Sequential network formation : formation of financial networks

Sequential Network Formation

Formation of Financial Networks

Bachelor thesis Nicole Gertsen

Studentnumber: 6173071 December 20, 2013

Bsc Econometrie en Operationele Research University of Amsterdam

Abstract

In literature of network formation theory it is frequently assumed that players make simultaneous announcements of intended links. All players choose at the same time. In previous research by Goyal and Vega-Redondo (2007) the authors show that in the case of simultaneous decision making the star structure is the most important pairwise stable and efficient network. In practice however, the assumption that players choose simultaneously, is not realistic. How one agent acts, depends on what another agent has done in de past. Therefore, in this thesis the assumption that all players make simultaneous announcements of intended links is rejected and hence, a new assumption is introduced: players choose in an ex ante known order – they choose sequentially. In order to evaluate what happens under rational decision-making when, instead of simultaneous choices, players choose sequentially, the Subgame Perfect Nash Equilibrium is determined. This thesis shows that there is always one unique equilibrium structure for any parameter value, while in the model of Goyal and Vega-Redondo the empty network can emerge as a second stable network (next to the star), for the same parameter values. In addition, for the sequential game with three players, a ‘first-player-advantage’ emerges. For low costs, the central player in the star will either be player 1 or player 2.

Samenvatting

In veel gevallen in de literatuur van netwerkformatietheorie wordt aangenomen dat spelers hun voorgenomen connecties simultaan bepalen. Alle spelers kiezen op hetzelfde moment. In vorig onderzoek van Goyal en Vega-Redondo (2007) laten de auteurs zien dat als spelers simultane beslissingen maken, de sterstructuur het belangrijkste paarsgewijs stabiele en efficiënte netwerk is. Echter, in de praktijk is het niet realistisch om aan te nemen dat spelers simultaan kiezen. Hoe een agent reageert hangt af van wat vorige agenten in het verleden besloten hebben. Daarom wordt in deze scriptie de aanname dat spelers voorgenomen connecties simultaan bepalen, verworpen. Spelers kiezen in een van te voren bekende volgorde; zij kiezen sequentieel. Om te zien wat er gebeurt wanneer spelers sequentieel in plaats van simultaan kiezen, wordt het Subgame Perfect Nash Evenwicht bepaald. Het blijkt dat er altijd een unieke structuur is voor een willekeurige parameterwaarde, waar in het model van Goyal en Vega-Redondo het lege netwerk zich voordoet als tweede stabiele uitkomst (naast de ster), voor dezelfde parameterwaarden. Ook doet een ‘first-player-advantage’ zich voor wanneer het spel slechts drie spelers bevat. Wanneer de kosten laag zijn, zal de centrale speler in de ster of speler 1 of speler 2 zijn.

Table of Contents

1. Introduction ... 4

2. The model ... 7

2.1 Basic definitions ... 7

2.2 Goyal and Vega-Redondo (2007) ... 7

2.3 The sequential formation game ... 8

2.4 Stability, Equilibrium and Efficiency ... 10

3. Analysis ... 12

3.1 Efficiency ... 12

3.2 Pairwise Stability ... 13

3.3 Pairwise Equilibrium ... 14

3.4 Subgame Perfect Nash Equilibrium ... 15

3.4.1 The value of c between 0 and 1₆ ... Error! Bookmark not defined. 3.4.2 The value of c between 1₆ and 1₂ ... Error! Bookmark not defined. 3.4.3 The value of c between 1₂ and 2₃ ... Error! Bookmark not defined. 3.4.4 The value of c between 2₃ and 5₆ ... Error! Bookmark not defined. 3.4.5 The value of c higher than 5₆ ... Error! Bookmark not defined. 3.5 Network structures in the sequential game ... 20

3.5.1 Comparison of Theorem 3 and Theorem 4 for 𝑛𝑛 = 3 ... 20

3.5.2 R program ... 21 4. Conclusion ... 22 References ... 23 Appendix ... 24 Proof Theorem 1.. ... 24 Proof Theorem 2.. ... 26 Proof Theorem 3.. ... 27

Example with arrows for 𝑛𝑛 = 3 ... 28

1. Introduction

In recent years there have been many shocks and crisis situations in financial markets. The extent and depth of these shocks seem to be strongly related to the topology of the web of relationships linking banks and financial institutions in the system (Caldarelli et al., 2013). The financial crisis that started in the summer of 2007, was initiated because packaged securities of bundled mortgages lost their value in a high pace caused by a stagnant housing market in the United States of America. Quickly, financial institutions were facing significant difficulties and hundreds of billions were depreciated or written off on purchased securities. However, there was considerable uncertainty in determining which financial institutions were in trouble. As a consequence of this crisis, liquidity in money markets came to a hold and banks stopped lending each other money – trust had evaporated. Because of this reputational uncertainty and illiquidity in the banking and related financial networks, many banks were either nationalized or declared bankrupt or were taken over by competitors. Many economists consider the events of the last few years as the worst financial crisis since the Great Depression. The credit crisis shows that network relationships between banks, insurance and re-insurance companies, and other financial institutions are very important factors in determining how financial stability and trust are determined. This is one of the main reasons why network formation has become such an active research area over the last few years.

One well-known network-formation model is the model of Goyal and Vega-Redondo (2007). In this model, every pair of players (banks) can undertake a transaction if the players involved have a connection, and it is assumed that this transaction creates one unit surplus. Next, a direct connection between two players splits the unit surplus equally, while through an indirect connection the intermediate players also get an equal share of the surplus. In summary, there are three main goals associated with forming links; to create surplus, to gain intermediation rewards, and to circumvent others who are trying to become an intermediate link. In their model the agents play a network formation game where every one of them makes a simultaneous announcement of intended links, such as Cournot in quantity oligopolies. The authors assume that all links are formed simultaneously, i.e. all at the same time. However, in financial markets’ practices this assumption can be considered to be unrealistic. A more realistic assumption would take into account that how one agent acts, depends on what another agent has done in de past.

This thesis focuses on what would happen if instead of making a simultaneous choice, the agents make their choice sequentially, similar to the Stackelberg model in quantity oligopolies. This assumption makes the model more realistic, because in practice the decisions agents make,

depend on past decisions made by other agents. Under this assumption, there are leaders and followers in network formation decision-making. A follower will choose the option that gives the highest outcome for him, given the choices of his leaders. A leader knows ex ante which option followers choose and will act in a way that his outcome is optimal. As such, the game is solved backwards. To illustrate this concept, consider the following example from standard game theory. There are two players, a leader and a follower, and both have two options. The first player, the leader, can choose ‘left’ or ‘right’ and the second player, the follower, can choose ‘up’ or ‘down’. Given the choice that player 1 chooses ‘left’, player 2 will decide to choose option ‘up’ (in this case ‘up’ gives a higher outcome than the option ‘down’) and given the choice that player 1 chooses ‘right’, player 2 chooses ‘down’. To decide which path will be chosen eventually, the outcomes of ‘left-up’ and ‘right-down’ are compared. The option with the highest outcome for player 1 will be the final path.

When there are reasonable linking costs in the model of Goyal and Vega-Redondo (2007), the equilibrium turns out to be a star structure. In this structure there is one central player connecting all of the other players. The main objective of this thesis is to explore whether this equilibrium is also reached if players choose sequentially instead of making simultaneous decisions.

This thesis considers a limited number of players, in this case three, because the number of options will increase rapidly when more players are added. Subsequently, the number of players could be increased and a solution could be found by utilizing a computer program. In the sequential game, the order is known and player 1 chooses first, player 2 chooses second and player 3 chooses last. The payoff function that Goyal and Vega-Redondo use, is considered to also be applicable to this model. Players who form a link will share a unit surplus, but it also creates an expense to form this link. This expense is based on a fixed cost and is paid by each player for each established link. Additionally, the model also includes a factor to account for the intermediate rents. These intermediate rents arise as a consequence of an indirect connection. This thesis is related to research that studies network formation in a financial and economic context. Jackson and Wolinsky (1996) studied the social and economic networks of self-interested individuals who can form or sever links. They characterize stable and efficient networks and show that there does not always exist a stable network that is efficient. Their conclusion is that no fixed allocation rule will ensure that at least one stable graph is efficient for every value function. Since this thesis describes an adjustment of the paper of Goyal and Vega-Redondo (2007), this paper is discussed more explicitly. In fact, Chapter 2.2 of this thesis consists of a description of their model; the outcomes are subsequently reviewed in Chapter 3.

Currarini and Morelli (2000) explore the effects of sequential demands on network formation. They analyze the formation process of a network as a game where players make their decisions sequentially. The authors are interested in situations in which it is impossible to pre-assign a fixed number of links to a network. In this situation, players propose links and demand payoff. The distribution of payoff is therefore endogenous and the formation of networks turns out to be a bargaining process, in which the demand for payoff of participation is a critical variable. They show that if the efficient networks connect all players in some way, the sequential network formation process with endogenous payoff division will lead all equilibria to be efficient. Under the made assumptions, this main result of this paper does not correspond to the initial expectations of this thesis, as the hypothesis is that not all equilibriums are efficient. In most cases it would be better earning no payoff than negative payoff. These findings also show that the sequential structure alone, without endogenous payoff division, would not be sufficient. Van der Leij, in ‘t Veld and Hommes (2013) discuss another interesting concept: best-response dynamics. There is also a known order and in this order players consecutively try to improve their position by unilateral deviations. After the last player has improved his position, the second round starts again with player 1. The process is stopped if all players but the last player cannot improve their position. Thereby they agree to the deviation made by the last player. The outcome is a unilaterally stable network. Here, individuals are myopic. They only consider the difference in payoff when evaluating the possibility of a new strategy profile. They do not anticipate any further change that could decrease their payoff, i.e. they are bounded rational. In the case of sequential formation players do anticipate further change and are therefore fully rational.

In the next part of this research (Chapter 2), the assumptions and model used in the article of Goyal and Vega-Redondo will be explained and the most important and notable results will be described. It also shows the changes of the model when some of the assumptions are modified and introduces efficiency and pairwise stability and equilibrium. Chapter 3 presents the main results on these concepts. Furthermore, the Subgame Perfect Nash Equilibrium is determined for three players. Chapter 4 draws the final conclusions.

2. The model

Goyal and Vega-Redondo (2007) introduced the essential players. Without these players an interaction cannot take place and they assume that only essential players will earn intermediate rents. In this chapter some basic definitions are given, followed by an explanation of their model. The model is then modified in such a way that instead of players making simultaneous choices, they choose sequentially. Finally the notions of stability, equilibrium and efficiency are introduced.

2.1 Basic definitions

The network is composed of a finite set of ex ante identical players, 𝑁𝑁 = {1, 2, … , 𝑛𝑛}, where 𝑛𝑛 ≥ 3. A graph (or network) 𝑔𝑔 is a set of links that is obtained by formed links between any pair of players in 𝑁𝑁. A graph is called a complete graph when every pair of players is linked to one another, and is denoted by 𝑔𝑔𝑁𝑁_{. The set {𝑔𝑔: 𝑔𝑔 ⊆ 𝑔𝑔}𝑁𝑁_{} is the set of all possible graphs and the set} 𝑁𝑁𝑖𝑖(𝑔𝑔) denotes the set player 𝑖𝑖 has links with. Because of the bilateral assumption, a link between two players 𝑖𝑖 and 𝑗𝑗 is formed if and only if 𝑔𝑔𝑖𝑖𝑖𝑖≡ 𝑔𝑔𝑖𝑖𝑖𝑖= 1. The absence of a link is denoted by 𝑔𝑔𝑖𝑖𝑖𝑖≡ 𝑔𝑔𝑖𝑖𝑖𝑖 = 0. A path between players 𝑖𝑖 and 𝑗𝑗 exist if either 𝑔𝑔𝑖𝑖𝑖𝑖= 1 or there is a set of players {𝑖𝑖1, 𝑖𝑖2, … , 𝑖𝑖𝑛𝑛} such that 𝑔𝑔𝑖𝑖,𝑖𝑖1 = 𝑔𝑔𝑖𝑖1,𝑖𝑖2 = 𝑔𝑔𝑖𝑖2,𝑖𝑖3= ⋯ = 𝑔𝑔𝑖𝑖𝑛𝑛,𝑖𝑖 = 1. All players with whom player 𝑖𝑖 has a direct or indirect connection define the component of 𝑖𝑖 in 𝑔𝑔, which is denoted by 𝐶𝐶𝑖𝑖.

2.2 Goyal and Vega-Redondo (2007)

In the model of Goyal and Vega-Redondo all players make a simultaneous announcement of intended links. An intended link implies that player 𝑖𝑖 intends to form a link with player 𝑗𝑗. A pair of players that undertakes a transaction creates a unit surplus. A transaction can only take place if the players have a direct or indirect connection. When the players are directly connected they split the surplus equally while when they are indirectly connected, the division of the surplus depends on the competition between the intermediaries. An intermediate player is essential in case transactions cannot take place without him. Essential players and players 𝑗𝑗 and 𝑘𝑘 all get an equal share, while non-essential players between players 𝑖𝑖 and 𝑘𝑘 get a zero share of the surplus. 𝐸𝐸(𝑗𝑗, 𝑘𝑘: 𝑔𝑔) is the set of players who are essential to connect players 𝑗𝑗 and 𝑘𝑘 in the network and 𝑒𝑒(𝑗𝑗, 𝑘𝑘: 𝑔𝑔) = | 𝐸𝐸(𝑗𝑗, 𝑘𝑘: 𝑔𝑔)|. This allocation can be supported as follows: for a transaction {𝑗𝑗, 𝑘𝑘} players 𝑗𝑗 and 𝑘𝑘 and every essential player to that transaction demand 1/(𝑒𝑒(𝑗𝑗, 𝑘𝑘: 𝑔𝑔) + 2). In the

model agents have to pay a fixed cost 𝑐𝑐 for each link they establish. Listed, the (net) payoffs for player 𝑖𝑖 are given by

∏𝑖𝑖(𝑔𝑔)= ∑𝑖𝑖𝜖𝜖𝐶𝐶𝑖𝑖(𝑔𝑔)_{e(𝑖𝑖,𝑖𝑖∶ 𝑔𝑔) + 2)}1 + ∑𝑖𝑖,𝑘𝑘 𝜖𝜖 𝑁𝑁_{e(𝑖𝑖,𝑘𝑘∶ 𝑔𝑔) + 2)} 𝐼𝐼{𝑗𝑗 𝜖𝜖 𝐸𝐸 (𝑗𝑗,𝑘𝑘)} − 𝜂𝜂𝑖𝑖(𝑔𝑔)𝑐𝑐. (1)

The first term represents the surplus for player 𝑖𝑖 from connecting with player 𝑗𝑗. The second term represents the surplus for player 𝑖𝑖 from being an essential player for players 𝑗𝑗 and 𝑘𝑘, where 𝐼𝐼{𝑖𝑖 𝜖𝜖 𝐸𝐸 (𝑖𝑖,𝑘𝑘)}∈ {0,1} denotes an indicator function which specifies whether player 𝑖𝑖 is essential for players 𝑗𝑗 and 𝑘𝑘. The 𝜂𝜂𝑖𝑖(𝑔𝑔) in the last term denotes the number of players with whom player 𝑖𝑖 has a connection.

2.3 The sequential formation game

In this thesis the assumption that every player makes a simultaneous announcement of intended links is rejected. For a sequential game, the order in which players choose is known. After a player has made his decision he sticks to it. This is called a pure strategy. Each graph 𝑔𝑔 ⊆ 𝑔𝑔𝑁𝑁 _{has its own payoff. The payoff division for each individual player is calculated by means} of (1) and the total payoff is denoted as (∏ ,1 ∏ , … , ∏ ).2 𝑛𝑛

Each player has 2𝑛𝑛−1 _{options. For example, in the case where 𝑛𝑛 = 3, a player (for example} player 1) has four options: {∅} = propose no links with the other players, {2} = propose a link with player 2, {3} = propose a link with player 3 and {2,3} = propose links with both players 2 and 3. The total set of strategy profiles is denoted as 𝑆𝑆 = ({𝑠𝑠1}, {𝑠𝑠2}, … , {𝑠𝑠𝑛𝑛}), where 𝑠𝑠𝑖𝑖 represents the choices made by player 𝑖𝑖. For example, in the case where 𝑛𝑛 = 3, a possible set is 𝑆𝑆 = ({2}, {1,3}, {∅}). A concept of game theory where the optimal outcome of a game is determined is Nash Equilibrium: no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. Definition 1 gives a standard Nash Equilibrium.

Definition 1.

A strategy profile 𝒔𝒔𝑁𝑁𝐸𝐸∗ = �𝑠𝑠𝑁𝑁𝐸𝐸,1∗ , 𝑠𝑠𝑁𝑁𝐸𝐸,2∗ , … , 𝑠𝑠𝑁𝑁𝐸𝐸,𝑛𝑛∗ � is a Nash Equilibrium if ∏ (𝑔𝑔�𝑠𝑠𝑖𝑖 𝑁𝑁𝐸𝐸,𝑖𝑖∗ , 𝑠𝑠𝑁𝑁𝐸𝐸,−𝑖𝑖∗ �) ≥ ∏ (𝑔𝑔�𝑠𝑠𝑖𝑖 𝑁𝑁𝐸𝐸,𝑖𝑖, 𝑠𝑠𝑁𝑁𝐸𝐸,−𝑖𝑖∗ �) for all 𝑠𝑠𝑁𝑁𝐸𝐸,𝑖𝑖∈ 𝑆𝑆𝑁𝑁𝐸𝐸,𝑖𝑖 and for all 𝑖𝑖 ∈ 𝑁𝑁.

A link is formed if and only if both players intend to form the link. It then follows directly, that if every player announces that he wants to form no links, then a best response of player 𝑖𝑖 is to

announce that he also wants to form no links. In other words, the empty network is always a Nash Equilibrium.

To decide which possible 𝑆𝑆 will arise in the case of sequential network formation, all Subgame Perfect Nash Equilibria (SPNE) are calculated. Given any history of actions, ℎ, the remaining game can be considered as an extensive game on its own. It is called a subgame after the history. A Subgame Perfect Nash Equilibrium is an equilibrium such that players’ strategies constitute a Nash equilibrium in every subgame of the original game1_{. It may be found by backward}

induction; an iterative process for solving finite extensive form or sequential games. First, one determines the optimal strategy of the player who makes the last move of the game. Then, the optimal action of the next-to-last moving player is determined taking the last player's action as given. The process continues in this way backwards in time until all players' actions have been determined.

Definition 2.

A strategy profile 𝒔𝒔∗ _{is a Subgame Perfect Nash Equilibrium (SPNE) if it represents a Nash}

Equilibrium of every subgame of the original game:

𝒔𝒔∗_{= (𝑠𝑠}

1∗, 𝑠𝑠2∗, … , 𝑠𝑠𝑛𝑛∗) is a Subgame Perfect Nash Equilibrium if ∏ (𝑔𝑔(𝒔𝒔𝑖𝑖|ℎ ∗|ℎ)) ≥ ∏ (𝑔𝑔(𝑠𝑠𝑖𝑖|ℎ 𝑖𝑖, 𝑠𝑠−𝑖𝑖∗ |ℎ))

for all 𝑠𝑠𝑖𝑖 ∈ 𝑆𝑆𝑖𝑖|ℎ and for all 𝑖𝑖 ∈ 𝑁𝑁 and all ℎ.

All choices a player makes are represented by means of arrows (see the example in the Appendix). A link is established if and only if there are arrows in both directions between two players. When there is only one arrow directed in one way, the proposed link will not be formed. Therefore the following definition holds true.

Definition 3.

A strategy profile 𝑠𝑠𝑖𝑖 = {… , 𝑖𝑖, … } = {𝑠𝑠𝑖𝑖𝑘𝑘<𝑖𝑖, 𝑖𝑖, 𝑠𝑠𝑖𝑖𝑘𝑘>𝑖𝑖} for player 𝑗𝑗 ∈ 𝑁𝑁, where 𝑖𝑖, 𝑘𝑘, … ∈ 𝑁𝑁 and 𝑖𝑖 < 𝑗𝑗 and 𝑖𝑖 = 1 if player 𝑗𝑗 intends to form a link with player 𝑖𝑖 and 𝑖𝑖 = 0 if he does not, is weakly dominated

by 𝑠𝑠𝑖𝑖= {𝑠𝑠𝑖𝑖𝑘𝑘<𝑖𝑖, 𝑠𝑠𝑖𝑖𝑘𝑘>𝑖𝑖} if the strategy profile for player 𝑖𝑖 does not include 𝑗𝑗: 𝑗𝑗 ∉ 𝑠𝑠𝑖𝑖.

The notion of weakly dominated strategies is useful because it induces the graph (see the example in the Appendix). Consider that for every proposed link a player pays a small cost, 𝜀𝜀. This cost 𝜀𝜀 is not explicitly incorporated in the model. When a player intends to form links with other players and one of these players has already chosen his strategy profile and does not intend to form a link with the player who’s turn it is now, the strategy profile for this player

1 _{Description by gametheory.net.}

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without a proposed link with the player who has already chosen, dominates the profile in which he does proposes a link to this player. His cost will be 𝜀𝜀 lower.

2.4 Stability, Equilibrium and Efficiency

Jackson and Wolinsky (1996) introduced pairwise stability. First, they let 𝑖𝑖𝑗𝑗 be denoted as the subset of 𝑁𝑁 containing 𝑖𝑖 and 𝑗𝑗 and is referred to as the link 𝑖𝑖𝑗𝑗. If 𝑖𝑖𝑗𝑗 ∈ 𝑔𝑔 then players 𝑖𝑖 and 𝑗𝑗 are directly connected, while 𝑖𝑖𝑗𝑗 ∉ 𝑔𝑔 means that players 𝑖𝑖 and 𝑗𝑗 are not directly connected. Further, they let 𝑔𝑔 + 𝑖𝑖𝑗𝑗 be denoted as the network obtained by adding link 𝑖𝑖𝑗𝑗 to the existing network 𝑔𝑔 and 𝑔𝑔 − 𝑖𝑖𝑗𝑗 as the network obtained by deleting the link 𝑖𝑖𝑗𝑗 from the existing network.

Definition 4.

A network 𝑔𝑔 is pairwise stable if:

i. for every 𝑖𝑖𝑗𝑗 ∈ 𝑔𝑔, ∏ (𝑔𝑔) ≥ ∏ (𝑔𝑔 − 𝑖𝑖𝑗𝑗) 𝑖𝑖 𝑖𝑖 and ∏ (𝑔𝑔) ≥ ∏ (𝑔𝑔 − 𝑖𝑖𝑗𝑗)𝑖𝑖 𝑖𝑖 , and

ii. for every 𝑖𝑖𝑗𝑗 ∉ 𝑔𝑔, ∏ (𝑔𝑔) < ∏ (𝑔𝑔 + 𝑖𝑖𝑗𝑗) 𝑖𝑖 𝑖𝑖 then ∏ (𝑔𝑔) > ∏ (𝑔𝑔 + 𝑖𝑖𝑗𝑗)𝑖𝑖 𝑖𝑖 .

A network is pairwise stable when (i.) having the link 𝑖𝑖𝑗𝑗 leads to higher payoffs for both player 𝑖𝑖 and player 𝑗𝑗 than not having the link, and when (ii.) not having the link 𝑖𝑖𝑗𝑗 one of the players is off strictly worse when adding the link while the other player strictly benefits from it. This means that no pair of players wants to add an extra link and no individual player wants to remove one. 2

When the concept of pairwise stability is used, a network is said to be stable if no player wants to remove a link, or no pair of players want to add one. In this case a player can only eliminate just one link. A set of possible deviations is ruled out. Only the deletion of a single link or the addition of a single link is considered. Consequently, a pairwise stable network is not a Nash Equilibrium as a player might want to delete more than one link. In some games the deviation in which several links are deleted together is profitable. This possibility is illustrated by the following example:

Suppose 𝑛𝑛 = 4. Assume that the payoffs satisfy (1). In a star network the payoffs for the central player are ∏ (𝑔𝑔∗₎

𝑖𝑖 =5₂− 3𝑐𝑐 and ∏ (𝑔𝑔𝑖𝑖 ∗)=7₆− 𝑐𝑐 for the peripheral players. In every network 𝑔𝑔 in which player 𝑖𝑖 has no links the payoff to player 𝑖𝑖 equals ∏ (𝑔𝑔)𝑖𝑖 = 0. In a network 𝑔𝑔 where one

2 _{In ‘Structural holes in social networks’ the authors Goyal and Vega-Redondo (2007) introduce bilateral}

equilibrium (BE) to study the architecture of networks that are strategically stable. It is a refinement of

pairwise stability.

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player has two links, two players have one link and one player has no links, the payoffs to the player with two links are 4₃− 2𝑐𝑐, the payoffs with one link are 5₆− 𝑐𝑐, while the payoffs with no links are 0. Assume 𝑐𝑐 = 1. The star network is pairwise stable, since no player has an incentive to delete just one link. However, the central player would strictly profit from deleting all links, because ∏ (𝑔𝑔∗₎

𝑖𝑖 = −1₂< 0. In this case he gets the highest possible payoff of 0.

To account for these deviations the notion of pairwise equilibrium is introduced. It addresses the concern of deleting several links directly by supplementing the idea of Nash equilibrium with the requirement that no pair of players wishes to form an additional link. The idea of pairwise stability is extended with this new concept.

Definition 5.

A network 𝑔𝑔∗ _{can be sustained in a pairwise equilibrium if}

i. There is a Nash equilibrium 𝑠𝑠∗ _{which yields 𝑔𝑔}∗_,

ii. For any 𝑔𝑔𝑖𝑖𝑖𝑖(𝑠𝑠∗) = 0, ∏ (𝑔𝑔(𝑠𝑠𝑖𝑖 ∗) + 𝑔𝑔𝑖𝑖𝑖𝑖) > ∏ (𝑔𝑔(𝑠𝑠𝑖𝑖 ∗)) ⇒ ∏ (𝑔𝑔(𝑠𝑠𝑖𝑖 ∗) + 𝑔𝑔𝑖𝑖𝑖𝑖)< ∏ (𝑔𝑔(𝑠𝑠𝑖𝑖 ∗)). Pairwise equilibria are robust to bilateral agreed one-link formation, and to unilateral multi-link severance. In other words, 𝑔𝑔∗ _{is a pairwise equilibrium if and only if it is both pairwise stable} and is a pure strategy Nash equilibrium outcome3_.

Finally, the notion of efficiency is introduced. Let 𝐺𝐺(𝑁𝑁) be denoted as the set of all possible networks of size 𝑛𝑛.

Definition 6.

A network 𝑔𝑔 is efficient if ∑𝑖𝑖∈𝑁𝑁∏ (𝑔𝑔) ≥𝑖𝑖 ∑𝑖𝑖∈𝑁𝑁∏ (𝑔𝑔𝑖𝑖 ′) for every 𝑔𝑔′ ∈ 𝐺𝐺(𝑁𝑁).

In words this means that a network is efficient if there is no other network structure that generates a higher payoff. When a network is stable but not efficient, or the other way around, it can be useful for higher authorities to take measures so that some players will form/remove links and a stable and efficient network is achieved.

3 _{Calvó-Armengol and Ilkılıç (2008).}

11

3. Analysis

In this chapter the values for 𝑐𝑐 for which a network structure is efficient, pairwise stable and in pairwise equilibrium, is analysed. Both simultaneous choices and the sequential formation game are studied. Three types of network structures are analysed: the complete network, the star network and the empty network (see Figure 1.). Every node indicates a player and a line between two nodes indicates a direct link. In the complete network all players are connected with all players, so that 𝑔𝑔𝑖𝑖𝑖𝑖= 1 ∀ 𝑖𝑖 ≠ 𝑗𝑗 and 𝜂𝜂𝑖𝑖 = 𝑛𝑛 − 1. In the star network there is one player who connects all the other peripheral players; he is the central player and has (𝑛𝑛 − 1) links. In the empty network there are no links between any of the players, so that 𝑔𝑔𝑖𝑖𝑖𝑖 = 0 ∀ 𝑖𝑖 ≠ 𝑗𝑗 and 𝜂𝜂𝑖𝑖 = 0.

Figure 1. Main architectures; complete, star and empty.

3.1 Efficiency

The conditions for which a network is efficient are analysed first. The conditions for the complete, the star and the empty network are summarised in Theorem 1. 4

Theorem 1.

Assume the payoff for every player is given by (1). The following network structures are efficient:

i. The complete network for no c, ii. The star network for 0 < 𝑐𝑐 <𝑛𝑛₄ iii. The empty network for 𝑐𝑐 >𝑛𝑛₄.

4 _{See the appendix for the proof of Theorem 1.}

12

As stated in Theorem 1 the complete network is never efficient. This can be argued intuitively as follows: For any costs to form a link, the benefits from forming a direct link won’t be compensated by the costs made to form the link. An indirect connection has an advantage because in this case there will only be an increase in return and no extra costs. This advantage offsets the extra gain in return when forming a direct link.

The star network is efficient for a large range of 𝑐𝑐. It turns out that as 𝑛𝑛 increases the range for which the star network is efficient also increases. Moreover, if 𝑐𝑐 >𝑛𝑛₄ the total return of the star network is negative. So when the number of players 𝑛𝑛 increases, the smaller the range of 𝑐𝑐 for which it has a negative return. Besides, the total payoff in this network is maximal. Every player, excluding the central player, has to pay the costs of forming just one link, while the distance between all players is at the most two links. Therefore, the total costs in a star structure are minimal. In this instance, the network is considered to be minimally connected (Goyal, 2007). This is why the star structure is the most important efficient network for low costs.

Finally, it turns out that the empty network is efficient for high values of 𝑐𝑐. When the costs for a direct link increases such that a star network has a negative total return, the empty network is the only efficient network.

The total payoffs in the sequential game do not differ from the payoffs in the model of Goyal and Vega-Redondo. That is why Theorem 1 also holds true in the case of the sequential formation game.

3.2 Pairwise Stability

The conditions for which a network is pairwise stable is analysed next. For the complete, the star and the empty network these conditions are summarised in Theorem 2. 5

Theorem 2.

Assume the payoff for every player is given by (1). The following network structures are pairwise stable:

i. The complete network for �0 < 𝑐𝑐 <16, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 = 3 𝑛𝑛𝑓𝑓 𝑐𝑐, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 ≥ 4

5 _{See the appendix for the proof of Theorem 2.}

13

ii. The star network for 1₆< 𝑐𝑐 <1₂+1₃(𝑛𝑛 − 2),

iii. The empty network for 𝑐𝑐 >1₂.

The complete network is a cycle network in the case of 𝑛𝑛 = 3. It is pairwise stable for low costs. This can be argued intuitively as follows: A player who removes a link has lower costs, but also gains a lower payoff. By removing the link, there is one player who becomes essential to the whole network and the other two players have to cede a part of their payoff to this player. The complete network is never pairwise stable for large 𝑛𝑛 (larger than 3). A player who removes a link has lower costs ((𝑛𝑛 − 2) instead of (𝑛𝑛 − 1)) but still gains the same profit (𝑛𝑛−1₂ is equal to 𝑛𝑛−2

2 + 1

2). By removing the link there are still no essential players. This is why in the complete network every player will want to remove a link and therefore this network can never be pairwise stable.

The star structure turns out to be pairwise stable for a large range of parameter values. The network is pairwise stable if the central player does not remove a link and a peripheral player does not add an extra link nor removes a link. In a star network, the payoffs to the central player are higher than if the player were to keep only 𝑘𝑘 of the (𝑛𝑛 − 1) links (∏ =𝑖𝑖 𝑛𝑛−1₂ +1₃(𝑛𝑛−1)(𝑛𝑛−2)₂ − (𝑛𝑛 − 1)𝑐𝑐 instead of ∏ =𝑖𝑖 𝑘𝑘₂+1₃𝑘𝑘(𝑘𝑘−1)₂ − 𝑘𝑘). The peripheral players in the star earn more than when they form an additional link (1₂+(𝑛𝑛−2)₃ − 𝑐𝑐 instead of 1₂+1₂+(𝑛𝑛−3)₃ − 2𝑐𝑐, which will be lower than the former if 𝑐𝑐 >1₆). The payoff for the central player is higher than the payoff for the peripheral players as a consequence of the introduction of the essential player. In the star structure this player connects all other players and is therefore essential to the whole network, which in turn results in a higher payoff. Also, when the number of players increases, the individual payoff for all players increases. The addition of another link with the central player only, increases his payoff as well as the payoffs for the peripheral players because they are now indirectly connected to an additional player.

3.3 Pairwise Equilibrium

The conditions for which a network can be sustained in a pairwise equilibrium is analysed next. For the complete, the star and the empty network these conditions are summarised in Theorem 3. 6

6 _{See the Appendix for the proof of Theorem 4.}

14

Theorem 3.

Assume the payoff for every player is given by (1). The following network structures can be sustained in pairwise equilibrium:

i. The complete network for � 0 < 𝑐𝑐 <16, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 = 3 𝑛𝑛𝑓𝑓 𝑐𝑐, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 ≥ 4

ii. The star network for 1

6< 𝑐𝑐 < 1 2+

1

6(𝑛𝑛 − 2),

iii. The empty network for 𝑐𝑐 >1₂.

The values for 𝑐𝑐 for which the several networks are sustainable in pairwise equilibrium can be interpreted as follows: The complete network is sustainable in pairwise equilibrium, when the complete network is also pairwise stable; as was stated in chapter 2.4, for a network to sustain in pairwise equilibrium it needs to be a pairwise stable network. If the network is not pairwise stable, the network can never be sustainable in pairwise equilibrium.

The star network is sustainable in pairwise equilibrium if the payoffs for the central player are positive. In the former case of pairwise stability, this payoff could be negative for some values of 𝑐𝑐. Because the player can now delete several links, he will do so if his payoff turns out to be negative. He would rather have a payoff of 0 than a negative one. This is why the star network arises less frequently when the concept of pairwise equilibrium is used instead of pairwise stability.

Therefore, the empty network is expected to arise more frequently in the case of pairwise stability. But as was determined above, for a network to sustain in pairwise equilibrium, it also has to be pairwise stable. So the minimum value for 𝑐𝑐 for which the network is sustainable in pairwise equilibrium equals the value for 𝑐𝑐 for which the network is pairwise stable.

3.4 Subgame Perfect Nash Equilibrium

In this thesis the star structure where player 1 is the central player is expected to be the most important network, because player 1 may have a ‘first-player-advantage’. Such an advantage may stem from the fact that the first player can gain control of resources that followers may not be able to match. In the following tables the Subgame Perfect Nash Equilibria are determined. In tables 1a-1c the best responses of player 3 are listed given the choices players 1 and 2 made. Table 2 (see Appendix) consists of an overview of all possible outcomes with 𝑛𝑛 = 3.

Table 1a. Best choice player 3, given the choices of players 1 and 2 and 0 < 𝑐𝑐 < 1₆ and 1₆ < 𝑐𝑐 < 1₂.

3.4.1 The value of c between 0 and 𝟏𝟏_𝟔𝟔

(Table 1a). For 𝑠𝑠1= {∅} player 2 will choose either {3} or {1,3}, because 1₂− 𝑐𝑐 > 0. By means of definition 1, 𝑠𝑠2= {3} weakly dominates 𝑠𝑠2= {1,3}. When player 1 wants to form a link with player 2, 𝑠𝑠1= {2}, player 2 chooses the outcome of 4₃− 2𝑐𝑐 over 1₂− 𝑐𝑐 and over 0; it is highest when 0 < 𝑐𝑐 <1₆. For 𝑠𝑠1= {3} player 2 will choose either {3} or {1,3} whereas 5₆> 0 and again 𝑠𝑠2= {3} weakly dominates 𝑠𝑠2= {1,3}. Finally when player 1 chooses option {2,3} player 2 will Player 1 Player 2 Best choice

Player 3 Payoff 0 < 𝒄𝒄 < 𝟏𝟏

𝟔𝟔 Best choice _{Player 3} Payoff 𝟏𝟏 𝟔𝟔 < 𝒄𝒄 < 𝟏𝟏 𝟐𝟐 {∅} {∅} Either (0, 0, 0) Either (0, 0, 0) {∅} {1} Either (0, 0, 0) Either (0, 0, 0) {∅} {3} ({2}, {1,2}) _(0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) ({2}, {1,2}) (0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) {∅} {1,3} ({2}, {1,2}) _(0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) ({2}, {1,2}) (0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) {2} {∅} Either (0, 0, 0) Either (0, 0, 0) {2} {1} Either ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) Either ( 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) {2} {3} ({2}, {1,2}) (0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) ({2}, {1,2}) (0, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) {2} {1,3} ({2}, {1,2}) ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) ({2}, {1,2}) ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) {3} {∅} ({1}, {1,2}) ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) ({1}, {1,2}) ( 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) {3} {1} ({1}, {1,2}) ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) ({1}, {1,2}) ( 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏 𝟐𝟐− 𝒄𝒄) {3} {3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {3} {1,3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {2,3} {∅} ({1}, {1,2}) (𝟏𝟏_𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏_𝟐𝟐− 𝒄𝒄) ({1}, {1,2}) (𝟏𝟏_𝟐𝟐− 𝒄𝒄, 𝟎𝟎, 𝟏𝟏_𝟐𝟐− 𝒄𝒄) {2,3} {1} ({1}, {1,2}) ₍𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) ({1}, {1,2}) ₍𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) {2,3} {3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {2,3} {1,3} {1,2} (𝟏𝟏 − 𝟐𝟐𝒄𝒄, 𝟏𝟏 − 𝟐𝟐𝒄𝒄, 𝟏𝟏 − 𝟐𝟐𝒄𝒄) {1} or {2} ( 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) or (𝟓𝟓_𝟔𝟔− 𝒄𝒄,𝟒𝟒_𝟑𝟑− 𝟐𝟐𝒄𝒄,𝟓𝟓_𝟔𝟔− 𝒄𝒄) 16

choose {1,3}. Now, which total outcome 𝑆𝑆 will be chosen for 0 < 𝑐𝑐 <1₆ is up to player 1. He can choose either an outcome of 0, 5₆− 𝑐𝑐 or 1 − 2𝑐𝑐 and in this case the latter one is the SPNE, so 𝑆𝑆 = ({2,3}, {1,3}, {1,2}).

3.4.2 The value of c between 𝟏𝟏_𝟔𝟔 and 𝟏𝟏_𝟐𝟐

(Table 1a). When 𝑠𝑠1= {∅} player 2 will choose either 𝑠𝑠2= {3} or 𝑠𝑠2= {1,3}. Again 𝑠𝑠2= {3} weakly dominates 𝑠𝑠2= {1,3}. The outcome for player 2 when 𝑠𝑠1 = {2} is highest when he chooses {1,3} and {3} or {1,3} when 𝑠𝑠1 = {3}. When player 1 chooses to form links with both other players, player 2 will choose {1,3}. In this case there are two outcomes possible. This requires the following assumption.

Assumption 1.

Assume players choose with equal probability between multiple optimal choices, if these are not weakly dominated.

By assumption 1 the probability player 2 will have a payoff of 4₃− 2𝑐𝑐 is 50% and if he does not get this payoff, he will get 5₆− 𝑐𝑐, also with a probability of 50%. This is the second highest payoff player 2 can receive, so he will choose this option. Then again it is up to player 1. When he chooses {2,3} his payoff will be 4₃− 2𝑐𝑐 or 5₆− 𝑐𝑐, both with a probability of 50%. These payoffs are higher or equal to any other possible choice (0 or 5₆− 𝑐𝑐). So with 50% 𝑆𝑆 = ({2,3}, {1,3}, {1}) or with the other 50% 𝑆𝑆 = ({2,3}, {1,3}, {2}).

3.4.3 The value of c between 𝟏𝟏_𝟐𝟐 and 𝟐𝟐_𝟑𝟑

(Table 1b). When𝑠𝑠1= {∅} player 2 will have a payoff of 0 in any case, so it does not matter which option he chooses. In this case 𝑠𝑠2= {∅} weakly dominates all of the other strategies. When player 1 wants to form a link with player 2, 𝑠𝑠1= {2}, player 2 will choose a payoff of 4

3− 2𝑐𝑐 over a payoff of 1

2− 𝑐𝑐 or 0, which is option {1,3}. When 𝑠𝑠1= {3} and 1 2< 𝑐𝑐 <

2

3 player 2 his outcome will be {3} or {1,3}, where {3} weakly dominates {1,3}. Finally, suppose player 1 chooses {2,3}, player 2 will choose {1} or {3} with equal probability. When he chooses one of these options he will have a payoff of 5₆− 𝑐𝑐 with 100% certainty, while if he chooses {1,3} he will have 5₆− 𝑐𝑐 or 4₃− 2𝑐𝑐, both with a probability of 50%. The payoff of 5₆− 𝑐𝑐 is larger than 4₃− 2𝑐𝑐 for the given value of 𝑐𝑐 and therefore player 2 will choose {1} or {3}. Player 1 will compare all

different payoffs; he can get 5₆− 𝑐𝑐 or 4₃− 2𝑐𝑐 or 0. Again, 5₆− 𝑐𝑐 is the highest payoff. He gains this payoff when he proposes to form a link with player 2 or with player 3 or, with a probability of 50%, when he proposes to form links with both other players. Consequently he will choose {2} or {3} and 𝑆𝑆 = ({2}, {1,3}, {2}) or 𝑆𝑆 = ({3}, {3},{1,2}).

Table 1b. Best choice player 3, given the choices of players 1 and 2 and 1

2 < 𝑐𝑐 < 2 3 and 2 3 < 𝑐𝑐 < 5 6.

3.4.4 The value of c between 𝟐𝟐_𝟑𝟑 and 𝟓𝟓_𝟔𝟔

(Table 1b). When 𝑠𝑠1= {∅} the option which player 2 chooses, will be the same as in the previous case where 1₆< 𝑐𝑐 <1₂ and 𝑠𝑠2= {∅} will weakly dominate all the other strategies. When player 1 wants to form a link with player 2, 𝑠𝑠1= {2}, player 2 will choose either {∅} or {3}, because 0 >1₂− 𝑐𝑐 and 0 >4₃− 2𝑐𝑐 when 2₃< 𝑐𝑐 <5₆. In this case 𝑠𝑠2= {∅} weakly dominates 𝑠𝑠2= {3}. For 𝑠𝑠1= {3} all payoffs for all players are 0 and again 𝑠𝑠2= {∅} weakly dominates all of Player 1 Player 2 Best choice

Player 3 Payoff 𝟏𝟏 𝟐𝟐 < 𝒄𝒄 <

𝟐𝟐

𝟑𝟑 Best choice _{Player 3} Payoff 𝟐𝟐 𝟑𝟑 < 𝒄𝒄 < 𝟓𝟓 𝟔𝟔 {∅} {∅} Either (0, 0, 0) Either (0, 0, 0) {∅} {1} Either (0, 0, 0) Either (0, 0, 0) {∅} {3} ({∅}, {1}) (0, 0, 0) ({∅}, {1}) (0, 0, 0) {∅} {1,3} ({∅}, {1}) (0, 0, 0) ({∅}, {1}) (0, 0, 0) {2} {∅} Either (0, 0, 0) Either (0, 0, 0) {2} {1} Either (𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) Either ( 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) {2} {3} ({∅}, {1}) (0, 0, 0) ({∅}, {1}) (0, 0, 0) {2} {1,3} ({2}, {1,2}) ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) ({2}, {1,2}) ( 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) {3} {∅} ({∅}, {2}) (0, 0, 0) ({∅}, {2}) (0, 0, 0) {3} {1} ({∅}, {2}) (0, 0, 0) ({∅}, {2}) (0, 0, 0) {3} {3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {∅} (0, 0, 0) {3} {1,3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {∅} (0, 0, 0) {2,3} {∅} ({∅}, {2}) (0, 0, 0) ({∅}, {2}) (0, 0, 0) {2,3} {1} ({1}, {1,2}) ₍𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) ({1}, {1,2}) ( 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) {2,3} {3} {1,2} ₍𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄) {∅} (0, 0, 0) {2,3} {1,3} {1} or {2} ₍𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) or (𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) {1} or {2} ₍𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) or (𝟓𝟓 𝟔𝟔− 𝒄𝒄, 𝟒𝟒 𝟑𝟑− 𝟐𝟐𝒄𝒄, 𝟓𝟓 𝟔𝟔− 𝒄𝒄) 18

the other strategies. Now when player 1 want to form links with both other players, player 2 can choose {1} the payoff will be 5₆− 𝑐𝑐 with a probability of 100% or he can choose {1,3} and then his payoff will be 4₃− 2𝑐𝑐 with a probability of 50% and 5₆− 𝑐𝑐 also, with a probability of 50%. So it is clear that player 2 will choose {1}, because then he will always have a positive payoff. Player 1 can get either a payoff of 0 or a negative payoff (in the case when 𝑠𝑠1= {2,3} and 𝑠𝑠2= {1}), so 𝑆𝑆 = ({∅}, {∅}, {∅}) where 𝑠𝑠1,2,3= {∅} weakly dominates all of the other strategies.

3.4.5 The value of c higher than 𝟓𝟓_𝟔𝟔

(Table 1c). For any choice of player 1, player 2 nor player 1 himself will ever have a positive payoff. So in this case 𝑆𝑆 = ({∅}, {∅}, {∅}). In other words, no one can ever have a strictly positive payoff in an SPNE.

Table 1c. Best choice player 3, given the choices of players 1 and 2 and 𝑐𝑐 > 5₆.

Player 1 Player 2 Best choice

Player 3 Payoff 𝒄𝒄 > 𝟓𝟓 𝟔𝟔 {∅} {∅} Either (0, 0, 0) {∅} {1} Either (0, 0, 0) {∅} {3} ({∅}, {1}) (0, 0, 0) {∅} {1,3} ({∅}, {1}) (0, 0, 0) {2} {∅} Either (0, 0, 0) {2} {1} Either (𝟏𝟏_𝟐𝟐− 𝒄𝒄, 𝟏𝟏_𝟐𝟐− 𝒄𝒄, 0) {2} {3} ({∅}, {1}) (0, 0, 0) {2} {1,3} ({∅}, {1}) ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) {3} {∅} ({∅}, {2}) (0, 0, 0) {3} {1} ({∅}, {2}) (0, 0, 0) {3} {3} {∅} (0, 0, 0) {3} {1,3} {∅} (0, 0, 0) {2,3} {∅} ({∅}, {2}) (0, 0, 0) {2,3} {1} ({∅}, {2}) ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) {2,3} {3} {∅} (0, 0, 0) {2,3} {1,3} {∅} ₍𝟏𝟏 𝟐𝟐− 𝒄𝒄, 𝟏𝟏 𝟐𝟐− 𝒄𝒄, 0) 19

3.5 Network structures in the sequential game

By means of the previous tables and the analysis it follows that for 𝑐𝑐 >2₃ the network is empty and that for 0 < 𝑐𝑐 <1₆ the network is complete, while for 1₆< 𝑐𝑐 <2₃ the star structure emerges. The following theorem is noted.

Theorem 4.

Assume the payoff for every player is given by (1), assumption 1 holds, the players choose sequentially and there are only three players, 𝑛𝑛 = 3, the following networks structures are a Subgame Perfect Nash Equilibrium:

i. The complete network for 0 < 𝑐𝑐 <1₆,

ii. The star network for �

1 6< 𝑐𝑐 < 1 2 𝑤𝑤ℎ𝑒𝑒𝑓𝑓𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒𝑓𝑓 1 𝑓𝑓𝑓𝑓 2 𝑤𝑤𝑖𝑖𝑝𝑝𝑝𝑝 𝑏𝑏𝑒𝑒 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑒𝑒𝑛𝑛𝑡𝑡𝑓𝑓𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒𝑓𝑓, 1 2< 𝑐𝑐 < 2 3 𝑤𝑤ℎ𝑒𝑒𝑓𝑓𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒𝑓𝑓 2 𝑓𝑓𝑓𝑓 3 𝑤𝑤𝑖𝑖𝑝𝑝𝑝𝑝 𝑏𝑏𝑒𝑒 𝑡𝑡ℎ𝑒𝑒 𝑐𝑐𝑒𝑒𝑛𝑛𝑡𝑡𝑓𝑓𝑝𝑝𝑝𝑝 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑒𝑒𝑓𝑓,

iii. The empty network for 𝑐𝑐 >2₃.

3.5.1 Comparison of Theorem 3 and Theorem 4 for 𝒏𝒏 = 𝟑𝟑

The first difference is that there is always one unique outcome. Where in the model of Goyal and Vega-Redondo when 𝑐𝑐 >1₂ both the empty network and star network can emerge, in the case of the sequential game there is only one network structure for any value of 𝑐𝑐. This result corresponds to one of the theorems Currarini and Morelli (2000) found in their research in which they argue that a sequential game always admits Subgame Perfect Equilibria. 7 _{This is also}

why the empty network occurs less frequently in the case of the sequential game; the empty network arises when 𝑐𝑐 >2₃, whereas in the model of Goyal and Vega-Redondo it arises when 𝑐𝑐 >1₂.

The results show that the star structure is an important network, just like in the simultaneous game. In the case of the simultaneous game every player can be the central player, but in the case of a sequential game it turns out that the identity of the central player depends on the value of 𝑐𝑐. When the costs are low, the payoffs are higher for the central player than for peripheral players. The central player takes on a structural holes position. The total payoffs in this network are maximal, whereas all players, with the exception of the central player, only have to pay the

7 _{Chapter 3 Equilbrium, section 3.1 Existence of equilibrium, Theorem 1, page 233.} 20

costs of one link, while the distance between players is at the most two links (Goyal, 2007). However, when the costs are high, it is more profitable to be one of the peripheral players than to be the central player. This is apparent in Theorem 4; player 1 can be the central player when the costs are low and he will be a peripheral player when the costs are high. Player 1 has a

‘first-player-advantage’. He can force the other players to take on a certain position, which actually is

not the optimal position for them.

3.5.2 R program8

To verify these results, a program in software R is written. R is a software programming language for statistical computing and graphics. First, the best actions for the last player given all histories are calculated. Histories are all possible strategies chosen by all players prior to the player in turn. The program has to take this into account. Next, the best actions for the penultimate player are calculated, given all histories and knowing what the player who chooses after him will choose given his choice. This continues until the first player is in turn. He knows what all other players will choose once he has chosen. Therefore, although he is the first player to move, he has the decisive move. More precisely, he can decide his best action from the available structures possible resulting from the best actions of the subsequent players.

For 0 < 𝑐𝑐 <1₆ the outcomes of the program correspond to the outcomes in Theorem 4; the complete network emerges. However, for higher costs, it appears there is a limitation to the program and a new assumption needs to be made.

Assumption 2.

A player’s best action is only given by the best action that appears first in the sequence.

The program gives just one optimal outcome, even though there are multiple outcomes optimal in some cases. For instance, when 1₆< 𝑐𝑐 <1₂ the program gives the outcome where player 1 is the central player and earns the highest payoff. However, as was mentioned in chapter 3.4.2, player 2 can also be the central player in this case. The same occurs when 1₂< 𝑐𝑐 <2₃; player 1 will propose a link to either player 2 or player 3. He does not have a preference and therefore player 2 or player 3 will be the central player. In this case it is detrimental to be the central player. The program, however, only returns the outcome where player 2 is the central player. For 𝑐𝑐 >2₃ the program also returns the empty network.

8

This is a concised description of a very elaborate program. I would like to thank my supervisor, Mr. D. in ‘t Veld. He helped me a lot with the program and the overall process of writing this thesis.

21

4. Conclusion

In the model of Goyal and Vega-Redondo (2007) the star structure is the most important efficient network. The star network is efficient for the largest range of 𝑐𝑐 and as 𝑛𝑛 increases the range for which the star network is efficient also increases. The total payoffs in the sequential game do not differ from the payoffs in the model of Goyal and Vega-Redondo. That is why Theorem 1 also holds true in the case of the sequential formation game.

The first main result is that in the sequential game there is one unique network for any parameter value of 𝑐𝑐, while in the model of Goyal and Vega-Redondo for some range of 𝑐𝑐 both the empty and star network can emerge. Another interesting difference is that in the former model of Goyal end Vega-Redondo, when a star structure emerges, all players can be the central player for the total range of 𝑐𝑐, while in the case of sequential network formation, the identity of the central player depends on the costs of forming a link. For low costs the first player can be at the centre of the star, while for higher costs the first player will never be the central player, which is the most profitable outcome. There is an advantage gained by the first-moving player. This advantage stems from the fact that the first player can gain control and he can force the other players to take on a certain position in the network. To verify these results a program in software R is written. Results of the program correspond to the former results, except for the limitation mentioned earlier.

This research is limited in some ways. First, it is assumed that all players are fully rational. They know what the players before them have chosen; it is perceptible what actions were performed in the past. However, in many cases of financial network formation this information is not available or not transparent. If players cannot observe this, they might base their actions on probabilities. Also, it could be the case that players are bounded rational and cannot anticipate any further change that could decrease their payoff. In this case it can be assumed to use the concept of best-response dynamics. In other cases players might deviate from the optimal in order to achieve other goals. This research only considers financial incentives. However, there can be other incentives to take on a certain position for which players are willing to make offers, such as perceived power and/or reputational enhancement.

Finally, the network formation in which there are only three players is considered. It is possible to extend this research to cases in which the number of players increases further, possibly to any general number. This can be achieved by developing an enhanced R program. The analysis of sequential network formation with an increased number of players is an interesting topic, which is left for further research.

References

Caldarelli, G., Chessa, A., Pammolli, F., Gabrielli, A.and Puliga, M., ‘Reconstructing a credit network’, Nature Physics, 2013, 9: 125–126.

Calvó-Armengol, A. & Ilkılıç, R., ‘Pairwise-Stability and Nash Equilibria in Network Formation’,

International Journal of Game Theory, 2009.

Chinazzi, M. & G. Fagiolo, , ‘Systemic Risk, Contagion, and Financial Networks: a Survey’, April 2013, http://ssrn.com/abstract=2243504.

Currarini, S. & Morelli, M., ‘Network formation with sequential demands’, Review of Economic

Design, 2000, 5: 229-249.

Goyal, S. Connections: An introduction to the economics of networks, Princeton: Princeton University Press, 2007.

Goyal, S. & F. Vega-Redondo, ‘Structural holes in social networks’, Journal of Economic Theory, 2007, 137: 460-492.

Jackson, M.O. & Wolinsky, A., ‘A Strategic Model of Social and Economic Networks’, Journal of

economic theory, 1996, 71: 44-74.

Houy, N., ‘Structural holes in social networks: A remark’, Journal of Economic Theory, 2009, 144: 422-431.

Van der Leij, M., in ’t Veld, D., Hommes, C., ‘The formation of a core-periphery structure in financial networks’, 2013.

Varian, H.L., Intermediate Microeconomics, a modern approach, 2010, eighth edition, Chapter 28: 522-536.

http://www.econ.ucla.edu/iobara/SPE201B.pdf

http://www.gametheory.net/dictionary/SubgamePerfect.html

Appendix

Proof Theorem 1.

Part i.

The complete network is efficient for no 𝑐𝑐.

Proof.

Assume the network is not complete. The maximum return for an indirect connection with a distance of two, when there are no essential players between this indirect connection, is 1₂. Forming a new direct link with the player who was at a distance of two links away, gains a return of 1₂− 𝑐𝑐. So forming a new direct link is profitable if 1₂<1₂− 𝑐𝑐. Simplified this means that if 𝑐𝑐 < 0 all not-direct connected players will want to form a direct link with each other and the network will become a complete network. However by definition 𝑐𝑐 is never smaller than zero, so therefore the complete network is never efficient.

Part ii.

The star network is efficient for 0 < 𝑐𝑐 <𝑛𝑛₄.

Proof.

Assume there is a subnetwork 𝑔𝑔′_{, with 𝑛𝑛 players, of the total network 𝑔𝑔, with 𝑁𝑁 players. The} number of direct links is 𝑘𝑘 ≥ 𝑛𝑛 − 1. Every direct link gains a return of 2 ∗ 1₂− 2𝑐𝑐 to the network, so in total all direct links gain �2 ∗ 1₂− 2𝑐𝑐� 𝑘𝑘. This leaves at most �𝑛𝑛₂� − 𝑘𝑘 =1₂𝑛𝑛(𝑛𝑛 − 1) − 𝑘𝑘 indirect links left and when there are no essential players between an indirect connection the maximum return of a indirect connection is 2 ∗1₂. The component 𝑔𝑔′ _{has a total maximum return} of ∏ (𝑔𝑔′₎

𝑡𝑡𝑡𝑡𝑡𝑡 = (1 − 2𝑐𝑐)𝑘𝑘 +1₂𝑛𝑛(𝑛𝑛 − 1) − 𝑘𝑘.

In the star structure there are (𝑛𝑛 − 1) direct links with a total return of (1 − 2𝑐𝑐)(𝑛𝑛 − 1). There are �𝑛𝑛−1₂ � =1₂(𝑛𝑛 − 1)(𝑛𝑛 − 2) indirect links left. Each indirect link between two peripheral players has a return of 2 ∗1₃ and the player who is essential to this connection has a return of 1₃. So the total return for a indirect link is 2₃+1₃= 1. The total return in a star structure network is 9 _{All parts of this theorem are proved accordingly to the analysis of Goyal and Vega-Redondo (2007) and} Jackson and Wolinsky (1996).

24

∏ (𝑔𝑔𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑡𝑡𝑠𝑠𝑠𝑠′ )= (1 − 2𝑐𝑐)(𝑛𝑛 − 1) +1₂(𝑛𝑛 − 1)(𝑛𝑛 − 2). Note that ∏ (𝑔𝑔′₎ 𝑡𝑡𝑡𝑡𝑡𝑡 − ∏ (𝑔𝑔𝑡𝑡𝑡𝑡𝑡𝑡 𝑠𝑠𝑡𝑡𝑠𝑠𝑠𝑠′ )= (1 − 2𝑐𝑐)𝑘𝑘 +1₂𝑛𝑛(𝑛𝑛 − 1) − 𝑘𝑘 − �(1 − 2𝑐𝑐)(𝑛𝑛 − 1) +1₂(𝑛𝑛 − 1)(𝑛𝑛 − 2)� = 𝑘𝑘 − 2𝑘𝑘𝑐𝑐 +1₂𝑛𝑛2₋1 2𝑛𝑛 − 𝑘𝑘 − �𝑛𝑛 − 1 − 2𝑛𝑛𝑐𝑐 + 2𝑐𝑐 + 1 2𝑛𝑛2− 𝑛𝑛 − 1 2𝑛𝑛 + 1� = −2𝑘𝑘𝑐𝑐 + 2𝑛𝑛𝑐𝑐 − 2𝑐𝑐 = 2𝑐𝑐(𝑛𝑛 − 𝑘𝑘 − 1),

which is at most zero since 𝑘𝑘 ≥ 𝑛𝑛 − 1 and 𝑐𝑐 > 0, and less than zero if 𝑘𝑘 > 𝑛𝑛 − 1. This equation is equal to zero if 𝑘𝑘 = (𝑛𝑛 − 1), which means that the total return of the component 𝑔𝑔′ _{equals the} total return of the star structure. Every other network with 𝑘𝑘 = (𝑛𝑛 − 1) must have at least one indirect connection which has a path longer than two. The return of this longer indirect connection is lower than the return of an indirect connection in the star network. The fact that this longer connection can result in de existence of other essential players has no influence on the total return. In this case the total return is divided differently over the two indirect connected players and the essential players in between, but the total return stays unchanged. Therefore, the component 𝑔𝑔′ _{with the star structure with 𝑘𝑘 = (𝑛𝑛 − 1) gains the highest returns} for 𝑐𝑐 > 0.

It is shown that if 𝑐𝑐 > 0 , then any component of an efficient network must be a star (Jackson and Wolinsky, 1996). Also a single star gains a higher return than several stars. The return of one single star with 𝑛𝑛 + 𝑚𝑚 players is equal to (1 − 2𝑐𝑐)(𝑛𝑛 + 𝑚𝑚 − 1) +1₂(𝑛𝑛 + 𝑚𝑚 − 1)(𝑛𝑛 + 𝑚𝑚 − 2), while the return of two separate stars is equal to (1 − 2𝑐𝑐)(𝑛𝑛 + 𝑚𝑚 − 2) +1₂(𝑛𝑛 + 𝑚𝑚 − 2)(𝑛𝑛 + 𝑚𝑚 − 4). Hence, the latter is always lower than the former. Besides, an efficient network consists of positive components. There are �𝑛𝑛₂� =1₂𝑛𝑛(𝑛𝑛 − 1) links in the star structure which all have a payoff of 1. Each link in the star costs 2𝑐𝑐 and there are 𝑛𝑛 − 1 links. The total payoff in this minimally connected (star) network is ∏ (𝑔𝑔∗_{) =}1

2𝑛𝑛(𝑛𝑛 − 1) − (𝑛𝑛 − 1)2𝑐𝑐

𝑡𝑡𝑡𝑡𝑡𝑡 . This is positive when

1

2𝑛𝑛 − 2𝑐𝑐 > 0 ⇒ 𝑐𝑐 < 𝑛𝑛 4. Part iii.

The empty network is efficient for 𝑐𝑐 >𝑛𝑛₄.

Proof.

As was mentioned in de proof of part 2, a non-empty network will have a negative total return if 𝑐𝑐 >𝑛𝑛₄. In this case the empty network with a total return of 0 is the network with the highest return and therefore the only efficient network.

Proof Theorem 2.

The complete network is pairwise stable for � 0 < 𝑐𝑐 <

1

6, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 = 3 𝑛𝑛𝑓𝑓 𝑐𝑐, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 ≥ 4.

Proof.

In the complete network every player earns ∏ =𝑖𝑖 𝑛𝑛−1₂ − (𝑛𝑛 − 1)𝑐𝑐 and when removing one of the links the payoff will be 𝑛𝑛−2₂ +1₂− (𝑛𝑛 − 2)𝑐𝑐. When there are only three players, the payoff in the complete network for every player is 1 − 2𝑐𝑐. If a player removes one link, the payoff for that player will be 5₆− 𝑐𝑐. The first payoff is higher than the latter if 1 − 2𝑐𝑐 >5₆− 𝑐𝑐 ⇒ 𝑐𝑐 <1₆. For higher 𝑛𝑛, the payoff in a complete network is higher than in a non-complete network when 𝑛𝑛−1

2 − (𝑛𝑛 − 1)𝑐𝑐 > 𝑛𝑛−2

2 +

1

2− (𝑛𝑛 − 2)𝑐𝑐 ⇒ 𝑐𝑐 < 0. This is by definition never the case, so therefore a complete network is never pairwise stable for 𝑛𝑛 ≥ 4.

Part ii.

The star network is pairwise stable for 1₆< 𝑐𝑐 <1₂+1₃(𝑛𝑛 − 2). Proof.

In a star network the central player earns ∏ =𝑖𝑖 𝑛𝑛−1₂ +1₃(𝑛𝑛−1)(𝑛𝑛−2)₂ − (𝑛𝑛 − 1)𝑐𝑐 and the payoff for adding one extra link is 1₂+1₃(𝑛𝑛 − 2) − 𝑐𝑐. This is profitable when 𝑐𝑐 <1₂+1₃(𝑛𝑛 − 2, ), so in this case he will have no incentive to remove a link. The payoff for a peripheral player is ∏ =𝑖𝑖 1₂+1₃(𝑛𝑛 − 2) − 𝑐𝑐 and when he forms one extra link with another peripheral player his payoff will be 1₂+1₂+1₃(𝑛𝑛 − 3) − 2𝑐𝑐. This payoff will be lower than the payoff when not adding the extra link when 1₂+1₃(𝑛𝑛 − 2) − 𝑐𝑐 >1₂+1₂+1₃(𝑛𝑛 − 3) − 2𝑐𝑐 ⇒ 𝑐𝑐 >1₆. In this case the peripheral player will have no incentive to form an extra link. Also a peripheral player will not remove the link he has with the central player when his payoff is higher than the costs, so when 𝑐𝑐 <1₂+1₃(𝑛𝑛 − 2).

Part iii.

10 _{All parts of this theorem are proved accordingly to the analysis of Goyal and Vega-Redondo (2007) and} Jackson and Wolinsky (1996).

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The empty network is pairwise stable for 𝑐𝑐 >1₂. Proof.

In an empty network the payoff for forming a link is ∏ =𝑖𝑖 1₂− 𝑐𝑐. For 𝑐𝑐 >1₂ none of the players will want to form a link and the network stays empty.

Proof Theorem 3.

Part i.

The complete network is sustainable in pairwise equilibrium for �0 < 𝑐𝑐 <16, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 = 3 𝑛𝑛𝑓𝑓 𝑐𝑐, 𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛 ≥ 4. Proof.

As was stated in the text in chapter 2.4, for a network to sustain in pairwise equilibrium in needs to be a pairwise stable network. And as proved in Theorem 2, part 1, a complete network is pairwise stable for 𝑛𝑛 = 3 and never pairwise stable for 𝑛𝑛 ≥ 4. Therefore the complete network is also sustainable in pairwise equilibrium when 𝑐𝑐 <1₆ and 𝑛𝑛 = 3 and for higher 𝑛𝑛 the network is never sustainable in pairwise equilibrium.

Part ii.

The star network is sustainable in pairwise equilibrium for 1₆< 𝑐𝑐 <1₂+1₆(𝑛𝑛 − 2). Proof.

A peripheral player in a star has just one link and will keep it if 𝑐𝑐 <1₂+1₃(𝑛𝑛 − 2), as was shown in the proof of Theorem 2, part 2. In this case any peripheral players has an incentive to directly access the central player; it is a dominant strategy for all peripheral players to directly or indirectly access every other player. Furthermore, the payoff for the central player is positive when 𝑛𝑛−1₂ +1₃(𝑛𝑛−1)(𝑛𝑛−2)₂ − (𝑛𝑛 − 1)𝑐𝑐 > 0 ⇒ 𝑐𝑐 <1₂+1₆(𝑛𝑛 − 2). So in this case the central player has no incentive to delete all links and the network is sustainable in pairwise equilibrium. Besides, as was also shown in the proof of Theorem 2, 𝑐𝑐 >1₆ implies that if there is a central player who

11 _{All parts of this proof assume all the already calculated values of 𝑐𝑐 for which the network is pairwise} stable. If the network is also a Nash equilibrium within these boundaries, it is said that the network is sustainable in pairwise equilibrium.

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has links with all peripheral players, then all the other players will form a direct link with this player and no direct links with other peripheral players.

Part iii.

The empty network is sustainable in pairwise equilibrium for 𝑐𝑐 >1₂. Proof.

If every player announces that he wants to form no links, then a dominant strategy of player 𝑖𝑖 is to announce that he also wants to form no links. In other words, the empty network is always a Nash equilibrium for any network formation game. Now the empty network is pairwise stable if 𝑐𝑐 >1₂, so the empty network is also sustainable in pairwise equilibrium if 𝑐𝑐 >1₂.

Example with arrows for 𝒏𝒏 = 𝟑𝟑

Player 1 has four choices (see figure 2); he can decide not to form any links with the other two players, just to form a link with player 2 or with player 3, or he can form links with both players 2 and 3. A link is established if and only if two players agree on the link; there have to be arrows in both directions.

Figure 2. Choices of player 1.

After player 1 has made his choices, player 2 gets to choose. For every given choice of player 1, he has the same choices player 1 had; form no links, a link with either player 1 or 3 or with both.