Strategic network formation subject to
the law of increasing marginal costs
Namju Yoon
10248560
Supervisors: Dr. Marco J. van der Leij & Daan in 't Veld
________________________________________________________
Abstract
This thesis studies a network formation model, based on Goyal and Vega-Redondo
(2007), where players are allowed to build or remove links for higher payoffs. The
purpose is to show in what way the law of increasing marginal costs affects previous
results found by Goyal and Vega-Redondo on network formation theory. Increasing
marginal costs in the number of links makes social actors in a network less likely to build
many connections. The analysis shows that, for this reason, networks that contain actors
with more than one connection become less stable. For a large amount of players in the
star network, taking the position of a structural hole as the central player brings costs
that exceed the gained benefits, which shows that structural holes are not always
profitable. Also, the cycle network becomes relatively more important among the stable
networks. Furthermore, this paper studies network efficiency. When players are subject
to the law of increasing marginal costs, for a network to be efficient, it has to be minimal.
The only possible network that can be simultaneously stable and efficient is the empty
network.
________________________________________________________
Bachelor Thesis
University of Amsterdam
December 20, 2013
Acknowledgements
I take out this opportunity to thank the people that made this thesis possible. In the first
place, I would like to express my sincere thanks to my thesis supervisors Dr. Marco J.
van der Leij and Daan in ‘t Veld. They were open and helpful regarding my initial ideas
for the thesis. They provided feedback on draft versions through each step of the way,
including corrections on structure, insightful suggestions on the research and assistance
with programming in R. My supervisors provided me with articles, essential R-scripts
and much needed encouragement to complete this thesis.
I also thank Dr. K.J. van Garderen, Programme Director of the Bachelor of Science
Econometrics, for the organization and necessary facilities and Drs. R. van Hemert for
providing me with valuable information on writing a research paper.
Last but not least, I am grateful to God, my parents and friends for their constant
encouragement, inspiration and support.
I place on record, my sincere gratitude towards the aforementioned and every one who,
directly or indirectly, have been a helping hand with the completion of this Bachelor
thesis.
1 Contents
1. Introduction
3
2. The model
5
2.1 Goyal and Vega-Redondo’s model
5
2.2 Adjusted model
7
2.3 Network formation concepts
8
3. Analysis
10
3.1 Stability
10
3.2 Efficiency
13
3.3 Efficiency vs. Stability
18
4. Discussion
19
5. Conclusion
21
References
23
Appendix
2 1
Introduction
Relationships between social actors (such as individuals, firms or countries) have a
large variety of aspects. To be able to describe and measure these in a useful manner it
is essential to find a conceptual framework (Goyal, 2007). This framework has to contain
a language that naturally shows the variations in relationships. The concept of a social
and economic network addresses these conditions. There has been previous literature
on social networks and the applications of social network analysis (Wasserman and
Faust, 1994). A network generally describes a collection of nodes and the links between
them. The nodes may be seen as individuals, firms or countries or a collection of such
entities. A link between two nodes signifies a direct relation between them. Since actors
can build various combinations of links, a network can take on many different structures.
The basic principle of building links in a network for social actors is to create potential
benefits from relationships. They receive assets by exchanging goods or information. It
is also important to take into consideration that there are costs involved in forming a link
in terms of time and effort.
Over the last two decades, network theory has increasingly been applied to
economic situations as a tool to analyze social relationships. Pastor-Satorras and
Vespignani (2007) discussed the role of networks on the Internet described through
statistical physics. Overall, network theory has been applied in many fields. Ellison,
Steinfeld and Lampe (2007) found in their empirical paper that the use of social network
website Facebook has great benefits in social capital. Goyal and Joshi (2006) studied
bilateral trade in domestic and foreign markets and found that bilateral trade is
consistent with global free trade. Because of this, the formation of network structures
and their stability is an important topic to be studied.
There have been several papers on the formation of network structures. Goyal
and Vega-Redondo (2007) studied what network structures emerge when players are
allowed to position themselves in structural holes (Burt, 1992), connecting players that
otherwise would not have been able to connect. They analyzed the interaction between
actors in the process of strategic network formation. In their model, an exchange
between two players produces a unit surplus that they share equally between them and
3 the essential intermediaries connecting them. Essential players are players without
whom an interaction cannot take place. Players try to maximize their utility, which is
evaluated by a payoff function that consists of three components. The first component
consists of the surpluses players get for transactions with other players. The second
component consists of intermediation payoffs that are gained when a player is essential
in connecting two other players. The last component is the costs, which are determined
by a fixed cost for each link and the number of links a player has.
Goyal and Vega-Redondo obtain two main insights from their analysis. First, it is
shown that when every player wants to maximize their utility, without any additional
constraints, it leads to a star network, where one player takes the position of the
structural hole and connects every other pair of players. This player is essential for the
network and has the highest payoff. The second insight shows that when players have a
small capacity to form links relative to the number of players it leads to a cycle network,
where payoffs are equal among every player.
In this thesis, the last component of Goyal and Vega-Redondo’s payoff function,
the costs, will be adjusted by implementing the law of increasing marginal costs. This is
done by replacing the fixed cost for every link by a new cost function that is marginally
increasing with the number of links a player has. The law of increasing marginal costs
implies that adding more links, while holding all other factors constant, will at some point
cost more per link.
The law of increasing marginal costs has been applied in many papers. Bils
(1987) analyzed the effects of marginal costs and price in American manufacturing
industries. Kahn (1986) found that a durable goods monopolist produces more slowly
when purchasing firms are subject to increasing marginal costs. The law of increasing
marginal costs also applies to the formation of financial networks. Banks or financial
institutions have certain factors (e.g. capital, time, employees) that have to be
considered in making a link with another bank. When these factors reach a certain
capacity, building more links will bring more costs than before. Goyal and VegaRedondo suggest in the discussion of their paper that, “[i]n some settings it seems more
natural to suppose that the costs per link are increasing in the number of links.” (p. 477).
4 I will use the model with the law of increasing marginal costs to analyze what effect this
law has on the formation of network structures. It will also be used to analyze the
division of trade surplus between players in the process of strategically forming
networks.
This thesis will use a research analysis based on economic network theory. The
original model by Goyal and Vega-Redondo shall be discussed in section 2.
Subsequently, an adjusted model will be proposed for the research. Section 3 presents
the analysis and gives the results and their interpretation. Afterwards, the method of
research and results will be discussed in section 4. Section 5 will conclude with the most
important outcomes of this research.
2
The model
Goyal and Vega-Redondo’s paper has been cited in several research papers on network
formation theory. They developed a model of network formation and analyzed the trade
between players in the process of forming strategic networks. Players form links with
each other that contain a trade-off of benefits and constant costs made for the forming of
links. This section will first describe their model and its theoretical notations. In section
2.2, this model will be adjusted by changing the costs to include the law of increasing
marginal costs. Section 2.3 will touch on several network formation concepts that will be
used for the analysis
2.1 Goyal and Vega-Redondo’s model
In Goyal and Vega-Redondo’s model the players are traders who can trade goods. The
model makes use of the idea of structural holes, introduced by Burt (2007), which
implies that players can strategically position themselves between players and connect
players that otherwise would not have been able to connect. Players are also allowed to
form or delete links to maximize their own utility, even if this breaks the connection
between two different players who are not direct neighbors.
The model considers a finite population of identical players 𝑁 = {1,2, … , 𝑛} with
𝑛 ≥ 3. Players in the network formation game make a simultaneous decision of intended
5 links. Their strategies are defined by 𝑠!" ∈ {0,1}, where a player 𝑖 wants to form a link
with 𝑗 if 𝑠!" = 1 and does not want to form a link if 𝑠!" = 0. A link between player 𝑖 and 𝑗
will be formed if both players intend to form the link, which is the case if 𝑠!" = 𝑠!" = 1. If a
link is formed between these two players, it will be given by 𝑔!" ≡ 𝑔!" = 1 and 𝑔!" ≡ 𝑔!" =
0 if there is no link. The component of 𝑖 in 𝑔, denoted by 𝐶! (𝑔) is defined by all players in
the network that player 𝑖 is connected to, directly or indirectly.
Each pair of players is able to trade if they are directly or indirectly connected. An
exchange between two players produces a unit surplus that they share between them
and the essential intermediaries in between them. The distribution of surpluses depend
on an implicit bargaining process. For this bargaining process the concept of essential
players is introduced. Essential players are players without whom an interaction cannot
take place. The concept is modeled by denoting 𝐸(𝑗, 𝑘; 𝑔) the set of players who are
essential to connect 𝑗 and 𝑘 in network 𝑔. It is assumed that non-essential players
between 𝑗 and 𝑘 get a zero share of the surplus, while the essential players and 𝑗 and 𝑘
divide the unit surplus equally. Additionally, players pay a fixed cost 𝑐 for each link they
establish. Then, for every strategy profile 𝑠 = (𝑠! , 𝑠! , … , 𝑠! ), the payoffs to player 𝑖 are
given by
Π! 𝑠! , 𝑠!! =
(2.1)
where 𝐼{!∈!
!,! }
!
!∈!! (!) ! !,!;! !!
+
!{!∈! !,! }
!,!∈! ! !,!;! !!
− 𝜂! 𝑔 𝑐
∈ {0,1} is the function that indicates whether 𝑖 is essential for 𝑗 and 𝑘,
𝜂! 𝑔 ≡ 𝑗 ∈ 𝑁: 𝑗 ≠ 𝑖, 𝑔!" = 1 stands for the number of players that player 𝑖 has link with
and 𝑒 𝑗, 𝑘; 𝑔 = 𝐸(𝑗, 𝑘; 𝑔) .
The model allows a player or a pair of players to deviate if at least one of them
benefits from this deviation. This principle is based on the original formulation of pairwise
stability by Jackson and Wolinsky (1996), which implies that pairs of players can agree
on forming or deleting links simultaneously. Goyal and Vega-Redondo’s analysis
focuses on network structures that have a strict bilateral equilibrium (SBE). This means
that when a player or a pair of players make a deviation, the payoffs for the players
involved in the network cannot increase. This SBE is a refined version of the bilateral
equilibrium (BE), which implies that if a player or a pair of players deviate, neither of the
6 players will benefit from the deviation. Both the SBE and BE are stronger stability
concepts than pairwise stability.
2.2 Adjusted model
The Goyal and Vega-Redondo model holds the assumption that the cost per link is
constant for an increasing number of links. In this thesis, Goyal and Vega-Redondo’s
model will be adjusted to include the effect of the law of increasing marginal costs. In
general, this means that the costs involved in forming a link will increase exponentially in
the number of links. The cost function in the model will be replaced by a new marginally
increasing cost function. This new marginal cost curve is therefore convex in the number
of links.
The cost function that will be used for the adjusted model in this thesis is
𝐶 𝜂 = 𝜂! 𝑔
!
∗ 𝑐 with a positive real number 𝑝 > 1. This 𝑝 represents the rate in which
the marginal costs increases. The total payoff function of the adjusted model is then:
(2.2)
Π! 𝑠! , 𝑠!! =
!
!∈!! (!) ! !,!;! !!
+
!{!∈! !,! }
!,!∈! ! !,!;! !!
− 𝜂! 𝑔
!
∗ 𝑐 𝑓𝑜𝑟 𝑝 ≥ 1
For 𝑝 = 1 the original model of Goyal and Vega-Redondo will be retrieved. Their model
will be used as a comparison to the model with higher values of 𝑝. For 𝑝 > 1 the cost
function is convex and satisfies the law of increasing marginal costs. For 𝑝 < 1 the cost
function is concave and will therefore not be considered in the model.
Goyal and Vega-Redondo also analyzed a model with capacity constraints, as an
extension of their main model. The function that Goyal and Vega-Redondo used is
𝐶 𝜂 = 𝜂! 𝑔 𝑐 for 𝜂! 𝑔 < 𝜂(𝑔) and 𝐶 𝜂 = ∞ for 𝜂! 𝑔 ≥ 𝜂(𝑔) with 𝜂(𝑔) the capacity
constraint for the number of links in network 𝑔. This implies that once a player reaches
the link capacity allowed in the model, the costs will become too large to form an
additional link.
The reason for making an adjustment to Goyal and Vega-Redondo’s capacity
constraints is because my adjusted model carries costs that increase in the number of
links. In Goyal and Vega-Redondo’s model, players have a capacity constraint where
the costs suddenly become unaffordable from a certain amount of links. Also, in the
7 adjusted model, the payoff of a player with a large number of links can be determined
even if it becomes far negative, whereas in Goyal and Vega-Redondo’s model, a
network with a player that has a significant amount of connections simply does not exist.
2.3 Network formation concepts
Different from the research done by Goyal and Vega-Redondo, this paper will focus on
the concept of pairwise stability introduced by Jackson and Wolinsky (1996) instead of
SBE networks for simplicity reasons. As stated earlier, pairwise stability allows for a pair
of players to delete or form a link as long as they benefit from the deviation. A network is
pairwise stable when none of the players involved have an incentive to deviate. Jackson
and Wolinsky described this concept of pairwise stability as shown in the following
definition.
Definition 1. A network 𝑔 is pairwise stable if
(i)
for every 𝑔!" = 1, Π! 𝑔 − 𝑔!" ≥ Π! (𝑔) and Π! 𝑔 ≥ Π! 𝑔 − 𝑔!"
(ii)
for 𝑔!" = 0, Π! 𝑔 + 𝑔!" > Π! 𝑔 ⟹ Π! 𝑔 + 𝑔!" < Π! 𝑔
The first condition implies that every link between any two players 𝑖 and 𝑗 must be
beneficial or indifferent for their payoffs. The second condition says that for every link
that is not present between two players, a player must be worse off from forming a link if
the other player benefits from it.
Another factor that has to be considered is the efficiency of a network. A network
is called efficient when there is no other network that generates a higher total surplus.
This can be an important factor to consider. If a formed network is stable but is not
efficient an intervention by a social planner like the government is preferable from a
social point of view. 𝑊 𝑔 ≡
!∈! Π! (𝑔)
is given as the total surplus generated and
𝒢 denotes the set of all possible networks. Then, a network 𝑔 is efficient if 𝑊 𝑔 ≥ 𝑊 𝑔
for all 𝑔 ∈ 𝒢.
The analysis of this thesis will focus on four important network structures. These
networks are illustrated on the following page in Figure 2.1. A network 𝑔 is called
connected if there is a path between every pair of players in the network. Furthermore, a
8 network is symmetric if every player has the same amount of links. An example of a
symmetric network is the complete network, denoted by 𝑔! , where every player has
𝑛 − 1 amount of links. Another symmetric network is the empty network, 𝑔! , where there
are no links between any of the players. The cycle network is also a symmetric network
where every player holds two links. The star network is a network where a single player
holds a link with every other player and where there are no further links between the
other players. This is an asymmetric network because one player has 𝑛 − 1 links and the
other players hold only a single link.
Lastly, the concepts of minimally connected and minimal networks are explained.
Minimally connected networks are networks where every player is connected directly or
indirectly and where two directly or indirectly connected players only have one possible
path between them. An example of this is the star network, where every player has one
path towards any other player. A minimal network follows the same concept, except for
the fact that players do not necessarily have to be connected.
Figure 2.1 Main network structures (with 𝑛 = 5)
Star network
Cycle network
Empty network
Complete network
9 3
Analysis
The adjusted payoff function (2.2) is used to analyze the effects of the convex cost
function (for different values of 𝑝 > 1 and 𝑐) on networks regarding pairwise stability and
efficiency. Different values for the amount of players 𝑛 in a network are considered in the
research. Finally, the efficiency of the pairwise stable networks are discussed. The
results are compared with the results of Goyal and Vega-Redondo’s model.
3.1 Stability
The following theorem shows the pairwise stability conditions obtained by using the
adjusted payoff function for the star, empty, complete and cycle networks.
Theorem 1 Pairwise stability of the four main network structures
!
!!!!
(i)
The star network is pairwise stable for !(!! !!) < 𝑐 < !( !!! ! ! !!! ! ) for all 𝑝.
(ii)
The empty network is pairwise stable for 𝑐 > ! for all 𝑝.
(iii)
The complete network is not pairwise stable for 𝑛 > 3 and for all 𝑝.
(iv)
The cycle network is pairwise stable for 𝑐 < !! !!
!
!
!!!
!
−
! !
!!! !
.
Proof See the Appendix
This theorem shows for which values of the costs the four main network structures are
pairwise stable. To be able to compare this with Goyal and Vega-Redondo’s model, we
look at their model and results for pairwise stability, which can be obtained by giving a
value of 𝑝 = 1 in Theorem 1 for all the network structures. The results for 𝑛 = 4 and
𝑛 = 6 are also shown graphically in the plots in Figure 3.1. They show the conditions for
𝑐 and 𝑝 in which case the star, empty and cycle networks are pairwise stable. In the
graphs these results are shown for each network by the symbols (∗), (-) and (0)
respectively.
10 Figure 3.1 Pairwise stability of the star, empty and cycle networks for 𝑛 = 4 and 𝑛 = 6.
Looking at the stability boundaries for the star network in Theorem 1, it appears that for
any given 𝑛 > 2 the range of 𝑐 for pairwise stability becomes smaller for increasing
𝑝 > 1. This can be explained by the fact that the central player will sooner have an
incentive to remove a link in the case of high marginal costs. This effect outweighs the
increase of the interval of 𝑐 in the lower bound, which is caused by the fact that
peripheral players will less likely have an incentive to form an extra link for high 𝑝. When
𝑛 increases, it is clear to see that the interval of 𝑐 for pairwise stability is larger for 𝑝 = 1,
but will become smaller more quickly for 𝑝 > 1. This is again due to the increasing
marginal costs, which implies that the central player will have higher costs to form a link
the more players he already has a link with. Consequently, when there are more players
(𝑛 increases), the star network is more stable for fixed marginal costs, but the star
network subject to increasing marginal costs will become less stable more quickly.
For the cycle network it also seems that for a set 𝑛 > 2 the interval of 𝑐 for
pairwise stability becomes smaller when 𝑝 > 1 is increasing. A player will not have an
incentive to add a link because it will not create additional benefits and only increase
11 costs. On the other hand, a player will sooner tend to remove a link if the effect of the
marginal costs increases. This is because for higher marginal costs, removing a link will
save costs that can exceed the loss of benefits by making the other players essential.
When 𝑛 increases, the range of 𝑐 for pairwise stability becomes larger. This can be
explained by the fact that when there are more players in the cycle network, removing a
link will lead to a higher loss of benefits because it makes more players essential. When
𝑛 is high, this loss surpasses the costs saved by removing a link for a larger range of 𝑐,
in which case players will not have an incentive to remove a link.
Figure 3.1 clearly shows that the range of 𝑐 for stability of both the star and the
cycle network decreases as 𝑝 increases. It appears that values for 𝑝 > 1 have a
negative effect on the stability of networks that contain players with more than one
connection. Note that for increasing 𝑝, the decrease in range of 𝑐 for the stability of the
cycle network is faster than for the star network. The cycle network becomes relatively
more important compared to the star network for higher values of 𝑝. This is because the
effect of increasing marginal costs has relatively low effect on the cycle network,
because this network consists of players with each two links. The star network suffers
from high effects of increasing costs, because the central player has a large amount of
connections when the number of players is large.
The empty and complete networks are shown to be constant for 𝑝 and 𝑛. In the
empty network, players will not have an incentive to form the first link if the costs exceed
!
the benefits. The graph shows that when there is a set 𝑐 > !, then from a certain value
for 𝑝, the empty network is the only pairwise stable network. The complete network can
never be pairwise stable and is therefore left out in the graphs.
Given the fact that the star network becomes less and less stable for higher
effects of increasing marginal costs and larger amount of players in the network, there
should be an 𝑛 big enough with certain 𝑝 where the star network stops being pairwise
stable. It appears that for 𝑝 = 2, 𝑛 can become infinitely large and the star network will
stay pairwise stable for some c. Given that 𝑝 = 2.5, it will stop being pairwise stable for
𝑛 > 17 and if 𝑝 = 3 this is the case for 𝑛 > 7. This suggests that as 𝑝 increases, the
maximum number of players for the star network to stay pairwise stable will decrease.
This intuition turns out to be true.
12 Proposition 1 For 𝑝 > 2, there exists some 𝑛 such that for every 𝑛 > 𝑛 there does not
exist a 𝑐 for which the star network is pairwise stable.
Proof See the Appendix
Intuitively, this is because when the number of players becomes large, the central player
becomes subject to a large effect of increasing marginal costs. If 𝑛 becomes high
enough, the costs for the central player becomes greater than the acquired benefits from
taking the position of the structural hole. This shows that taking this position, as the
central player, may lead to a lower payoff for a large number of players.
3.2 Efficiency
The adjusted payoff function (2.2) is used to analyze the efficiency of the four main
network structures (Figure 2.1). Goyal and Vega-Redondo found in their paper that for
their model (𝑝 = 1), efficient networks must be either minimally connected or empty. This
is not the case for the adjusted model used for this thesis. For this model, it still applies
that a link between two players that are already indirectly connected will not bring any
aggregate benefits and will increase costs. This means that an efficient network has to
be minimal like in Goyal and Vega-Redondo’s model. However, an efficient network is
not limited to be only minimally connected or empty if we consider increasing marginal
costs in the number of links. It turns out efficient networks can be minimal non-empty,
but not connected. This can be shown by an example of a network of four players.
Figure 3.2 Possible networks with four players
Minimally connected (line)
Minimally connected (star)
13 Connected pairs
In the figure above three possible networks with 𝑛 = 4 are shown. The first two networks
are minimally connected, and have the same aggregate net payoff in Goyal and VegaRedondo’s model. However, in the adjusted model with increasing marginal costs, the
payoffs of these two networks are different. The net payoff for the first network from
Figure 3.2 is 6 − 2 + 2 ∗ 2! ∗ 𝑐 and 6 − 3 + 3! ∗ 𝑐 for the second. It is clear to see that
the payoffs are equal for Goyal and Vega-Redondo’s model (𝑝 = 1), but for higher 𝑝 the
first minimally connected network has a higher net payoff.
The third network has a net payoff of 2 − 4 ∗ 𝑐. Comparing the first network with
!
the third, we can see that for a set 𝑐, for example 𝑐 = !, their respective payoffs will be
!"#!
5 − 2! and 0. Clearly, the first network has higher payoff for 𝑝 < !"#! = 2.3219 and lower
payoff for 𝑝 > 2.3219. This shows that a network does not necessarily have to be
minimally connected to be efficient depending on values of 𝑐 and 𝑝.
Proposition 2 Efficient networks must be minimal, but do not have to be connected.
Proof Like in Goyal and Vega-Redondo (2007) a link between players that are already
indirectly connected lowers the aggregate net payoff. The aforementioned example
shows that a not connected minimal network can have a higher payoff than a minimally
connected network.
The fact that an efficient network must be minimal shows that a cycle network is never
efficient. Removing a link from a cycle network will save costs, but will not take away
from the profits. Concerning minimally connected networks, the previous example of
networks with four players shows that different minimally connected networks have
different payoffs.
It appears that a line network of four players has a higher payoff than the star with
𝑝 > 1. The minimally connected star network generally has aggregate net costs of
𝑛−1 + 𝑛−1
!
∗ 𝑐. This consists of the costs for the central player 𝑛 − 1
!
∗ 𝑐 and
for the peripheral players 𝑛 − 1 ∗ 𝑐. For some 𝑝 > 1, the costs for the central player will
grow to be very large as 𝑛 increases. As for the minimally connected line network the
costs are 2 + 𝑛 − 2 ∗ 2! ∗ 𝑐. Comparing the aggregate net costs of the star network
14 to that of a line network, it is clear to see that the line network has considerably lower
costs for 𝑝 > 1. This is because the star network has large costs due to the effect of
increasing marginal costs on the central player. This suggests that when the ratio of the
costs of every player in a minimally connected network is more equally divided, it leads
to lower net costs. Therefore, the ratio of the number of links for every player has to be
as low as possible where the network is still connected, to eliminate large effects of
increasing marginal costs on players with more links. The line network is optimal for this
case.
Proposition 3 For 𝑝 > 1, the minimally connected network with the highest aggregate
net payoff is the line network.
Proof See the Appendix
Proposition 2 however, shows that the line network is not necessarily efficient. Following
this proposition and the previous example of networks with four players, it is known that
other minimal networks can also be efficient. Networks with lines of length 2 (connected
pairs), as in the third network in Figure 3.2, serve as an example for this. It is difficult to
analyze all the possible efficient networks for general 𝑛, since there could be
exponentially many minimal networks that could be efficient.
For the following, the focus will lie on networks of lines of length 𝑘. For a set 𝑛, if
𝑝 becomes large enough a network with connected pairs will have higher payoff than a
network with higher 𝑘. Effects of increasing marginal costs do not affect the network with
connected pairs, since each player has only one link. The law of increasing marginal
costs affects players with more than one link and therefore the costs in a network with
lines with higher length will become larger if 𝑝 increases. If 𝑝 goes closer to 1, networks
with lines of higher length become preferable, because the benefits from having lines
with larger length will outweigh the marginal costs of additional links.
To illustrate this with an example, the payoffs of networks with lines of length 2
and 3 are determined considering a set number of players 𝑛 = 12. For simplicity
reasons, a multiple of 𝑘 is taken for 𝑛, such that the aggregate net payoff of a network
15 !!! !
with lines of length 𝑘 is
!
− 2 + 𝑘 − 2 ∗ 2! ∗ 𝑐
!
!
. The payoffs are 6 − 12 ∗ 𝑐 and
!
12 − 8 + 4 ∗ 2! ∗ 𝑐 respectively. When 𝑐 > !(!! !!), the network with connected pairs will
have a higher payoff than the network with lines of 3. Now it is clear to see that as 𝑝
increases, a network with connected pairs will be preferred more quickly over a network
with lines of larger length. As 𝑝 goes closer to 1, the network with lines of length 𝑘 > 2
will be preferred for a larger range of 𝑐.
The aggregate payoff of the line network in this case is 66 − 2 + 10 ∗ 2! ∗ 𝑐.
!!
!
!
This network has a higher net payoff than the network with lines of length 3 for 𝑐 < !! !!.
This interval of 𝑐 contains the range for which the network with lines of length 3 is
!.!
preferable over the connected pairs 𝑐 < !! !! . This suggests that when 𝑝 becomes
small enough for the network with lines of length 𝑘 > 2 to be preferred over the network
with connected pairs, it is even more preferable to form a line network. This turns out to
be true.
Proposition 4 When the network with lines of 𝑘 > 2 is preferred over the network with
lines of 2, the line network has higher aggregate payoffs than the network with lines of 𝑘.
Proof See the Appendix
The following graphs in Figure 3.3 show the efficiency for networks with three, four, five
and six players, depending on values for 𝑐 and 𝑝. It appears there are three possible
efficient network structures: the line network, network with connected pairs and the
empty network. These networks are shown in the graphs by the symbols (0), (∗) and (-)
respectively.
!
The aggregate net payoff of the network with connected pairs is 1 − 2𝑐 ∗ ! . It is
now clear that empty network has a higher aggregate net payoff than the network with
!
connected pairs for 𝑐 > !. This is also apparent in the graphs. Players do not have the
incentive to form a connected pair if the cost of forming the link is higher than the gained
16 !
!
benefits, which is the case for 𝑐 > !. For 𝑐 = !, the net payoff of the network with
connected pairs is indifferent to that of the empty network.
Looking at the efficiency of the line network, it appears that for a set 𝑛 the range
of 𝑐 decreases as 𝑝 increases. For increasing 𝑝, the line network eventually becomes
!
non-efficient for a value of 𝑐 > 0. For a set 𝑐 < !, the network with connected pairs
!
eventually becomes efficient over the line network and for 𝑐 > ! this is the case with the
empty network. This is because when the effect of the increasing marginal costs
becomes bigger, the costs for the line network increase exponentially, since the line
network contains players with more than one link. As 𝑝 becomes larger, the increasing
marginal costs grow faster for higher 𝑐. Therefore for higher values of 𝑐, the line network
becomes non-efficient more quickly as 𝑝 increases.
For the Goyal and Vega-Redondo model (𝑝 = 1), it is clear that the interval of 𝑐
for efficiency of the line network becomes larger when 𝑛 increases. For higher 𝑛, it holds
that for each value of 𝑐 for which the line network can be efficient, it stops being efficient
at a higher level of 𝑝. This is because the net payoff of the line network with more
players becomes larger as the aggregate benefits gained outweigh the cost of one extra
connection.
Figure 3.3 Efficiency for networks with 𝑛 = 3,4,5 and 6
17 3.3 Efficiency vs. Stability
So far we have separately analyzed the efficiency and pairwise stability of networks
based on the adjusted payoff model with increasing marginal costs as in equation (2.2).
Yet, it is possible for a pairwise stable network to not have a high aggregate net payoff
or an efficient network to not be stable. A preferable situation would obviously be where
a network is pairwise stable and efficient. In the adjusted model with structural holes,
even if a network is efficient, players have an incentive to deviate if they can get higher
payoff from it. At the same time, a stable network is not always efficient, in which case
an intervention by a social planner might be more desirable. Therefore, this section will
analyze the potential efficiency of the networks studied in paragraph 3.1 that are already
pairwise stable.
The analysis will concern the stable star, cycle and empty networks, because the
complete network is never pairwise stable for any values of 𝑝 and 𝑐. As stated before, a
cycle network cannot be efficient, since it is not minimal. A star network is a minimally
connected network and following Proposition 1, is therefore potentially efficient.
18 However, Proposition 2 implies that the minimally connected network with highest
aggregate net payoff is the line network for 𝑝 > 1. This means that also the star network
cannot be efficient if 𝑝 > 1, which leaves the empty network as the only potentially
efficient network out of the three that were taken into account.
!
For all 𝑝, the empty network is pairwise stable and efficient for 𝑐 > ! if 𝑛 = 2.
From 𝑛 ≥ 3, the efficiency depends on the value for 𝑝. For example, for 𝑛 = 3, the line
networks with length 2 and 3 have aggregate net payoffs of 1 − 2𝑐 and 3 − 2 + 2! ∗ 𝑐
respectively. The cycle network is not efficient, because it has lower net payoffs. The
empty network is pairwise stable and has higher payoffs than the line network with
!
length 2 for 𝑐 > !. For the empty network to also be efficient the following inequality has
!
!
to stand: 0 > 3 − 2 + 2! ∗ 𝑐 with 𝑐 > !. This leads to 𝑐 > !!!! in which case the empty
network is both pairwise stable and efficient. In this case 𝑝 > 2 has to hold, because
!
!
𝑐 > !!!! ≥ !.
For the case of 𝑛 = 4, the line network of length 4 and the network with two lines
of length 2 are also potentially efficient. The empty network has higher payoffs than the
!
network with connected pairs for 𝑐 > !. The aggregate payoffs for the line network of
length 4 are 6 − 2 + 2 ∗ 2! ∗ 𝑐. It follows that the empty network has higher payoffs for
!
!
𝑐 > !!!! ≥ !!!! , which implies that the empty network is pairwise stable and efficient for
!
𝑐 > !!!! with 𝑝 > 2. Figure 3.3 shows that for increasing 𝑛, the empty network becomes
!
relatively less important regarding efficiency. However, for high values of 𝑝 and 𝑐 > !,
the empty network is the only possible simultaneously efficient and pairwise stable
network.
4
Discussion
This section touches on the results and tries to make an interpretation of the findings. It
will also raise questions for discussion on the method of research used for the analysis.
First, note that there is an unlimited amount of other network structures that can be
19 pairwise stable and/or efficient, which have been taken out of account for the research.
In this thesis, the focus lays on the main network structures (star, empty, cycle and
complete) that are discussed in Jackson and Wolinsky (1996) and Goyal and VegaRedondo (2007).
As mentioned in the introduction, Goyal and Vega-Redondo already suggested
the subject for this thesis by suggesting costs per link that are increasing in the number
of links. The law of increasing marginal costs is relevant for research on strategic link
formation. When financial institutions or banks have many relationships with other
banks, these relationships will become harder to maintain and prolong. Despite this, it
raises the question whether the increasing marginal cost function that is used for the
adjusted model is practical. It can be argued that the cost function should not be strict
convex for 𝑝 > 1, because the marginal cost curve is generally ‘u’-shaped due to the law
of variable proportions. Due to increasing returns for relatively few connections, the
marginal cost curve decreases until the stage of constant returns (and constant cost),
before the cost starts increasing with more connections. This may also be the case for
financial institutions forming links in a network.
The results for stability show that as 𝑝 increases, the range of 𝑐 for pairwise
stable star and cycle networks becomes smaller. This shows that the effect of increasing
marginal costs has clear consequences for the stability of networks that contain players
with more than one link. The results also show that there are values of 𝑝 and 𝑐 for which
more than one network can be pairwise stable. For these cases, it will be preferred to
form the network with higher payoffs. To illustrate, the graph (Figure 3.1) shows that
there is an area in which the star and empty network are both pairwise stable. These two
networks will have a different aggregate net payoff for the range of values for 𝑝 and 𝑐 in
this area. If the star network has a higher payoff, a social planner like the government
could make an intervention and attempt to encourage banks and financial institutions to
form links by using monetary and fiscal policies. This is done to stimulate the economy
and because it is preferable from a social point of view if the star network has higher net
payoff than the empty network.
For a following step on this thesis in the future, it may be possible to focus on
other stability concepts, such as bilateral equilibrium and strict bilateral equilibrium as in
20 Goyal and Vega-Redondo (2007) instead of pairwise stability. Another suggestion is to
adjust Goyal and Vega-Redondo’s payoff function to satisfy the law of diminishing
marginal returns with benefits decreasing in the number of links and perhaps to combine
this law with the law of increasing marginal costs.
5
Conclusion
This thesis discussed Goyal and Vega-Redondo’s strategic network formation model
based on structural holes. An adjustment to this model was introduced where players
are subject to the law of increasing marginal costs for the number of links. In this
network formation model, players are allowed to deviate if this provides them a higher
payoff. The adjusted model of network formation analyzed the star, cycle, empty and
complete network on the their pairwise stability. Furthermore, efficiency for this model
was analyzed and the pairwise stable star, cycle and empty networks were studied on
their potential efficiency.
The first main finding is that the new payoff function subject to the law of
increasing marginal costs has a negative effect on the stability of networks that carry
players with more than one connection. This means that if the effect of increasing
marginal costs is high, the range of the cost per link for which the networks are pairwise
stable decreases. For the star network, taking the position of structural holes may not be
beneficial when the number of players is large. This conclusion is different from Goyal
and Vega-Redondo’s, because in the adjusted model the cost per link increases
exponentially, which exceeds the benefits gained from taking the position if many
connections are formed.
Secondly, efficient networks are not bound to minimally connected and empty
networks as found in Goyal and Vega-Redondo, but can also be often minimal, not
connected networks. Increasing marginal costs eliminates Goyal and Vega-Redondo’s
finding that every minimally connected networks has the same aggregate net payoff.
Players with many links are not preferred, since they are subject to larger effects of
increasing marginal costs. Therefore, the minimally connected network with the highest
net payoff is the line network. A graphical analysis has shown that the line network, the
21 network with connected pairs and the empty network are the only possible efficient
networks. When there are more players, the line network becomes relatively important
regarding efficiency. When the costs and the effect of increasing marginal costs become
high, the empty network is efficient. Therefore, the empty network is the only possible
simultaneously efficient and pairwise stable network.
Previous literatures that use the law of increasing marginal costs reaffirmed the
importance of studying this law. The findings in this thesis confirm that this law also has
an impact on network formation theory and is an important factor to consider.
22 References
Bils, M., 1987. The Cyclical Behavior of Marginal Cost and Price. The American
Economic Review 77-5: 838-855
Burt, R.S., 1992. Structural holes. Academic Press, New York.
Ellison N.B., C. Steinfield & C. Lampe. 2007. The Benefits of Facebook “Friends:”
Social Capital and College Students’ Use of Online Social Network Sites. Journal
of Computer-Mediated Communication 12: 1143–1168.
Goyal, S., 2007. Connections, Princeton University Press. Princeton
Goyal, S. & S. Joshi. 2006. Bilateralism and Free Trade, International Economic Review
47: 749–778.
Goyal, S. & F. Vega-Redondo, 2007. Structural holes in Social Networks. Journal of
Economic Theory 137: 460–492
Jackson, M.O. & A. Wolinsky, 1996. A Strategic Model of Social and Economic
Networks. Journal of Economic Theory 71: 44–74.
Kahn, C., 1986. The durable goods monopolist and consistency with increasing costs.
Econometrica: Journal of the Econometric Society 54-2: 275-294
Pastor-Satorras, R. & A. Vespignani, 2007. Evolution and Structure of the Internet: A
Statistical Physics approach. Cambridge University Press
Wasserman, S. & K. Faust, 1994. Social Network Analysis: Methods and Applications
(Structural Analysis in the Social Sciences). Cambridge University Press.
23 Appendix
Proof of Theorem 1 (i) In the star network the central player has a payoff of
! (!!!)(!!!)
!
!
− 𝑛−1
!
!!!
!
+
∗ 𝑐. For the star network to be pairwise stable this has to be larger
than the payoff of the central player when it deviates by removing one of the links with a
peripheral player. This payoff is
!!!
!
! (!!!)(!!!)
+!
!
player has no incentive to deviate for 𝑐 < !
!
have a payoff of ! +
!!!
!
− 𝑛−2
!!!!
!!! ! ! !!! !
!
∗ 𝑐. Therefore the central
. The peripheral players each
− 𝑐. When a peripheral player forms a link with another
!
!
peripheral player his payoff will be ! + ! +
!!!
!
− 2! ∗ 𝑐 and when he removes his link with
the central player his payoff will be zero. For the star network to be pairwise stable the
payoff of a peripheral player has to be larger than the payoffs for when the player adds
!
!
or removes a link. This results in ! !! !! < 𝑐 < ! +
!
+
!
!!!
!
!!!
!
!!!!
. Since for 𝑝 > 1, !( !!! ! ! !!! ! ) <
for a network with at least 𝑛 > 2 it is proven that the star network is pairwise
!
!!!!
stable so long as !(!! !!) < 𝑐 < !( !!! ! ! !!! ! ).
(ii) In the empty network a player will only have an incentive to form a link if the
costs of forming one link are lower than the benefits. This shows that an empty network
!
is pairwise stable as long as 𝑐 > !.
(iii) In a complete network a deviation will always be profitable, since removing a
link will only reduce costs but no profits will be shared, since none of the players become
essential.
(iv) Players in a cycle network each have a payoff of
!!!
!
− 2! ∗ 𝑐. A player will not
create additional benefits if he adds a link. If a player removes a link it will make other
players essential for all transactions that the player makes. This means that the cycle
network can only be pairwise stable if the costs saved by removing one link exceed the
loss of benefits due to other players becoming essential. This is the case only if
!!!
!
− 2! ∗ 𝑐 >
! !
!!! !
!
− 𝑐, which leads to 𝑐 < !! !! (
24 !!!
!
−
! !
!!! ! ).
Proof of Proposition 1 To prove there is a maximum 𝑛 for 𝑝 > 2, I have to show that
the upper bound of the pairwise stability conditions for the star network becomes lower
than the lower bound for large 𝑛. Since the lowerbound does not depend on 𝑛 and the
upperbound is continuous in 𝑛, it is sufficient to prove that the upper bound goes to zero
!!!!
in the limit of 𝑛 to infinity, that is, lim!→! !( !!! ! ! !!! ! ) = 0 for 𝑝 > 2.
2𝑛 − 1
!→! 6 𝑛 − 1 ! − 𝑛 − 2
lim
!
=
1
2𝑛 − 1
lim
6 !→! 𝑛 − 1 ! − 𝑛 − 2
!
This limit goes to 0 if the denominator goes to ∞. To show this, we first define 𝑝 the
floor of 𝑝 and 𝑒 the remaining fractional part of 𝑝 − 𝑝 .
𝑛−1
!
− 𝑛−1
!
= 𝑛−1
> 𝑛−2
Because 𝑛 − 1
!
!
𝑛−1
!
𝑛−1
− 𝑛−1
!!!!
!!! ! ( !!! ! ! !!! ! )
!
!
!
!
− 𝑛−2
− 𝑛−2
!
𝑛−2
> 𝑛 − 2 !( 𝑛 − 1
!
𝑛−2
!
!
= 𝑛 − 2 !( 𝑛 − 1
− 𝑛−2
!
!
− 𝑛−2
), showing that
goes to zero in limit of n to infinity indicates that
!!!!
!!! ! ! !!! !
to zero in limit of n to infinity.
lim 𝑛 − 2
!→!
𝑛−1
!
!
=∞
− 𝑛−2
!
! !!
= 𝑛 ! − 𝑝 𝑛 ! !! +
(𝑎! 𝑛! ) + −1
!
!!!
! !!
− 𝑛 ! − 2 𝑝 𝑛 ! !! +
𝑏! 𝑛! + −2
!
!!!
! !!
= 𝑝 𝑛 ! !! +
(𝑐! 𝑛! ) +
−1
!
!
− −2
!!!
2𝑛 − 1
!→! ( 𝑛 − 1 ! − 𝑛 − 2
lim
!
)
= lim
!→!
2𝑛 − 1
𝑝 𝑛 ! !! +
! !!
!
!!! (𝑐! 𝑛 )
+
−1
!
− −2
!
0 𝑖𝑓 𝑝 > 2
= lim
𝑙 𝐻𝑜𝑝𝑖𝑡𝑎𝑙 = ∞ 𝑖𝑓 𝑝 < 2
! !!
!→! 𝑝 ( 𝑝 − 1)𝑛 ! !! +
!)
(𝑑
𝑛
!
1 𝑖𝑓 𝑝 = 2
!!!
2
!
25 !
goes
)
for some 𝑎! , 𝑎! , … , 𝑎 ! !! , 𝑏! , 𝑏! , … , 𝑏 ! !! , 𝑐! , 𝑐! , … , 𝑐 ! !! and 𝑑! , 𝑑! , … , 𝑑 ! !!
It is now shown that lim!→!
lim!→!
!!!!
!!! ! ! !!! !
!!!!
!!! ! ( !!! ! ! !!! ! )
= 0 𝑖𝑓 𝑝 > 2. From this follows that
= 0 𝑖𝑓 𝑝 > 2.
This proves that for 𝑝 > 2 there is some maximum for 𝑛 for the star network to be
pairwise stable.
Proof of Proposition 3 In minimally connected networks, the aggregate benefits are
equal, but the costs depend on the distribution of links over the players. There are
always at least two players that have one connection. Looking at the remaining players
and their number of links, I have to show that the optimal case is that each player then
has two links, which forms a line network. The costs for the number of links have to be
minimized under the conditions that there are 2(𝑛 − 2) links for the remaining players
and that the number of links is a positive integer.
!!!
min
!
!!!
𝑥!!
𝑠. 𝑡.
𝑥! = 2 𝑛 − 2 𝑎𝑛𝑑 𝑥! ∈ ℕ, 𝑝 > 1
!!!
!!!
!!!
!!!
𝑥!!
𝐿 𝑥, 𝜆 =
−𝜆
!!!
𝑥! − 2 𝑛 − 2
(𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒)
!!!
!!!
!!!
𝑥!!
∇
= 𝜆∇
!!!
𝑝𝑥!!!!
𝑥!
!!!
= 𝜆 𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑛 − 2
𝑥! = 𝑥! = ⋯ = 𝑥!!!
!!!
𝑥! = 2 𝑛 − 2 ⇔ 𝑥! = 2 ∀𝑖
!!!
This minimization problem gives 𝑥 = 2 as a solution, which means the costs are
minimized when every player has two links, apart from two edge players with each one
link. This is the case only for the line network, which shows that the minimally connected
network with the highest aggregate net payoff is the line network.
26 Proof of Proposition 4 To be shown is that when the network with lines of length 𝑘 > 2
has a higher aggregate net payoffs than the network connected pairs, it always holds
that the line network is preferred over the network with lines of length 𝑘. Firstly, the
values of 𝑐 are shown for which the network with lines of 𝑘 > 2 has a higher net payoff
than the network with connected pairs.
𝑘−1 𝑘
𝑛
𝑛
− 2 + 𝑘 − 2 2! ∗ 𝑐
> 1 − 2𝑐
2
𝑘
2
𝑘𝑛
−𝑛 >
2
𝑐<
𝑛
2 + 𝑘 − 2 2!
𝑘
2
− 𝑛 ∗ 𝑐
𝑘
, 𝑘 > 2
−1
2!
Subsequently, the values of 𝑐 for which the line network has a higher net payoff than the
network with lines of length 𝑘 are shown.
𝑛−1 𝑛
− 2 + 𝑛 − 2 2! ∗ 𝑐 >
2
𝑛! − 𝑘𝑛
>
2
2 + 𝑛 − 2 2! −
𝑐<
!"
𝑘−1 𝑘
𝑛
− 2 + 𝑘 − 2 2! ∗ 𝑐
2
𝑘
𝑛
2 + 𝑘 − 2 2!
𝑘
∗𝑐
𝑘𝑛
, 𝑘 > 2
4 2! − 1
!
Since ! !! !! > ! !! !! for 𝑘 > 2 and 𝑛 ≥ 𝑘, it is clear to see that when 𝑐 is in the range
where lines of length 𝑘 > 2 are preferred over the network with connected pairs, it is
always in the range where the line network is preferred over networks with lines of
length 𝑘.
27