WHEN DOES EVERYONE CONTRIBUTE IN THE PRIVATE
PROVISION OF LOCAL PUBLIC GOODS?
GUANG-ZHEN SUN
University of Macau
Abstract
When does everyone genuinely contribute in the private
provision of a local public good? We first introduce a monotonic condition to characterize the relationship between the
structure of the network that underlie the noncooperative
game of private provision of local public goods on the one
hand, and the preferences of the agents on the other, showing that the monotonic condition is a sufficient and necessary condition of existence of a distributed Nash equilibrium (DNE) in which each agent exerts a positive amount of
effort to provision of the public good (Theorem 1). We then
study the number of equilibria, and, by using the monotonic
condition, characterize the condition under which the DNE
set is a singleton, a continuum, or null (Theorem 2). As it
turns out, the structure of the network and the agents’ preferences jointly shape the effort profile in the provision of
local public goods.
1. Introduction
Ever since Bergstrom, Blume, and Varian’s (1986) seminal study of the private provision of public goods, there has emerged a large literature on this
fundamental issue in public economics (see, e.g., the special issue of the
Guang-Zhen Sun, Department of Economics, Faculty of Social Sciences and Humanities,
University of Macau, Macau Special Administrative Region, China ([email protected]). The
author would like to thank the two referees and the associate editor for their insightful and
constructive comments on earlier versions of the paper, and to gratefully acknowledge the
support from the Research Council of University of Macau (RG 096/09 – 10S/SGZ/FSH),
and the Logan Fellowship from Monash University, Australia as well, as part of this research
was done when the author held the Senior Logan Fellow at Monash University. The usual
disclaimer applies.
Received March 9, 2009; Accepted February 28, 2011.
C
2012 Wiley Periodicals, Inc.
Journal of Public Economic Theory, 14 (6), 2012, pp. 911–925.
911
912
Journal of Public Economic Theory
Journal of Public Economics, 2007, 91(7), celebrating the 20th anniversary of
publication of the paper), and a few studies characterizing the equilibrium
existence and uniqueness in particular (see, e.g., Bergstrom, Blume, and
Varian 1992, Fraser 1992, Cornes, Hartley, and Sandler 1999, Kotchen 2007).
However the postulate adopted in these studies that the public good is
global, that is, that the effort made by anyone toward the supply of the public
good benefits all members of the economy, appears too restrictive. For instance, technological innovations often spill over from innovating firm(s) to
other firms, but such innovation diffusion is often local and hence dictated
in a great measure by social and/or geographical networks in the industry.
Localness of information sharing, in turn, informs the incentive to provide
the public good. Such examples of local public goods (LPG) abound indeed.
In addition to local diffusion of innovations (see, e.g., Jaffe, Trajtenberg, and
Henderson 1993 for empirical evidence), localness of public goods is also
found in R&D activities (Goyal and Moraga-Gonzalez 2001, Andersson and
Ejermo 2005), information sharing among limited agents in the labor market (Granovetter 1995, Calvó-Armengol and Jackson 2004), and public goods
supply in ethnic communities (Dasgupta and Kanbur 2003).
The localness of public goods under the various circumstances has at
least two important implications for the effort each individual exerts in providing the public goods. First, the localness of consumption of a public good
implies that its provision, on the supply side, is decentralized, because each
agent is only concerned about local circumstances when deciding on how
much if any to contribute. The economy in which the game of private provision of the public good is played is thus essentially made up of local communities that are of course interrelated with one another, despite that dividing
lines between the communities can be drawn only in a rather loose manner;
that is, the provision of the public good is decentralized. As a consequence,
the localness provides the agents greater incentive than in the game of a
global public good to exert effort toward the provision of the public good. It
becomes possible that under certain condition(s) everyone has incentive to
make a genuine contribution (i.e., everyone exerts a strictly positive amount
of effort) to the provision of LPG. In contrast, in the game of a global public
good there exists a unique set of contributors of the public good (Theorem
3 in Bergstrom et al. 1986) and the rest free ride, and in the particular case of
identical preference across all the agents each contributing agent allocates
all his resource above a certain level to the provision of the public good and
the rest free ride (Bergstrom et al. 1986, Theorem 5).
Second, related to decentralization of the provision of the public good,
any combination of compatible supplies in all the interconnected local communities constitutes an equilibrium for the game played in the entire economy, hence multiple equilibria being rendered possible. In particular, decentralization of the provision renders it possible for that in certain games
there exist multiple equilibria, in each of which everyone genuinely contributes. To see this, one may consider the provision of a public good in a
When Does Everyone Contribute to a Public Good?
913
three-agent economy, wherein agent B is linked to both agents A and C but
A and C are not linked with each other. The preference of all the agents is
the same and described by the utility function ui = xi Gi , where Gi is the (local) public good accessible to agent i and xi the amount of the private good
he consumes, i = A, B, C. The efforts of the neighboring agents’ efforts are
perfectly substitute. For simplicity, the production technology of the public
good is postulated as being described by an identity function of the aggregate
supply of the neighboring agents’ efforts. Denote by g i the resource agent
i allocates to the provision of the public good, and hence GA = g A + g B ,
GB = g A + g B + g C , and GC = g√
B + g C . The endowed resources of the agents
A, B, and C are respectively 1, 2, and 1. It can be shown that for a global
public good game, in which GA =√GA = GC = g A + g B + g C , the unique Nash
equilibrium is (g A∗ , g B∗ , g C∗ ) = (0, 22 , 0), wherein agent B is the only contributor of the public good. In contrast,
in the LPG game,√there exists a con√
√
2
tinuum of equilibria (x, 2(1 − 2 − x), x), x ∈ [0, 1 − 22 ]. Apart from the
√
two equilibria (when x = 0, and x = 1 − 22 ), each of the rest is a profile of
strictly positive efforts by all the three agents.
In this paper, we characterize the condition under which everyone contributes in the provision of LPG in equilibrium, and study multiplicity of
the equilibria as well. In such games, intuitively, the further one is distanced
from the provider of the public good, the less likely one may benefit. Under
many circumstances, only those who are close enough to the public-goods
provider benefit. The well-known Marshallian knowledge spillover in an industrial cluster, as mentioned earlier, is often confined to the local community of firms. In order to precisely capture this local effect, we consider a
game of public good provision in a network, wherein the contributions to
the provision of the public good by the agents who are connected by links
are strategic substitutes.1 The public good provided as such is nonexcludable
and nonrivalrous only along links in the network. Such a network game of
LPG has been studied by Bramoullé and Kranton (2007) and Yuan (2005),
but with quite different focuses from ours. Bramoullé and Kranton (2007)
characterize the Nash equilibrium in which some agents are specialists in
providing LPG and all the neighbors of such an agent contributes nothing (free-riders), and which is thus termed as specialized Nash equilibrium
(SNE) by the authors. Making use of the notion and analysis of maximal independent sets (MIS) in graph theory,2 they show that any SNE corresponds
to one MIS of the graph underlying the network game, and consequently,
1
A R&D collaboration model wherein collaborating firms’ innovation efforts are strategically complementary is studied in Goyal and Moraga-Gonzalez (2001).
2
A collection of vertices in which no two elements are connected is called an independent
set of the graph, hence any MIS, an independent set that is not a proper subset of any
other independent set, induces a partition of all vertices of the graph: the MIS and its
complement (each element of which is connected to at least one vertex in the MIS).
914
Journal of Public Economic Theory
existence of MIS for any graph guarantees the existence of SNE in any network game of LPG (Bramoullé and Kranton 2007, Theorem 1). The authors
analyze welfare implications of SNE as well. Equilibrium private provision
of LPG in networks under strategic substitute between adjacent agents’ efforts is also examined by Yuan (2005). Using a few interesting examples, the
author demonstrates the possible multiplicity of Nash equilibria. Differing
from these two studies, this paper focuses on the condition under which
there exists a strictly positive effort profile in equilibrium (i.e., everyone genuinely contributes to the public good in a network), as well as the number of
equilibria, especially the number of the strictly positive effort equilibria. We
obtain two novel results. First, a sufficient and necessary condition of the existence of an equilibrium in which everyone contributes to the public good is
given. For that purpose, we introduce a monotonic condition to characterize
the relationship between the structure of the network underlying the game
of LPG on the one hand, and the preferences of the agents (reflected by
the minimal level of consumption of the LPG by each agent) on the other.
Second, we show that the set of equilibria in a LPG network game is either
finite, bounded from above by an exponential function of the number of the
agents (the order of the graph), or possessing a continuum of elements. In
particular, we study the set of those equilibria, in each of which every agent
contributes a positive amount of effort to LPG provision, showing that the
number of the so-called distributed equilibria (everyone contributes to LPG)
in any network game is either very small (unique or simply nonexisting), or
very large (a continuum).
The rest of the paper is organized as follows. The next section describes
the basic model of network games of LPG. Section 3 presents the main results
and Section 4 concludes.
2. The Model of γ -Degree LPG Game in Networks
Consider an undirected network of n vertices (agents). For any agent i,
she exerts a nonnegative effort, denoted as xi , in providing LPG, which
only benefits those who are no more than γ steps (γ ≥ 1) away. In the
special case that γ = 1, only the LPG provider’s neighbors (those linked
to her) are benefited. A public good in this very scenario may be termed
as an ordinary LPG. The efforts by agents are (perfect) substitutes across
the agents no more than γ steps away from one another. That is, the benefit one receives from LPG provision that is jointly produced by agents
is dependent on the sum of her own effort and the efforts of all those
linked to her by no more than γ steps, but independent of the efforts
of agents further away. For the sake of convenience, we use the maximal distance of the affected neighborhood of the public good to measure
the localness of the public good, and refer to the corresponding game of
private provision of such LPG in a network as a γ –degree LPG network
game.
When Does Everyone Contribute to a Public Good?
915
The graph supporting the network game is described by the graph’s adjacency matrix, A = [ai j ]n×n . For any pair of agents, i and j , ai j = a j i = 1 if
agents i and j are linked and ai j = a j i = 0 if unlinked. For reasons to be
seen shortly, we set aii = 1, i = 1, . . . , n, for the adjacency matrix. For the
network structure is assumed throughout to be exogenously given, that is,
no agent can add or sever a link with another, the adjacency matrix serves as
the basic structural parameter of the game. Based on the adjacency matrix
of the graph, we can now define a new matrix for the γ -degree LPG network
game on the graph,
γ
(s )
(ai j − e i j )
(1)
A(γ ) = [ai j (γ )]n×n , with ai j (γ ) ≡ sign ai j +
s =2
= j
, hence I ≡ [e i j ]n×n is the identity matrix of nwhere e i j = { 1,0,wheni
otherwise
dimension. The term (ai j − e i j )(s ) represents the (i, j ) entry of the
matrix (A − I )(s ) , s = 2, . . . , γ . The sign function is here defined as
0, whenx = 0
. Note aii (γ ) = 1, ∀i. By a long-known result in combisign(x) = { 1,
whenx > 0
natorial algebra, (ai j − e i j )(s ) equals the number of walks of length s starting
from vertex i and terminating at vertex j , s = 2, . . . , γ (see, e.g., Cvetković,
Doob, and Sachs 1980, p. 44). As such, ai j (γ ) = 1 if agent i is within the
γ − neighborhood of agent j. That is, if and only if agent i is connected to
agent j through no more than (γ − 1) other agents, ai j (γ ) = 1; otherwise,
ai j (γ ) = 0. For ease of expression, the matrix A(γ ) defined in Formula (1)
will be referred to as the γ -degree LPG matrix.3
For any agent i, i ∈ {1, . . . , n}, the marginal cost of effort in providing LPG is a constant, equaling c i .4 The monetary utility, designated
3
Alternatively, one may see A(γ ) as the adjacency matrix of an ordinary LPG game (i.e.,
γ = 1) in a network that is generated from the original underlying network with any pair
of agents who are no more than γ steps away from each other being linked. Such an interpretation is perfectly in line with the two theorems to be presented in the next section. But
this interpretation obscures the degree of localness of the public good, described by the
parameter γ , which, as is shown elsewhere (Sun 2010), plays a crucial part in determining
the set of equilibria.
4
In formulating the preference of the agent we assume quasi-linearity in the agent’s effort
cost, in line with the practice in the network game literature that deals with the “payoffs
depending on the sum of actions” (see, e.g., Bramoullé and Kranton 2007, p. 481, Galeotti
et al. 2010, p. 227). This considerably simplifies all the analysis, but admittedly at the cost
of assuming away convexity in private and public goods and hence rendering it possible
for multiple equilibria even in the conventional game of a global public good. Idealistically,
one should get to grips with a LPG network game for any convex preference (in consumption of the private and public goods) that is employed in the conventional public good
literature (e.g., Bergstrom et al. 1986). Unfortunately, we have not yet been able to obtain
analytical results along this line of inquiry and have to leave it for future study. On the
other hand, even for the case of quasi-linear preferences, as shown below, topologically
different networks accommodate quite different profiles of efforts toward the provision
916
Journal of Public Economic Theory
by ui , that agent i derives from LPG is a strictly concave and differentiable function of the aggregate efforts
exerted by all her γ -degree neigh
bors and herself. Thus, ui = ui ( { j |ai j (γ )=1} x j ), ∀i. To avoid the rather
trivial situation, we assume ui (0) > c i . Thus, for any given effort vec∗
T
tor
x = [x1 , . . . , xn ] , agent is optimal effort xi = arg maxz∈[0,∞) {ui (z +
{ j | j =i,ai j (γ )=1} x j ) − c i z}. But it follows from ui (0) > c i and the concavity of the utility function that there exists a unique value, denoted as y i ,
such that ui (y i ) = c i . For the sake of convenience in exposition, y i may
be referred to as the minimal or survival level of LPG of agent i. Apparently a nonnegative effort vector x = [x1 , · · · , xn ]T is Nash equilibrium
provision of LPG if and only if (A(γ )x)i ≥ y i for all i and (A(γ )x)i = y i
when xi > 0. It is easy to see that there always exists at least one Nash
equilibrium.5
LEMMA 1: There exists at least one equilibrium in the γ -degree LPG network
game.
3. The Main Results
The problem of (pure) free-riding, that is, the problem that some agents
do not make any contribution to the provision of public goods, is simply too typical of the games of private provision of a global public good,
whereas in network games of LPG, as analyzed in Section 1, localness of
the public goods implies decentralization of the provision, hence leaving
less room for free-riding and more room for possible equilibrium(s) other
than SNE. An interesting question to ask is then when everyone does contribute in the network game of LPG. We shall first characterize the condition under which there exists a distributed equilibrium, and then come
to grips with the problem of how “large” the equilibrium set, especially
the set of distributed equilibria, is. Before doing so, it is useful to formally
introduce6
of the public good, making a much richer set of equilibrium than that which is found in
the conventional (global) public good game. The author is indebted to an anonymous referee for the insightful observation on the possible limitation of the quasi-linear preference
postulate.
5
As is succinctly stated in Bramoullé and Kranton (2007, p. 483), the existence of Nash
equilibria in LPG games can be verified by invoking the standard fixed-point argument.
6
The author is grateful to an associate editor of this journal for suggesting possible use of
the term of Positive Effort Nash Equilibrium to reflect accurately what it really is. To call
the equilibrium in which everyone genuinely contributes as such is arguably more informative than all alternatives. To keep consistency of terms used in the literature, however,
we perhaps would better adopt distributed Nash equilibrium (DNE), a term that has been
used by other authors (Bramoullé and Kranton 2007, p. 482, for instance).
When Does Everyone Contribute to a Public Good?
917
DEFINITION 1: A Nash equilibrium x = [x1 , . . . , xn ]T is a distributed Nash
equilibrium (DNE) if for any i ∈ {1, . . . , n}, xi > 0.
We shall offer, in Theorem 1 below, a characterization of the condition
under which there exists at least one DNE. For the sake of illustration of that
theorem, we need to first introduce an index function as follows. For any i ∈
{1, 2, . . . , n}, let I ({i}) signify the collection of the indices of the agents who
benefit from the contribution of agent i, that is, I ({i}) = { j |ai j (γ ) = 1}. For
instance, for an ordinary LPG network game in a path network of n vertices
(agents), I ({i}) = {i − 1, i, i + 1} for i = 1 or n, I ({1}) = {1, 2} and I ({n}) =
{n − 1, n}. For any i τ ∈ {1, 2, . . . , n}, τ = 1, . . . , k, likewise I ({i 1 , i 2 , . . . , i k })
is defined as the collection of the indices of the agents who benefit from
the contribution of agent i 1 , agent i 2 , . . . , or agent i k , allowing for multiple occurrence of the index in such collection if the corresponding agent
happens to benefit from the contribution of more than one agent among
{i 1 , i 2 , . . . , i k }. Note that i 1 , i 2 , . . . , i k do not have to be different from one
another. We introduce a monotonicity condition using the index function.
The Monotonicity Condition (MC). For any i τ ∈ {1, 2, . . . , n}, τ =
1, . . . , k, and jμ ∈ {1, 2, . . . , n}, μ = 1, . . . , s , if I ({i 1 , i 2 , . . . , i k }) is a subset
of I ({ j1 , j2 , . . . , js }), then kτ =1 y iτ ≤ sμ=1 y jμ ; and if I ({i 1 , i 2 , . . . , i k }) is a
proper subset of I ({ j1 , j2 , . . . , js }), then kτ =1 y iτ < sμ=1 y jμ .
Let Ai be the ith column of the matrix A(γ ), that is, Ai ≡
[a1i (γ ), a2i (γ ), . . . , ani (γ )]T , i = 1, . . . , n. Then, based on the above analysis, MC may
nbe formally defined as follows: For any integers ki , i = 1, . . . , n,
ki Ai is nonnegative
(nonnegative
with at least one positive
such that i=1
n
n
ki y i ≥ 0 ( i=1
ki y i > 0). For the sake of forcomponent), it holds that i=1
mulation, however, it would be desirable if the above definition could be
generalized to any real numbers ki , i = 1, . . . , n. Apparently, if the condition
holds for any real vector K ≡ (k1 , . . . , kn )T , it holds true when the vector
is restricted to only rational or integer numbers. Now suppose the condition holds for integer vectors, and hence apparently for rational vectors as
well. We can then show that the condition
actually remains valid for real vecn
ki Ai ≥ 0, then there exist a sequence
tors too. For any real vector K , if i=1
(s )
(s )
of rational vectors K (s ) ≡ (k1 , . . . , kn ), s = 1, 2, . . . , such that K (s ) → K
n (s )
as s → ∞ and i=1 ki Ai is nonnegative for any s . Then, by the assumpn (s )
tion,
for any s , i=1
ki y i ≥ 0. By continuity, we obtain by letting s → ∞
n
k
y
≥
0.
Note
that the reasoning applies to the scenario in which
that
i=1 i i
n
k
A
is
nonnegative
with at least one positive component. We are thus
i=1 i i
led to the following formulation of MC:
DEFINITION
2 (MC): For any real numbers ki , i = 1, . . . , n, such that
n
is nonnegative (nonnegative
and at least one component is positive), it
i=1 ki Ai n
n
holds that i=1
ki y i ≥ 0 ( i=1
ki y i > 0).
918
Journal of Public Economic Theory
We are now ready to introduce
THEOREM 1: For the γ -degree LPG game in a network there exists at least one DNE
if and only if MC is satisfied.
Proof : The “only if” part. Since there exists x 0 such that A(γ )x = y ,
, in view of
where y = [y 1 , . . . , y n ]T for any real vector K ≡ (k1 , . . . , kn )T
n
ki AiT )x.
symmetry of the matrix A(γ ), we have K Ty = (K T A(γ ))x = ( i=1
n
Thus, as a consequence of the fact x 0, i=1 ki y i is nonnegative (positive)
n
ki Ai is nonnegative (nonnegative and at least one component is
when i=1
positive).
For the “if” part, we first establish the fact that (∃x ≥ 0)(A(γ )x = y ).
Suppose not, then by Farkas’s Lemma (see, e.g., Sydsæter, Strøm, and
Berck 2000, p. 96), (∃z)(A(γ )z ≥ 0)(z T y < 0). But that contradicts MC.
We then show that there actually exists a strictly positive equilibrium. Suppose not, that is, some components of x are zeros. When A(γ ) is nonsingular, that is, |A(γ )| = 0, it readily follows x = [A(γ )]−1 y . Without loss of
generality, assume x1 = 0. Denote byb i j the element of [A(γ)]−1 of the
n
n
b 1i y i = x1 = 0 and i=1
b 1i Ai =
position (i, j ), i, j = 1, . . . , n. Then, i=1
T
(1, 0, . . . , 0) , due to symmetry of the matrix A(γ ). But that contradicts
MC.
We now consider the case that |A(γ )| = 0. Suppose that some components of the nonzero vector x such that A(γ )x = y are zeros, and assume,
without loss of generality, that x1 = · · · = xl = 0, and ∀i > l, xi > 0. For the
sake of simplicity in notations, assume that each of the first s columns of the
matrix A(γ ) is not linearly independent of {As +1, · · · , Al , Al+1 , . . . , An } and
any A j ∈ {As +1, · · · , Al } is independent of {As +1, · · · , Al , Al+1 , . . . , An }\{A j }.
Let t = l − s . If t = 0, that is, any A j ∈ {A1, · · · , Al } is not independent of {Al+1 , . . . , An }, then there exist real numbers βl+1 , . . . , βn such
that A(γ )z = 0 where z ≡ (1, . . . , 1, βl+1 , . . . , βn )T . Let xε ≡ x + εz. Obviously for any real number ε, A(γ )xε = y and for sufficiently small and
positive ε, xε 0, that is, uncountably many DNE are obtained. On
the other hand, if s = 0, then A1 is independent of {A2 , . . . , An }. Consequently, r ank(A(γ )) = 1 + r ank([A2 A3 · · · An ]), hence the kernel (null
space) of A(γ ) is a proper subspace of the kernel of [A2 A3 · · · An ]T , that
is, ker(A(γ )) ⊂ ker([A2 A3 · · · An ]T ) and ker([A2 A3 · · · An ]T ) = ker(A(γ )),
from which it follows that there exists a n-dimensional vector τ such that
A(γ )τ = (1, 0, . . . 0)T . But τ T y = (τ T A(γ ))x = (A(γ )τ )T x = 0, contradicting MC.
Now turn to the situations in which s ≥ 1 and t ≥ 1. Since any
A j ∈ {As +1, · · · , As +t }, where s + t = l, is independent of {As +1, · · · , Al ,
Al+1 , . . . , An }\{A j }, similar analysis to that which is conducted for the case
of s = 0 above leads to the observation that there exist τ1 , . . . , τt such
that ATj τ j −s = 1, AkT τ j −s = 0, for any j ∈ {s + 1, · · · , s + t},k ∈ {s + 1, · · · , n},
j = k. Let αi j = AiT τ j , ∀i ∈ {1, · · · , s }, ∀ j ∈ {1, · · · , t}. Then
When Does Everyone Contribute to a Public Good?
⎤
⎡
A1T
α11 . . . α1t
⎢ .. ⎥
⎢ .. . . ..
⎢ . ⎥
⎢ .
. .
⎢ T ⎥
⎢
⎢ As ⎥
⎢ αs 1 · · · αs t
⎢ T ⎥
⎢
⎥
⎢A
⎢ 1 ··· 0
⎢ s +1 ⎥
⎢
⎢ .. ⎥
⎢
A(γ )[τ1 . . . τt ] = ⎢ . ⎥ [τ1 . . . τt ] = ⎢ ... . . . ...
⎥
⎢
⎢
⎢ AT ⎥
⎢ 0 ··· 1
⎢ s +t ⎥
⎢
⎢ AT ⎥
⎢ 0 ··· 0
⎢ l+1 ⎥
⎢
⎢ . ⎥
⎢ . . .
⎣ .. ⎦
⎣ .. . . ..
⎡
919
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(2)
0 ··· 0
AnT
Now consider the matrix à ≡ (αi j )s ×t . For any nonpositive vector b ∈
R t with at least one component being strictly negative, by Farkas’s Lemma,
either ∃μ ∈ R s , μ ≥ 0 such that ÃT μ = b, or ∃z ∈ R t such that Ãz ≥ 0 and
z T b < 0.
In the light of MC, however, it is impossible that for any ∀b ∈ R t , b ≤ 0
with at least one strictly negative component there exists a vector z ∈ R t such
that Ãz ≥ 0 and z T b < 0. To see that, we may consider the two convex sets,
one being all the nonpositive vectors in R t , that is, {b ∈ R t , b ≤ 0} and the
other the convex cone of the vectors {(αi1 , . . . , αit )T , i = 1, . . . , s }, namely
c one {(αi1 , . . . , αit )T , i = 1, . . . , s }. Note that no interior point of the former
belongs to the latter. For otherwise
negative
sthere exists a strictly
s vector in
σi (αi1 , . . . , αit )T , where i=1
σi = 1,
R t , denoted as b ∗ , such that b ∗ = i=1
and σi ≥ 0, ∀i ∈ {1, . . . , s }. Then, the fact that for some z ∈ R t , z T b < 0 requires that at least one of {(αi1 , . . . , αit )z, i = 1, . . . , s } is strictly negative,
hence it is impossible to have Ãz ≥ 0. Then, by the separation theorem of two
convex sets (refer to, e.g., Sydsæter et al. 2000, p. 80), there exists a hyperplane that separates {b ∈ R t , b ≤ 0} and c one {(αi1 , . . . , αit )T , i = 1, . . . , s }.
That is, there exists a nonzero vector β ∈ R t such that β T w ≥ 0 ≥ β T b,
∀w ∈ c one {(αi1 , . . . , αit )T , i = 1, . . . , s }, and ∀b ∈ R t , b ≤ 0. Apparently for
any b ≤ 0, β T b ≤ 0 requires that β ≥0. That is, there exists a nonzero vector β = (β1 , . . . , βt )T ≥ 0 such that tj =1 αi j β j ≥ 0, ∀i ∈ {1, · · · , s }. It then
follows, in view of Equation (2), that
⎡ t
⎤
j =1 α1 j β j
⎢
⎥
..
⎢
⎥
⎢ t .
⎥
⎢
⎥
α
β
j =1 s j j ⎥
⎢
⎛
⎞ ⎢
⎥
β
1
t
⎢
⎥
⎢
⎥
.
A(γ ) ⎝
βjτj⎠ = ⎢
..
⎥
⎢
⎥
j =1
⎢
⎥
βt
⎢
⎥
⎢
⎥
0
⎢
⎥
⎢
⎥
..
⎣
⎦
.
0
920
Journal of Public Economic Theory
100
102
5
3
3
Figure 1: Connections, LPG survival levels and the monotonic condition (MC).
is a nonzero nonnegative vector. However ( tj =1 β j τ j )T y =
(( tj =1 β j τ j )T A(γ ))x = (A(γ )( tj =1 β j τ j ))T x = 0, contradicting MC.
Now turn to the case that there exists some nonnegative
μ = (μ1 , . . . , μs )T such that ÃT μ = (b 1 , . . . , b t )T with b j ≤ 0, ∀ j ∈ {1, · · · , t}
and at least one of {b 1 , · · · , b t } is strictly negative and one of
(2), we obtain
{μ
1 ,s . . . , μsT} is strictly positive. In view of Equation
μi Ai )(τ1 , . . . , τt ) = (b 1 , . . . , b t ) = ( tj =1 b j AsT+ j )(τ1 , . . . , τt ),
( i=1
s
hence [( i=1
μi AiT ) − ( tj =1 b j AsT+ j )](τ1 , . . . , τt ) = 0. Since each of
not linearly independent
of {As +1, · · · , Al , Al+1 , . . . , An },
{A1, · · · , As } is s
μi AiT ) − ( tj =1 b j AsT+ j ) falls within the span of
the vector ( i=1
exist real numbers ϕs +1 , . . . , ϕn
{As +1, · · · , A
l , Al+1 , . . . , An },namely there s
n
T
such that ( i=1
μi AiT ) − ( tj =1 b j AsT+ j ) = i=s
+1 ϕi Ai . But ∀ j ∈ {1, · · · , t},
n
s
t
ϕs + j = ( i=s +1 ϕi AiT )τ j = [( i=1 μi AiT ) − ( j =1 b j AsT+ j )]τ j = 0.
Thus,
s
t
n
T
T
T
( i=1 μi Ai ) − ( j =1 b j As + j ) = i=l+1 ϕi Ai . Denote by z again the vector
(μ1 , . . . , μs , b 1 , . . . , b t , ϕl+1 , . . . , ϕn )T then, A(γ )z = 0. Let xε ≡ x + εz.
Obviously for any real number ε, A(γ )xε = y and for sufficiently small
and positive ε, xε is a nonnegative equilibrium with at most (l − 2) zero
components such that A(γ )xε = y . If necessary, repeat the reasoning for
the new equilibrium xε obtained as such until s or t equals zero. Our proof
for that there exists a strictly positive vector x such that A(γ )x = y is thus
completed.
It may be worth pointing out that although MC essentially links the connections of each individual agent (vertex) in the network to her survival level
of the public goods, this condition does so not in a context-free monotonic
manner; that is, when MC holds an agent with more connections does not
necessarily have a higher survival level than one with less connections. To
whom one may be connected also matters, in rendering MC valid, and hence
in guaranteeing existence of at least one DNE. To see this, consider the following example in which there are five agents, whose survival levels of the
public goods are found in Figure 1. The unique DNE in this LPG network
game is (99, 1, 2, 1, 1), that is, in the equilibrium agent one’s effort is 99,
agent two’s effort is 1, and so on. There is remarkable heterogeneity among
the agents’ survival levels of LPG, even among the neighboring agents (the
two agents in the middle of the figure) on the one hand. The two agents
whose LPG survival levels are much higher than the rest of the community
are somewhat “clustering” on the other, to ensure that one equilibrium be
guaranteed in which everyone genuinely contributes.
When Does Everyone Contribute to a Public Good?
921
Theorem 1 describes a sufficient and necessary condition for existence
of an equilibrium in which everyone contributes a strictly positive amount
of effort to LPG. Apparently for any DNE effort vector x, we must have
A(γ )x = y . A slightly more general equilibrium than DNE is that in which
all the components of x are nonnegative (i.e., some allowed to be zeros) and
A(γ )x = y holds. When does the equilibrium in which some agents free ride
(i.e., some components of the equilibrium x equal zeros), while A(γ )x = y
remains valid? As an immediate consequence of Theorem 1, we have
Corollary of Theorem 1. Under the condition that MC is satisfied, there
exists an equilibrium x, that is, A(γ )x = y , such that some components of x
are zeros if and only if the matrix A(γ ) is singular.
Proof: The “only if” part. Suppose A(γ ) is nonsingular. It can be seen from
the proof of Theorem 1 that MC implies that any equilibrium vector is strictly
positive, hence a contradiction.
The “if” part. It is already shown in the proof of Theorem 1, by invoking Farkas’s Lemma, that there exists an nonnegative x > 0, such that
A(γ )x = y . If some components of the vector x are zeros, we are done. Suppose otherwise. Since the matrix A(γ ) is singular, there exists a nonzero vector z such that A(γ )z = 0. Consider xε ≡ x + εz. It is then obvious that for
any real number ε, A(γ )xε = y , and that for some nonzero number ε, at least
one component of xε is zero and the rest are either positive or zeros.
Now turn to the multiplicity of equilibria in network games of LPG. We
posit
THEOREM 2:
(i) The collection of Nash equilibria in a LPG network game is either a finite
set, upper bounded by an exponential function of the size of the network, or a
continuum.
(ii) The DNE set is a singleton (if and only if MC holds and |A(γ )| = 0 ), a
continuum (if and only if MC holds and |A(γ )| = 0 ) or null (if and only if
MC does not hold).
Proof: (i) By Lemma 1, the collection of equilibria is nonempty. Suppose
(k) ∞
(k)
that there are countably many equilibria, {x
n }k=1 , that is, A(γ )x ≥
T
(k)
y ≡ [y 1 , . . . , y n ] ,∀k. Note for any k, x ∈ i=1 [0, y i ]. By the BolzanoWeierstrass Theorem (see, e.g., Vohra 2005, p. 4) we may, without loss of
generality, assume x (k) → x = [x1 , . . . , xn ]T as k → ∞. Thus, either (ia) for
any i xi > 0 or (ib) there is some i such that xi = 0.
In Case (ia), there must exist a sufficiently large number k̄ such that
for any k1 > k̄ and k2 > k̄, x (k1 ) 0 and x (k2 ) 0, which in turn imples
that A(γ )x (k1 ) = A(γ )x (k2 ) = y . Assume, without loss of generality, x (k1 ) =
x (k2 ) . Then, for any real number ρ ∈ (0, 1), A(γ )[ρx (k1 ) + (1 − ρ)x (k2 ) ] =
ρA(γ )x (k1 ) + (1 − ρ)A(γ )x (k2 ) = y , that is, γ x (k1 ) + (1 − γ )x (k2 ) is an
922
Journal of Public Economic Theory
equilibrium, there hence existing a continuum of equilibria, contradicting
that there are only countably many equilibria.
In Case (ib), let K0 ≡ {i|xi = 0} and K+ ≡ {i|xi > 0}. For a sufficiently
(k)
large number k̄, xi > 0 for any k > k̄ and any i ∈ K+ . For any i ∈ K0 we
(k)
(k)
have, of course, either xi > 0 or xi = 0. Thus, we obtain a partition of the
(k)
set {x }k>k̄ , consisting of at most 2#(K0 ) subsets, where #(K0 ) is the number
of elements contained in the index set of K0 . At least one of these subsets
contains two (infinitely many indeed) different points, of which any convex
combination is a Nash equilibrium. Again, a contradiction is obtained.
The analysis made for Case (ib) can also be used to estimate the upper
bound of the number of equilibria when the latter is finite. One may classify all the equilibria into (2n − 1) categories, depending on whether each
component of the equilibrium is zero or positive (it is impossible that an
equilibrium is a zero vector). If there exist two equilibria in any category,
then any convex combination of the two equilibria must be a Nash equilibrium as well. That is, there exist a continuum of equilibria, hence a contradiction. We can thus conclude that the set of equilibria is either a continuum,
or finite and upper bounded by (2n − 1).7
(ii) It suffices, in the light of Theorem 1, to observe that if there exist
more than two DNE then any convex combination of these two DNE is also
a DNE, and therefore there is a continuum of DNE.
Unlike SNE, which always exists in LPG games in networks, the set of
DNE may or may not be null, and when it is not null it is either a singleton
or a continuum. If A(γ ) is nonsingular, then either there exists a unique
DNE when [A(γ )]−1 y 0 and x ∗ = [A(γ )]−1 y is the DNE, or no DNE exists
when at least one component of [A(γ )]−1 y is nonpositive (MC does not hold
under this circumstance). On the other hand, when A(γ ) is singular, the
DNE set is either a continuum (when MC holds), or null (when MC does not
hold). That is, the size of the DNE set is jointly determined by the topological
structure of the network and the degree of localness of the public good on
the one hand (characterized by the matrix A(γ )), and the distribution of the
agents’ survival levels of LPG on the other (described by the vector y ).
Networks of different structure vary in exploiting the benefits of the
public goods. Apparently a LPG game in a complete network turns out
to be a conventional game of a global pubic good. The same is true for
7
Among all the equilibria, one may be particularly interested in the upper bound of SNE.
Since any SNE corresponds to a MIS in the graph underlying the LPG network game
(Bramoullé and Kranton 2007, Theorem 1 and Proposition 4), the number of SNE is no
more than the number of MIS. In answering a question raised by Erdös, Moon and Moser
(1965) prove that the maximal number of MIS of a graph with n vertices equals 3n/3 when
n = 0(mod 3), 4 × 3(n−1)/3 when n = 1(mod 3), and 2 × 3(n−2)/3 when n = 2(mod 3), and
identify the particular graphs that attain the above numbers of MIS. Thus, the number
of SNE in a network of order n is upper bounded by 3n/3 . A generalization of Moon and
Moser’s result for large networks is made by Füredi (1987).
When Does Everyone Contribute to a Public Good?
4
7
923
4
Figure 2: A LPG game in a path (also a bipartite).
γ -degree LPG games in a nontrivial complete bipartite network Kn1 ,n2 (n1 ≥ 2
or n2 ≥ 2) when γ ≥ 2. In ordinary LPG games (γ = 1), however, it is not so,
for there then exists at most one DNE for the game in bipartite networks, including the star network Kn,1 (n ≥ 2) wherein one central agent is connected
to each of the other n agents, which are unconnected with one another.8
The structural difference of the DNE sets across networks of different densities of links may merit a bit further discussion. When the links
(connections) between agents become sufficiently dense, the network in
which the LPG game is played is functionally close to, if not becomes de
facto, a complete graph. For instance, bipartite graphs contain not only some
“structural holes” (Goyal and Vega-Redondo 2007, Buskens and van de Rijt
2008), but also fairly dense links (especially so for the balanced bipartite).
For any γ ≥ 2, the γ − degree LPG network game in a bipartite graph turns
out to be a game of public goods in a de facto complete graph. For the sake
of illustration, we may consider the simple case of a three-agent LPG game
in a path (also a bipartite graph), in which the survival level of LPG of the
middle agent is 7 and that for the other two agents are both 4 (see Figure 2).
The unique DNE is x ∗ = (3, 1, 3).9 The middle agent exploits the rent of the
relational position of a structural hole. In the γ -degree LPG games for any
γ ≥ 2, however, the game effectively turns out to be one in a complete network, with the structural hole having disappeared, and the only equilibrium
is a SNE, (0,7,0) in which only the middle agent contributes.
4. Concluding Remarks
We examine in this paper the condition under which everyone really contributes to the provision of a LPG. For that purpose we consider a formal
model of what we call γ − degree LPG network game and introduce a monotonic condition characterizing the relationship between the γ − degree LPG
matrix A(γ ), which reflects the localness of benefits of the public good and
the social/geographical links among the agents, and the survival LPG vector y , which reflects the agents’ preference profile. We show that a DNE
8
Making use of a long known formula in spectral graph theory (Cvetković et al. 1980, p.
72), we can obtain that the spectrum of the ordinary LPG matrix in a bipartite graph
√
√
Kn1 ,n2 , Sp e c (A(Kn1 ,n2 )) = { n1 n2 + 1, − n1 n2 + 1, 1(with multiplicity(n1 + n2 − 2))}. Obviously, 0 ∈
/ Sp e c (A(Kn1 ,n2 )). Thus the DNE set is either null or a singleton, according to
Theorem 2.
9
It may be noticed that the unique DNE is not a convex combination of the SNEs, (4, 0, 4)
and (0, 7, 0).
924
Journal of Public Economic Theory
(in which everyone contributes to the LPG) exists if and only if the monotonic condition holds. We also study the set of equilibria in LPG network
games, especially the DNE set, showing that the latter is strikingly simple, either very small (a singleton or null), or very many (a continuum). Given the
preferences of the agents, the structure of the network (represented by its
adjacency matrix) and the localness of the public good (described by the parameter γ ), combined together into the γ − degree LPG matrix A(γ ), shape
the effort profiles in LPG provision.
That the network structure affects the distribution and the number of
equilibria may merit a few more remarks. As is briefly analyzed in the preceding section, a complete network, as well as a complete bipartite network
for LPG games, permits at most one DNE, and often does not possess any
DNE at all. The reason is that in the complete network the public goods
game turns out to be one of effectively global public goods, and therefore
free-riding behavior prevails, resulting in some agents contributing and the
others free riding. In networks with far less dense links than the complete
network the public goods are only locally public (nonexcludable only along
links). Consequently, the provision of public goods is decentralized. Whether
or not the topological structure of the graph that supports the LPG network
game allows for decentralization of the provision is thus crucial to existence
and possible multiplicity of DNE. For instance, apart from the numerical
example of a game in a path of order three analyzed in Section 1, we may
consider the ordinary LPG network game in a circle of any order n. Suppose
the efforts of two agents, say agents 1 and 2, for some reason, slightly deviate
from their equilibrium levels. Then their neighbors (agents 3 and n), may
accordingly adjust their efforts to counter balance the effects of such deviations for the local communities each comprising three successive agents,
namely {agents n, 1, 2} and {agents 1, 2, 3}. Apparently, to make a counterbalance, the adjustment made in the effort of agent 3 must be equal to that
of agent n. By the same token, to neutralize the “disturbance” in the local
communities {agents 2, 3, 4} and {agents 3, 4, 5} the adjustment in the effort
of agent 4 (5) equals the deviation in the effort of agent 1 (2), and so on.
Clearly, when, and only when n(mod 3) = 0, it is possible for the effect of the
deviation to be locally neutralized as such. That is, only if n(mod 3) = 0 is it
possible that there exists a continuum of DNE. A more complete analysis of
how the DNE set is informed by the topology of the network is found in Sun
(2010).
References
ANDERSSON, M., and O. EJERMO (2005) How does accessibility to knowledge
sources affect the innovativeness of corporations? Evidence from Sweden, The
Annals of Regional Science 39, 741 – 765.
BERGSTROM, T., L. BLUME, and H. VARIAN (1986) On the private provision of
public goods, Journal of Public Economics 29, 25 – 49.
When Does Everyone Contribute to a Public Good?
925
BERGSTROM, T., L. BLUME, and H. VARIAN (1992) Uniqueness of Nash equilibrium in private provision of public goods: An improved proof, Journal of Public
Economics 49, 391 – 392.
BRAMOULLÉ, Y., and R. KRANTON (2007) Public goods in networks, Journal of Economic Theory 135, 478 – 494.
BUSKENS, V., and A. VAN DE RIJT (2008) Dynamics of networks if everyone strives
for structural holes, American Journal of Sociology 114, 371 – 407.
CALVÓ-ARMENGOL, A., and M. O. JACKSON (2004) The effects of social networks
on employment and inequality, American Economic Review 94, 426 – 454.
CORNES, R., R. HARTLEY, and T. SANDLER (1999) Equilibrium existence and
uniqueness in public good models: An alternative proof via contraction, Journal
of Public Economic Theory 1, 499 – 509.
CVETKOVIĆ, D. M., M. DOOB, and H. SACHS (1980) Spectra of Graphs: Theory and
Application. New York: Academic Press.
DASGUPTA, I., and R. KANBUR (2003) Bridging communal divides: Separation, patronage and Integration, Mimeo, Cornell University.
FRASER, C. D. (1992) The uniqueness of Nash equilibrium in the private provision
of public goods: an alternative proof. Journal of Public Economics 49, 389 – 390.
FüREDI, Z. (1987) The number of maximal independent sets in connected graphs,
Journal of Graph Theory 11, 463 – 470.
GALEOTTI, A., S. GOYAL, M. O. JACKSON, F. VEGA-REDONDO, and L. YARIV
(2010) Network games, Review of Economic Studies 77, 218 – 244.
GOYAL, S., and F. VEGA-REDONDO (2007) Structural holes in social networks, Journal of Economic Theory 127, 460 – 492.
GOYAL, S., and J. L. MORAGA-GONZALEZ (2001) R&D networks, Rand Journal of
Economics 32, 686 – 707.
GRANOVETTER, M. S. (1995) Getting a Job. Chicago: University of Chicago Press.
JAFFE, A., M. TRAJTENBERG, and R. HENDERSON (1993) Geographic localization
of knowledge spillovers as evidenced by patent citations, Quarterly Journal of Economics 108, 577 – 598.
KOTCHEN, M. J. (2007) Equilibrium existence and uniqueness in impure public
good models, Economics Letters 97, 91 – 96.
MOON, J. W., and L. MOSER (1965) On cliques in graphs, Israel Journal of Mathematics
39(1), 23 – 28.
SUN, G.-Z. (2010) The number of equilibria in some network games of local public
goods, Mimeo, University of Macau.
SYDSÆTER, K., A. STRØM, and P. BERCK (2000) Economists’ Mathematical Manual,
3rd ed. Berlin: Springer.
VOHRA, R. V. (2005) Advanced Mathematical Economics. London: Routledge.
YUAN, K. (2005) Local public goods and private equilibrium provision, Mimeo,
National Taiwan University.