Within the context of games on networks S. Goyal (Goyal (2007), pg. 39) posed the following problem. Under any arbitrary but fixed topology, does there exist at least one pure Nash equilibrium that exhibits a positive relation between the cardinality of a player's set of neighbors and its utility payoff? In this paper we present a class of games/topologies in which pure Nash equilibria with the above property do not exist.Social sharing of information, Game theory, Social network

Ali Kakhbod

Demosthenis Teneketzis

English

Research Papers in Economics

     
 
 
  
  
Volume 31, Issue 3 
  
Games on Social Networks: On a Problem Posed by Goyal 
  
 
 
Ali Kakhbod  
University of Michigan, Ann Arbor 
Demosthenis Teneketzis  
University of Michigan, Ann Arbor
Abstract 
Within the context of games on networks S. Goyal (Goyal (2007), pg. 39) posed the following problem. Under any 
arbitrary but fixed topology, does there exist at least one pure Nash equilibrium that exhibits a positive relation between 
the cardinality of a player's set of neighbors and its utility payoff? In this paper we present a class of games/topologies 
in which pure Nash equilibria with the above property do not exist.
 
Citation: Ali Kakhbod and Demosthenis Teneketzis, (2011) ''Games on Social Networks: On a Problem Posed by Goyal'', Economics 
Bulletin, Vol. 31 no.3 pp. 2177-2184. 
Submitted: Jul 14 2010.   Published: July 27, 2011. 
 
     Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
1 Introduction-Motivation
Games on social networks is a rapidly developing discipline the importance of which
has been extensively discussed in recent publications Goyal (2007), Jackson (2008)
and Vega-Redondo (2007). One of the goals of this discipline is the development of
a framework within which the eect of the structure of relationships on individual
behavior and well-being as well as on aggregate outcomes can be examined system-
atically.
The social sharing of information (Bramoulle and Kranton (2007)) which is pri-
vately costly to collect is one of the economic contexts within which the eect of
network structure on individual behavior and aggregate outcomes are currently be-
ing studied. Within this context Goyal posed a question that addresses the eect of
ones's neighbors on his/her payo at equilibrium. In this paper we provide an answer
to Goyal's problem.
The paper is organized as follows: In section 2 we present the social network model
and Goyal's problem. In section 3 we present preliminary results that are essential in
determining a solution to Goyal's problem. In section 4 we present an example that
provides a negative answer to Goyal's problem. We conclude in section 5.
2 The Model and Goyal's Problem
We consider a set N := f1;2;3 ;Ng of players. These players are connected,
directly or indirectly, with one another through a network whose topology is denoted
by g; g is xed. Since g is xed the set Ni(g), i 2 N, of neighbors of player i is
well-dened; two players i;j are called neighbors if they are directly connected with
a link in the graph that corresponds to the topology g. Each player can choose a
strategy/eort si 2 S, where S is compact and convex in R+. Let s 2 S
N, s :=
(s1;s2; ;sN) denote a strategy prole. Given s 2 S, the payo of player i;i 2 N;
is given
i(sjg) = f
2
4si +
X
sj2Ni(g)
sj
3
5   csi; (1)
where c is the marginal cost of eort and it is a xed positive constant. The function
f is such that f(0) = 0;f0() > 0, f00() < 0; and there is a constant  such that
f0() = c. The function f and the constants c and  are common knowledge among
the players. Consider a game played by the N players, with strategy space and utility
functions described above. Assume for the moment that there exist Nash equilibria in
pure strategies, called pure Nash equilibria (PNE), for this game. Denote by PNEg
the set of PNE of the game when the network topology is g. Denote by PNE
+
g the
1
2178Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
subset of PNEg that have the following property:
PNE
+
g =
n
s
 2 PNEg : 8 i;j 2 N;jNi(g)j > jNj(g)j ) i(s
jg)  j(s
jg)
o
; (2)
where jNi(g)j denotes the cardinality of the set Ni(g). PNE
+
g is called the set of
PNEg that exhibit for every player i 2 N a positive relation between the cardinality
of Ni(g) and the player's utility/payo under g.
Let G denote the set of the all possible topologies of the N players' network. Goyal
(2007), pg. 39, posed the following problem:
Goyal's Problem: Under any arbitrary but xed topology g 2 G is the set PNE
+
g
nonempty?
One would expect a positive answer to the above question. In the model under
consideration the players' payo functions are identical. Thus, it is reasonable to
expect that players with a higher number of neighbors would earn a higher payo at
equilibrium. In this paper we provide a negative (thus, in our opinion counterintu-
itive) answer to Goyal's problem. We present, via an example, a class of topologies
for which PNE
+
g is empty.
3 Preliminaries
First we assert that for every topology g 2 G the corresponding game possesses PNE.
This assertion follows from the compactness of the space S, continuity of f() in the
players' strategies, the concavity of f() in each player's strategy, and a result of
Debreu, Glicksberg and Fan (Fudenberg and Tirole (1991) pg. 34).
For any g 2 G all PNEg can be characterized by the following result.
Theorem 1 (1, pg. 35) For a given g 2 G, a strategy prole s = (s
1;s
2; ;s
N) 2
PNEg, if and only if for every i 2 N
s

i =

  
P
j2Ni(g) s
j; if
P
j2Ni(g) s
j < ;
0 if
P
j2Ni(g) s
j  : (3)
The following lemma reveals a property of PNE
+
g. This property along with the result
of Theorem 1 are critical in establishing our answer to Goyal's problem.
Lemma 2 Let s := (s
1;s
2; ;s
N) 2 PNE
+
g . Then for any u;v 2 N, if jNu(g)j <
jNv(g)j and s
u = 0, then s
v = 0.
Proof: We prove the result by contradiction. Consider s 2 PNE
+
g. Let u;v 2
N, assume that jNu(g)j < jNv(g)j, s
u = 0, and suppose s
v = " > 0: Then, since
2
2179Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
s 2 PNEg, by Theorem 1 we must have
X
j2Nu(g)
s

j  : (4)
Furthermore, since " > 0, f is increasing, and
P
j2Nu(g) s
j  , it follows that
v(s
jg) = f()   c" < f
0
@
X
j2Nu(g)
s

j
1
A = u(s
jg); (5)
which contradicts the fact that s 2 PNE
+
g. Consequently, we must have
s

v = 0: (6)
2
Remark 3 Using the arguments in the proof of Lemma 2 we can also establish the
following results:
 Independently of an agent's degree, if s 2 PNEg and s
i < s
j then i(sjg) >
j(sjg). Equivalently, if at equilibrium agent i is no better o than agent j
then agent i contributes no more than agent j.
 Let s 2 PNEg be such that
P
k2Ni(g) s
k + s
i =  for all i 2 N: Then,
s

i  s

j , i(s
jg)  j(s
jg):
Using the results on the existence of PNE, Theorem 1 and Lemma 2, we present an
example that provides a negative answer to Goyal's problem.
4 An Example
Consider the network depicted in Figure 4, and denote it by g. Assume there exists
s 2 PNE
+
g. Restrict attention to node v2. We show that no matter what value
s
v2 takes, s can not be in PNE
+
g, thus PNE
+
g is empty. We consider six cases that
exhaust all possibilities.
 Case 1: s
v2 = 0.
Then s
v2 = 0 along with Lemma 2 imply that
s

v3 = s

v4 = s

v5 = 0; (7)
therefore,
s

v2 + s

v3 + s

v4 + s

v5 = 0 (8)
3
2180Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
Since by assumption s 2 PNE
+
g, s 2 PNEg. Therefore, s must satisfy the
condition of Theorem 1. This further implies that
s

v3 +
X
vj2Nv3(g)
s

vj = s

v3 + s

v2 + s

v4 + s

v5  : (9)
But (9) contradicts (7), thus we can not have any s 2 PNE
+
g with s
v2 = 0:
 Case 2: 0 < s
v2 <  and  := s
v4 + s
v5 = 0.
Consider node v6. Since jNv6(g)j = 5 > 4 = jNv5(g)j and s
v5 = 0, Lemma 2
implies that
s

v6 = 0: (10)
By the same argument, s
v6 = 0 implies that
s

v16 = s

v17 = s

v18 = s

v19 = 0; (11)
On the other hand, by Theorem 1 we must have
s

v6 +
X
vj2Nv6(g)
s

vj   (12)
Equation (12) contradicts Eq. (11).
 Case 3: 0 < s
v2 < , 0 <  := s
v4 + s
v5 <  and 0 <  + s
v2 < .
Since s 2 PNE
+
g, by Theorem 1 we must have
s

v3 + s

v4 + s

v5 + s

v2 = s

v3 +  + s

v2  : (13)
Furthermore, by assumption
 + s

v2 < : (14)
Then Eqs. (13), (14) and Theorem 1 imply that s
v3 =    ( + s
v2); or,
equivalently,
s

v2 + s

v3 =     < ; (15)
because  > 0:
On other other hand, Theorem 1 applied to node v2 gives
s

v2 + s

v3  : (16)
Equations (15) and (16) contradict one another.
 Case 4: 0 < s
v2 < , 0 <  := s
v4 + s
v5 <  and  + s
v2  .
4
2181Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
In this case, Theorem 1 applied to node v3 gives
s

v3 +
X
vj2Nv3(g)
s

vj   (17)
Inequality (17) combined with the assumption  + s
v2   result in
s

v3 = 0: (18)
Equation (18) and Lemma 2 in turn imply that
s

v5 = s

v6 = s

v7 = s

v8 = 0: (19)
On the other hand, by Theorem 1 we must have
s

v5 +
X
vj2Nv5(g)
svj  : (20)
which contradicts (19).
 Case 5: 0 < s
v2 < ,  := s
v4 + s
v5  .
The result of Theorem 1 for node v3 along with the above conditions imply
s

v3 = 0: (21)
Then, by Eq. (21) and Lemma 2 we must have
s

v4 = s

v5 = 0; (22)
which contradicts the assumption
s

v4 + s

v5  : (23)
 Case 6: s
v2  .
Then, the result of Theorem 1 for node v2 and s
v2   imply
s

v3 = 0: (24)
Equation (24) and lemma 2 result in
s

v5 = 0; (25)
which along with Lemma 2 imply that
s

v6 = s

v7 = s

v8 = 0: (26)
5
2182Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
v6
v7
v8
v9
v10
v11
v12
v13
v14
v15
v16
v17
v18
v19
v20
v21 v21
v22
v23
v24
v25
v26
v27
v28
v29
v30
v31
v32 v33
v34
v35
v36
v37
v38
v39
v40
v41
v42
v43
v44
v45
v46
v47
v48
v49
v50
v51
v52
v53
v54
v55
v56
v57
v58
v59
v60
v61
v62
v63
v64
v65
v66
v67
v68
v69
v70
v71
v72
v73
v74
v76 v75 v77 v78 v79
v80 v81 v82 v83 v84 v85 v86 v87 v88 v89
v90 v91 v92 v93 v94
v95
v96
v97
v98
v99
v100
v101
v102
v103
v104
v105
v106
v107
v108
v109
v110
v111
v112
v113
v114
v115
v116
v117
v118
v119
v120
v121
v122
v123
v124
v125
v126
v127
v128
v129
v130
v131
v132
v133
v134
v135v136v137 v139 v138
v140v141v142v143v144v145v146v147v148v149
v150v151v152v153v154
v2
v3
v4
v5
∆ := v4 + v5
Figure 1: g: An example
On the other hand, the result of Theorem 1 for node v5 requires that
s

v5 +
X
vj2Nv5(g)
svj   (27)
which contradicts (26).
The analysis of cases 1-6 demonstrates that no matter what value s
v2 takes, s = 2
PNE
+
g. Consequently, the set PNE
+
g is empty.
5 Conclusion/Discussion
Note that the example presented here is not necessarily minimal, but it introduces a
class of topologies (games) with nite tree structure such that
6
2183Economics Bulletin, 2011, Vol. 31 no.3 pp. 2177-2184
 (i) the degree of nodes (except the terminal/leaf nodes) increases monotonically
as their distance from the root of the tree increases.
 (ii) the depth of the tree is at least 5.
For any game on a network with topology g that belongs to the above described class,
the set PNE
+
g is empty. This can be proved by exactly the same arguments (Cases
1-6) as those presented in the tree structure of Section 4.
Furthermore, consider any topology/game that results in from arbitrary exten-
sions along the leaf nodes of the tree structure of Section 4. By the same arguments
as in Section 4 we can show that PNE
+
g is empty for those topologies/games.
The form of the utility function, namely the fact that the function f has all the
properties stated in the model and is the same for all players, is critical in establishing
a negative answer to Goyal's problem. The class of network topologies g for which
PNE
+
g is nonempty remains unknown.
References
Bramoulle, Y. and R. Kranton (2007), "Public Goods in Networks", Journal of Eco-
nomic Theory 135, 478-494.
Goyal, S. (2007) "Connections, An Introduction to the economics of Networks",
Princeton University Press.
Jackson, M. (2008) "Social and Economic Networks", Princeton University Press.
Vega-Redondo, F. (2007) "Complex Social Networks", Cambridge University Press.
Fudenberg, D. and Tirole, J. (1991) "Game Theory", MIT Press.
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Games on Social Networks: On a Problem Posed by Goyal

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