Recently a variety of link-based stability concepts have emerged in the literature on game theoretic models of social network formation.We investigate two basic formation properties that establish equivalence between some well known types of stable networks and their natural extensions.These properties can be identified as convexity conditions on the network payoff structures.social networks;network formation;pairwise stability

Gilles, R.P.

Sarangi, S.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2005–84 
 
 
STABLE NETWORKS AND CONVEX PAYOFFS 
 
By Robert P. Gilles, Sudipta Sarangi 
 
July 2005 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Stable Networks and Convex Payoffs∗
Robert P. Gilles† Sudipta Sarangi‡
July 2005
Abstract
Recently a variety of link-based stability concepts have emerged in the literature
on game theoretic models of social network formation. We investigate two ba-
sic formation properties that establish equivalence between some well known
types of stable networks and their natural extensions. These properties can be
identiﬁed as convexity conditions on the network payoff structures.
Keywords: Social networks; network formation; pairwise stability.
JEL classiﬁcation: C72, C79.
1 Of stability and efﬁciency in network formation
Link-based stability is founded on the actions undertaken by individual decision
makers—usually called “players”—with regard to the creation and breaking of links.
Under the requirement of consent in link formation (Jackson and Wolinsky 1996,
Gilles, Chakrabarti, and Sarangi 2005), a link can only be created if both players
involved agree. On the other hand, links can be deleted by decisions to do so by
individual players, i.e., deletion of links is accomplished without consent. Link-based
stability requires that no player wants to delete one or more of her links to other play-
ers, and there is no pair of players that would like to establish a new link between
them. Variations in the precise formalization of link addition and link deletion has
resulted in a variety of link-based stability concepts in the literature.
∗We would like to thank Hans Haller and Tim Chakrabarti for elaborate discussions on the subject
of this paper and related work.
†Address: Department of Economics, Virginia Tech (0316), Blacksburg, VA 24061, USA. Email:
rgilles@vt.edu. Part of this research was done at the Center for Economic Research at Tilburg University,
Tilburg, the Netherlands. Their hospitality is gratefully acknowledged.
‡Corresponding author. Address: Department of Economics, Louisiana State University, Baton
Rouge, LA 70803, USA. Email: sarangi@lsu.edu
1Jackson and Wolinsky (1996) introduced this link-based stability approach. They
developed the seminal notions of link addition and link deletion. They consider the
creation (under consent) and deletion of a single link at a time. This resulted in their
notion of pairwise stability. Since this foundational contribution there have been
developed numerous alternative stability concepts. We refer to, e.g., Belleﬂamme
and Bloch (2004), Bloch and Jackson (2004), Goyal and Joshi (2003), Gilles and
Sarangi (2005) and Gilles, Chakrabarti, and Sarangi (2005) for extensions of this
initial concept of pairwise stability. However, there is almost no work that examines
the relationship between these different link-based stability concepts themselves. Our
investigation makes a ﬁrst step in this direction.
We limit our analysis to two notions of link addition and two notions of dele-
tion. As mentioned, a network is link deletion proof if there is no player who wants
to delete exactly one of her links. We also consider strong link deletion proof net-
works in which there is no individual who wants to deletion one or more of her links.
Strong link deletion proofness is equivalent to imposing Nash equilibrium conditions
in a non-cooperative game theoretic model of network formation as shown by Gilles,
Chakrabarti, and Sarangi (2005). We show here that link deletion proofness results
in exactly the same networks as the requirement of strong link deletion proofness
if and only if the network payoffs satisfy a convexity property. This result has al-
ready been hinted at in Calv´ o-Armengol and Ilkilic ¸ (2004), but has not been properly
developed until now.
Further, we consider the link addition proofness condition introduced seminally by
Jackson and Wolinsky (1996), which states that there is no link which will make both
participating constituents better off. In Gilles and Sarangi (2005) we show that the
implementation of belief systems in every player’s behavior results in networks with
stronger properties. We introduce there the notion of strict link addition proofness,
which requires that both constituent players of a new link become strictly worse off
if this link would be formed. Here we show that these two link addition proofness
conditions result into the same class of networks if and only if the payoff function
satisﬁes a sign-uniformity condition on the marginal payoffs.
Through these results we are able to clearly delineate the various link-based sta-
bility concepts clearly. These insights might be useful for future investigations of dif-
ferent equilibrium and stability concepts in these game theoretic models of network
formation.
22 Preliminaries
Next we formally introduce the main tools of our analysis. Throughout we let N be
some ﬁnite set of players or individuals.
2.1 Social networks
In our subsequent discussion we use established mathematical notation from Jackson
and Wolinsky (1996), Dutta and Jackson (2003), and Jackson (2005). The reader
may refer to these sources for a more elaborate discussion.
We limit our discussion to non-directed networks on N. Formally, if two players
i,j ∈ N with i 6= j are related we say that there exists a link between players i and j.
Now, if players i and j make up a single link, both players are equally essential; links
have a bi-directional nature. Formally such a link can be expressed as a binary set
{i,j}. We use the shorthand notation ij to describe the link {i,j}. We deﬁne
gN = {ij | i,j ∈ N, i 6= j} (1)
as the set of all potential links.
A network g on N is any set of links g ⊂ gN. Particularly, gN denotes the complete
network and g0 = ∅ is known as the empty network. The collection of all networks is
deﬁned as GN = {g | g ⊂ gN}.
The set of (direct) neighbors of a player i ∈ N in the network g ∈ GN is given by
Ni(g) = {j ∈ N | ij ∈ g} ⊂ N. (2)
Similarly we introduce
Li(g) = {ij ∈ gN | j ∈ Ni(g)} ⊂ g (3)
as the link set of player i in the network g. These are exactly the links with i’s direct
neighbors in g.
For every pair of players i,j ∈ N with i 6= j we denote by g + ij = g ∪ {ij} the
network that results from adding the link ij to the network g. Similarly, g−ij = g\{ij}
denotes the network resulting from removing the link ij from network g.
2.2 Link-based stability concepts
We complete the preliminaries on network theory with the deﬁnition and discussion
of the link-based stability conditions already mentioned in the introduction to this
paper.
3Within a network g ∈ GN, beneﬁts for the players are generated depending on
how they are connected to each other. Formally, for each player i ∈ N the function
ϕi: GN → R denotes her network payoff function which assigns to every network
g ⊂ gN a value ϕi(g) that is obtained by player i when she participates in g. The
composite network payoff function is now given by ϕ = (ϕ1,...,ϕn): GN → RN.
We emphasize that these payoffs can be zero, positive, or negative and that the empty
network g0 = ∅ generates (reservation) values ϕ(g0) ∈ RN that might also be non-
zero.1
For a given network g ∈ GN we now deﬁne the following concepts:
(a) For every ij ∈ gN, the marginal beneﬁt of the link ij to player i in the network
g for payoff function ϕ is given by
Di(g,ij) = ϕi(g) − ϕi(g − ij) ∈ R. (4)
(b) For every player i ∈ N and link set h ⊂ Li(g) the marginal beneﬁt of link set h
to player i in the network g for payoff function ϕ is given by
Di(g,h) = ϕi(g) − ϕi(g − h) ∈ R. (5)
Using these additional tools we can give a precise description of the various link-
based stability concepts:
(a) A network g ⊂ gN is link deletion proof for ϕ if for every player i ∈ N and
every neighbor j ∈ Ni(g) it holds that Di(g,ij) > 0.
Denote by D(ϕ) ⊂ GN the set of link deletion proof networks for ϕ.
(b) A network g ⊂ gN is strong link deletion proof for ϕ if for every player i ∈ N
and every h ⊂ Li(g) it holds that Di(g,h) > 0.
Denote by Ds(ϕ) ⊂ GN the set of strong link deletion proof networks for ϕ.
(c) A network g ⊂ gN is link addition proof if for all players i,j ∈ N: ϕi(g+ij) >
ϕi(g) implies ϕj(g + ij) < ϕj(g).
Denote by A(ϕ) ⊂ GN the set of link addition proof networks for ϕ.
(d) A network g ∈ GN is strictly link addition proof for φ: GN → R if for all
i,j ∈ N it holds that ij 6∈ g implies that ϕi(g + ij) < ϕi(g).
Denote by As(ϕ) ⊂ GN the set of strict link addition proof networks for ϕ.
1We remark that ϕ can be viewed as gross beneﬁts ϕ minus the link maintenance costs c
m. Hence,
we can reformulate ϕi(g) = ϕi(g) −
P
j∈N: ij∈g c
m
ij. However, explicit modeling of these maintenance
costs is only essential in a dynamic model of network formation processes.
4The two link deletion proofness notions are based on the severance of links in a
network by individual players. The notion of link deletion proofness considers the
stability of a network with regard to the deletion of a single link while strong link
deletion proofness considers the possibility that a player deletes any subset of her
existing links. Clearly, strong link deletion proofness implies link deletion proofness.
Adding a link on the other hand is considered for a pair at a time and also requires
consent. A network is link addition proof if every pair of non-linked players as a
whole does not have the incentive to add this link. Strict link addition proofness
requires that every individual player in the pair being considered has a loss from
adding a link. This formulation makes the consent requirement somewhat moot since
neither player has an incentive to add the link. This is a signiﬁcant strengthening of
the link addition proofness requirement.
We conclude our discussion with an example which delineates the different link-
wise stability concepts.
Example 1 Consider the network payoffs given in the following table:
Network ϕ1(g) ϕ2(g) ϕ3(g) Stability
g0 = ∅ 0 0 0 Ds
g1 = {12} −1 −1 −1
g2 = {13} −1 −1 −1
g3 = {23} 5 3 3 Ds, A
g4 = {12,13} 1 1 1 Ds
g5 = {12,23} 0 4 0
g6 = {13,23} 0 0 4
g7 = gN 1 5 5 D, As
In the table D stands for link deletion proofness and Ds for strong link deletion proof-
ness. Similarly, A stands for link addition proofness and As for strict link addition
proofness.
The main features here are that the complete network g7 is link deletion proof, but
not strong link deletion proof and that network g3 is link addition proof, but not strict
link addition proof. To make the differences between the various possibilities more
clear we provide an overview of the marginal beneﬁts:
5Network D(g,12) D(g,13) D(g,23)
g0 = ∅ — — —
g1 = {12} −1,−1,−1 — —
g2 = {13} — −1,−1,−1 —
g3 = {23} — — 5,3,3
g4 = {12,13} 2,2,2 2,2,2 —
g5 = {12,23} −5,1,−3 — 1,5,1
g6 = {13,23} — −5,−3,1 1,1,5
g7 = gN 1,5,1 1,1,5 0,4,4
In g7 player 1 is stuck with bad company if she could delete only a single link at the
time; she would like to break links with both players 2 and 3 and improve her payoff
from 1 unit to 5 units. However, deleting either of these two links separately would
make her only worse off. In this regard network convexity requires that no player is
in such a bad company situation. 
Using the basic components of link formation we now deﬁne two more link-based
stability concepts.
(a) A network g ∈ GN is pairwise stable for ϕ if g is link deletion proof as well
as link addition proof.
Denote by P(ϕ) = D(ϕ) ∩ A(ϕ) ⊂ GN the family of pairwise stable networks
for ϕ.
(b) A network g ∈ GN is strictly pairwise stable for ϕ if g is strong link deletion
proof as well as strict link addition proof.
Denote by P?(ϕ) = Ds(ϕ) ∩ As(ϕ) ⊂ GN the family of strict pairwise stable
networks for ϕ.
Jackson and Wolinsky (1996) seminally introduced the notion of pairwise stabil-
ity. This requirement combines link deletion proofness and link addition proofness.
Given that these two proofness conditions can be strengthened in various ways it is
possible to deﬁne a variety of modiﬁcations depending on the context. In Gilles and
Sarangi (2005) we introduce two modiﬁcations: (1) one where an agent can delete
any subset of their links and (2) a variation of link addition proof where no agent in
the pair wishes to add the link. The notion of strict pairwise stability combines these
two features making it a natural link-based stability concept.
63 Equivalence and convexity
In this section we state the conditions that establish equivalence between the dif-
ferent stability concepts. We identify two related conditions under which the main
proofness conditions result in the same networks.2
Equivalence Theorem
(a) It holds that Ds(ϕ) = D(ϕ) if and only if ϕ is network convex in the sense that
for every network g, every player i ∈ N and every link set h ⊂ Li(g):
X
ij∈h
Di(g,ij) > 0 implies Di(g,h) > 0. (6)
(b) It holds that As(ϕ) = A(ϕ) if and only if ϕ is link uniform on A(ϕ) in the
sense that for every network g ∈ A(ϕ) and all players i,j ∈ N with ij / ∈ g:
Di(g + ij,ij) > 0 implies Dj(g + ij,ij) > 0. (7)
(c) It holds that P?(ϕ) = P(ϕ) if and only if ϕ is network convex as well as link
uniform on A(ϕ).
A proof of this equivalence theorem is relegated to Section 4 of this paper.
To illustrate the two payoff conditions, network convexity and link uniformity, we
return to the example discussed previously.
Example 2 Consider N = {1,2,3} and the payoff structure ϕ described in Example
1. We show that this payoff structure is neither network convex nor link uniform.
Indeed, ﬁrst note that D(g7,12) + D(g7,13) = (2,6,6) and that D(g7,{12,13}) =
(−4,2,2). Hence, the case of the removal of the links 12 and 13 from network g7
shows that ϕ is not network convex.
With regard to link uniformity of ϕ we remark that for network g6 = {13,23} we have
that D1(g6,13) = −5 < 0, while D3(g6,13) = 1 > 0. This violates link uniformity of
the marginal payoffs with regard to adding the link 13 to the network g3 = {23}. That
is also the reason why g3 is link addition proof, but not strict link addition proof. 
Further, from the Equivalence Theorem it follows immediately that
2Calv´ o-Armengol and Ilkilic ¸(2004) introduced the concept of α-convexity on a network payoff struc-
ture. This concept requires that the marginal beneﬁts of link formation are related in a supermodular
fashion. Note that the notion of “network convexity” deﬁned here is weaker than their α-convexity
condition.
7Corollary: If ϕ is (link) monotone in the sense that D(g,h) > 0 for all networks
g ⊂ gN and link sets h ⊂ g, then P?(ϕ) = P(ϕ).
This equivalence result in fact follows from the observation that monotonicity of the
value function ϕ implies that ϕ is both network convex as well as link uniform.
4 Proof of the equivalence theorem
PROOF OF ASSERTION (A):
Obviously from the deﬁnitions it follows that in general Ds(ϕ) ⊂ D(ϕ).
Only if: Suppose that g ∈ D(ϕ) and that ϕi is not network convex on g for some
i ∈ N and some link set h ⊂ Li(g). We show that g 6∈ Ds(ϕ).
Indeed, from the hypothesis that g is link deletion proof, we know that Di(g,ij) >
0 for every ij ∈ Li(g). Then for h it has to be true that since
P
h Di(g,ij) > 0,
Di(g,h) < 0. But then this implies that player i would prefer to sever all links in h.
Hence, g cannot be strong link deletion proof, i.e., g 6∈ Ds(ϕ).
If: Let g ∈ D(ϕ) and assume that ϕ is network convex on g. Then for every player
i ∈ N and link ij ∈ Li(g) it has to hold that Di(g,ij) > 0 due to link deletion
proofness of g. In particular, for any link set h ⊂ Li(g):
P
h Di(g,·) > 0. Now by
network convexity this implies that Di(g,h) > 0 for every link set h ⊂ Li(g). In other
words, g is strong link deletion proof, i.e., g ∈ Ds(ϕ).
This completes the proof of the assertion.
PROOF OF ASSERTION (B):
First suppose that ϕ is link uniform on A(ϕ). Since As(ϕ) ⊂ A(ϕ) we only have to
show that A(ϕ) ⊂ As(ϕ).
Let g ∈ A(ϕ). Then it follows by deﬁnition of link uniformity that for all i,j ∈ N with
ij / ∈ g:
ϕi(g) 5 ϕi(g + ij) implies ϕj(g) 5 ϕj(g + ij)
since Di(g + ij,ij) = ϕi(g + ij) − ϕi(g). Hence, equivalently,
ϕi(g) > ϕi(g + ij) implies ϕj(g) > ϕj(g + ij).
Now since g is link addition proof it holds that
ϕi(g + ij) > ϕi(g) implies ϕj(g) > ϕj(g + ij),
8which in turn implies that ϕi(g) > ϕi(g + ij).
This is a contradiction. Therefore, we conclude that ϕi(g + ij) 5 ϕi(g) as well as
ϕj(g + ij) 5 ϕj(g). Hence, g ∈ As(ϕ).
Next suppose that A(ϕ) = As(ϕ). Let g ∈ A(ϕ) be such that there are i,j ∈ N
with ij / ∈ g. Then by strict link addition proofness of g—resulting from the assumed
equivalence—it follows that
ϕi(g + ij) 5 ϕi(g) as well as ϕj(g + ij) 5 ϕj(g).
This in turn implies that ϕ is link uniform for g.
PROOF OF ASSERTION (C):
This is a direct consequence of assertions (a) and (b) of the Equivalence Theorem
proven above.
This completes the proof of the Equivalence Theorem stated in Section 3.
5 Conclusion
We have identiﬁed conditions under which strictly pairwise stable networks coincide
with pairwise stable networks. This has useful applications since we can now apply
the more natural notion of strict pairwise stability to the connections model of Jack-
son and Wolinsky (1996). However this equivalence does not hold for the co-author
model also developed in Jackson and Wolinsky (1996).3
More importantly, we also conjecture that the network convexity condition can
be extended to coalitions by identifying one such condition for each player. This will
enable us to build establish the equivalence of pairwise stability and Jackson and
van den Nouweland (2005)’s notion of strong stability.
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Stable Networks and Convex Payoffs

Stable Networks and Convex Payoffs

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