction).

Given that γμ(GI(Y))<1, we have X≤Y by direct application of lemma 1.

1. We prove that the equilibriumX∗is locally stable and its basin of attraction encompasses region(V). From stage 3.2 in the preceding proof, we know that γμ(GI(Y))<1. Thus the condition 3 holds, which guarantees local stability. The size of the basin of attraction contains region (V) by standard arguments of supermodular games. Indeed, consider a SBRA {Xt} starting from X0 in region (V). It is easily shown that for all t≥1, Xt≥x01. Since the equilibrium is unique in region (E), the SBRA converges to X∗.

2. We show that the efforts of interior agents in activity 1 are characterized by the Bonacich centrality of the network restricted to interior agentsGI. Consider the variable substitution ti=xi−xem. Agent i’s first-order condition becomes

   [18]ti∗−∑j∈I(X,G)gijtj∗=1+γ(1−xe)mdi1(X∗,G)

   That is, if H=1+(1−xe)γD1(X∗,G), ti∗=bH,i(GI;γ). Hence, xi∗=x0+m(bH,i(GI;γ)−1).

3. We show that, for anyγ<γc, there is a unique equilibrium, and that it is globally stable. We will first consider the case γμ(G)<1, and, second, the case γ<γf.

• • Case 1: we suppose that γμ(G)<1. The point follows from the following two lemmas:

Lemma 2Let G be non-negative andγμ(G)<1. For any vectors A and Y (they could have negative entries):

Lemma 3Whenγμ(G)<1, any equilibrium Y satisfiesY≥x01.

Proof of lemma 3. Given that γμ(G)<1, we have that (I−γG)−1≥0. Let Z(Y) be the set of agents such that their efforts are less than 1 (they could be zero). Any agent i∈Z(Y) satisfies

By lemma 2, we obtain

Given that x0<1, we have x0GZ(Y)1Z(Y)+DZ(Y)1≥x0GZ(Y)1Z(Y)+x0DZ(Y)1=x0DZ(Y). Recalling that (I−γG)−1≥0, we obtain

where the last inequality obtains because m≥0, (I−γG)−1≥0 and DZ(Y)≥0. We conclude that yi∗≥x0>0 for all i. ∎

From lemma 3, we have that any equilibrium belongs to region (E). Now, by statement (i), this equilibrium is unique.

• Case 2: we suppose that γ<γf. Consider two equilibria X,Y. Following stage 3 of the proof of proposition 1, it is without loss of generality to assume that X>Y with I(X)⊂I(Y) with strict inclusion. Then, we find that (I|I(Y)|−γGI(Y))(X−Y)I(Y)≤0 (we omit the proof, which is identical to stage 3.1 of the proof of proposition 1). Now, remind first that R(γ)>>0, i.e. the first equation in system [3] admits a positive right-hand side, and second that γμ(GI(Y))<1. We can therefore apply lemma 1 (replacing Z with Y and G with GI(Y)), which implies that (X−Y)I(Y)≤0, a contradiction.

The global stability for γ<γc follows directly from uniqueness. Indeed, in supermodular games, a large class of adaptive processes (including SBRA) converge to a value in-between the minimal and the maximal equilibrium (see Milgrom and Roberts 1990, 1270). ∎

Proof of theorem 2.

We show that every stable equilibrium, distinct fromX∗, contains at least one agent specialized in activity 2. Suppose the existence of a stable equilibrium Y distinct from X∗, which does not contain a 0’s. We know that Y does not lie in region (V). Since X∗ is the maximal equilibrium, we have Y<X∗. Simple computation entails (I|I(Y)|−γGI(Y))(X∗−Y)I(Y)≤0 (see the proof of theorem 1). Moreover, by condition 3, local stability requires that γμ(GI(Y))<1. Applying then lemma 1, we get that X∗≤Y, a contradiction.

We prove that whenγ≥γ0, a stable equilibrium distinct from0contains at least one agent specialized in activity 1. Assume that γ≥γ0. We know that a stable equilibrium X distinct from X∗ contains some 0’s. Suppose that it does not contain any 1’s. Since X does not contain agents at 1, we have

Again, by condition 3, stability implies γμ(GI(X))<1. Then, lemma 1 entails XI(X)≤0, a contradiction. ∎

Proof of proposition 2. Consider any stable equilibrium X with interior agents, i.e. I(X)≠∅. The first-order conditions given in eq. [3] restricted to the subset of interior agents are written

We differentiate eq. [19] with respect to parameter c2:

Since 1≥X, the right-hand side of eq. [20] is non-negative. Since X is stable, we have γμ(GI(X))<1. Lemma 1 applies and the sign of ∂XI(X)∂c1 is positive. That is, a decrease of the cost of activity 2 leads to an decrease of all efforts in activity 1.

We differentiate now eq. [19] with respect to parameter c1. We obtain

The right-hand side of eq. [21] is thus negative. Using lemma 1 again, an increase of the cost of activity 1 leads to an decrease of all efforts in activity 1. ∎

Appendix B: basic properties of the elasticities of Bonacich centralities with respect to the intensity of interaction

The previous analysis suggests that the elasticity of Bonacich centralities with respect to decay is crucial to assess how the network reacts to a policy affecting activity costs. In this appendix, we examine some of their basic properties.

We have ɛi(G;γ)=γbi(G;γ)−1∂bi(G;γ)∂γ. Let us denote di,q=[GqJ]i, for all q≥1. Basically,