BiGSEM
Seventh Doctoral Workshop
on Economic Theory
BiGSEM
A Connections Model with Negative Externalities
Philipp Möhlmeier, BiGSEM, Bielefeld University, [email protected]
Agnieszka Rusinowska, CES, University of Paris I, [email protected]
Emily Tanimura, CES, University of Paris I, [email protected]
1. Introduction
3. The Model
The seminal connections model by Jackson and
Wolinsky (1996) is an example of social communication between individuals where benefits
and costs for each individual are determined by
the direct and indirect connections among them.
The authors focus on the identification of pairwise stable and strongly efficient networks. The
original model incorporates only positive externalities due to link formation. Despite numerous extensions of the connections model in the
literature, the issue of negative externalities in
this framework has not received much attention.
We develop a modification of it that accounts for
negative externalities while staying close to the
functional form of the established connections
model.
In the original connections model, the utility of i
from g is defined as
uiJW (g) =
where 0 < δ < 1 is the undiscounted valuation
of a connection, tij counts the number of links in
the shortest path between i and j (with tij = ∞,
if there is no path connecting i and j) and c > 0
denotes the costs of a direct connection.
G := g | g ⊆ g N
o
The network obtained by adding (deleting) a link
ij to (from) g is denoted by g + ij (g − ij).
Some basic structures that often play an important role are the empty network g ∅, the complete
network g N , the star g S and the line g L :
Figure 1: Sample architectures
The value of a graph is determined by v : G → R
and we assume that it is an aggregate of individual utilities:
X
v(g) =
ui (g)
i ∈N
where ui : G → R.
A network g ∈ G is strongly efficient (SE) if
v(g) ≥ v(g 0) for all g 0 ∈ G.
A network g ∈ G is said to be pairwise stable
(PS) if the following two conditions hold:
1. ∀ ij ∈ g, ui (g) ≥ ui (g − ij) and uj (g) ≥ uj (g − ij)
2. ∀ ij < g, if ui (g) < ui (g + ij) then uj (g) > uj (g + ij).
Theorem 3 g S with n ≥ 3 is PS iff δ 23 − δ ≤
2c ≤ δ and n2 + (n − 2)δ ≥ 2c
δ.
The following figure illustrates the stability regions for the parameters δ ∈ [0, 1], c ∈ [0, 0.5]
and n = 9:
Obviously, forming an additional connection may
only induce a positive externality on other players because it eventually shortens the distance
to other agents. In contrast to this approach, we
suggest a modification that generates also negative externalities by increasing connectivity:
2. Network Notation
n
δtij − c ηi (g)
j ,i
uiMRT (g)
Let N = {1, 2, ... , n} denote the set of players,
agents or nodes. A network g is described by
a set of pairs {i, j } (denoted ij), with i, j ∈ N and
i , j. A link ij indicates the presence of a relationship between i and j. The nodes i and j are
directly connected in g iff ij ∈ g. The degree ηi (g)
counts the number of links i has in g and is formally described by ηi (g) = |{j ∈ N | ij ∈ g }|. The
set of all possible networks g on N is
X
The star architecture g S is PS for a rather intermediate range of parameters:
=
1
X
j ,i
1 + ηj (g)
δtij − c ηi (g)
Figure 2: PS networks
By this, the dispersion of information is negatively influenced by the number of direct links
a (distant) neighbor has. This means that the
busyness of a potential neighbor influences the
valuation to form a link with him. To provide a
simple example, consider a co-author relationship: On the one hand, the more co-authors
an author has, the more knowledge may spill
over from the projects he is involved in. On
the other hand, the more connections an author
maintains, the busier he is so that he may not be
able to dedicate enough time to every co-author
to transfer all information. Hence, the model incorporates positive and negative externalities of
link formation. Morrill (2011) introduced a model
which applies this perception to direct neighbors.
Our framework covers in addition externalities
from neighbors of neighbors and so on.
Regarding PS we have the following results:
Theorem 1 g ∅ is PS iff δ ≤ 2c.
Hence, the empty network g ∅ is PS for rather
large costs.
Theorem 2 g N with n = 2 is PS iff δ ≥ 2c. For
n ≥ 3, g N is PS iff 4c ≤! nn−21 and δ1 ≤ δ ≤ δ2 with
δ1 =
δ2 =
n −1
2n
q
1−
1−
4cn2
n −1
≥
1
3
1−
√
1 − 18c and
!
q
1+
5. Conclusions and Outlook
In comparison with Jackson and Wolinsky
(1996), our model incorporates besides positive
also negative externalities. A slight change in
the functional form of the utility induces for example an overlap of PS architectures that does
not occur in the original version. Furthermore,
the sets of PS and SE networks seem to be
much larger and not always analytically tractable
so that we are currently using computational
methods for further analysis.
References
4. Results
n −1
2n
The green area indicates the stability region of
g ∅, the red area the one for g N and the yellow
area the one for g S . Striking are the overlapping
orange parameter region in which g N and g S are
simultaneously PS and the white area in which
none of the three simple sructures is PS.
1−
4cn2
n−1
<
n −1
n
< 1.
This simply means that the complete network g N
is PS for very small costs.
Buechel, B., and T. Hellmann (2012): “Underconnected and over-connected networks: the
role of externalities in strategic network formation,” Review of Economic Design, 16, 71–87.
Goyal, S., and S. Joshi (2006): “Unequal Connections,” International Journal of Game Theory,
34, 319–349.
Jackson, M. O., and A. Wolinsky (1996): “A Strategic Model of Social and Economic Networks,”
Journal of Economic Theory, 71, 44–74.
Morrill, T. (2011): “Network formation under
negative degree-based externalities,” International Journal of Game Theory, 40, 367–385.
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