Games on Networks Yves Zenou (Stockholm University and IFN) Whos - - PowerPoint PPT Presentation

Games on Networks Yves Zenou (Stockholm University and IFN)

Who’s Who in Networks. Wanted: the Key Player

Coralio Ballester, Antoni Calvó- Armengol and Yves Zenou Econometrica 2006

When σij > 0, an increase in the effort xj of agent j creates an incentive for i to increase his level of activity

• xi. We then talk of strategic complementarity in efforts.

When σij < 0, instead, an extra effort from j triggers a downards shift in i’s effort in response. We say that efforts are strategic substitutes.

The General Model

• We can decompose bilateral influences like

where G represents a network of local complementarities, 0 ≤ gij ≤ 1

∑ 

Net Self- Substitutability  −I Global Substitutability  −U Local Complementarity  G

I is the n−square identity matrix and U the n−square

matrix of ones.

Σ = − βI − γU + λG

with β > 0, γ ≥ 0 and λ > 0. The pattern of bilateral influences results from the com- bination of an idiosyncratic effect, a global interaction effect, and a local interdependence component. The idiosyncratic effect reflects (part of) the concavity of the payoff function in own efforts. The global interaction effect is uniform across all players (matrix U) and corresponds to a substitutability effect across all pairs of players with value −γ ≤ 0. The local interaction component captures the (relative) strategic complementarity in efforts that varies across pairs of players, with maximal strength λ and population pattern reflected by G.

The decomposition is depicted in Figure 1. This is just a centralization (β and λ are defined with respect to γ) followed by a normalization (the gijs are in [0, 1]) of the cross effects. The figure in the upper panel corresponds to σij of either sign (the case σij ≤ 0, for all i 6= j is similar) while the figure in the lower panel corresponds to σij ≥ 0, for all i 6= j (recall that we assume σ < 0).

β λ λgij γ σij σ σ σ β λ γ = 0 λgij σij σ σ σ

The General Model

• Three players

∑ 

−6 1/2 −1 1/2 −6 1 −1 1 −6 u1x 1,x 2,x 3  x 1 − 3x 1

2  1

2 x 1x 2 − x 1x 3 u2x 1,x 2,x 3  x 2 − 3x 2

2  1

2 x 1x 2  x 2x 3 u3x 1,x 2,x 3  x 3 − 3x 3

2 − x 1x 2  x 1x 3

1 2 3

∑ 

−

 −5 1 0 0 0 1 0 0 0 1

−

 −1 1 1 1 1 1 1 1 1 1



 2 3/4 0 3/4 1 1

Explanation of this example: Σ =

⎛ ⎜ ⎝

−6 1/2 −1 1/2 −6 1 −1 1 −6

⎞ ⎟ ⎠

σ = min{σij} = −1 and σ = max{σij} = 1 OBS: σ and σ do not include σii = σ. σii = σ = −6 γ = − min{σ, 0} = − min{−1, 0} = 1 λ = σ + γ = 1 + γ = 2

gij = σij + γ λ but 0 ≤ gij ≤ 1 Thus

G =

⎛ ⎜ ⎜ ⎜ ⎝

max

n−6+1

2

, 0

• = 0

1/2+1 2

= 3

4 −1+1 2

= 0

1/2+1 2

= 3

4

max

n−6+1

2

, 0

• = 0

1+1 2

= 1

−1+1 2

= 0

1+1 2

= 1 max

n−6+1

2

, 0

• = 0

⎞ ⎟ ⎟ ⎟ ⎠

As a result

Σ = −βI − γU + λG

= −5 × I − 1 × U + 2 × G

In our example, we have: α = 1 , γ = 1 , λ = 2 , β = 5 Since for all i = 1, 2, 3 ui = αxi − 1 2 (β − γ) x2

i − γ 3

X

j=1

xixj + λ

3

X

j=1

gijxixj we have: u1 = αx1 − 1 2 (β − γ) x2

1 − γ 3

X

j=1

x1xj + λ

3

X

j=1

g1jx1xj = x1 − 1 24 x2

1 − x1x1 − x1x2 − x1x3 + 23

4x1x2 = x1 − 3x2

1 + 1

2x1x2 − x1x3 Similarly u2 = x2 − 3x2

2 + 1

2x1x2 + x2x3 u3 = x3 − 3x2

3 − x1x2 + x1x3

The network Bonacich centrality To each network g, we associate its adjacency matrix G = [gij]. Symmetric zero-diagonal square matrix that keeps track of the direct connections in g. The kth power Gk = G(k times) ...

G of the adjacency matrix G keeps

track of indirect connections in g. The coefficient g[k]

ij in the (i, j) cell of Gk gives the number of paths of

length k in g between i and j.

Example Network g with three individuals (star)

t t t

2 1 3 Figure 1

Adjacency matrix :

G =

⎡ ⎢ ⎣

0 1 1 1 0 0 1 0 0

⎤ ⎥ ⎦

G2k =

⎡ ⎢ ⎣

2k 2k−1 2k−1 2k−1 2k−1

⎤ ⎥ ⎦

and

G2k+1 =

⎡ ⎢ ⎣

2k 2k 2k 2k

⎤ ⎥ ⎦ , k ≥ 1

G3 =

⎡ ⎢ ⎣

0 2 2 2 0 0 2 0 0

⎤ ⎥ ⎦

G3: two paths of length three between 1 and 2: 12 → 21 → 12 and

12 → 23 → 32. no path of length three from i to i

For all integer k, define: bk

i (g) = n

X

j=1

g[k]

ij

This is the sum of all paths of length k in g starting from i. Next, let φ ≥ 0, and define: bi(g,φ) =

+∞

X

k=0

φkbk

i (g)

This is the sum of all paths in g starting from i, where paths of length k are weighted by the geometrically decaying factor φk.

For φ small enough, this infinite sum takes on a finite value.

b(g, φ) =

+∞

X

k=0

φkGk · 1 = [I−φG]−1 · 1, (2) where 1 is the vector of ones.

b(g, φ) Bonacich network centrality of parameter φ in g .

bi(g,φ) as the Bonacich centrality of agent i in g. To each agent, it associates a value that counts the total number of direct and indirect (weighted) paths in the network stemming from this agent.

Example Consider the network g in Figure 1.

t t t

2 1 3 Figure 1

When φ is small enough, the vector of Bonacich network centralities is:

b (g, φ) =

⎡ ⎢ ⎣

b1 (g, φ) b2 (g, φ) b3 (g, φ)

⎤ ⎥ ⎦ =

1 1 − 2φ2

⎡ ⎢ ⎣

1 + 2φ 1 + φ 1 + φ

⎤ ⎥ ⎦ .

The Bonacich centrality of node i is bi(g, a) = Pn

j=1 mij(g, a),

and counts the total number of paths in g starting from i. It is the sum of all loops starting from i and ending at i, and all outer paths that connect i to every other player j 6= i: bi(g, a) = mii(g, a)

| {z }

self−loops

+

X

j6=i

mij(g, a)

| {z }

• ut−paths

. Note that, by definition, mii(g, a) ≥ 1, and thus bi(g, a) ≥ 1.

Example 2. Consider the network g in Figure 1.

t t t

2 1 3 Figure 1

Largest eigenvalue: 21/2. When d

³

21/2´ < c, the unique Nash equilib- rium is: x∗

1 = a c + 2d

c2 − 2d2 and x∗

2 = x∗ 3 = a c + d

c2 − 2d2.

Dyads No social interactions. Then, the utility of each agent i would be given by: ui(xi) = αxi − 1 2x2

i

The unique symmetric equilibrium is: x∗

ni = α

Now, in order to understand the general model and to see the role of λ and γ, let us take the simplest possible network, that is n = 2 and each player has a link with the other, that is g12 = g21 = 1. The adjacency matrix

G =

Ã

0 1 1 0

!

Two eigenvalues: 1, −1. Thus µ1(G) = 1. The network locations in g are interchangeable. In this case, the utility is now given by: ui(x1, x2) = αxi − 1 2x2

i − γ

³

x2

i + xixj

´

+ λxixj where 0 ≤ γ < 1. Compared to our utility function β = 1 + γ.

The first order condition are: ∂ui ∂xi = α − (1 + 2γ) xi − (γ − λ) xj = 0 Since we have a dyad, the unique symmetric equilibrium is given by: x∗ = α 1 − λ + 3γ Observe that since β = 1 + γ and µ1(G) = 1, β > λµ1(G) ⇔ λ < 1 + γ Guarantees this solution to be strictly positive.

Check with Theorem

x∗ =

α β + γb(g, λ/β)b(g, λ/β) Here

b(g, a) =

"

I−λ

βG

#−1

· 1 =

"Ã

1 0 0 1

!

− λ β

Ã

0 1 1 0

!#−1 Ã

1 1

!

=

Ã

1 −λ/β −λ/β 1

!−1 Ã

1 1

!

=

⎛ ⎜ ⎝

β2 β2−λ2 λβ β2−λ2 λβ β2−λ2 β2 β2−λ2

⎞ ⎟ ⎠ Ã

1 1

!

=

⎛ ⎝

β β−λ β β−λ

⎞ ⎠

Thus

Ã

x∗

1

x∗

2

!

= α β + γb(g, λ/β)

⎛ ⎝

β β−λ β β−λ

⎞ ⎠

where b(g, λ/β) = b1(g, λ/β) + b2(g, λ/β) = 2β β − λ We have

Ã

x∗

1

x∗

2

!

=

Ã

α β−λ+2γ α β−λ+2γ

!

Now since β = 1 + γ, we have:

Ã

x∗

1

x∗

2

!

=

Ã

α 1−λ+3γ α 1−λ+3γ

!

Suppose first that γ = 0, i.e. there is no global substitu-

• ability. We obtain

x∗ = α 1 − λ In the dyad, agents rip complementarities from their part- ner, and choose an effort level above the optimal value for an isolated agent (x∗ = α). The factor 1/(1 − λ) > 1 is often referred to as the social multiplier. Suppose now that λ = 0. We obtain x∗ = α 1 + 3γ Equilibrium efforts are decreasing in γ. Indeed, global substituabilities add to the idiosyncratic concavity in one’s efforts, an exhaust decreasing marginal returns below the

• ptimal value for an isolated agent. The general expres-

sion results from a combination of both effects.

In words, the denser the pattern of local complementari- ties, the higher the aggregate outcome, as players can rip more complementarities in g0 than in g. The geometric intuition for this result is clear. Recall that b(g, λ∗) counts the total number of weighted paths in g. This is obviously an increasing function in g (for the inclusion ordering), as more links imply more such paths.

Application 1 : Crime networks

There are n criminals, each exerting a level of crime xi that results from a trade off between the costs and ben- efits of criminal activities. The expected utility of criminal i is: ui(x, r) = yi(x) − pi(x, r)f, (1) yi(x) are the proceeds, pi(x, r) the apprehension proba- bility, and f the corresponding fine. The cost of committing crime pi(x, r)f increases with xi, as the apprehension probability increases with one’s involvement in crime, hitherto, with one’s exposure to deterrence.

Also, and consistent with standard criminology theories, criminals improve illegal practice through interactions with their direct criminal mates. Formally, criminals are connected through a friendship network r, where rij = 1 when i and j are directly related to each other. For instance, let:

⎧ ⎨ ⎩

yi (x) = xi

h

1 − η Pn

j=1 xj

i

pi(x, r) = p0xi

h

1 − ν Pn

j=1 rijxj

i .

The expected utility then becomes: ui(x, r) = (1−π)xi−η

n

X

j=1

xixj+πν

n

X

j=1

rijxixj, (2) where π = p0f is the marginal expected punishment cost for an isolated criminal, and −η < 0 captures a congestion in the crime market. The utility function (2) coincides with the expression our general utility with α = 1 − π, β = γ = η, λ = πν and

g = r.

When πνµ1(r) < η, the unique Nash equilibrium of the crime game with payoffs (2) is:

x∗ = 1 − π

η 1 1 + b(r, πν

η)b(r, πν

η).

Application 2 : R&D collaboration networks

Consider a standard Cournot game with n (ex ante) iden- tical firms, each of them choosing the quantity qi. As in Goyal and Moraga-González (2001) and Goyal and Joshi (2003), firms can form bilateral agreements to jointly invest in cost-reducing R&D activities. We set rij = 1 when firms i and j set up a collaboration link. Firm i’s marginal cost is λ0 − λ P

j6=i rijqj. Here, λ0 >

0, represents the marginal cost of an isolated firm, while λ > 0 is the cost reduction induced by each link it forms. With a linear inverse demand, the profit function of firm i is:

ui(x, r) =

⎡ ⎣φ −

n

X

j=1

qj

⎤ ⎦ qi − ⎡ ⎣λ0 − λ X

j6=i

rijqj

⎤ ⎦ qi

(3) = (φ − λ0) qi −

n

X

j=1

qiqj + λ

X

j6=i

rijqiqj. Again, this objective function is a particular case of our general utility, where α = φ − λ0 > 0, β = γ = 1 and

g = r. We conclude that the Cournot game with payoffs

(3) has a unique Nash equilibrium in pure strategies:

q∗ =

φ − λ0 1 + b(r, λ)b(r, λ), when 1 > λ√g + n − 1. In particular, the comparative statics Theorem implies that the overall industry output increases when the net- work of collaboration links expands, irrespective of this network geometry and the number of additional links.

For the case of a linear inverse demand curve, this gener- alizes the findings in Goyal and Moraga-González (2001) and Goyal and Joshi (2003), where monotonicity of indus- try output is established for the case of regular collabo- ration networks, where each firm forms the same number

• f bilateral agreements.

For such regular networks, links are added as multiples of n, as all firms’ connections are increased simultaneously.

Application 3 : Conformism and social norms

Each individual has a utility that depends on the dif- ference between her behavior and that of her reference group. Each individual chooses an action xi ≥ 0 and loses utility when failing to conform to the social norm of her refer- ence group, which is equal to the average action of its members. The network N = {1, . . . , n} is a finite set of indi-

• viduals. Individuals are connected by a network of social

connections. We represent social connections by a graph/network f. To any network f, we can associate its adjacency matrix, that we denote by F.

The coefficients of the matrix F are the fijs, 1 ≤ i, j ≤

• n. When i and j are friends we set fij = 1. Let also

fii = 0 for all i. Thus, by definition, each cell in F takes

• n values zero or one. Given our convention that fii = 0,

the diagonal of F consists on zeros. Since fij = fji, the matrix F is symmetric. Each player i has fi = Pn

j=1 fij direct links in f, and

thus the average action of her friends, that is the action

• f her reference group, is given by:

xi = 1 fi

n

X

j=1,j6=i

fijxj

Preferences Each individual i = 1, ..., n selects an ef- fort/action xi ≥ 0, and obtains a payoff ui(x, f), given by the following utility function, with ξ, α, θ, d > 0: ui(x, f) = ξ + αxi − θx2

i − d(xi − xi)2

(4) The utility function (4) is such that each individual wants to minimize the social distance between herself and her reference group, where d is the parameter describing the taste for conformity. Indeed, the individual loses utility d(xi−xi)2 from failing to conform to others. The average action of the reference group of agent i, xi, explicitly depends on the underlying network structure, and thus each agent has a different x depending on her location in the network.

Bilateral influences of this utility function. ∂2ui(x, f) ∂xi∂xj =

⎧ ⎪ ⎨ ⎪ ⎩

−2(θ + d), when i = j 0, when i 6= j and fij = 0 2d/fi > 0, when i 6= j and fij = 1 . Since, when i 6= j, σij > 0, an increase in effort from j triggers a downwards shift in i’s response and thus efforts are strategic complements from i’s perspective within the pair (i, j). This utility function (4) thus coincides with ui = αxi − 1 2 (β − γ) x2

i − γ n

X

j=1

xixj + λ

n

X

j=1

gijxixj with β = 2(θ + d), γ = 0, λ = 2d and gij = fij/fi. Note that g is a row-normalization of the initial friendship network f, as illustrated in the following example, where

F and G are the adjacency matrices of, respectively, f

and g.

Example 1 Consider the following friendship network f:

t t t

2 1 3

Then,

F =

⎡ ⎢ ⎣

0 1 1 1 0 0 1 0 0

⎤ ⎥ ⎦

and

G =

⎡ ⎢ ⎣

0 1/2 1/2 1 1

⎤ ⎥ ⎦

Observe that G is a stochastic matrix, that is gij ≥ 0 and

P

j gij = 1. This implies that µ1(G) = 1 and Gk is also

a stochastic matrix, that is g[k]

ij

≥ 0 and P

j g[k] ij

= 1, ∀k. Applying our Theorem, it is easy to see that this con- formity game with payoffs (4) has a unique Nash equi- librium in pure strategies and, whatever the structure of the network, this equilibrium is always symmetric, that is x∗ = x∗

1 = ... = x∗ n and x∗ = x∗ 1 = ... = x∗ n, and is

given by: x∗ = x∗ = α 2θ (5)

In particular, the equilibrium Bonacich-centrality measure is the same for all individuals and is equal to: b1(g, d θ + d) = ... = bn(g, d θ + d) = θ + d θ To prove this result, one has to calculate the Bonacich vector since it is the only source of heterogeneity between

• players. In a conformist game, we have:

bi(g, a) = mii(g, a) +

X

j6=i

mij(g, a) = a

n

X

j=1

gij + ... + ak

n

X

j=1

g[k]

ij + ...

=

+∞

X

j=1

ak = 1 1 − a The equilibrium value (5) is exactly the value found by Akerlof (1997), page 1010.

Even if individuals are ex ante heterogeneous because of their location in the network, in a conformist equilibrium where each individual would like to conform as much as possible to the norm of her reference group, all individuals will exert the same effort level. The distribution of population does not matter in equi- librium even if it matters ex ante. The only relevant statistics is the average.

The Bonacich centrality of player i counts the number of paths in g that stem from i. The intercentrality counts the total number of such paths that hit i; It is the sum of i’s Bonacich centrality and i’s contribu- tion to every other player’s Bonacich centrality. Holding bi(g,a) fixed, ci(g,a) decreases with the propor- tion of i’s Bonacich centrality due to self-loops mii(g,a)/bi(g,a).

Ex ante Heterogeneity in Games on Networks

More general network game with linear quadratic payoffs. N = {1, . . . , n} is a finite set of agents. Each agent i ∈ N selects zi ≥ 0. Payoffs are: ui(z) = αizi + 1 2σiiz2

i +

X

j6=i

σijzizj.

Let α = (α1, ..., αn) and Σ = [σij]. Game Γ (α, Σ) with players in N such that α > 0 (that is, αi > 0, for all i ∈ N) and σii < min{0, min{σij : j 6= i}}, for all i ∈ N. We further assume that σii = σ11, for all i ∈ N. This is without loss

• f generality.

Indeed, let D = diag(1, σ11/σ22, ..., σ11/σnn). This is a diagonal ma- trix with a strictly positive diagonal. It is readily checked that the Nash equilibria of Γ (α, Σ) and that of Γ (Dα, DΣ) coincide, where the di- agonal terms of DΣ are all equal to σ11. Let I be the identity matrix and J the matrix of ones.

Additive decomposition of the interaction matrix:

Σ = −βI − γJ + λG.

Own-concavity effects −βI Global substitutability effects −γJ Local (network) complementarity effects +λG. Following this decomposition, payoffs can now be rewritten as: ui(z) = αizi − 1 2 (β − γ) z2

i − γ n

X

j=1

zizj + λ

n

X

j=1

gijzizj

Definition 0.1 Given a vector u ∈ Rn

+, and a ≥ 0 a small enough

scalar, we define the vector of u-weighted centrality of parameter a in the network g as:

wu (g, a) =

³

I − aG−1´ u =

+∞

X

p=0

apGpu. Katz-Bonacich centrality b (g, a) corresponds to the u-weighted cen- trality with u = 1 (where 1 is the vector of ones)

Denote by ω (G) the largest eigenvalue of G. For all vector u ∈ Rn, let u = u1 + ... + un. Theorem 0.1 Consider a game Γ (α, Σ) with α > 0 and Σ is decom- posed additively. (a) Suppose first that α = α1. Then, Γ (α, Σ) has a unique Nash equilibrium in pure strategies if and only if β > λω (G). This equilibrium

z∗ is interior and given by: z∗ =

α β + γw1 (g, λ/β)w1 (g, λ/β) . (1)

(b) Suppose now that α 6= α1. Let αmax = max {αi | i ∈ N} and αmin = min{αi | i ∈ N}, with αmax > αmin > 0. If β > λω (G) + nγ (αmax/αmin − 1), then Γ (α, Σ) has a unique Nash equilibrium in pure strategies z∗, which is interior and given by:

z∗ = 1

β

"

wα (g, λ/β) −

γwα (g, λ/β) β + γw1 (g, λ/β)w1 (g, λ/β)

#

. (2)

When α = α1, the equilibrium existence, uniqueness (and interiority) condition is independent of γ, the global level of substitutabilities, and

• nly depends on the own concavity term β and the network of local

complementarities λG. For general α, a necessary and sufficient condition for equilibrium ex- istence and uniqueness is that −Σ has all its principal minors strictly positive, that is, −Σ is a P−matrix in the language of the linear com- plementarity problem. The P−matrix condition does not guarantee that the equilibrium is inte-

• rior. Besides, the P−matrix property is computationally very demanding

and economically nonintuitive.

This motivates the sufficient condition in Theorem 0.1(b), which is de- rived from that in Theorem 0.1(a), but imposes a more stringent re- quirement on β, λ, G as the right-hand side of the inequality is now augmented by nγ (αmax/αmin − 1) ≥ 0. In words, everything else equal, the higher the discrepancy αmax/αmin of marginal payoffs at the origin, the lower the level of network complemen- tarities λω (G) compatible with a unique and interior Nash equilibrium. To summarize, the condition in Theorem 0.1(b) bounds local comple- mentarities λω (G), global substitutabilities γ and marginal payoff dif- ferences αmax/αmin such that players have no incentives to increase their effort level without bound, neither to free-ride on their network peers by decreasing them down to zero. A unique and interior equilibrium is then achieved.

The next example illustrates Theorem 0.1. When n = 2, symmetric cross effects correspond either to substitutabil- ity or to complementarity, but not both. Formally, γλ = 0. We analyze the cases γ = 0 and λ = 0 separately.

General utility for player i: ui(z) = αizi−1 2 (β − γ) z2

i −γ n

X

j=1

zizj+λ

n

X

j=1

gijzizj For n = 2 and players 1 and 2 connected (dyad), we have: u1(z) = α1z1 − 1 2 (β − γ) z2

1 − γz1z2 + λz1z2

u2(z) = α2z2 − 1 2 (β − γ) z2

2 − γz1z2 + λz1z2

Case γ = 0 (no global substituability): u1(z) = α1z1 − 1 2βz2

1 + λz1z2

u2(z) = α2z2 − 1 2βz2

2 + λz1z2

Thus ∂u2

i (z)

∂z2

i

= −β for i = 1, 2 ∂u2

i (z)

∂zizj = λ for i = 1, 2 Thus the interaction matrix is: Σ =

Ã

−β λ λ −β

!

FOC: ∂u1(z) ∂z1 = α1 − βz1 + λz2 = 0 ∂u2(z) ∂z2 = α2 − βz2 + λz1 = 0

• r

βz1 − λz2 = α1 βz2 − λz1 = α2 In matrix form:

Ã

β −λ −λ β

! Ã

z1 z2

!

=

Ã

α1 α2

!

Example with n = 2 and γ = 0 The interaction matrix is:

Σ =

"

β −λ −λ β

#

. The equilibrium existence and uniqueness necessary and sufficient con- dition in Theorem 0.1(a) becomes β > λ.

Indeed, equilibrium conditions for an interior equilibrium (i.e., zero mar- ginal payoffs for both players) are:

"

β −λ −λ β

# "

y1 y2

#

=

"

α1 α2

#

. It is readily checked that this system has a unique positive solution if and only if β > λ, given by:

"

z∗

1

z∗

2

#

= 1 β2 − λ2

"

βα1 + λα2 λα1 + βα2

#

, which corresponds to (2) when γ = 0 and

G =

"

0 1 1 0

#

.

Case λ = 0 (no local interactions): u1(z) = α1z1 − 1 2 (β − γ) z2

1 − γz1z2

u2(z) = α2z2 − 1 2 (β − γ) z2

2 − γz1z2

Thus ∂u2

i (z)

∂z2

i

= − (β − γ) for i = 1, 2 ∂u2

i (z)

∂zizj = −γ for i = 1, 2 Thus the interaction matrix is: −Σ =

Ã

β − γ γ γ β − γ

!

FOC: ∂u1(z) ∂z1 = α1 − (β − γ) z1 − γz2 = 0 ∂u2(z) ∂z2 = α2 − (β − γ) z2 − γz1 = 0

• r

(β − γ) z1 + γz2 = α1 γz1 + (β − γ) z2 = α2 In matrix form:

Ã

β − γ γ γ β − γ

! Ã

z1 z2

!

=

Ã

α1 α2

!

Solution is:

Ã

z1 z2

!

=

⎛ ⎜ ⎝

α1β(β−2γ)−γ[α2(β−γ)−γα1] β(β−2γ)(β−γ) α2(β−γ)−γα1 β(β−2γ)

⎞ ⎟ ⎠

Equilibrium existence, uniqueness and interiority α1 α2 < β − γ γ

Sufficient condition from Theorem: β > λω (G) + nγ (α/α − 1) Here λ = 0, n = 2, and α1 > α2: α1 α2 < β + 2γ 2γ

Example with n = 2 and λ = 0 The interaction matrix is now:

Σ =

"

β + γ γ γ β + γ

#

. When α1 = α2, the equilibrium condition in Theorem 0.1(a) is trivially satisfied. Suppose that α1 > α2. The sufficient condition for equilibrium exis- tence, uniqueness and interiority in Theorem 0.1(b) is (β + 2γ) /2γ > α1/α2. Instead, we show that the equilibrium existence, uniqueness and interi-

• rity is obtained here if and only if α2 (β + γ) /γ > α1/α2, highlighting

the fact that Theorem 0.1(b) is only sufficient but not necessary.

Beyond this simple example with only n = 2 players, we believe that the fact that the condition in Theorem 0.1(b) is too stringent is compensated by its full generality and economic appeal. An effort profile z∗ =

³

z∗

1, z∗ 2

´

∈ R2

+ is a pure strategy Nash equilibrium

• f Γ (α, Σ) if and only if:

∂ui ∂zi (z∗) = 0, for all i = 1, 2 such that z∗

i > 0

∂ui ∂zi (z∗) ≤ 0, for all i = 1, 2 such that z∗

i = 0.

Notice that marginal payoffs are: ∂ui ∂zi (z) = Σz. The equilibrium conditions thus boil down to a system of inequalities.

Straight algebra leads to the following equilibrium characterization, when α1 > α2:

z∗ =

⎧ ⎨ ⎩ ³ α1

β+γ, 0

´

, if (β + γ) /γ > α1/α2

1 (β+γ)2−γ2 [(β + γ) α1 − γα2, −γα1 + (β + γ) α2] , otherwise .

The case when α1/α2 ≥ (β + γ) /γ corresponds to (2) for λ = 0. The next figure shows the regions for corner and interior equilibria. The dashed line corresponds to the sufficient condition in Theorem 0.1(b).

α1 α2

(z1*,z2*)>(0,0) (z1*,0) (0,z2*)

(β+γ)/γ=α1/α2 (β+2γ)/2γ=α1/α2 (β+γ)/γ=α2/α1 (β+2γ)/2γ=α2/α1