Stability and Equilibrium Selection in Link Formation Games

Rodrigo Harrison and Roberto Muñoz * April 2003. This version: December 2006.

Abstract

In this paper we use a non cooperative equilibrium selection approach as a notion of stability in link formation games. We apply an extension of Carlsson and van Damme (1993) to study the robustness, to the introduction of incomplete information, of the set of Nash equilibria for a class of link formation games with supermodular payo¤ functions. Interestingly, the equilibrium selected is in con‡ict with those predicted by traditional coalitional re…nements. Moreover, we

…nd a con‡ict between stability and e¢ ciency even when no such con‡ict exists with the coalitional re…nements. JEL codes: C70, D20, D82 Keywords: Global Games, Networks, Equilibrium Selection.

1 Introduction

In the network literature it is well known that a link formation game in strategic form can lead to the formation of multiple networks supported by multiple Nash equilibria. Traditional non cooperative re…nements are not always useful and consequently, some stability notions have been proposed to re…ne the set of equilibria.

* We are specially grateful to Deborah Minehart and Roger Laguno¤ for their advice and help. We thank Daniel Vincent, Peter Cramton, Luca Anderlini, Axel Anderson, Felipe Zurita, Francis Bloch and three anonymous referees for valuable comments that helped us to improve the paper. We also thank participants at The European Meetings of the Econometric Society, Stockholm 2003 and the IUSC conference, Columbia University, 2003. Harrison: Instituto de Economía Ponti…cia Universidad Católica de Chile. E-mail: [email protected]; Muñoz: Departamento de Economía, CIDE and Departamento de Industrias, Universidad Técnica Federico Santa María. E-mail: [email protected].

1 The stability notions used so far have been based on cooperative game theory. The most prominent of them have been Strong Nash Equilibrium (SNE),1 Coalition Proof Nash Equilibrium (CPNE) and Pairwise Strong Nash Equilibrium (PSNE).2 However, the applicability of these re…nements relies critically on the feasibility of cooperation among agents. This assumption seems too strong for a link formation game when, by de…nition, the network has not been formed. Several authors have studied the theoretical foundations of network formation and its properties (Myerson (1977), Aumann and Myerson (1988), Dutta et al. (1998), Slikker and van den Nouweland (2000), Gilles and Sarangi (2005), Chakrabarti and Gilles (2006) among others). Particular emphasis has been given to study the link formation process and the con‡ictbetween stability and e¢ ciency in networks (Jackson and Wolinsky (1996), Dutta and Mutuswami (1997), Jackson and van den Nouweland (2005), Gilles et al. (2005)). In this paper we focus on the "Consent Game" introduced by Myerson (1991, p. 448) as a normal form game, and studied by Dutta et al. (1998) and Gilles et al. (2006).3 In a recent article, Chakrabarti and Gilles (2006) have identi…ed two di¤erent branches of theories of network formation under consent. One is the link-based approach, pioneered by Jackson and Wolinsky (1996), where the central elements are the links rather than strategy pro…les. The other one is the network-based re…nements of the set of Nash equilibria. This approach has been followed by Gilles and Sarangi (2005) and Chakrabarti and Gilles (2006) among others. Although our paper could be considered as part of this second venue, we depart from it in that we use a non cooperative equilibrium selection approach. Speci…cally, we follow the Global Games approach, pioneered by Carlsson and van Damme (1993),4 where a particular kind of perturbation is introduced in the payo¤ functions and the robustness of the set of Nash equilibria, to this way of introducing incomplete information, is studied. The global games’ results have been extended by

1 There are slightly di¤erent de…nitions for SNE in Dutta and Mutuswami (1997) and Jackson and van den Nouweland (2005). We will discuss the di¤erences later on, but in our context they generate the same re…nement. 2 Jackson and Wolinsky (1996) introduced the concept of Pairwise Stability, which is understood as a necessary condition for a network to be stable against deviations by pairs of agents. The extension of this concept as a re…nement of the set of Nash equilibria arising in a link formation game has been called Pairwise Strong Nash Equilibrium (see Goyal and Vega-Redondo (2005) and Gilles et al. (2006)). 3 The consent game is a static link formation game where each player establishes a list of desired connections with the other players. A link is formed under mutual agreement, but it can be blocked unilaterally. The main problem with this game is that it usually admits a multitude of Nash equilibria. 4 For a survey of the ensuing literature see Morris and Shin (2002).

2 Frankel et al. (2003) for games with many players and many actions, but it is limited to the case of games with strategic complementarities where the action space is a compact subset of the real numbers. Even though these authors obtain a quite general result, it cannot be applied to games where agents’strategies are vector valued. In this sense, our paper extends an important uniqueness result in the global games literature to a more general class, games with vector-valued action spaces where each component of the agents’ strategy vector represents a binary decision. This decision can be seen as player’s intention of establishing a link with another player. A link will be formed if and only if both players want to form the link. In this extension, however, we have to constrain payo¤ functions to be supermodular. We show that, under a plausible set of additional assumptions, in our link formation game of incomplete information there exists a unique strategy pro…le surviving iterated elimination of strictly dominated strategies. In the limit when the incomplete information tends to disappear, the implied outcome is in con‡ict with those arising from the application of traditional coalitional re…nements: SNE, CPNE and PSNE. This di¤erence shows that the set of equilibria under the coalitional notions of stability are not robust to incomplete information in the form we introduce it. Moreover, we show that the stability notions based on coalitional re…nements do not con‡ict with e¢ ciency in our class of payo¤ functions, while the equilibrium selected under the global games approach does. These di¤erences raise some practical questions about which criterion should be satis…ed by network formation games. The paper is organized as follows. In section 2 we provide a simple example where we show intuitively the main …ndings of the paper. Section 3 extends the main results to a class of supermodular games of network formation. The main conclusions are contained in section 4. Finally, the proofs are relegated to the appendix.

2 An Illustrative Example

The idea of this section is to provide a simple example and an intuitive explanation of the main results developed in the paper. We are not going to be strictly formal, and technical details are postponed to the next sections.

Consider a static link formation game of complete information Gx with a set of players

= 1; 2; 3 , where the action set for each player i is given by Ai = (aij)j=i aij 0; 1 N f g f 6 j 2 f gg

3 and x IR is a variable scaling the level of revenues accruing to the players. A strategy for 2 player i is a two component vector of zeros and ones which identi…es the set of players he wants to form links with. The players simultaneously choose strategies and a link between two players will be formed if and only if both of them want to form the link as in the Myerson’sConsent Game. For example, if the strategy pro…le a is composed the strategies ai = (aij = 1; aik = 1); aj = (aji = 0; ajk = 1), ak = (aki = 0; akj = 1), then only the link jk is created. The payo¤ function for player i is de…ned by:

i(ai; a i; x) = aijaji (x + ajkakj x) + ( x c)aij + aikaki (x + akjajk x) + ( x c)aik (1) Where we denote by c a …xed parameter representing the cost incurred by agent i for each link he wants to form. Note that if agent i wants to connect agent j, then even if j does not behave reciprocally, and consequently the link ij is not formed, agent i has to incur the cost c and receives a net return ( x c). In this sense we say that the source of bene…ts x is independent of other players’actions. On the other hand, if j also wants to form a link with i then the return to agent i changes to (x+ x c). In other words, there is an extra direct bene…t x from connection with each potential partner. Finally, agent i “pro…ts”from the relation between j and k when they are connected and provided that i is connected with at least one of them. Note that if i is connected with both of them, this indirect bene…t x is duplicated. For example, in the complete network the total payo¤ for player i is given by i = 2[x (1 + + ) c]. It seems natural to assume that 0 < < 1, 0 < < 1, because we are scaling the bene…ts in relation to those obtained from reciprocity (x). A case where this payo¤ function can arise is in a variation of the connection model of Jackson and Wolinsky (1996) where we interpret links as friendship relations. Bene…ts come from direct and indirect connections to individuals as usual but, di¤erently from the standard model, there also exist bene…ts and costs of requiring links, even if these links are not …nally formed. For example, it is plausible that a unilateral friendly behavior of agent i with agent j generates bene…ts ( x) and costs (c) to agent i no matter whether there is reciprocity.

4 2.1 The Nash Equilibria

Part (a) of …gure 1 provides a summary of the di¤erent network structures supported by a strategy pro…le in the set of Nash equilibria, NE(Gx), for di¤erent values of x. Several cases were considered: Case (i): In the case x < x = c=(1 + + ) a dominant strategy for any player i is to play ai = (0; 0) 0 and the empty network emerges. Case (ii): Suppose that c=(1 + + ) x < c=(1 + ): It is easy to see that the best response correspondence of player i depends on the existence of the link between players j and k. It is possible to prove that in this case only the empty and the complete network can be supported by strategy pro…les in NE(Gx). Case (iii): Suppose that c=(1 + ) x c= : In this case the strategies of agents j and k in relation to their connection do not a¤ect the best response correspondence of agent i. The intuition is that direct connections are enough to guarantee pro…tability. This characteristic leads to a multiplicity of Nash equilibria and, even more, it is possible to prove that every feasible network among the three agents can be supported by a pro…le in NE(Gx). Case (iv): Finally, when x > x = c= a dominant strategy is to require links with all the other players: ai = (1; 1) 1, leading to the complete network.

2.2 Equilibrium Selection using Coalitional Re…nements

A strategy pro…le is called a Strong Nash Equilibrium (SNE) if it is a Nash equilibrium and there is no coalition of players that can strictly increase the payo¤s of all its members using a joint deviation (Aumann (1959), Dutta and Mutuswami (1997)).5 On the other hand, a strategy pro…le is called a Coalition Proof Nash Equilibrium (CPNE) if, as in a SNE, no coalition can deviate to a pro…le that strictly improves the payo¤s of all the players in the coalition, but the set of admissible deviations is now smaller, because they have to be stable with respect to further deviations by subcoalitions (Dutta and Mutuswami (1997)). Finally, a strategy pro…le is a Pairwise Strong Nash Equilibrium (PSNE) if it is a Nash equilibrium immune to deviations of coalitions of size 2, such that no player

5 Unfortunately this de…nition does not imply PSNE, because in the latter a feasible deviation occurs when one player in the coalition can be better o¤ while the other is not worse o¤. Jackson and van den Nouweland (2005) provides an alternative de…nition of SNE which implies PSNE. In our context, however, both notions of SNE coincide.

5 ·

· · (a) NE · ·

· · · ·

· (b) SNE and CPNE · ·

· · (c) PSNE · · · ·

· · · (d) BNE · · · · · ·

c c 4c c x 1++ab 1+a 2 + 4a + 4 b 3 a

Figure 1: Networks supported by Nash equilibria and alternative equilibrium selection approaches.

in the coalition is strictly worse o¤ and at least one of them is strictly better o¤. The application of these coalitional re…nements to our three players game Gx is very direct and a summary of results can be observed in …gure 1: in part (b) are networks supported by elements in SNE(Gx), CPNE(Gx) and in part (c) are networks supported by elements in PSNE(Gx).

First, it is possible to prove that, in this particular example, SNE(Gx) coincides with

CPNE(Gx). Second, the analysis has to be performed in separated areas. It is easy to see that the strategy pro…le a = (0; 0; 0) [0] is the unique pro…le in SNE(Gx) when x < c=(1 + + ), because for this range of values each agent plays 0 as a dominant strategy and, consequently, no coalition of agents can improve upon.6 On the other hand, a = (1; 1; 1) [1] is the unique pro…le in SNE(Gx) when x > c=(1 + + ). In order to see this, consider any other given strategy pro…le a = [1] (Nash equilibrium or not). 6 Because of the complementarities involved in the payo¤ function (1), the grand coalition, b 6 In what follows [0] and [1] represents a matrix full of zeros or ones respectively. The dimensionality is given by the pro…le they are representing. For example, in this three players case, a = [0] is a 2x3 matrix of zeros representing a complete strategy pro…le and a i = [0] is a 2x2 matrix of zeros representing a strategy pro…le that excludes player i’s strategy.

6 by playing a = [1], can strictly increase the payo¤s of all its members and then a is not in

SNE(Gx). Additionally, starting from a = [1] which is trivially a Nash equilibrium, it is b not possible to increase the payo¤ of any player in a deviation. Finally, if x = c=(1+ + ), then SNE(Gx) = [0]; [1] . f g We have to be more careful in the analysis of PSNE. If c=(1+ + ) x c=(1+ ) then the empty and the complete networks are supported by Pairwise Strong Nash Equilibria and consequently, this concept does not re…ne the set of Nash equilibria.7 The reason is that for these low values of x the indirect connections are needed to make any connection pro…table so, if nobody is making links, an agreement of two players to form a link is not enough to obtain a pro…table relationship. On the other hand, if everybody is making links then no agent bene…ts from cutting a link. When c=(1 + ) < x c= any pair of agents which are not connected can pro…tably make a link and, consequently, only the complete network can be supported by a Pairwise Strong Nash Equilibrium. Finally, if x < c=(1 + + ) then action ai = 0 is a dominant strategy for player i and then the unique Pairwise Strong Nash Equilibrium is [0] forming the empty network. Analogously, if x > c= then action ai = 1 is a dominant strategy and then the unique Pairwise Strong Nash Equilibrium is [1] and the complete network is formed.

2.3 Equilibrium Selection using the Global Games Approach

Suppose now we introduce incomplete information in the payo¤ structure such that player i’spayo¤ function depends on a private value xi, observed just by i, having the following structure: xi = x + "i; where > 0 is a scale factor, x is drawn from X; X with uniform density and "i is an independent realization of a random variable with density 1 1 8 9 and support in [ ; ]: We assume "i is i:i:d: across the individuals. A pure strategy 2 2 for player i is a function si :[X ; X + ] Ai; the set of all such functions is denoted Si 2 2 ! and s = (s1; s2; s3) is a pure strategy pro…le, where si Si. Call this game of incomplete 2 information G(): Let us de…ne an essentially unique switching strategy between 0 and 1 with threshold

7 The exception is when x reaches the upper limit of the interval where some strategy pro…les are selected. This is, however, a technicality because this is a zero measure set. 8 Note that, in this way to introduce incomplete information, the signal received by player i coincides with his/her private value. The structure of the values is important, because it permits player i to obtain the conditional distribution of xj given xi j = i. 9 Note that does not need to be symmetric8 6 around the mean nor even have zero mean.

7 ki 10 ki as a pure strategy s ( ) satisfying: i 1 if xi > ki ki si (xi) = 8 0 if x < k < i i We establish the following result.11 :

Proposition 1 For any > 0, the essentially unique switching strategy pro…le s =

k si ( ) i is the only strategy pro…le that survives iterated elimination of strictly domi- 2N 4c nated strategies in G(), where k = 2+4 +4 =3 .

It is important to notice that the equilibrium pro…le s selected in G() does not depend on the size of the noise, . Since the values’ structure is xi = x + "i; if

0 then xi x; thus the unique equilibrium selected implies that in the limit ! ! 8 x < k all the agents select action 0, so the empty network is formed, and x > k all 8 the agents select action 1 and, consequently, the complete network is formed. Part (d) of …gure 1 shows the networks supported by this Bayesian Nash Equilibrium (BNE), as the noise goes to zero, as a function of x.

2.4 E¢ cient Allocation of the Game

A strategy pro…le a is e¢ cient for the game Gx if it maximizes the sum of the payo¤s of the players. The set of e¢ cient strategy pro…les for the game Gx is denoted by E(Gx). It is easy to check that in our example:

[0] if x < c=(1 + + ) f g E(Gx) = 8 [0] ; [1] if x = c=(1 + + ) > > f g < [1] if x > c=(1 + + ) f g > :> and as a result, for given x, the networks supported by pro…les in the set E(Gx) coincide with those supported by the sets SNE(Gx) and CPNE(Gx) described in part (b) of …gure 1. The example illustrates the main results of the paper. First, the traditional coalitional re…nements do not con‡ict with the e¢ cient allocation at any level of x. However, the actions mandated by the equilibrium selected using the global games approach when

10 The essentially unique quali…cation arises because any action may be played if the private value is exactly equal to ki (see Morris and Shin (2003)) 11 This proposition is a particular case of Proposition 2 below, so we are not going to give a formal proof here.

8 c 4c goes to zero clearly con‡icts with e¢ ciency when 1+ + < x < 2+4 +4 =3 . Second, in the c 4c interval 1+ < x < 2+4 +4 =3 all the coalitional re…nements predict the formation of the complete network, however, our selected equilibrium predicts the empty network. This means that, for these values of x, it is impossible to satisfy the two stability conditions simultaneously.12 In what follows we are going to show that these …ndings hold in a much more general setting.

3 Stability and Equilibrium Selection in a Class of Link Formation Games

3.1 The Link Formation Game

Consider a given set of players = 1; 2; :::; N + 1 . We denote by gN the complete N f g network among players in : gN = ij i; j ; i = j and the collection of all the N f j 2 N 6 g possible networks on is denoted by GN = g g gN : N j Player i’s strategy set is given by Ai = (aij)j aij 0; 1 . A strategy for j2N=i j 2 f g 6 player i satis…es that aij = 1 if he wants to form a link with player j while aij = 0 if player i does not want the link with j.

We assume symmetric players with a payo¤ function given by i(ai; a i; x) where 13 ai Ai, a i A i = j=iAj; and x [X; X] IR is an exogenous variable. We also 2 2 6 2 14 denote A = Ai A i: Players simultaneously choose strategies and the resulting static link formation game is denoted Gx: A link between two players will be formed if and only if both players want to form the link. The resulting network g is a list of pairs of players linked to each other. Formally, g(a) = ij : i; j such that aij = aji = 1 : f 2 N g For any network g GN , we de…ne: 2 (a) g + ij = g ij and g ij = g ij : [ f g n f g (b) N(g) = i = j such that ij g f 2 N 9 2 N 2 g 12 For example, PSNE implies that a pair of agents can be strictly better o¤ if they cooperate, however the strategies required to support this behavior do not survive the iterated elimination of strictly dominated strategies under any level of incomplete information in the parameter x. 13 Ai is a partially ordered set, that is: (i) ai ai if aij aij j = i ,and (ii) ai > ai if aij aij 8 6 j = i and aij > aij for some j: 8 6 In the same way A i is a partially ordered set: (i) a i a i if aj aj j = i , and (ii) a i > a i if b b 8 6 b b aj aj j = i andb aj > aj for some j: 14 8 6 In an analogous way we denote a i;j as the strategy pro…le excluding the actions chosen by agents i b b b and jb. Of course, .a i;j b k=i Ak. 2 k=6 j 6

9 We assume a separable payo¤ function given by:15

i(ai; a i; x) = f(x)vi(g(a)) + h(ai; x)

where:

f(x): D IR IR+ such that [X; X] D. ! vi(g(a)) is the network component of the payo¤ function. Note that it takes the same value for di¤erent strategy pro…les generating the same network. We also assume that it 16 takes the value zero if ai = 0 or a i = [0], i.e. vi(g(0; a i)) = vi(g(ai; [0])) = 0: h(ai; x) is the independent component of the payo¤ function. It depends exclusively on i0s actions and the parameter x. We assume that it takes the value zero if ai = 0, i.e. h(0; x) = 0:

Note that the previous normalizations over vi(g(a)) and h(ai; x) imply that i(0; a i; x) = 0, a i A i. 8 2 The independent and the network components of the payo¤ function will play an important role in what follows. We introduce the following set of assumptions about them:

(A1). Increasing Di¤erences (a). The greater the other players’action pro…le, the

greater is player i’s network incentive to choose a higher action: i , ai Ai 8 2 N 8 2 and a i; a0 i A i; if ai ai0 and a i a0 i then: 8 2

vi(g(ai; a i)) vi(g(ai0 ; a i)) vi(g(ai; a0 i)) vi(g(ai0 ; a0 i)): (b). The greater the value of the exogenous parameter x, the greater is player

i’s independent incentive to choose a higher action: i , ai; a0 Ai and 8 2 N 8 i 2 x; x0 [X; X]; if x > x0 and ai > a0 then h(ai; x) h(a0 ; x) > h(ai; x0) h(a0 ; x0): 8 2 i i i

(A2). Link Monotonicity i; j with i = j, g GN if ij g then vi(g) < 8 2 N 6 2 62 vi(g + ij):

15 Our results also hold for a more general setting where i(ai; a i; x) is pre-multiplied by '( aij )(l(a i)) where ' and are increasing and concave functions and l(a i) is the number of j=i X6 players other than i connected under a i: This generalization permits to introduce scale returns to own required links and rival’s existing links. 16 In the language of the network literature (Jackson (2003)), the symmetry assumption among players guarantees that the network component is an anonymous allocation rule. The corresponding value function, obtained as the sum of all the allocations, is in fact also anonymous.

10 (A3). Continuity and Monotonicity h(ai; x) and f(x) are continuous functions of x. Moreover, the function f(x) is strictly increasing in x:

(A4). Additive Link Request i , ai Ai we have: h(ai; x) = j=i aij h(ai0 ; x) 8 2 N 8 2 6 for any ai0 Ai such that j=i aij0 = 1: P 2 6

For given x; the agent i´s independentP payo¤ associated to a strategy ai is equal to

the number of links requested by him/her, j=i aij, times the agent i´s independent 6 payo¤ of requesting just one link. In otherP words, there are constant returns to scale in the number of links requested.17

(A5). Additive Link Formation Consider any partition of = 1 2; with 1 N N [N jN j 18 1 2, such that there exists a complete network gN among players in 1 and an empty N network among players in 2. For all i 2 and k IN 0 with k 1 , if N 2 N 2 [ f g jN j 1 1 j1; j2; :::; jk 1 then: vi(gN + ij1 + ij2 + ::: + ijk) = kvi(gN + il) for any l 1: 2 N 2 N

Agent i’s network payo¤ of establishing k links with players in 1 is equal to the N number of links formed, k, times agent i’s network payo¤ of forming just one link with any player in the complete network. In other words, there are constant returns to scale in the number of links formed between the isolated individual i and the

1 players in 1 forming the complete network gN . N (A6). Dominance Regions For su¢ ciently low (resp. high) value of the exogenous parameter x; requesting a link with nobody (resp. everybody) is a strictly dominant

strategy: x and x satisfying X < x < x < X such that i and a i A i: 9 8 2 N 8 2

if x < x then i(0; a i; x) > i(ai; a i; x) ai Ai 0; 8 2 n

if x > x then i(1; a i; x) > i(ai; a i; x) ai Ai 1: 8 2 n The previous assumptions have important implications over the total incentive to

0 0 choose higher actions i(ai; ai; a i; x) i(ai; a i; x) i(ai; a i; x), which are contained in the following Lemma.

Lemma 1 Under the set of assumptions (A1) to (A3) the following properties hold:

i. Increasing Di¤erences (ID): i , ai; ai0 Ai and a i; a0 i A i; 8 2 N 8 2 8 2 17 For example, in our leading example we have: h(ai; x) = j=i aij ( x c): 6 18 For any set S we denote S the cardinality of S. j j P

11 if ai ai0 and a i a0 i; then i(ai; ai0 ; a i; x) i(ai; ai0 ; a0 i; x) x [X; X]: 8 2 ii. Strict Monotonicity (SM): The incentive to choose a higher action strictly increases in x, that is: i , ai; ai0 Ai; a i A i and x; x0 [X; X] if ai > ai0 and x > x0 8 2 N 8 2 8 2 8 2 0 0 then i(ai; ai; a i; x) > i(ai; ai; a i; x0). iii. Continuity (C): For all i , ai; ai0 Ai and a i A i the function 2 N 8 2 8 2 i(ai; ai0 ; a i; x) is continuous in x .

Finally, Assumption (A6) is a necessary condition for the applicability of the global games approach. Furthermore, given continuity and monotonicity properties of the payo¤ functions, without loss of generality x and x can be considered as points where the dominance regions start. In order to see this, let us consider the lower dominance region.19 In this case:

Consider any ai Ai 0 and let us de…ne x(ai) such that i(ai; [1]; x(ai)) = 0 i 2 n 8 2 . By Lemma 1 (ID) and (SM) we have a i A i: i(ai; a i; x) i(ai; [1]; x) N 8 2 i(ai; [1]; x(ai)) i , x < x(ai). It is possible to prove that these values (x(ai)) do 8 2 N 8 20 not depend on ai: As a consequence, x can be de…ned by the condition i(ai; [1]; x) = 0

ai Ai 0. Note that this is the unique sensible way to de…ne it in order to satisfy A6. 8 2 n On the one hand, if i(ai; [1]; x) were strictly positive ai Ai 0, then, by Lemma 1 (C), 8 2 n we got a contradiction with the de…nition of x in A6. On the other hand, if i(ai; [1]; x) were strictly negative ai Ai 0, then the dominance region can be extended by …nding 8 2 n x > x such that i(ai; [1]; x) = 0 and x can be replaced by x in A6.

3.2e Equilibrium Selectione using the Global Gamese Approach

Suppose now that the game is one of incomplete information in the payo¤ structure. The amount of incomplete information will be parameterized by a variable , so we call this game G() and let BNE(G()) denote its set of Bayesian Nash equilibria: We introduce incomplete information in the same way as we did previously in the illustrative example, thus player i’s payo¤ function depends now on a private value xi instead of x. These values are, however, related so that xi also constitutes a noisy signal of xj for all j = i. 6 Each player’s value has the following structure: xi = x + "i , where > 0 is a scale factor, x is drawn from the interval [X; X] with uniform density, and "i is a random variable

19 The argument for the upper dominance region is analogous. 20 A formal proof of this statement is contained in the proof of Proposition 2.

12 distributed according to a continuous density with support in the interval [ 1 ; 1 ]: We 2 2 assume "i is i:i:d: across the individuals. This noise structure has been used in the global games literature to model in a simple way the conditional distribution of the opponents’values x i; given player i’sown value, xi: In this context of incomplete information, a pure strategy for player i is a function si :[X ; X + ] Ai. The set of all these functions is denoted Si and we also de…ne 2 2 ! S i = j=iSj, so that a complete strategy pro…le is given by s = (si; s i) S Si S i. 6 2 21 Formally, if a player i has a private value xi, his interim expected payo¤ associated to play action ai; when he faces an opposing strategy pro…le s i is Ex i [i(ai; s i(x i); xi) j xi]: In particular, and with the goal of simplify notation in what follows, if the strategy k pro…le of players other than i is given by s i = s i we will denote:

k (ai; xi; k) Ex i [i(ai; s i(x i); xi) xi] j We also assume:

(A7). Single Indi¤erence Point There exists a unique value k [X; X] of the exoge- 2 nous variable x such that if any player i has received a private value xi = k and all the other players are using essentially unique switching strategies sk ( ) then player j i is indi¤erent between the actions 1 and 0. In other words ! k [X; X] such that 9 2 i : 8 2 N (1; k; k) (0; k; k) 0 The main result of the paper is contained in the following proposition.

Proposition 2 Consider the game G(). Under assumptions A1 to A7, for any > 0,

k the essentially unique switching strategy pro…le s = si ( ) i is the only strategy 2N pro…le that survives iterated elimination of strictly dominated strategies, where k is given by A7 and satis…es x < k < x.

Even though the proposition establishes that each player is using, in equilibrium, 22 a switching strategy si independent on the size of the noise , the network formed 21 Players take decisions at interim stage when the uncertainty about players types is partially solved, players just know their own type. They still face uncertainty about their opponents’types. 22 Note that si is also independent of the noise density : In addition, we have assumed that the

13 depends on it. In general, if some xi > k + then every player receives a value greater than k and therefore the complete network is formed. Equivalently if some xi < k the empty network is formed, but if all xi [k ; k + ] then any network can be 2 formed depending on the realization of each player’s value. Following this analysis it is easy to see that as goes to zero just two possibilities remain, the empty and the complete network. Noting that a = [0] is trivially a Nash equilibrium of Gx for x < x; we can see that equilibrium selection is obtained under the global games approach when goes to zero. In e¤ect, for all x X; X the action pro…le determined by s is included in 2 NE(Gx).

3.3 Equilibrium Selection using Coalitional Re…nements

In this section, we apply the three most commonly used stability concepts in order to re…ne the multiplicity of Nash equilibria that arise in Gx. They are Pairwise Strong Nash Equilibrium (PSNE), Coalition Proof Nash Equilibrium (CPNE) and Strong Nash Equilibrium (SNE). The two last concepts can be found in Dutta and Mutuswami (1997), but an stronger de…nition of SNE is provided by Jackson and van den Nouweland (2005).23 A formal de…nition of PSNE is the following:

De…nition 1 (Gilles et al. (2006)) An action-tuple a A is a Pairwise Strong Nash 2 Equilibrium (PSNE) of the game Gx if and only if a is a Nash Equilibrium of the game

Gx and for every pair of players i; j it holds that 2 N i(bi; bj; a i;j; x) > i(a; x) implies that j(bi; bj; a i;j; x) < j(a; x) for all bi Ai and bj Aj 2 2

The set of E¢ cient Allocations for the game Gx, E(Gx), is de…ned as follows:

E(Gx) = a A, such that a arg max i(a; x) (2) 2 2 a A ( 2 i ) X2N parameter x is distributed accordingb to a ‡at prior,b but it is possible to prove that any prior can be treated as a ‡at prior when goes to zero. 23 The main di¤erence between these alternative de…nitions is that a SNE a la Dutta and Mutuswamy (SNEDM ) is not necessarily a PSNE, while a SNE a la Jackson and van den Nouweland (SNEJV ) is always a PSNE. The relations between the sets of strategy pro…les for a general game G are: SNEJV (G) SNEDM (G) CPNE(G) and SNEJV (G) PSNE(G) For the particular link formation game Gx, however, the sets SNEDM (Gx) and SNEJV (Gx) coincide.

14 Under this general set of assumptions it is not our purpose to give a detailed description 24 of the set of Nash equilibria of the game Gx. However, it is possible to show that some particular pro…les, as those in E(Gx), are indeed Nash Equilibria that survive under the traditional coalitional re…nements.

Proposition 3 Consider the link formation game Gx. Under assumptions A1 to A6 we have:

a. The set E(Gx), satis…es:

[0] if x < x E(Gx) = f g 8 [1] if x > x < f g : b. If a E(Gx) then a is a Pairwise Strong Nash Equilibrium (PSNE), Coalition 2 Proof Nash Equilibrium (CPNE) and Strong Nash Equilibrium (SNE).

Proposition 3 shows that the set of strategy pro…les leading to e¢ cient allocations is contained in the set of equilibria selected under the di¤erent coalitional re…nements. As a consequence, no con‡ict exists between e¢ ciency and stability under these re…nements.

4 Conclusion

The goal of this paper was to introduce the use of a non cooperative equilibrium selection approach as a notion of stability in link formation games. Speci…cally, we considered a link formation consent game, introduced by Myerson (1991) and further studied by Dutta et al. (1998) and Gilles et al. (2006). We constrained the payo¤s to a class of supermodular functions de…ned by assumptions A1 to A7, and we applied a global games approach as a stability concept to select equilibria. The equilibrium selected with this stability notion was (essentially) unique. Moreover, it was not only di¤erent, but also con‡icts with those predicted by the traditional coalitional re…nements. As a consequence, an important insight of this paper was to show that the equilibria selected under the cooperative notions of stability are not robust to the incomplete information structure considered.

24 We refer the reader to Gilles et al (2006) to see a complete characterization of the set of Nash Equilibria in closely related consent games.

15 In Proposition 3 we showed that, in the class of link formation games studied, there is no con‡ict between stability and e¢ ciency when coalitional re…nements are used. On the contrary, we showed in Proposition 2 that the con‡ict exists when the global games technique is used, because the actions played in lim 0 BNE(G()) are not e¢ cient ! 25 (k > x). From an applied point of view, the paper highlights the importance of two standard assumptions in the link formation literature. First, the assumption of complete information can be the origin of the multiplicity of networks supported by Nash Equilibria in link formation games. Proposition 2 shows that this multiplicity disappears when we introduce incomplete information. Second, the possibility of cooperation among coalitions of agents seems to be a strong assumption in a link formation game, specially in the presence of coordination problems. This observation, and the con‡ict between the equilibria selected under a coalitional and a global games approach, raise some doubts about which criterion is satis…ed by networks in reality.

5 Appendix

Proof of Lemma 1 (i) ID is implied by (A1a) and f(x) 0 x [X; X]: 8 2 (ii) (SM) is implied by (A1b), (A2) and (A3). (iii) (C) is trivially implied by (A3)

Proof of Proposition 2 The assumption of the existence of dominance regions directly implies that if a player’s private value belongs to these regions, he will select the corresponding dominant action.

For example, if player i receives a private value xi > x he will decide to connect to everybody regardless other players’strategies. On the other hand, if a player receives a private value outside the dominant regions, the action selected will depend on his beliefs about other players’private values. We will show that in the class of games under analysis, players’beliefs will permit to

25 This con‡ict between e¢ ciency and stability does not come as a surprise. In fact Carlsson and van Damme (1996) considered a 2 2 coordination game where the risk dominant equilibrium selected is not necessarily e¢ cient. In this paper we extend the same kind of conclusion to a class of link formation games with a …nite number of players.

16 successively “extend” these dominance regions.26 Each extension is obtained through a round of elimination of strictly dominated strategies. For example, in the …rst round of elimination we prove that there exist x1 > x and x1 < x such that each surviving strategy 1 1 satis…es si(xi) = 0 if xi < x and si(xi) = 1 if xi > x : Moreover, in general, in round t of t t 1 t t 1 deletion of strictly dominated strategy will exist x > x and x < x such that each t t surviving strategy satis…es si(xi) = 0 if xi < x and si(xi) = 1 if xi > x : In what follows, we will prove that these processes of “extension”of the dominance re- t t gions not only converge, but also have the same limit point, i.e. limt x = limt x = !1 !1 k: Therefore k represents a switching point that describes the essentially unique strategy played in equilibrium: for signals less than k players request links with nobody and for signals greater than k players request links with everybody. t De…ning Si as player i’s set of strategies that survive t rounds of deletion of interim strictly dominated strategies in the link formation game G();under assumptions A1 to t A7, we will argue by induction that the set Si is given by:

t t 1 t t S = si S : si(xi) = 0 if xi < x and si(xi) = 1 if xi > x : i 2 i and

0 S = si Si : si(xi) = 0 if xi < x and si(xi) = 1 if xi > x : i f 2 g In each round, xt and xt are obtained recursively as follows: xt corresponds to the minimum private value such that the agent is indi¤erent between connecting with every- t t body (si(x ) = 1; the highest action) and with nobody (si(x ) = 0; the lowest action) when he believes that all opponents are using a switching strategy between the lowest and t 1 27 t the highest action with threshold x . The value x ; is obtained analogously. The following lemma describes the …rst round of elimination, showing that both dominance regions strictly increase, and x1 and x1 are the same for all agents.

Lemma 2 For all i x1 > x and x1 < x such that 9 1 1 1 si S if and only if si(xi) = 0 if xi < x and si(xi) = 1 if xi > x ; where 2 i 26 Strictly speaking, these extensions are not dominance regions, but a set of private values for which players’beliefs leads to select the same dominant action. 27 From lemma 1 part i, we know that there are increasing di¤erences in the total payo¤s, i; then if t si Si such that si(xi) = 0 if xi < x is a best response to this pro…le of switching strategies, it will be 2 t 1 a best response to any s i S i . 2

17 x1= max x : (1; x; x) = 0 and f g x1= min x : (1; x; x) = 0 f g Moreover x1 and x1 are the same for all agents.

Common knowledge of rationality allows players to update their information about other players’dominance regions. This fact permits to repeat the process described in the previous lemma. The last lemma can be easily modi…ed to prove by induction that:

t t 1 t t 1 For all i x > x and x < x such that 9 t t t S = si : si(xi) = 0 if xi < x and si(xi) = 1 if xi > x i f g where:

t t 1 x = max x : (1; x; x ) = 0

t t 1 x = min x : (1; x; x ) = 0 : This process generates a strictly increasing sequence xt and a strictly decreasing se- f g quence xt : Note that both sequences are bounded, the increasing sequence xt is f g bounded above by the beginning of the upper dominance region, x; and the decreasing sequence, xt is bounded below by the beginning of the lower dominance region, x:

Therefore both sequences converge. Let us call these limits points x1 and x1, then

(1; x1; x1) = 0; and

(1; x1; x1) = 0:

From A7 we know that both equations are the same and have a unique solution: k.

Therefore it must be the case that x1 = x1 = k: Finally we can conclude that the only strategy that survives a repeated process of elimination of strictly dominated strategies is a switching strategy between the lowest and

k the highest action with threshold k, s ( ). Then the set containing the surviving pro…les i 1 t k N+1 is a singleton, S1 = S = (si ( ))i=1 t=0\ Proof of Lemma 2 We want to extend the lower dominance region to the right, x1 > x (the analysis

18 for the upper dominance region is analogous). Given (A1) the worst scenario to achieve such an extension is given by when the other players are using a switching strategy with 1 threshold in x. In other words if ai = 0 is a best response action for signals xi < x when x he is facing the pro…le s i, then ai = 0 must also be the best response action to any other 0 s i S i: 2 Accordingly, we are interested in characterizing the expected payo¤ (ai; xi; k): In order to do that it is convenient to introduce the following de…nitions:

h De…nition 2 Ai is player i’ set of homogenous actions (the lowest and the highest ac- h h tion), i.e. A 1; 0 Ai. If ai A , then ai is an homogenous action. i f g 2 i

De…nition 3 Pr(a i s i; x) represents player i’s beliefs about the action pro…le a i; j conditional on other players’strategy s i and that he is receiving the private value x:

h h Denoting A i = j=iAj , player i’interim expected payo¤ can be written as: 6

k (ai; xi; k) = i(ai; a i; xi) Pr(a i s i; xi): h j a i A i X2

In addition we also need to de…ne player i’spayo¤ di¤erence as

De…nition 4 i(ai; ai0 ; a i; x) i(ai; a i; x) i(ai0 ; a i; x) is the payo¤ di¤erence between choosing action ai and the action ai0 :

Then player i’sexpected payo¤ di¤erence is

k (ai; ai0 ; xi; k) = i(ai; ai0 ; a i; xi) Pr(a i s i; xi) h j a i A i X2 In what follows we will be interested in solving (ai; ai0 ; xi; k) = 0. In order to h characterize the solutions to this equation it is convenient to introduce a partition of A i.

h;l h; De…nition 5 Let A i be an strict subset of A i such that the elements are action pro…les where exact l individuals are playing the homogenous action 1 and the rest, (N l); are N h;l h; playing the homogenous action 0. The set A i constitutes a partition of A i, that l=0 h N h;l n o h;l h;l0 is, A i = l=0A i and for l; l0 0; 1; :::; N , l = l0, A i A i = . [ 2 f g 6 \

19 h;l In simple words if a i is an elements of A i then the pro…le a i contain exactly l individuals choosing to connect to everybody and N l choosing to connect to nobody. We can rewrite player i’sexpected payo¤ di¤erence as:

N k (ai; ai0 ; xi; k) = i(ai; ai0 ; a i; xi) Pr(a i s i; xi) (3) j l=0 h;l a i A i X X2

h;l k Noting that, by symmetry, if a i and a0 i A i then Pr(a i s i; xi) = Pr(a0 i 2 j j k s i; xi) we can de…ne: l k De…nition 6 Pr(a i s i; xi) represents player i’s beliefs about any action pro…le in j h;l k A i ; conditional on other players’ switching strategy pro…le s i and that he is receiving the private value xi:

Lemma 3 There exist functions and ' such that h;l i(ai; ai0 ; a i; xi) = (ai; ai0 )'(l; xi) a i A i 2 From Lemma (3)and de…nition (6) we got: P

N l k (ai; ai0 ; xi; k) = (ai; ai0 ) '(l; xi) Pr(a i s i; xi) (4) j l=0 X We conclude that any xi solving (ai; ai0 ; xi; k) = 0 does not depend on ai, ai0 and i. As a result, we can …nd such x by choosing ai0 = 0 and ai = 1 and solving (1; x; k) = 0. Moreover, we are going to prove that (1; x; x) < 0 and (1; x + ; x) > 0 then we can de…ne x1= min x : (1; x; x) = 0 : f g

First, let us prove that (1; x; x) < 0. When xi = k it is possible to prove the following lemma.

l k 1 Lemma 4 Pr(a i s i; k) = (N )(N+1) > 0 for all k [X 2 ; X + 2 ] and for all j l 2 l 0; :::; N . 2 f g Accordingly, we can write:

N l x (ai; x; x) = i(1; a i; x) Pr(a i s i; x) j l=0 h;l a i A i X X2 l x From Lemma 4 we know that Pr(a i s i; x) > 0. From assumption A6, the exis- j tence of dominance regions and using the fact that the payo¤ function satis…es increasing

20 28 di¤erences, it is also true that i(1; a i; x) i(1; [1]; x) 0: But, by assumption h 3, link monotonicity, a i A i such that i(1; a i; x) < i(1; [1]; x) 0; therefore 9 2 (1; x; x) < 0:

Second, let us prove that (1; x + ; x) > 0. In this case player i is receiving a su¢ - ciently high private value such that he knows with probability one that all his opponents are receiving values greater than x; so they are playing the action pro…le a i = [1]. That is:

N l x (1; x + ; x) = i(1; a i; x + ) Pr(a i s i; x + ) = i(1; [1] ; x + ) > 0 j l=0 h;l a i A i X X2

where the inequality arise by Lemma 1ii. Given continuity of the expected utility function and using the intermediate value theorem then > 0; x1 > x such that x1 = min x (1; x; x) = 0 :In other words 8 9 f j g 1 ai = 0 is a best response action for signals xi < x when player i is facing any pro…le 0 s i S i: 2 For the upper dominance region we use again the result that any xi solving (ai; ai0 ; xi; k) =

0 does not depend on ai, ai0 and i. Then, we can …nd such x by choosing ai0 = 0 and ai = 1 and solving (1; x; k) = 0. In the upper region we have that (1; x; x) > 0 and (1; x ; x) < 0 then we can de…ne x1 = max x (1; x; x) = 0 : f j g

Proof of Lemma 3 Using the payo¤ de…nition we get:

i(ai; ai0 ; a i; xi) = [f(xi)vi(ai; ai0 ;a i)) + h(ai; ai0 ; xi)] h;l h;l a i A i a i A i X2 X2 where vi(ai; ai0 ;a i) vi(g(ai; a i)) vi(g(ai0 ; a i)) and h(ai; ai0 ; xi) h(ai; xi) h(ai0 ; xi):

28 Recall that requesting links with nobody gives zero payo¤, i(0; a i; x) = 0 for all a i A i 2

21 From A4, and de…ning J j=i aij and J 0 j=i aij0 6 6 P P h(ai; a0 ; xi) = (J J 0)h(ai; xi) with ai Ai such that aij = 1 i 2 j=i X6 b b b then

N i(ai; ai0 ; a i; xi) = f(xi) vi(ai; ai0 ;a i) + (J J 0)(l )h(ai; xi): 2 3 h;l h;l a i A i a i A i X2 6 X2 7 4 5 b h;l Given that a i A i implies that g(0; a i) is a complete subnetwork connecting l individ- 2 l uals other than i, g i; in this symmetric environment the value h;l vi(g(ai; a i)) a i A i 2 will depend directly on the potential number of networks to beP formed conditional on l: Therefore we can write:

l J N J l vi(g(ai; a i)) = (k )(l k )vi(g i + ij1 + ij2 + ::: + ijk) h;l k=0 a i A i X2 X l where j1; j2; :::; jk are any agents in N(g i): Using assumption A5:

l J N J l l vi(g(ai; a i)) = (k )(l k )kvi(g i + ij) for any j N(g i) 2 h;l k=1 a i A i X2 X l l J N J l = vi(g i + ij) (k )(l k )k for any j N(g i) 2 k=1 X

J J 1 29 Using the combinatorial identity k(k ) = J(k1 ), we know that

l l N J J 1 l vi(g(ai; a i)) = vi(g i + ij)J (l k )(k1 ) for any j N(g i) 2 h;l k=1 a i A i X2 X

30 l N J J 1 N 1 and using Vandermonde convolution formula we get J k=1(l k )(k1 ) = J(l 1 ): 29 See Riordan (1968) page 12. P 30 See Riordan (1968) page 8.

22 Therefore

l N 1 l vi(g(ai; a i)) = vi(g i + ij)J(l 1 ) for any j N(g i) h;l 2 a i A i X2 then the term under analysis is

l N 1 N l i(ai; ai0 ; a i; xi) = (J J 0)f(x)vi(g i+ij)(l 1 )+(J J 0)(l )h(ai; x) for any j N(g i) h;l 2 a i A i X2 b Which can be written as

h;l i(ai; ai0 ; a i; xi) = (ai; ai0 )'(l; xi) a i A i 2 Pwith

(ai; ai0 ) = j=i aij j=i aij0 and 6 6 l N 1 N l '(l; xi) = vPi(g i + ij)(Pl 1 ) + (l )h(ai; x) for any j N(g i) 2

Proof of Lemma 4 b Given symmetry a player observing a private value k believes that the probability that N l k 31 exact l individuals that have private signals greater than k is (l ) Pr(a i s i; k): , and j

1 N l k N N l l (l ) Pr(a i s i; k) = () [()] [1 ()] d (5) j 0 N l 1 Z @ A 1 k x with = and is the density ( the cumulative) of the noise "i: Consider a more general expression:

1 N N l+k l k () [()] [1 ()] d (6) 0 N l + k 1 Z @ A 1 So that the original integral is obtained when k = 0. l k Consider any k with 0 k l 1. Integrating by parts with u = [1 ()] and N l+k dv = () [()] we got that (6) is equal to:

1 N N+1 l+k l k 1 () [()] [1 ()] d (7) 0 N + 1 l + k 1 Z @ A 1 31 This proof use the same argument showed in appendix B in Morris and Shin (2002)

23 And then iterating until k = l 1 we obtain that (6) is equal to:

1 () [()]N d (8) Z 1 Integrating by parts again, but now using u = [()]N and dv = () we got:

1 1 () [()]N d = 1 N () [()]N d Z Z 1 1 and then:

1 1 () [()]N d = (9) N + 1 Z 1 1 l As a consequence, the original integral in (5) is equal to N+1 : Therefore Pr(a i j k 1 s i; k) = (N )(N+1) for all k [X 2 ; X + 2 ] and for all l 0; :::; N : l 2 2 f g

Proof of Proposition 3

(a) By de…nition, for given x, the set of E¢ cient Allocations of the game Gx is given by the strategy pro…les solving: max i(ai; a i; x): From the proof of part (b) below, a A 2 i X2N we know that the strategy pro…le a = [1] satis…es: i(1;[1]; x) i(ai; a i; x) x > x 8 , a A, i . Consequently, a = [1] E(Gx) for all x > x. 8 2 8 2 N 2 Suppose that there exists other strategy pro…le a = [1] such that a E(Gx) for x > x. 6 2 In such a case, there exists j such that aj = 1 and for this agent, using Lemma 1 2 N 6 (ID), (A4) and (A5) we have: j(aj; a j; x) j(aj;[1]; x) < j(1;[1]; x) x > x; 8 and then the sum of pro…ts would not be maximized in the original pro…le. As a result, the unique E¢ cient Allocation when x > x is given by the strategy pro…le a = [1]. An analogous argument leads us to prove that the unique E¢ cient Allocation when x < x is given by the strategy pro…le a = [0] (b) The …rst step is to prove that a = [0] and a = [1] are indeed Nash Equilibria of the game when x < x and x > x respectively.

Consider …rst x > x and any i such that a i = [1]. If ai = 1 then by Lemma 1 2 N 6 (SM) and (A6) we have:32

32 In what follows we use Lemma 1 with ai0 = 0 and the implied properties over i(ai; a i; x).

24 i(ai; [1]; x) i(ai; [1]; x) = i(0; [1]; x) = 0 where the inequality is not necessarily strict because ai can takes the value 0. Using (A4) and (A5) we have:

i(1; [1]; x) > i(ai; [1]; x) 0 ai = 1; x > x: (10) 8 6

In other words, when the others are playing a i = [1], then play ai = 1 is a strict best response. As a result, a = [1] is a strict NE of the game when x > x. Now we are going to prove that a = [1] is a SNE of the game when x > x: Consider any strategy pro…le a A; a = [1] and any x > x. By Lemma 1 (ID) we have: 2 6 i(ai; a i; x) i(ai; [1]; x) i ; and using equation (10) we have i(ai; a i; x) 8 2 N i(1;[1]; x) i ; ai Ai and a i A i. In particular, no coalition of players 8 2 N 8 2 8 2 can be strictly better o¤ by using a coalitional deviation, which completes the proof that a = [1] is a SNE a la Dutta and Mutuswami (1997) of the game when x > x. Noting that there is no coalitional deviation where some players are strictly better o¤ and others weakly better o¤, a = [1] is also a SNE a la Jackson and van den Nouweland (2005). As a consequence, the strategy pro…le a = [1] is also a CPNE and a PSNE of the game Gx for x > x:

Now in the case that x < x, by (A6) the strategy ai = 0 is strictly dominant for player i and then a = [0] is a strict NE of the game Gx when x < x. Moreover, given any strategy pro…le a = [0] (Nash equilibrium or not) and any x < x we have: i(0; [0]; x) 6 i(ai; a i; x) i and then a = [0] is a SNE of the game Gx under the alternative 8 2 N de…nitions of Dutta and Mutuswami (1997) and Jackson and van den Nouweland (2005).

As a result, a = [0] is also a CPNE and a PSNE of the game Gx for x < x:

6 References

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