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
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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.
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Bachelor Thesis University of Amsterdam
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.
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
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
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
Vega-Redondo 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).
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
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
(2.1) Π! 𝑠!, 𝑠!! = ! !,!;! !!! + !{!∈! !,! }
! !,!;! !!− 𝜂! 𝑔 𝑐 !,!∈!
!∈!!(!)
where 𝐼{!∈! !,! } ∈ {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
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
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
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
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.
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
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.
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
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 Vega-Redondo’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
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
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
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.
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.
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
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 Vega-Redondo (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
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
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.
References
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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.
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 !!!! +!!(!!!)(!!!)! − 𝑛 − 2 !∗ 𝑐. Therefore the central
player has no incentive to deviate for 𝑐 < ! !!!!!!!!! !!!! . The peripheral players each have a payoff of !
!+ !!!
! − 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 <!!!!!( !!! ! − ! ! ! !!! ).
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.
lim !→! 2𝑛 − 1 6 𝑛 − 1 !− 𝑛 − 2 ! = 1 6!→!lim 2𝑛 − 1 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 − 1 ! − 𝑛 − 2 ! 𝑛 − 2 !
> 𝑛 − 2 ! 𝑛 − 1 ! − 𝑛 − 2 ! 𝑛 − 2 ! = 𝑛 − 2 !( 𝑛 − 1 ! − 𝑛 − 2 ! )
Because 𝑛 − 1 !− 𝑛 − 1 ! > 𝑛 − 2 !( 𝑛 − 1 ! − 𝑛 − 2 ! ), showing that !!!!
!!! !( !!! !! !!! !) goes to zero in limit of n to infinity indicates that
!!!!
!!! !! !!!! goes 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 ! lim !→! 2𝑛 − 1 ( 𝑛 − 1 ! − 𝑛 − 2 ! )= lim!→! 2𝑛 − 1 𝑝 𝑛! !!+ (𝑐 !𝑛!) ! !! !!! + −1 ! − −2 ! = lim !→! 2 𝑝 ( 𝑝 − 1)𝑛! !!+ (𝑑 !𝑛!) ! !! !!! 𝑙!𝐻𝑜𝑝𝑖𝑡𝑎𝑙 = 0 𝑖𝑓 𝑝 > 2 ∞ 𝑖𝑓 𝑝 < 2 1 𝑖𝑓 𝑝 = 2
for some 𝑎!, 𝑎!, … , 𝑎! !!, 𝑏!, 𝑏!, … , 𝑏! !!, 𝑐!, 𝑐!, … , 𝑐! !! and 𝑑!, 𝑑!, … , 𝑑! !!
It is now shown that lim!→! !!! !( !!!!!!!!! !!! !) = 0 𝑖𝑓 𝑝 > 2. From this follows that lim!→! !!!!!!!!! !!!! = 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 𝐿 𝑥, 𝜆 = 𝑥!! !!! !!! − 𝜆 𝑥! !!! !!! − 2 𝑛 − 2 (𝐿𝑎𝑔𝑟𝑎𝑛𝑔𝑒) ∇ 𝑥!! !!! !!! = 𝜆∇ 𝑥! !!! !!! 𝑝𝑥!!!! = 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.
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 2 − 2 + 𝑘 − 2 2! ∗ 𝑐 𝑛 𝑘> 1 − 2𝑐 𝑛 2 𝑘𝑛 2 − 𝑛 > 𝑛 𝑘 2 + 𝑘 − 2 2! − 𝑛 ∗ 𝑐 𝑐 < 𝑘 2 2!− 1 , 𝑘 > 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 − 2 + 𝑛 − 2 2! ∗ 𝑐 > 𝑘 − 1 𝑘 2 − 2 + 𝑘 − 2 2! ∗ 𝑐 𝑛 𝑘 𝑛!− 𝑘𝑛 2 > 2 + 𝑛 − 2 2! − 𝑛 𝑘 2 + 𝑘 − 2 2! ∗ 𝑐 𝑐 < 𝑘𝑛 4 2!− 1 , 𝑘 > 2
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.
