Dynamic Network Formation with Incomplete

Information∗

Yangbo Song† Mihaela van der Schaar‡

November 20, 2013

Abstract

How do networks form and what is their ultimate topology? Most of the literature that addresses

these questions assumes complete information: agents know in advance the value of linking to other

agents, even with agents they have never met and with whom they have had no previous interaction

(direct or indirect). This paper addresses the same questions under what seems to us to be the

much more natural assumption of incomplete information: agents do not know in advance – but

must learn – the value of linking to agents they have never met. We show that the assumption

of incomplete information has profound implications for the process of network formation and the

topology of networks that ultimately form. Under complete information, the networks that form

and are stable typically have a star, wheel or core-periphery form, with high-value agents in the core.

Under incomplete information, the presence of positive externalities (the value of indirect links) implies

that a much wider collection of network topologies can emerge and be stable. Moreover, even when

the topologies that emerge are the same, the locations of agents can be very different. For instance,

when information is incomplete, it is possible for a hub-and-spokes network with a low-value agent

in the center to form and endure permanently: an agent can achieve a central position purely as the

result of chance rather than as the result of merit. Perhaps even more strikingly: when information

is incomplete, a connected network could form and persist even if, when information were complete,

no links would ever form, so that the final form would be a totally disconnected network. All of this

can occur even in settings where agents eventually learn everything so that information, although

initially incomplete, eventually becomes complete.

Keywords: Network Formation, Incomplete Information, Dynamic Network Formation, Link

Formation, Formation History, Externalities

JEL Classification: A14, C72, D62, D83, D85

∗We are grateful to William Zame, Ichiro Obara, Moritz Meyer-ter-Vehn, Sanjeev Goyal, Luca Canzian,

Simpson Zhang and a number of seminar audiences for suggestions which have significantly improved the paper.†Department of Economics, UCLA. Email: darcy07@ucla.edu.‡Department of Electrical Engineering, UCLA, and Director of UCLA Center for Engineering Economics,

Learning, and Networks. Email: mihaela@ee.ucla.edu.

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1 Introduction

How do social and economic networks form and what is their ultimate shape (topology)? The lit-

erature, which includes work by Jackson and Wolinsky (1996)[20], Barabasi and Albert (1999)[4]

Bala and Goyal (2000)[2], Jackson and Watts (2002)[19] and Ballester et al. (2006)[3], has ad-

dressed these questions in both static and dynamic contexts. The central conclusion of this

work is that special shapes of networks can occur and persist. However, this literature makes

the strong assumption of complete information: agents know in advance the value of linking

to other agents - even agents they have never met and with whom they have had no previous

interaction (direct or indirect). The present paper addresses the same questions under what

seems to us to be the much more natural assumption of incomplete information: agents do not

know in advance – but must learn – the values of linking to agents they have never met1. As

is usual in environments of incomplete information, agents begin only with beliefs about the

values of linking to other agents, make choices on the basis of their beliefs, and update their

beliefs (learn the true values) on the basis of their experience (history).

We show that the assumption of incomplete information has profound implications for both

the process of network formation and the topology of networks that ultimately form. When

information is complete, the networks that form and persist typically have a star or core-

periphery form, with high-value agents in the core. By contrast, when information is incomplete,

a much larger variety of networks and network shapes can form and persist. Indeed, the set of

networks that can form and persist when information is incomplete is a superset (typically a

strict superset) of the set of networks that can form and persist when information is complete.

Moreover, even when the network shapes that form are the same or similar, the locations of

agents within the network can be very different. For instance, when information is incomplete,

it is possible for a star network with a low-value agent in the center to form and persist

indefinitely; thus, an agent can achieve a central position purely as the result of chance rather

than as the result of merit. Perhaps even more strikingly, when information is incomplete,

a connected network can form and persist even if, when information were complete, no links

would ever form so that the final form would be a totally disconnected network.

However, the most important consequence of incomplete information is not that a larger

variety of network shapes (topologies) might emerge, but that the particular shape that does

emerge depends on the history of link formation and of link formation opportunities. For

instance, when information is incomplete, agents i, j might choose to form a link because each

expects the value of the link to exceed the cost of forming it. Having formed the link, the agents

may learn that their expectations were wrong and so might wish to sever it. However, before

the agents have the opportunity to sever the link, each of them may have formed other links,

so that the indirect value of the link between i and j - the value of the connection to other

1We emphasize that we study learning during the network formation process, rather than learning in an

exogenously given and fixed network structure. For a study on the latter, see for example Acemoglu et al.

(2008)[1]

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agents - may be sufficiently large that they prefer to maintain the link between them after all.2

However, whether these other links have formed will depend not only on the values of those

links but also on the random opportunities presented to form or not form them.

We stress that all of this can occur even in settings where agents eventually learn everything

so that information, although initially incomplete, eventually becomes complete. Incomplete-

ness of information may eventually disappear but its influence may persist forever.

2 Literature Review

The literature on network formation studies the characteristics of networks emerging as a

result of the strategic interaction of self-interested agents. This literature includes renowned

early papers in this area, including Jackson and Wolinsky (1996)[20], Bala and Goyal (2000)[2],

Watts (2001)[32], etc. These papers and subsequent works building on them (for instance

Johnson and Giles (2000)[21]) assume that agents are homogeneous, i.e. the value that each

agent in the group provides to and receives from another agent is the same. This is the strongest

form of complete information, in the sense that agents do not only know their exact payoffs

from linking to others, but are also aware that the payoffs are solely determined by the network

topology, and that the agents’ identities play no role in affecting payoff characteristics. As a

result, a prominent feature in network topologies that emerge and/or become stable is that they

are either empty or connected. By contrast, in our paper, agents are of different types and thus

connecting to them results in different (heterogeneous) payoffs; moreover, there is incomplete

information: an agent does not know the types of agents that he has never connected with,

but he is able to form beliefs based on which he chooses the optimal action. Networks which

result under this, more realistic, assumption are strikingly different from those obtained in the

model assuming homogeneous agents and complete information. On one hand, connectedness

is no longer a key property of stable networks - the formation process converges to connected

networks in some range of parameters and to multiple components in other ranges. On the

other hand, even if the network stays empty forever under complete information, a non-empty,

even connected network may emerge and be stable with positive probability under incomplete

information.

The literature studying how agent heterogeneity affects network formation defines ”hetero-

geneity” in various manners: as differentiated failure probabilities for different links (Haller

and Sarangi (2003)[15]), as differences in values obtained from links and costs for forming links

across agents (Galeotti (2006)[11], Galeotti et al. (2006)[13]), as different amounts of valuable

information produced endogenously (Galeotti and Goyal (2010)[12], Zhang and van der Schaar

(2012)[34]), as different endogenous effort levels (Koenig et al. (2012)[22]), or as agent-specific

resource holding amounts (Zhang and van der Schaar (2013)[35]). Despite such diverse views

on heterogeneity, complete information is still a common assumption, and the predictions in

2Indirect linking provides a positive externality when information is complete as well, but the effect is

completely different: i, j will never wish to form a link at some point in time and sever it later.

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the above papers are often restricted to a few, specific, types of network topologies such as

stars, wheels or core-periphery networks. Moreover, in networks with asymmetric connectivity

degree distribution, for instance stars and core-periphery networks, agents with high values

or low costs always stay at the center or core, whereas agents with low values or high costs

remain in the periphery. We differ from these works in four aspects. First, as mentioned above,

agents have no precise knowledge about their exact payoffs due to incomplete information;

instead, they choose an optimal action according to their beliefs about the payoffs that they

will obtain from connecting to others. Secondly, in most of the above papers an agent’s payoff

is affected by a single factor, either his own characteristics or other agents’ characteristics;

our model allows both factors to jointly determine the payoffs, for instance, a type a agent’s

payoff from linking to another type a agent may differ from his payoff from linking to a type

b agent, and vice versa. Thirdly, we show that the interaction of incomplete information and

agent heterogeneity produces a much wider range of network topologies, which includes stars,

wheels, core-periphery networks, etc. Finally, the topology that emerges and becomes stable

strongly depends on the formation history : an agent may exhibit a high degree of connectivity

in equilibrium not necessarily because he is of a special (high) type but also because initially

he was fortunate to obtain sufficiently many links by chance, which in turn attracted others to

form and maintain links with him due to the large indirect benefits that he can offer. Therefore,

unlike most existing literature that only emphasizes what topologies can be formed, we argue

that how a certain topology comes into being is equally important.

Among the empirical literature on network formation games, works such as Falk and Kosfeld

(2003)[8], Corbae and Duffy (2008)[5], Goeree et al. (2009)[14] and Rong and Houser (2012)[30]

have conducted experimental studies on the types of emerging topologies. In particular, Goeree

et al. (2009)[14] raises similar concerns to ours regarding the assumption of complete infor-

mation being unrealistic in practice, and introduces incomplete information about the private

types of agents in their experiments. The experimental results indicate that (1) typical equilib-

rium network topologies predicted by the existing theoretical analysis are not always consistent

with the empirical observations; especially, stars are formed only in a proportion of the total

number of experiments conducted([5],[30]), and such proportions, under some treatments such

as a two-way flow of payoffs, are rather low ([8]); (2) even in the experiments where equilibrium

network topologies do emerge with high frequency, such topologies are developed rather than

born([14]), which we believe suggests a dynamic network formation process of a sufficiently

long duration as a more appropriate environment for stable networks to emerge, compared

with a static one. Moreover, the study of networks which actually get formed in large social

communities, as presented for instance by Mele (2010)[25] and Leung (2013)[23], shows that in

environments where agents are heterogeneous and withhold certain private information in their

payoffs from links, numerous phenomena which are not predicted by the existing theoretical

literature can happen. For instance, networks can have multiple components, agents with dif-

ferent attributes can cluster over time, agents can remain singletons even in the long run, etc.

Our model does not explain all these phenomena; however, we do believe that incorporating

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incomplete information in the dynamic network formation game represents a first and impor-

tant step towards understanding why several previously seemingly irregular network topologies

can emerge and remain stable in practice.

One assumption in our model that does stay in conformity with the above theoretical works

is agent myopia, i.e. agents only care about their current period payoffs when making decisions.

In contrast to this assumption, there is a literature including Page Jr. et al. (2003)[29],

Deroian (2003)[6] and Dutta et al. (2005)[7], that considers farsighted agents, in the sense

that agents take the effect that their current actions have on future payoffs into account. The

major objective of these papers is to construct equilibrium network topologies that are strongly

efficient. We would like to point out that in our setting, determining the topology of a strongly

efficient network would be by itself a challenging task since such topologies may not be unique

(to be discussed in more detail later). We leave a thorough analysis on this topic as a promising

direction for future research.

Finally, regarding learning in social networks, the usual approach as in Gale and Kariv

(2003)[10] and Acemoglu et al. (2008)[1] is to assume a fixed network topology or an exogenous

process determining the topology, and study the evolution of actions and beliefs as information

gets updated over time. Essentially, the learning process in our paper goes in line with the

above literature in the sense that agents are learning about a payoff relevant state which is

the type vector; however, in our model the network formation process is endogenous, which

is a key difference to the works existing in this area. In other words, in our environment two

processes evolve simultaneously : the network formation process, which pins down the network

topology, and the belief formation process, which determines the information each agent learns

about others’ types. On one hand, network formation serves as the medium of belief formation

since agents acquire knowledge on others’ types via connection. On the other hand, belief

formation provides the basis for network formation because agents choose actions according to

their current beliefs.

The rest of the paper is organized as follows. Section 3 introduces the model. Section 4

analyzes the model in detail and interprets the results. Section 5 analyzes a special case in

the context of large networks. Section 6 discusses some extensions of the model. Section 7

concludes and introduces relevant future research topics.

3 Model

Our notation mainly follows Jackson and Wolinsky (1996)[20], Bala and Goyal (2000)[2] and

Watts (2001)[32].

3.1 Networks with Incomplete Information

Networks and the Agents’ Types

Let I = {1, 2, ..., N} denote a group of N agents. Each agent can either be of type a or

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type b. Most of our analysis in a two-type environment carries over to one with more than two

types, yielding similar conclusions with slightly modified conditions. Let ki denote agent i’s

type, and let na and nb denote the number of type a and type b agents. Let κ = {ki}Ni=1 denote

the type vector of the agents. The realization of this type vector is drawn i.i.d. for each agent,

with Prob(a) = p and Prob(b) = 1− p.A network is denoted by g ⊂ {ij : i, j ∈ I, i 6= j}, and a sub-network of g on I ′ ⊂ I, denoted

gsub(I′), is defined as a subset of g such that ij ∈ gsub(I

′) if and only if i, j ∈ I ′ and ij ∈ g.

ij is called a link between agents i and j. We assume throughout that links are undirected, in

the sense that we do not specify whether link ij points from i to j or vice versa. A network g

is empty if g = ∅.

We say that agents i and j are connected, denoted ig↔ j, if there exist j1, j2, ..., jn for some

n such that ij1, j1j2, ..., jnj ∈ g. Let dij denote the distance, or the smallest number of links

between i and j. If i and j are not connected, define dij := ∞. An agent i in a network is a

singleton if ij /∈ g for any j 6= i.

Let N(g) = {i|∃j s.t. ij ∈ g}. A component of network g is a maximal connected sub-

network, i.e. a set C ⊂ g such that for all i ∈ N(C) and j ∈ N(C), i 6= j, we have iC↔ j, and

for any i ∈ N(C) and j ∈ N(g), ij ∈ g implies that ij ∈ C. Let Ci denote the component that

contains link ij for some j 6= i. Unless otherwise specified, in the remaining parts of the paper

we use the word ”component” to refer to any non-empty component.

A network g is said to be empty if g = ∅, and connected if g has only one component which

is itself. g is minimal if for any component C ⊂ g and any link ij ∈ C, C − ij is no longer a

component. g is minimally connected if it is minimal and connected.

Payoff Structure

Following the assumption in literature on networks that explicitly model non-local external-

ities3, such as Jackson and Wolinsky (1996)[20], Watts (2001)[32] and Galeotti et al. (2006)[13],

we assume that once agents i and j form a link, i not only obtains payoffs from his immediate

neighbor j, but also from the agents that he is indirectly connected to via that particular link.

As stated before, the payoff of an agent also depends on the type vector. Specifically, an agent

i’s payoff from a network g is given by

ui(ki, k−i,g) = ui(ki, k−i, Ci) :=∑jCi↔i

δdij−1f(ki, kj)−∑j:ij∈Ci

c

where f : {a, b} × {a, b} → R++ is the payoff function for an agent from a link with another

agent. Therefore, f(ki, kj) > 0 denotes the payoff to an agent i of type ki by linking to an

agent j of type kj. c > 0 is the cost of maintaining a link, which is assumed to be bilateral and

homogeneous across agents. δ ∈ [0, 1] denotes a common decay factor4, such that the payoff of

i from j with a distance of dij is δdij−1f(ki, kj). Note that c is assumed to be independent of

3As opposed to papers that assume pure local externalities, i.e. agents only obtain payoffs from their

immediate neighbors. See for example Galeotti and Goyal (2010)[12].4In this paper, unless otherwise specified, we assume that δ ∈ (0, 1).

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the agents’ types5. Without loss of generality, we assume that f(a, a) ≥ f(b, b), but we impose

no additional assumptions on the relative magnitude among f(a, a), f(a, b), f(b, a) and f(b, b).

In particular, the payoffs may be asymmetric, i.e. f(ki, kj) may be different from f(kj, ki)

when ki 6= kj. As we will see in the analysis, the relative magnitudes of these parameters have

decisive effects on the resulting convergence of the formation process.

Note that, given the considered payoff structure, there are increasing returns to link for-

mation: linking with an agent is more valuable if that particular agent connects to a larger

number of agents. Moreover, when agents are heterogeneous, not only does the (discounted)

number of connections matter, but also the types of agents connected.

From the formulation above, two forces are at work in determining whether a certain link

is valuable for an agent. On one hand, the payoff an agent gets from a link not only depends

on his own type, but also the type of the counterparty. As a result, the private information an

agent possesses–his type–determines his willingness to link to others as well as his preference

on what types of agents to link to. On the other hand, the belief of an agent over the types

of others plays an equally crucial role. In contrast to works in the literature which assume

complete information on payoffs, the same realization of a type vector may result in rather

distinct network formation processes given different beliefs. We provide the following notations

to emphasize the above two points, which will be discussed throughout the analysis.

We define the self effect by fself = |f(a, a)− f(b, a)|+ |f(a, b)− f(b, b)|, and the peer effect

by fpeer = |f(a, a)−f(a, b)|+ |f(b, a)−f(b, b)|.6 As the names indicate, the self effect measures

how much an agent’s payoff depends on his own type, and the peer effect measures how much

his payoff depends on the types of agents he connects to. For example, a strong self effect may

be found in a network among workers collaborating on a job, where agents need to exchange

homogeneous messages from time to time but some agents are introverted/extroverted and

thus have different benefits from social interaction, regardless of the type of the neighbor. In

this case, the payoff of an agent would only depend on his own characteristics. Strong peer

effects are often present in networks dedicated to information and resource sharing (e.g. peer

to peer networks), such as data, knowledge, news as well as computational and processing

capabilities, since in such networks the resource acquirer’s characteristics would not affect the

quality of resources it obtains from others. The two effects can both be strong in social networks

– not only do different appearances or personalities leave different impressions, but different

people may have different preferences even towards the same person. Note that when both

effects are zero, the model essentially becomes one with homogeneous agents and no incomplete

information, since there is no payoff heterogeneity whatsoever. In the analysis, we will show

that the magnitude of these two effects will greatly influence the formation process and will

5We make this assumption to economize on notations. In the case where costs are also heterogeneous, our

analysis and results hold with slightly modified conditions.6The effect-relevant results we derive are mostly about pure self effect or pure peer effect, i.e. one of the

effects is zero. Thus we tacitly assume the particular forms of the two terms as above, and even if we defined

the effects alternatively by putting positive coefficients before the absolute values, for instance a weighted sum,

the results would not be affected.

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lead to the emergence of very different network structures.

Let fa = pf(a, a) + (1 − p)f(a, b) denote a type a agent’s expected payoff from linking to

a singleton agent, and let fb = pf(b, a) + (1− p)f(b, b) denote that of a type b agent. Because

information is incomplete, the expected payoff and the actual payoff are different, which can lead

to a very different formation process from the one under complete information. For example,

consider a group consisting of only type b agents where f(b, b) < c. Under complete information,

no link will ever form since no agent has the incentive to form the first link. However, under

incomplete information, if fb is large enough, agents would be incentivized to make initial links,

and it is possible that a number of agents would be connected after a few periods. Due to the

assumption that externalities are not purely local, i.e. agents get payoffs from other indirectly

connected agents, the positive externality may become large enough to maintain a non-empty,

even connected network. We will derive general properties and provide more examples in the

analysis.

3.2 Dynamic Network Formation Game

The Game

We model the dynamic game in a similar fashion to Watts (2001)[32] and Jackson and Watts

(2002)[19]. Time is discrete and the horizon is infinite: t = 0, 1, 2, .... The game is played as

follows: agents start with an empty network ∅ in period 0. In each following period, a pair of

agents (i, j) is randomly selected to update the link between them. For simplicity, we assume

that the matching probability is uniform, so (i, j) is selected with probability 2N(N−1) . The two

agents then play a simultaneous move game, where each can choose to sever the link between

them if there is one, and if there is not, whether to agree to form a link with the other agent.

Let aij = 1 denote the strategy that i agrees to form a link with j (if there is no existing link)

or not to sever the link (if there is an existing one), and aij = 0 otherwise. A link is formed or

maintained after bilateral consent (i.e. aij = aji = 1). Let γ(t) := {(i, j)t′}tt′=1 (t ≥ 1) denote

a selection path up to time t, or the set of selected pairs of agents, ordered from 1 to t. The

agents are assumed to be myopic, i.e. they only care about their current period payoffs.

In addition, we make the following assumptions throughout the paper:

Assumption 1. (Component observation) When agents i and j are selected, i can observe Cj,

and vice versa.

Assumption 2. (Linking is valuable in expectation) The expected payoff fk ≥ c ∀k ∈ {a, b}.

Assumption 1 is common in the literature on network formation. It means that there is

no incomplete information on the network topology regarding the link to be updated, so our

analysis focuses on the effect of incomplete information about types. Assumption 2 works as a

participation incentive: since agents are myopic and the network is empty to start with, we let

their expected payoff from a first link be non-negative, so that they are willing to participate

in this network formation game from the beginning. Note that even if fk ≥ c, it could still be

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the case that f(k, k′) < c for some type k′. For example, suppose that p = 0.5, f(a, a) = 10,

f(a, b) = 2, c = 5. Then fa = 6 > c but we still have f(a, b) < c.

Updating Rule

With incomplete information, agents maximize their expected payoffs, rather than actual

payoffs, when deciding the optimal action. Therefore, the belief of an agent on the types of

the other agents plays a crucial role in shaping his behavioral patterns. For any i, we let

Bi ∈ ∆{a, b}N denote agent i’s belief on the type vector of the group, and B = (B1, ..., BN)

define a belief vector for the group of agents. Note that for any i, Bi must put zero probability

on any type vector where i’s type differs from the true type. Moreover, we assume the following

simple updating rule for the agents:

1. If two agents are ever connected, they know each other’s type.

2. Otherwise, their belief on each other’s type remains at the prior.

In other words, agents can only observe their own formation history: they do not observe

the actions by agents in other components and thus make no relevant inferences on their

types. A similar assumption appears in McBride (2006)[24], which is denoted as imperfect

monitoring and describes agents’ inability to observe all other agents’ strategies in a static

network formation game. A plausible alternative updating rule is to allow agents to observe

each other’s past actions and perform Bayesian updating accordingly, which results in very

complicated belief formation. We will discuss this in a later section. Throughout the paper,

we assume that the updating rule is common knowledge among the agents.

4 Analysis

In this section, we analyze the nature of the network formation process and show a clear contrast

between results under complete information and incomplete information. We begin by defining

the solution concept for the two-player game each period, and the notion of a stable network.

4.1 Stable Optimistic Equilibrium and Stable Network

Stable Optimistic Equilibrium

Let C denote the set of all possible components (including the empty component), i.e. when

agents i and j are selected, for agent i, C is the set of all possible observations of Cj. Let B denote

the set of all possible beliefs for an agent. Denote K = {a, b}×C×B as the set of information7

for an agent who is selected to update a link. The interpretation of knowledge is the following:

the first argument is the agent’s own type, the second argument is the component containing

7Technically speaking, agent i’s information also includes Ci, but omitting such information does not affect

the existence and any property of the equilibrium we define below.

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his counterparty, i.e. the other agent selected, and the third argument is his current belief on

the type vector.

Now we define an agent’s (pure) strategy in the two-player game after a pair of agents is

selected, followed by our proposed solution concept, which we call a stable optimistic equilibrium

(SOE):

Definition 1. Agent i’s (pure) strategy towards j is sij : K → {0, 1}, a function from agent i’s

set of information to his set of actions.

Definition 2. A strategy profile s = (sij, sji) is a stable optimistic equilibrium (SOE) if: (1) it

is mutual best response; (2) sij = 1 if

E[ui(ki, k−i, (Ci ∪ Cj) + ij)|Bi] ≥ ui(ki, k−i, Ci)

In words, a SOE is an equilibrium where any agent would choose to agree to form a link (if

there is no existing link) or choose not to sever the link (if there is an existing one) as long as

the expected payoff from the link is non-negative. It is stable because the prescribed strategy is

robust to small probabilistic changes in the counterparty’s strategy when the above inequality

is strict. More specifically, if some agent j other than i changes sji with a sufficiently small

probability ε, i’s best response would not change. It is optimistic in the sense that it excludes

the “pessimistic”, or null equilibrium in which no agent agrees to form any link. In other words,

agents choose to link when they are indifferent. If two selected agents know each other’s type,

then they simply play a weakly dominant strategy; otherwise, by Assumption 2 the expected

payoff from linking is non-negative, so it is a mutual best response for each agent to agree.

This solution concept can be viewed as an extension of pairwise stability introduced by

Jackson and Wolinsky (1996)[20], with the new element of incomplete information and belief

formation. In their definition, a link between two agents is pairwise stable if it brings non-

negative payoffs to both, and there is no need to distinguish between payoffs before and after

link formation because information is complete. In our case, we define the equilibrium in

conformity with the idea that a link is formed, or is not severed, if it brings bilateral non-

negative payoffs; however, we highlight the clear difference between expected payoff, i.e. the

payoff before link formation and realized payoff, i.e. the payoff after.

We would like to emphasize the connection between the existence of SOE and Assumption

2. By Assumption 2, the expected payoff from a link to some unknown agent is non-negative;

when two agents who have never been connected before, say i and j, are selected, even though i

may not know j’s belief about the types of other agents in Ci (and vice versa), i does know that

j would play the equilibrium strategy of agreeing to form a link, no matter what j’s type is. As

a result, also agreeing would be i’s best response. Assumption 2 ensures that an agent makes

no inference on the counterparty’s type from equilibrium strategies, and thus the prescribed

strategy in an SOE indeed constitutes an equilibrium.

Lemma 1. In every period, for any pair of selected agents, any network topology and any belief

vector according to the simple updating rule, a SOE exists and is unique.

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Lemma 1 together with the above robustness property of a SOE ensures that the outcome

of the formation process is unique and robust to small perturbation.

Stable Network

Let g(γ(t)) denote the unique network formed after period t, following a selection path of

γ(t). Let B(γ(t)) denote the associated belief vector after period t. By Lemma 1 we know

that both g(γ(t)) and B(γ(t)) are well-defined. Let σ(t) := (γ(t), {g(γ(t′))}tt′=1) denote a

formation history up to time t. We say that:

1. A network together with the associated beliefs (g, B) can emerge if there exists a selection

path γ(t) for some t, such that g = g(γ(t)), B = B(γ(t)).

2. (g, B) is a stable network if no link is formed or broken given any subsequent selection path.

3. The formation process can converge to g if there exists a selection path γ(t) for some

t, such that g = g(γ(t)) and (g(γ(t)), B(γ(t))) is stable. Mathematically, we denote it as

limt→∞ g(γ(t)) = g for some fixed γ(∞).

4.2 Contrast with Complete Information

Before discussing the differences between the network topologies that can emerge and be stable

under complete information and incomplete information, we first inspect how long incomplete

information can persist. We say that information is complete when every agent’s belief is the

degenerate belief on the true type vector κ, i.e. Prob(κ|Bi) = 1 for any i, and that information

is incomplete otherwise.

Proposition 1. In the formation process, for every κ:

1. Information becomes complete within finitely many periods almost surely.

2. Information is complete in any stable network.

In essence, everyone eventually learns the true type vector with probability 1 over time.

Indeed, as agents are willing to form links with others of unknown type, after sufficiently many

periods the probability of unconnected agents still existing in the group would be arbitrarily

small. In other words, the effect of incomplete information and thus the difference with complete

information only occurs in an early stage of the formation process. Moreover, we can now

suppress the notation for beliefs in (g;B) and only use g when referring to a stable network.

Nevertheless, even though Proposition 1 may leave the impression that incomplete infor-

mation is not so important as it only takes effect in the short run, we will emphasize in the

following analysis that such short-term influence is actually persistent over time.

Let GC(κ) = {g : g = g(γ(t)) for some γ(t)} denote the set of networks that can emerge

under complete information given κ, and GIC(κ) that under incomplete information. Similarly,

let G∗C(κ) = {g ∈ GC(κ) : g = limt→∞ g(γ(t)) for some γ(∞)} denote the set of networks

11

that can emerge and be stable under complete information given κ, and G∗IC(κ) that under

incomplete information.

Theorem 1. For every κ, GC(κ) ⊂ GIC(κ).

Theorem 1 states that if some network can emerge under complete information, then it can

also emerge under incomplete information. Intuitively, if a link could be formed under complete

information, then given Assumption 2 and the simple updating rule, it can also be formed under

incomplete information, whether or not the relevant agents know each other’s type. Note that

the reverse is not necessarily true: even if a link could be formed under incomplete information,

it may not form under complete information because the expected payoffs are sufficiently higher

than the realized payoffs, i.e. in the incomplete information setting, were the agents to know

each other’s type beforehand, the link may never be formed.

Our next result shows a similar relation between complete and incomplete information when

the formation process converges.

Theorem 2. For every κ, G∗C(κ) ⊂ G∗IC(κ).

Theorem 2 states that if the formation process can converge to some network under complete

information, then it can converge to the same network under incomplete information as well.

Again, the reverse may not be true: under incomplete information, high expected payoffs can

initialize link formation such that even though agents would “regret” the links they form after

knowing each other’s type, more links have formed before they are selected again to update

their initial links. Then due to increasing returns to link formation, the positive externalities

would in turn ensure that the initial links are maintained. The following example illustrates

this point.

Example 1. Assume the following parameter values: N = 5, ki = b ∀i = 1, ..., 5, and the other

parameters are such that f(b, b) < c, fb ≥ c, (1 + δ − δ2 − δ3)f(b, b) ≥ c.

Consider the following selection path: (1, 2), (2, 3), (3, 4), (4, 5), (1, 5). Under complete

information, it is clear that the network remains empty regardless of the selection path, as in

Figure 1(A). Under incomplete information, the SOE can be explicitly computed in each period.

For example, in period 1, agent 1’s expected payoff from the link with agent 2 is fb ≥ c and vice

versa, and thus the link is formed. The formation process is shown in Figure 1(B).

According to the assumptions on parameters, one can then easily show that the network

formed in period 5 is stable.

From the above example, we can see the stark contrast between the two environments:

even if no agent would make any single link under complete information, a connected network

can emerge under incomplete information, and finally be stable due to the sufficient positive

externalities created. With the same setting as Example 1, if we consider a larger group

of agents and an appropriate selection path, the formation process converges to the network

shown in Figure 2(B)8.

8For instance, under the following selection path: (1, 2), (1, 3), (2, 6), (3, 7), (6, , 7), (1, 5), (5, 9), (6, 9), (1, 4),

(4, 8), (7, 8), (8, 9).

12

Figure 1: Empty under Complete Info. vs. Connected under Incomplete Info.

Figure 2: Empty under Complete Info. vs. ”Hub-and-Spokes” under Incomplete Info.

The network exhibits a certain “law of the few”, in the sense that only agent 1 in the

center has the highest connectivity degree. However, unlike in Galeotti and Goyal[12] where

the few agents with high connectivity degree stand out because they are of a more valuable

type (i.e. they are the only ones producing valuable resources), in the above network agents are

identical in type, and some agent has a higher connectivity degree than others merely because

the resulting convergence is history dependent. In other words, even if agents turn out to be

homogeneous, it may well be the case that some agent gets ”lucky” in the selection process

and becomes linked with a considerable number of other agents. In subsequent periods, this

agent attracts even more links to be formed because linking to him grants access to large

indirect benefits. Though occurring by pure chance initially, such a topology in which agents

are homogeneous in type but heterogeneous in connectivity may finally be stable, due to the

existence of increasing returns to link formation.

In fact, such history dependence may generate an even stronger result: even if a type is more

valuable or preferable than the other, under incomplete information it is not necessary that an

agent of that type ends up with a higher connectivity degree. Consider the following example:

assume the same values for fb, f(b, b), c and δ as in Example 1, and in addition that fa ≥ c,

13

f(a, b) ≥ c, (1− δ)f(a, b) < c, and consider a group of agents consisting of 1 type a agent and 8

type b agents. Under complete information, the formation process will only converge to a star

network with the type a agent as the center (Figure 3(A)); under incomplete information, the

formation process may again converge to a ”hub-and-spokes” network (Figure 3(B)).

Figure 3: Different Connectivity Degree Distributions

Under complete information, the type a agent has to be the center of the star network

because the only possible type of link that agents would be willing to form is one between a

type a agent and a type b agent. By contrast, under incomplete information, it first becomes

possible for two type b agents to form a link; and then, as it turns out in this particular topology,

each type b agent’s distance with the type a agent is sufficiently small. Even though the type

a agent has a low connectivity degree, the other agents do not find a new link with the type a

agent attractive, because it does not offer sufficient indirect benefits. Hence, the one agent with

the more valuable type – type a – ends up with the lowest connectivity degree in the network.

This is in stark contrast with the existing results in the literature, which often show that a

more valuable agent is better connected.

Furthermore, under incomplete information, the probability that the formation process con-

verges to a different network topology from those under complete information can be significant.

The following figure shows simulation results9 under an environment with a pure peer effect

(i.e. f(a, a) = f(b, a), f(b, b) = f(a, b)). We make this assumption only to economize on pa-

rameters. Under complete information, no type b agent would ever get linked; under incomplete

information, the probability that some type b agent remains linked by the end rises to more

than 0.8 for a range of the group size N .

The above examples also indicate that, unlike in most existing literature that specifies only

a few types of network topologies as the only possible ones in equilibrium, under incomplete

information the formation process can converge to potentially more types of networks, as there

9We assume for the simulation that δ = 0.8 and p = 0.5. For each value of N , we run 500 simulations and

average the results. Each simulation consists of a fixed long run of 5N(N − 1) periods, so that each link is

selected for 10 times on average.

14

Figure 4: Simulation: Significant Probability of Difference

may exist some network g which can emerge and be stable under incomplete information, but

not under complete information. We define some typical network structures in the literature

and the empirical works below:

1. g is complete if ij ∈ g ∀i, j such that i 6= j.

2. g is a star network if there exists i ∈ I such that ij ∈ g ∀j 6= i, j ∈ I and i′j /∈ g ∀i′, j 6= i.

3. g is a core-periphery network if there exists non-empty I ′ ( I, such that ij ∈ g ∀i, j ∈ I ′, i 6=j, and that ∀j′ ∈ I \ I ′, ij′ ∈ g for some i ∈ I ′ and jj′ /∈ g ∀j 6= i. Note that a star network is

a special case of a core-periphery network.

4. g is a tree network if there exists a partition of I, I1, ..., In, such that (1)#(I1) = 1;

(2)∀n′ = 2, ..., n, each agent in In′ has one and only one link with some agent in In′−1; (3)no

other link exists.10

5. g is a wheel network if there exists a bijection π : I → I such that g = {π−1(1)π−1(2),

π−1(2)π−1(3), ..., π−1(N − 1)π−1(N)}.

Theorem 3. Given κ, if a connected network g is stable and belongs to one of the following

categories:

1. Empty network;

2. Minimally connected network (i.e. tree network, including star network);

10Essentially, a tree network is equivalent to a minimally connected network, and a star network is a special

case of a tree network.

15

Figure 5: Some typical networks

3. Fully connected network;

4. Core-periphery network;

5. Wheel network.

then g ∈ G∗IC.

Theorem 3 explicitly characterizes types of connected networks that can emerge and be

stable under incomplete information, and most typical networks in both the literature and

empirical studies are included. However, note that there may be some stable networks that

cannot emerge under complete or incomplete information – e.g. networks with links such that

(1) the benefit from any one link cannot cover the maintenance cost without the existence of

the other links and (2) the network would still be connected if these links were severed. Such

network topologies may never be formed since only one pair of agents is selected in each period,

and the agents are myopic.

We end this section with an observation when δ = 1, i.e. the value from connection does

not decay with distance.

Proposition 2. Given κ and δ = 1, every g ∈ GIC(κ) is minimal.

Indeed, when there is no decay of indirect benefits, it is never optimal for agents to form

any “redundant” links. Note that according to Theorems 1 and 2, the above result applies to

GC(κ), G∗C(κ) and G∗IC(κ) as well. It also applies to situations where link formation and the

associated costs are directed (one-sided), as long as the benefit flow is undirected (two-sided).

See for instance Zhang and van der Schaar (2013)[35] for such a model which also predicts

minimal equilibrium topologies.

16

4.3 Long-Term Phenomena

We have shown that even though incomplete information only exists for finitely many periods,

its impact on the formation process can be persistent. Therefore, the long-term structure of a

network is determined by two factors: the influence of incomplete information accounts for the

formation, and the payoff characteristics indicate properties of the network once information

becomes complete. In this section, we focus on teasing apart the influences of these factors.

Our findings are in conformity with the empirical works by Newman (2002, 2003)[26][27],

Holme et al. (2004)[16], Hu and Wang (2009)[17] and Fricke et al. (2013)[9], who analyze data

on a variety of social, biological, technological and business networks and show that assortative

mixing (i.e. agents with high connectivity degree tend to link to each other) prevails in some

cases and neutral or disassortative mixing in others. In one of our applications where agents

are divided into subgroups, we propose particular network formation processes that essentially

exhibit disassortative mixing locally and assortative mixing across subgroups. However, we

diverge from the above studies in arguing that besides connectivity degree, assortativity can

also exist in types, which may in turn alter the network topology considerably.

Proposition 3. If types k, k′ (k and k′ may be the same type) are such that min{f(k, k′),

f(k′, k)} ≥ c, then

1. Every type k agent and every type k′ agent are connected for infinitely many periods almost

surely.

2. In every network g ∈ G∗IC(κ), every type k agent is connected to every type k′ agent.

Proposition 3 reveals a pattern of connection over time: as long as two agents both have an

incentive to link to each other when they are not connected, they would be connected (though

may not be linked) repeatedly starting from any period. Moreover, they are connected in every

stable network.

Note that even if min{f(k, k′), f(k′, k)} ≥ c, an existing link between a type k agent and

a type k′ agent may be severed. For instance: suppose fa, fb ≥ c f(a, b), f(b, a) ≥ c, (1 −δ)f(a, a) ≥ c, (1− δ)f(b, a) < c. Consider agents 1, 2, 3 such that k1 = b, k2 = k3 = a and the

selection path (1, 2), (1, 3), (2, 3), (1, 2). By computing the SOE for each period, it is clear that

every link is formed in periods 1− 3. In period 4, the payoff of agent 1 from the link between

agents 1 and 2 is (1− δ)f(b, a) < c, and therefore the link would be severed.

Corollary 1. If different types k, k′ are such that min{f(k, k′), f(k′, k)} ≥ c and there is at

least one agent of each type in I, then every g ∈ G∗IC(κ) is a connected network.

The implication of the above corollary is straightforward: if both types are willing to link

with each other, then connectedness is guaranteed once the network topology becomes stable.

Next, we analyze how self effect and peer effect, and a mixture of the two, influence the

formation process and the resulting network topologies.

Proposition 4. The following are true:

17

1. With pure self effect, given κ and γ(∞), the formation process is the same under complete

information and incomplete information.

2. With pure peer effect, every g ∈ G∗IC(κ) has at most one component.

3. If f(b, b) ≥ c > max{f(a, b), f(b, a)} and there are at least two agents of each different

type, then there exists δ > 0 such that for any δ ∈ [0, δ], every g ∈ G∗IC(κ) has exactly two

components, each one consisting of all the agents of one type.

Different effects impose different influences on the network structure. When an agent’s

payoff depends solely on his own type, information does not play a role and agents know their

exact payoffs as long as they know the pattern of connection; on the other extreme, when agents

only care about others’ types, which can be interpreted as the acquisition of others’ resources,

either there is always incentive to connect between any components, or the whole network

would stay empty in the long run; when both effects are non-trivial, clustering according to

types could emerge and be sustained.

Furthermore, the relative significance of self and peer effects determines the influence of

incomplete information as well. As shown by Proposition 4, the more significant the self

effect is, the smaller the difference between network topologies that result from complete and

incomplete information. At the extreme, where there is pure self effect, the formation processes

under complete and incomplete information, given a fixed selection path, become identical. On

the other hand, the more significant the peer effect is, the more likely that such differences are

large, as illustrated in Example 1.

4.4 Long-Term Connectedness of Large Networks

In this section we discuss asymptotic properties of the formation process, i.e. what types of

networks can emerge as a result of convergence when the group size goes to infinity. We focus

on the ex ante probability, which takes the initial choice of nature on the type vector κ into

account.

Proposition 5. If for some type k, k′ 6= k, min{f(k, k), f(k, k′)} ≥ c and f(k′, k) > (1− δ)c,then

limN→∞

Prob(every g ∈ G∗IC(κ) is connected) = 1

Intuitively, since when f(k, k) > c all the type k agents must be connected in any stable

network, as the group size grows the sub-group of type k agents could be sufficiently large to

“attract” any other agent. In other words, the increasing returns to link formation create incen-

tives for link formation between type k and k′ agents. Comparing this result with Proposition

4, we see that even if clustering occurs almost surely given almost every realization of the type

vector in a group of fixed size, clustering may eventually disappear and a connected network

can be sustained if the group size grows.

18

Our next result proves the existence of an upper bound on the diameter of a network, which

is defined to be the longest distance between any two agents in the network, as the group size

enlarges.

Proposition 6. If for some type k, k′ 6= k, f(k, k) > c and f(k, k′) ≥ c, then there exists

d∗ ∈ N+11 such that

limN→∞

Prob(every g ∈ G∗IC(κ) has a diameter of at most d∗) = 1

In other words, as long as agents of some type tend to cluster ”densely”, i.e. within a

bounded distance, as the group size grows large, agents of the other type would be incentivized

to stay close enough to this clustered sub-group. This “small-world” property has been con-

firmed in both the folk wisdom such as the “six degrees of separation” between any two persons

on the planet, and renowned research papers such as Watts and Strogatz (1998)[33], Newman

and Watts (1999)[28] and Jackson and Rogers (2007)[18].

We would like to emphasize that there can still be significant differences between network

structures under complete information and incomplete information, even in the long run and

asymptotically. Consider for example a case similar to Example 1, where if information were

complete no agent would be willing to form a first link with a type b agent. Even asymptotically,

the type b agents would stay singletons. Yet under incomplete information, if sufficiently many

agents are connected to a type b agent because of the initial incentive of link formation, then

the positive externalities thus created may ensure a type b agent stays connected even in the

long run and in a group of arbitrarily large size.

In the following section, we discuss some of network structures where the diameter of the

network does not increase with N , which enables an agent’s payoff to go up unboundedly as

the group size is enlarged. More concise conditions and more precise assertions can be obtained

as a result.

5 A Special Case: Preferential Attachment

So far we have assumed that opportunities for link formation are completely and uniformly

random. However, it is easy to imagine situations in which link formation is not entirely

random but is at least partially directed. In this section, we construct special selection processes

to illustrate how some particular types of networks could form asymptotically, and how to

determine the corresponding probabilities in the limit. In the selection processes we consider

below, an agent is always paired with the agent with the highest connectivity degree (i.e. the

number of links an agent has) globally or locally. Formally, if a network (or a sub-network) is

empty, then any pair of agents can be selected with equal probability; if there is one link ij,

then either ij is selected, or i (or j) and some agent other than i, j are selected.

11In fact, we have a closed form characterization of this upper bound. Please see the proof of this proposition

in the Appendix.

19

This characterization reflects the idea of preferential attachment in Barabasi and Albert

(1999)[4] and Vasquez (2003)[31], where agents are assumed to prefer to connect with other

well-connected agents. The importance of a “key player” with high centrality is also discussed

in Ballester et al. (2006)[3].

5.1 Centralized Preferential Attachment

We define centralized preferential attachment as a selection process in which any agent can only

be selected together with the agent (or one of the agents) who has the highest connectivity

degree (we call such an agent the center of the group) in the network. The following result

shows that for large N , there is a complete dichotomy.

Proposition 7. Under centralized preferential attachment, define the following events:

A. The formation process converges to a network with a star sub-network. A type a agent is

the center and all type k agents such that f(a, k) ≥ c are periphery agents.

B. The formation process converges to a network with a star sub-network. A type b agent is the

center; all type k agents such that f(b, k) ≥ c are periphery agents.

We have limN→∞ Prob(A) = p, limN→∞ Prob(B) = 1− p.

Now we will further explore the difference between complete and incomplete information in

large networks. Consider the following example.

Example 2. Let the parameters be such that fa ≥ c, f(a, a) ≥ c, f(a, b) < c, fb ≥ c, f(b, a) ≥c, f(b, b) < c. Consider centralized preferential attachment as the selection process. Under

complete information, only a type a agent can be the center (Figure 6(A)), since a type b agent

can never form a single link. Under incomplete information, however, as shown in Proposition

7, the probability of a type a agent being the center and that of a type b agent being the center

go to p and 1− p as N goes to infinity (Figure 6(B)).

Figure 6: Contrast between Complete and Incomplete Info in Large Networks

20

In the above example, even though a star forms almost surely in both environments, it is only

under incomplete information that a ”bad” agent–an agent of type b–may end up as the center,

due to the difference between expected payoff and realized payoff. Even more interestingly, this

example exhibits a certain inverse relation between type and connectivity degree: in any stable

network among a sufficiently large group of agents, the only possible place for a non-singleton

“bad” agent is the center of a star – even though the agent himself provides less value to others

than the periphery agents he enjoys the best-connected position in the whole network, and no

agent is willing to unilaterally sever their link with him!

5.2 Regional Preferential Attachment

Assume that the group of agents can be partitioned into M subgroups, with each group con-

taining n agents. One may regard such an environment as one with hierarchy in communica-

tion: preferential attachment only occurs locally such that periphery agents can only link with

the agent with the highest connectivity degree in their corresponding subgroup. We define

regional preferential attachment as a selection process in which (1) only the agent(s) with the

highest discounted connectivity degree in his own subgroup (we call such an agent the center of

the corresponding subgroup) can be selected together with an agent from another subgroup; (2)

any non-center agent in a subgroup can only be selected with the corresponding center agent.

As before, we fully characterize the resulting network.

Proposition 8. Under regional preferential attachment, define the following event for integer

m ∈ [0,M ]:

A(m). The formation process converges to a network with a core-periphery sub-network. m

type a agents and M −m type b agents are in the core, and in any subgroup with a type a (b)

center, all type k agents such that f(a, k) ≥ c (f(b, k) ≥ c) are periphery agents.

We have limn→∞ Prob(A(m)) =(Mm

)pm(1− p)M−m.

Furthermore, as we have emphasized before, network formation in our model exhibits strong

history dependence, which can be typically represented by different selection rules. Consider

the following example.

Example 3. Consider the following environment: fa, fb, f(a, a), f(b, b) ≥ c, f(b, a)+f(b, b) nδ1−nδ , f(a, b)+

f(a, a) nδ1−nδ < c for some positive integer n > 1. We will show by the following example that

under one selection rule, there is a positive probability that some type a and type b agents are

connected in a large stable network, whereas under the other selection rule, such an event is

impossible no matter how large the network is.

Consider the following two selection rules:

1. Regional preferential attachment: By Proposition 8, we know that the formation process

converges to a core-periphery sub-network almost surely. According to the payoff characteristics,

in each subgroup the periphery agents are of the same type as the center agent. Since the

21

probability that there are both type a and type b agents as centers is positive, we know that with

positive probability some type a and type b agents (the number of such agents goes to infinity

as the group size goes to infinity) are connected. Moreover, some agents of each type are left

as singletons with positive probability.

2. Anti-preferential attachment: Assume that no agent can be selected with any agent with more

than n− 1 links (i.e. with connectivity degree of more than n− 1). By an argument similar to

the proof of Proposition 4, we can show that in any stable network of any size no type a agent

can be connected to any type b agent. Moreover, no agent would be left as singleton almost

surely, as long as there are at least two agents of each type.

6 Discussions

6.1 Alternative Updating Rules

The results we have derived so far are based on the simple updating rule, which assumes that

every agent’s posterior belief on another agent’s type is binary: either it is the degenerate

belief on the true type, or the prior. Note again that such an updating rule implicitly assumes

that agents can only observe their own formation history. If the agents adopt a different

updating rule, which reflects either more or less available information, the formation process

can exhibit a much different pattern. We discuss one such alternative in detail, which we call

Bayesian learning by formation history.

We assume that agents can observe the entire formation history, i.e. the pair of agents

selected and the resulting network structure each period, in addition to knowing the types

of agents connected to themselves. They then apply Bayesian updating in forming posterior

beliefs. Formally, let p′ij(σ(t)) denote the posterior probability in i’s belief after σ(t) that j is

of type a. The following result highlights the key difference between the simple learning rule

and this alternative.

Proposition 9. Under Bayesian learning by formation history, assume that f(b, b) < c and

that there is a pure peer effect12. Then for any type a agent i, there is a sufficiently large nb

and a formation history σ(t) for some t such that p′ji(σ(t))f(kj, a) + (1− p′ji(σ(t)))f(kj, b) < c

for any other agent j and any kj ∈ {a, b}.

A major implication here is that Bayesian learning by formation history makes it possible

for the posterior probability of an agent being of high type to fall close to 0. As a result, even if

making a link with some agent i is incentivized with the simple learning rule, it may no longer

be the case under Bayesian learning by formation history, given some particular selection path

that would drag posterior beliefs towards i being of type b. The following example illustrates

this difference.

12It is easy to check that, with pure peer effect, the SOE still exists and is unique under Bayesian learning

by formation history.

22

Example 4. Assume the following parameter values in an environment with pure peer effect

(i.e. f(a, a) = f(b, a), f(a, b) = f(b, b)): N = 5, k1 = k2 = a, k3 = k4 = k5 = b, and the

other parameters are such that fa = fb ≥ c, f(b, b) = f(a, b) < c, (1 + δ − δ2 − δ3)f(b, b) ≥ c,p(1−p)1−p2 f(a, a) + p(1−p)+(1−p)2

1−p2 f(a, b) < c. Consider the following selection path in period 1-9:

(1, 3), (1, 3), (2, 4), (2, 4), (1, 2), (3, 4), (4, 5), (2, 3), (1, 5). Note that in this environment with

pure peer effect, the existence of a SOE (and thus the uniqueness) is still valid under Bayesian

learning by formation history, as any agent’s expected payoff from any link is not affected by

his own type.

Under the simple learning rule, the formation process is shown in Figure 7(A). The network

formed in period 9 is stable; under Bayesian learning by formation history, the formation process

is shown in Figure 7(B). The network formed in period 4 is stable.

Figure 7: Connected under Simple Updating Rule vs. Empty under Bayesian Learning by

Formation History

Here with the simple learning rule, agents hold the prior belief each time they are selected

with another agent with an unknown type, and thus the given selection path induces a connected

network at last. Yet with Bayesian learning by formation history, each agent updates from their

observation to conclude that others are of type b with a sufficiently large probability, and thus

are unwilling to make any link.

One implication of the above proposition and example is that more learning can sometimes

be “bad”, i.e. it may lead to inefficient outcomes. Despite the specific differences brought about

by an alternative updating rule, our general results still hold under a range of parameters. For

instance, it can be shown that in an environment with pure peer effect, if any typical network

as depicted in Theorem 3 can emerge and be stable under complete information, then it can

under incomplete information and Bayesian learning by formation history as well.

23

6.2 Robustness to Link Failure

Consider the following variation of the model: in each period, before a pair of agents is selected

to update the (potential) link between them, each existing link may fail independently with

probability ε ∈ (0, 1). One could think of this scenario as introducing an exogenous environ-

mental error, such as a possible system breakdown for a social network website or software.

One particular influence of possible link failure on the formation process is that some stable

network is not “robust”, in the sense that if some link fails by accident, the formation process

will never converge to the previous network again. The following example illustrates this point.

Example 5. Assume the same parameter values and the same selection path as in Example

1. We know that if there were no link failures, the connected network formed in period 5 is

stable. If there is a positive probability of link failure, and in some period two adjacent links

(for instance the link between agents 1 and 2, and that between 2 and 3) of the five fail, then

since agents know each other’s types already, the same connected network can never be formed

thereafter. Moreover, given that f(b, b) < c, any network with some agent with only one link

is never stable because the link would be severed once selected for update. As a result, the

formation process would converge to an empty network almost surely.

The intuition behind this phenomenon is simple. Should there be no incomplete information,

no link would ever form between any two agents; the connected network emerges due to high

expected payoffs from potential links, and without link failure it is stable because each link carries

sufficient indirect benefits. Now if two adjacent links fail, which leaves one agent a singleton, it

is clear that no agent would be willing to link to the singleton agent again. Moreover, there is

no network structure among the remaining four type b agents that could sustain enough indirect

benefit to maintain any one link.

From the above example it is rather obvious that a stronger concept for a stable net-

work is needed in the presence of link failure. One feasible candidate is what we denote as

a robust stable network: a network g is a robust stable network if (1) g is a stable network

without link failure; (2) with link failure, given any number of failing links in any period, g

will emerge again in some future period almost surely. The second condition guarantees that

no matter how serious the link failure is, g would be formed again in the future, and hence the

name “robust stable network”. Formally, let G∗∗C (κ) and G∗∗IC(κ) denote the set of robust stable

networks that can emerge and be stable under incomplete information given the type vector κ.

Proposition 10. Given any κ, G∗∗IC(κ) = G∗∗C (κ) = G∗C(κ).

We see from the above result that once robustness is imposed as an additional constraint,

there is no essential difference between complete and incomplete information regarding the

resulting convergence under possible link failure. Note that condition (2) in the definition of

a robust stable network is a strong condition implying that a robust stable network does not

always exist, and it remains a future research direction to characterize the network formation

process under possible link failure in detail.

24

7 Conclusion and Future Research

In this paper we analyzed the network formation process under agent heterogeneity and in-

complete information. Our results are in stark contrast with the existing literature: instead

of restricting the equilibrium network topologies to fall into one or two specific categories, our

model generates a great variety of network types. Besides what networks can emerge as a result

of convergence, we argue that it is also important to understand how a network gets formed,

since we want to know, for instance, why some agents become central and others do not. While

link formation and belief formation are usually treated as two independent processes to be

studied separately, we combine them in our model and show that belief formation is in fact a

key factor that could facilitate or deter link formation. Even if incomplete information vanishes

in the long run, its impact on shaping the network topology is persistent.

Several future research topics can be built up on the basis of our model. One natural

extension of the model is to assume that there are more than two, or even a continuum of types

of agents. As long as the initial incentive to form links is guaranteed for each type, the contrast

between complete and incomplete information, as reflected by Proposition 1 and Theorems 1-3,

would remain. If the agents adopted a different updating rule such as Bayesian learning by

formation history, then more types would induce a more complicated updating process and

results would also be different from those in the two type environment. In terms of the network

structure, adding more types creates more possibilities for the realized payoffs, and thus adds to

the variety of emerging topologies. For example, a stable network may have more components,

or the pattern of clustering across types may be more complicated.

Another future research topic is to pin down the structure of an efficient network and

implement it in a game-theoretic setting. The usual definition on efficiency in networks adopted

in the literature is strong efficiency, i.e. a network is strongly efficient if it maximizes the sum

of agents’ payoffs. In general, we know that a strongly efficient network must exist (though

not necessarily be unique) because the set of possible network structures is finite. By Theorem

2, we also know that more networks can emerge and be stable under incomplete information,

so maximal social welfare is at least as large as that under complete information. However,

since payoffs are heterogeneous across agents according to the type vector, the exact topology

of an efficient network becomes difficult to characterize; moreover, the efficient network may

not be unique because in an agent-heterogeneous environment there could be multiple ways of

generating the same level of social welfare.

Most importantly, we have assumed throughout, as does most of the literature, that agents

are myopic rather than forward-looking. If it is assumed otherwise that agents are foresighted

and are concerned about both their current and future welfare, then the aim of analysis es-

sentially becomes solving an agent’s dynamic optimization problem in the presence of other

similarly foresighted agents. One can then easily anticipate a very different evolution pattern

of network topologies as well as very different stable network topologies in the limit, for now

link formation does not only serve as an action of maximizing the current expected payoff, but

also as a way of acquiring information for potential future benefit. It is our conjecture that

25

link formation would be more frequent at least in an early stage of the formation process, since

on top of the incentives already discussed in this paper, agents may also be willing to endure

current payoff losses in return for valuable information about the type vector.

26

Appendix

Proof of Lemma 1. Assume that agents i and j are selected in some period. Based on the

simple updating rule, either i and j know each other’s type, or their belief on each other’s type

remains at the prior. If they know each other’s type, then since actions are binary for each

agent, the strategy described in a SOE exists and is unique for each agent. It is also clear that

the strategy is a best response for each agent since it is a weakly dominant strategy.

If the agents do not know each other’s type, and hence their belief remains at the prior,

then by Assumption 2 i’s expected payoff from the link with j is non-negative regardless of

i’s type, and vice versa. Therefore, the strategy profile described in a SOE forms mutual best

responses for i and j and thus is indeed an equilibrium. Finally, it is clear that such a profile

is unique.

Proof of Proposition 1. 1: It suffices to show that any two agents are connected for at least

one period within finitely many periods almost surely. By Assumption 2, it further suffices to

show that any two agents are selected at least once within finitely many periods almost surely.

Consider any two agents i and j; the probability of the event that they are not selected in one

period is 1 − 2N(N−1) < 1, and thus the probability of this event occurring for infinitely many

periods is 0.

2: If g is connected, then clearly there is complete information. If g is unconnected and

information is not complete, then there must exist two unconnected agents such that their

beliefs on each other’s type remain at the prior. When they are selected, by Assumption 2 they

would form a link, which implies that g, B is not stable, a contradiction.

Proof of Theorem 1. For any g ∈ GC(κ):

If g is empty: since g ∈ GC(κ), there must exist two agents i, j such that f(ki, kj) < c or

f(kj, ki) < c. Consider the selection path γ(2) = ((i, j)1, (i, j)2) under incomplete information.

It is clear that a link would form between i and j in period 1, but the link would then be

severed in period 2, and thus g = g(γ(2)), which implies that g ∈ GIC(κ).

If g is non-empty: consider any selection path γC(t) such that g emerges for the first time in

period t. By the definition of GC(κ), we know that such γC(t) exists. Let γIC(t′) be a selection

path constructed from γC(t) such that the pairs of agents in γC(t) between whom there is no

existing link, but a new link is not formed either, are deleted.

Consider the formation process under incomplete information given γIC(t′). By Assumption

2 and the simple updating rule, we know that given the same network structure, if a link is

formed between i and j under complete information, it will also be formed under incomplete

information regardless of whether i and j know each other’s type. Also, it is clear that the

decision of severing a link by any agent is the same under complete information and incomplete

information, since such a decision is based on the realized payoff. Therefore, the formation

process yields the same link formation and severance results under complete information given

γC(t) and under incomplete information given γIC(t′). Hence given γIC(t′), g emerges for the

27

first time in period t′ under incomplete information, which implies that g ∈ GIC(κ). Therefore

GC(κ) ∈ GIC(κ).

Proof of Theorem 2. By Theorem 1, we already know that any network that can emerge

under complete information can also emerge under incomplete information. Thus it suffices to

show that, for any network g that can emerge and be stable under complete information, there

exists a subsequent selection path under incomplete information after g’s first appearance that

would make g stable. We prove the result by construction.

If g is empty: consider the selection path γ(N(N−1)2

) such that every pair of agents is selected

exactly twice consecutively. Since by assumption g ∈ G∗C(κ), we know that for every pair of

agents a link would first form and then be severed in the next period. In period N(N−1)2

+ 1,

information is complete and the empty network becomes stable.

If g is non-empty: denoting the number of components in g as q(g), under incomplete in-

formation let the subsequent selection path after g’s first appearance be such that in the first

q(g)(q(g)− 1) periods, two agents from different components are selected exactly twice consec-

utively and every two components are involved. Since by assumption g ∈ G∗C(κ), which means

that g is stable under complete information, when a pair of agents from different components is

selected for the second time, either there is no existing link between them and no link would be

formed, or an existing link would be severed. In either case, the agents know each other’s type

as well as the types of agents in the counterparty’s component. Therefore, after q(g)(q(g)− 1)

periods, every agent knows κ and essentially there is no incomplete information. Again by the

assumption that g ∈ G∗C(κ), we can conclude that g is such that no link would be formed or

severed in any later period given any selection path. Thus g ∈ G∗IC(κ), which completes the

proof.

Proof of Theorem 3. By the assumption that g is connected and stable, it suffices to show

that when g belongs to any of the categories there exists a selection path such that g can

emerge in the formation process. We discuss case by case and prove by construction below.

1: See the proof of Theorem 2.

2: let L be the total number of links in g. Let the selection path be such that the pair of

agents for each link in g is selected once and only once in the first L periods. By Assumption

2, we know that each link will be formed, and thus g emerges in period L.

3: Since g is fully connected and stable, we know that for any two agents i and j, δf(kikj) ≥c. Therefore regardless of the selection path g would emerge.

4: Let the selection path be such that: first each periphery agent is selected once and only

once with their corresponding core agent, then every two core agents are selected once and only

once before any other pair of agents is selected. By Assumption 2 and the assumption that g

is stable, we know that each link will be formed, and thus g emerges after the last pair of core

agents is selected.

5: Let the selection path be such that the pair of agents for each link in g is selected once

and only once in the first N −1 periods. By Assumption 2 and the assumption that g is stable,

28

we know that each link will be formed, and thus g emerges in period N − 1.

Proof of Proposition 2. For any non-minimal network g, consider agents i and j such that

ij ∈ g and ig−ij↔ j. For g to emerge, either ij is formed before some i′j′ belonging to the path

connecting i and j in g − ij, or such i′j′ is formed before ij. In the first case, since δ = 1,

the formation of i′j′ would only bring additional costs to i′ and j′, but no additional benefits,

and thus i′j′ would never be formed. A similar argument would show that ij would never be

formed in the second case. Therefore, we can conclude that g /∈ GIC(κ), which completes the

proof.

Proof of Proposition 3. Without loss of generality we assume that types k and k′ are dif-

ferent. The case where they are the same can be proved analogously.

1: Consider any type k agent i and any type k′ agent j. By Assumption 2, when i and j are

selected for the first time a link would be formed between them. After that, whenever i and j

are selected and they are not connected, by the assumption that min{f(k, k′), f(k′, k)} ≥ c, a

link would be formed between them. Then we know that for any m,n ∈ N+, n > m:

Prob(i and j are connected for m periods before period n, and unconnected thereafter)

≤Prob(i and j are unconnected after period n)

≤Prob(i and j are never selected after period n)

=(N(N − 1)− 2

N(N − 1))∞ = 0

which implies that

Prob(i and j are connected for m periods before period n, and unconnected thereafter) = 0

Therefore, we have

Prob(i and j are connected for finitely many periods)

≤∞∑m=0

∞∑n=1

Prob(i and j are connected for m periods before period n, and unconnected thereafter)

=∞∑m=0

∞∑n=1

0 = 0

We can then conclude that the probability that i and j are connected for finitely many periods

is 0, so i and j are connected for infinitely many periods almost surely.

2: From the above argument, whenever a type k agent and a type k′ agent are selected, they

would form a link. Therefore a network with unconnected type k and type k′ agents is never

stable, which implies that the formation process would never converge to such a network.

Proof of Corollary 1. Consider any two unconnected agents i and j in any unconnected

network g. If i and j are of different types, then according to Proposition 3 we know that they

have an incentive to form a link once selected, and thus g /∈ G∗IC(κ). If they are of the same

29

type, without loss of generality assumed to be type k, then since there is at least one agent of

each type, by Proposition 3 we know that for g to be stable there exists some agent i′ of type

k′ who is connected to either i or j. Without loss of generality, assume that i′g↔ i. But then

if i′ and j are selected, again by Proposition 3 they have an incentive to form a link, which

implies that g /∈ G∗IC(κ). This completes the proof.

Proof of Proposition 4. 1: Clear, since with pure self effect the realized payoff coincides

with the expected payoff.

For 2 and 3, by Lemma 2 we can restrict our attention to the case of complete information

without loss of generality.

2: We prove by contradiction that a network with more than one component is never stable,

which is a sufficient condition for 2. Suppose that a network with two components C1, C2 is

stable. If there is a type a agent in C1 and a type a agent in C2, since fa ≥ c by Assumption

2 and f(a, a) ≥ c, when they are selected they would form a link regardless of whether they

know each other’s type. Thus without loss of generality we only need to discuss the case where

there are only type b agents in C1. For any agent i ∈ C1, the assumption that some other agent

j ∈ C1 is willing to link to i in a stable network together with f(b, a) = f(a, a) ≥ c implies

that any agent l outside C1 is willing to link to i whether or not he knows i’s type, as both

l’s expected payoff and realized payoff from a link with i are strictly larger than j’s realized

payoff from the link with i. The same argument would establish that any agent outside C2 is

willing to link to any type b agent in C2. Finally, it is clear that any agent outside C2 is willing

to link to any type a agent in C2, since both the expected payoff and the realized payoff are

non-negative. Therefore, there is always one agent in C1 and one agent in C2 who are willing

to form a link, and thus a network with more than one component is never stable.

3: Let δ = c−max{f(a,b),f(b,a)}(N−2)f(a,a) > 0. We first prove by contradiction that a network with at

most one component is never stable. By Assumption 2 and the assumption that f(a, a) ≥f(b, b) ≥ c, it is clear that an empty network is never stable, and thus we only need to discuss

the case of a network with only one component. Since there are at least two agents of each

type, the largest possible payoff that a type a agent gets by linking to a type b agent is

(1 + δ)f(a, b) + (N − 3)δf(a, a). By the above construction, we know that this payoff is strictly

less than c. A similar argument would show that the largest possible payoff that a type b agent

gets by linking to a type a agent is also strictly less than c. Therefore, suppose that a network

with only one component is stable, then the component can only consist of one type of agent.

Without loss of generality assume that it consists of only type a agents. Since fb ≥ c and

f(b, b) ≥ c, we know that two type b agents would form a link, regardless of beliefs. Therefore

a network with only one component, which consists of only type a agents, cannot be stable,

which is a contradiction.

Now assume that the network formation process converges. We know that in any stable

network there cannot be a link between any type a agent and type b agent; also, since there

are at least two agents of each type, there cannot be any singleton agent. Therefore any stable

network is partitioned by components consisting only of type a or type b agents. Suppose

30

that there are two components consisting of the same type of agent; we know from the above

construction that any agent in one component would be willing to link with any agent in the

other, regardless of beliefs. Therefore such a network cannot be stable, which implies that in

any stable network there is exactly one component consisting of all the type a agents, and

another component consisting of all the type b agents.

Proof of Proposition 5. Without loss of generality we only consider the case of complete

information. First we find N ′ such that∑N ′

n=1 δn−1f(k′, k) ≥ c. By the assumption that

f(k′, k) > (1− δ)c we know that such N ′ exists. Then for any ε > 0, we find N∗ such that for

all N ≥ N∗, the ex ante probability that there are at least N ′ type k agents in the group is

at least 1 − ε. By the assumptions that p ∈ (0, 1) and agents’ types are i.i.d., we know that

such N∗ exists. We assume in the remaining part of the proof that there are at least N ′ type

k agents in the group.

Assume that the network formation process converges. The construction of N ′ is such that

the payoff of a type k′ agent from connecting to a subgroup of N ′ connected type k agents is

non-negative. Since f(k, k) ≥ c, in any stable network all type k agents must be connected.

Then since there are at least N ′ connected type k agents in the group, we know that each type

k′ agent must be connected to the subgroup of type k agents. Therefore, any stable network

must be connected, which completes the proof.

Proof of Proposition 6. We prove the result in the following steps. Without loss of gener-

ality we only consider the case of complete information.

Step 1: Define d′ := min{d : (1− δd−1)f(k, k) ≥ c} − 1. By the construction of d′, we know

that if two type k agents with a distance of at least d′ + 1 are selected, they both have the

incentive to form a link. It immediately follows that for any κ, in any g ∈ G∗IC(κ), the distance

between any two type k agents is at most d′.

Step 2: We find N ′ such that (1− δd′)f(k′, k) + (N ′ − 1)(δd′ − δd′+1)f(k′, k) ≥ c. Then for

any ε > 0, we find N∗ such that for all N ≥ N∗, the ex ante probability that there are at least

N ′ type k agents in the group is at least 1− ε. By the assumptions that p ∈ (0, 1) and agents’

types are i.i.d., we know that such N∗ exists. We assume in the remaining part of the proof

that there are at least N ′ type k agents in the group.

Consider a sub-group of N ′ type k agents, among which the distance between any two agents

is at most d′, and consider any type k′ agent i. Let j be the type k agent in the sub-group

whose distance with i is the shortest, and suppose that dij = d′ + 2. If i and j are selected, for

j it is clear that he has an incentive to form a link since f(k, k′) ≥ c; for i, a link with j would

generate an additional benefit of (1 − δd′)f(k′, k) from j, and an additional benefit of at least

(δd′ − δd′+1)f(k′, k) from every other type k agent in the sub-group. By the construction of N ′,

we know that i also has an incentive to form a link. According to the conclusion in step 1, we

know that in any g ∈ G∗IC(κ), for any type k′ agent, the shortest distance between himself and

any type k′ agent is at most d′ + 1. It then follows that the distance between any type k agent

and any type k′ agent is at most 2d′ + 1.

31

Step 3: Consider any two type k′ agents i′ and j′, and let i′′ and j′′ denote the type k

agents with the shortest distance to i′ and j′ respectively. From the above two steps, we

know that in any g ∈ G∗IC(κ), di′i′′ ≤ d′ + 1, dj′j′′ ≤ d′ + 1 and di′′j′′ ≤ d′, which imply

di′j′ ≤ di′i′′ + di′′j′′ + dj′j′′ = 3d′ + 2. Therefore d∗ = 3d′ + 2 is one of the desired upper bounds

on the network diameter. This completes the proof.

Proof of Proposition 7. We first assume that two agents of the same type are selected in

the first period. The case where two agents of different types are selected can be analyzed anal-

ogously. As N →∞, in the second period almost surely an agent among the two (denote this

agent as i) and some singleton agent are selected. LetN∗ = max{ c−min{f(a,a),f(b,b),f(a,b),f(b,a)}δmin{f(a,a),f(b,b),f(a,b),f(b,a)} }+1.

As N →∞, the probability that apart from i there are at least N∗+1 agents of type ki goes to

1. Hereby we prove that the probability that at least N∗ + 1 agents of type ki would be linked

to i before any of the existing links between i and type ki agents is selected to be updated goes

to 1 as N → ∞. For notational convenience, let P (n) denote the probability that n agents of

type ki are linked to i, and let P ′(n+ 1|n) denote the probability that given n agents of type ki

are linked to i, another agent of type ki is linked to i before any of the existing n links is ever

selected to be updated.

We know that limN→∞ P (1) = 1 since fki ≥ c and p ∈ (0, 1). Given that n agents of type

ki are linked to i and none of the existing n links is ever selected to be updated, we know that

almost surely i and another agent of type ki would be selected in some period in the future;

given this event, the ratio of the probability that a singleton type ki agent is selected with i,

and the probability that a non-singleton type ki agent (i.e. one that has already been linked to

i) is selected with i, ismax{nki

−n−1}n

. The value of this ratio goes to∞ almost surely as N →∞,

and thus limN→∞ P′(n+ 1|n) = 1 ∀n > 0. Now we discuss the following two cases for k′ 6= ki:

(1) If min{f(ki, ki), f(ki, k′)} ≥ c: by the construction of N∗, once i is linked with at least

N∗ + 1 agents, then every new link would be formed and no existing link would be severed.

From the above analysis, the probability that at least N∗ + 1 agents would be linked to i is

bounded below by P (1)×N∗i=1P′(i+1|i), which goes to 1 as N →∞. Thus the formation process

would converge to a star network.

(2) If min{f(ki, ki), f(ki, k′)} < c: without loss of generality we only discuss the case where

f(ki, ki) < f(ki, k′). From the assumption fki ≥ c we know that f(ki, ki) < c and f(ki, k

′) > c.

From above we know that with probability going to 1 as N → ∞, at least N∗ + 1 agents of

type ki would be linked to i before any of the existing links between i and type ki agents is

selected to be updated. By the same argument, with probability going to 1 as N →∞, at least

N∗ + 1 agents of type k′ 6= ki would be selected before the existing links between i and type ki

agents is selected to be updated, and a link would be formed upon each selection. Afterwards,

by the construction of N∗, every link between i and a type k′ 6= ki agent would be formed,

and since f(ki, ki) < c every link between i and a type ki agent would be severed. Thus the

formation process would converge to a star sub-network on the set containing i and all type

k′ 6= ki agents, and all other type ki agents would remain singletons.

Finally, note that by the Glivenko-Cantelli Theorem, the empirical distribution of the agents’

32

types converges uniformly to the prior almost surely as N →∞. Then we know that as N →∞,

(1) a pair of type a agents is selected in the first period with probability converging to p2, (2) a

pair of type b agents is selected in the first period with probability converging to (1− p)2, (3) a

type a agent and a type b agent are selected in the first period with probability converging to

2p(1− p), and (4) given that a type a agent and a type b agent are selected in the first period,

each is selected in the next period with equal probability.

It is clear that asN →∞, in the second period a new agent other than the two selected in the

first period would be selected almost surely. Then with probability converging to p2+ 2p(1−p)2

= p

((1− p)2 + 2p(1−p)2

= 1− p), a star with a high (low) type agent as the center and two periphery

agents would be formed. Then we can apply the above argument to show the result.

Proof of Proposition 8. Let im denote the center agent of subgroup m. By Proposition 7 we

know that as n→∞, with probability converging to 1 a star sub-network would form in each

subgroup. Given that, when there are sufficiently many periphery agents in subgroup m (which

happens with probability converging to 1 as n→∞), we know that with probability converging

to 1 as n→∞: (1) all other center agents would form a link with im if selected; (2) a star sub-

network would emerge in each group; (3) any two unlinked center agents would form a link once

selected; (4) any type k′ agent in group m′ such that f(kim′ , k′) < c would ultimately remain a

singleton. Finally, the limiting probabilities are derived using Glivenko-Cantelli Theorem.

Proof of Proposition 9. We prove the result by construction. Consider the selection path

that i is selected twice consecutively with a type b agent, then selected twice consecutively

with another type b agent, and so on. We know that initially a link forms and then breaks

each time i is selected with a different type b agent. Let p′m be the posterior probability for

some other type a agent j that i is a type a after the link between i and the mth type b

agent breaks. We know that by Bayesian updating, p′m+1 = p′m(1−p)1−p′mp

with the initial condition

p′0 = p. Thereforep′m+1

p′m= 1−p

1−p′mp≤ 1−p

1−p2 < 1, and thus there exists nb such that p′nbsatisfies

p′nbf(k, a) + (1− p′nb

)f(k, b) < c ∀k ∈ {a, b}.

Proof of Proposition 10. We first prove that G∗∗IC(κ) = G∗∗C (κ). Given any κ, it is first clear

that G∗∗C (κ) ⊂ G∗∗IC(κ), since by Proposition 1 we know that information must be complete in

any stable network under incomplete information.

Consider any network g ∈ G∗∗IC(κ). Again by Proposition 1 we know that information must

be complete in g. By the definition of a robust stable network, g can still emerge even if each

link in g fails, which implies that g can emerge from an empty network. Therefore g ∈ G∗∗C (κ),

and thus G∗∗IC(κ) ⊂ G∗∗C (κ). Therefore, G∗∗IC(κ) = G∗∗C (κ).

Now we prove that G∗∗C (κ) = G∗C(κ). By the definition of G∗∗C (κ), it is clear that G∗∗C (κ) ⊂G∗C(κ). For any g ∈ G∗C(κ), fix a large enough number of periods T such that there is a

positive probability PT that starting with an empty network, g emerges at least once within

T periods. Starting with some period that any link(s) fails in g, we know that almost surely

the network will become empty in some future period; starting from then, there is a positive

probability PT that g emerges within T periods; if that does not happen, which is an event of

33

probability 1− PT , again almost surely the network will become empty in some future period,

and an identical argument follows. Therefore, the probability of g emerging again in some

future period is bounded below by∑∞

i=0 PT (1 − PT )i = 1. Thus by the definition of G∗∗C (κ),

g ∈ G∗∗C (κ), which implies that G∗C(κ) ⊂ G∗∗C (κ). This completes the proof.

34

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