On the asymptotic behaviour of strategic interaction on random undirected graphs

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On the asymptotic behaviour of

strategic interaction on random

undirected graphs

by

Diego Alejandro Murillo Taborda Adviser: Luis Jorge Ferro Casas

A thesis submitted in partial fulfillment for the degree of Mathematician

Universidad de los Andes Departamento de Matem´aticas

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Declaration of Authorship

I, Diego Alejandro Murillo Taborda, declare that I am the author of this thesis titled, ”On the asymptotic behaviour of strategic interaction on random undirected graphs”. I confirm that:

• This work was written wholly or mainly while my undergraduate studies. • Where I have consulted the published work of others, this is always clearly

attributed.

• Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is my own work.

• I have acknowledged all main sources of help.

Signed:

Adviser:

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The study of mathematics, like the Nile, begins in minuteness but ends in mag-nificence.

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Abstract

This thesis is a review of the literature of strategic interaction on graphs for sit-uations in which the results not only depend on the strategic behaviour, but also on randomness. In particular, we examine the asymptotic behavior of a stochastic process of network formation for which players have some positive probability of perturbation or error in the process of network formation. Later, graphical games will be presented, and we will characterize the asymptotic behaviour of the number of Nash equilibria in pure strategies (PNE) on graphical games with random util-ities defined on a big family of graph structures. Finally, we will characterize the asymptotic probability of existence of PNE in graphical games with random payoffs, sampled from an Erd¨os and R´enyi graph, which is a random graph model in which all links occurs independently with the same probabilityp>0. Our results suggest that in almost all analysed graph structures, the number of PNE converges in dis-tribution to a random variable Poisson(1) or the probability of existence of some PNE converges to 0, when the number of players tends to infinity if the utilities are distributed independently and uniformly on [0,1]. Furthermore, for deterministic connected graph structures, the number of PNE converges to a random variable Poisson(1) if the number of strategies per player tends to infinity, and again we assume that utilities are distributed independently and uniformly on[0,1].

Keywords: Stochastic network formation, stochastically stable networks, graph-ical games, random utilities, random graphs.

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Acknowledgements

Firstly, I want to acknowledge all people that in some way have contributed to my personal and academic improvement. I am so grateful to my family for its unconditional support in all moments of my life, and its contribution which allow me to get where I am. Also, I want to thank the program of scholarships of Universidad de los Andes, Quiero Estudiar, which allow me to finance all my undergraduate studies. Furthermore, I want to acknowledge to all people with I have had some academic jobs as teaching assistant of some courses, who increased my knowledge in some areas. In particular, I want to thank Alexander Getmanenko, Leopoldo Fergusson, and Paula Jaramillo. Finally, I want to thank my adviser Luis Jorge Ferro Casas, with I had moments of academic and non academic conversation, which enriched me, and also Luis Jorge’s help was important for the development of this thesis.

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List of Figures

2.1 Example of pairwise stability but not efficiency or efficiency but not pairwise stability . . . 19 2.2 Relation between stochastically stability and pairwise stability . . . . 26 3.1 Example of a tree graph. . . 42 3.2 Example of Kearns,Littman and Singh algorithm . . . 45

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Introduction

There is some mathematical literature in which graph theory have been combined with game theory for the representation of games of different types, in which there are restrictions to strategic interaction, such geographic, spatial, altruism restric-tions, etc, so games on graphs allow a more real approach to the modelling on some situations. Almost all literature of this topic has been focused on the use of deter-ministic graphs to represent interaction among players. However, there are some real situations in which the relations among players not only depend on strategic choices, but also on some random process. With this purpose, this thesis explores and extends some results of the literature about games on random graphs, in which utilities also can be random. The theory of random graphs has its origin in 1950 with the pioneer works of Solomonoff and Rapoport (1951), and Erd¨os and R´enyi (1959), and allows the connection between probability theory and graph theory, because random graphs are graphs generated by some probability distribution or random process.

In the first chapter, I provide some mathematical preliminaries of game theory and graph theory for an optimal understanding of the thesis. The second chapter is about network formation process, which is the process by which individuals form relationships with other individuals. For this goal, we will model the process of network formation as a dynamic process, and we will expose a stochastic process of network formation, introduced by Jackson and Watts in [9], who consider a prob-ability of error or perturbation in every stage of the process of network formation, which can be understood as the probability of establishing non benefic relationships, or not forming benefic ones. We will model this process of network formation by Markov chains, and we will characterize stationary states of the system with positive probability when the probability of error or pertubation tends to 0.

In the second chapter, graphical games will be presented. A graphical game is a non cooperative game defined on a fixed graph, which allows to model situations in which the utility of every player depends only on strategies of his neighbours and its strategy. We will present an algorithm developed by Kearns, Littman and Singh in [10], which in at most exponential time in the number of players, finds exact Nash equilibria in mixed strategies for a graphical game. Later, in this same chapter, we

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will explore the asymptotic behaviour of the number of PNE in graphical games with payoffs distributed independently and uniformly in [0,1], and a big family of graph structures. The idea of Nash equilibrium is the main solution concept in non cooperative game theory.

In the last chapter, we will explore the asymptotic behaviour of the number of PNE for graphical games with random payoffs, defined on an Erd¨os and R´enyi graph, which is a model away from some real situations, for which it is somewhat simple to characterize the asymptotic behaviour of the number of PNE. Results suggest that the threshold for the probability of link formation, determines the asymptotic behaviour of the number of PNE, according with three regimes on the level of con-nectivity of the graph.

The last two chapters are motivated by the fact that Nash proved in 1951, the existence of some Nash equilibria in mixed strategies for any strategic game with fi-nite sets of strategies, but some games of this type could not have some PNE, which can be observed in some elementary games as rock, paper, and scissors, among oth-ers. Then study the existence of PNE in games with different characteristics is an interesting problem.

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Chapter 1 Mathematical preliminaries

1.1 Graph theory and random graphs

A network (or graph) is a pair (N, g) where N is a set and g is a set of edges over

N. There are two standard ways in which networks are represented: by their adja-cency matrices as well as by listing pairs of nodes that are connected. Thus, if N

is finite, g will sometimes be an ∣N∣ × ∣N∣ matrix, where entry gij denotes whether

iis linked to j or include the intensity of that relationship. Also, g denotes the set of all relationships that are present in the network, and so we use notation ij∈g to indicate thati is linked to j.

A network (N, g) is undirected if for all i, j ∈ N, ij ∈ g ⇐⇒ ji∈g, or gij = gji,

when we use the adjacency matrix to represent g. We say that (N, g) is directed if this network is not undirected. Usually, we use {i, j} to denote a link between nodesi and j on an undirected graph.

Definition 1.1.1. A network (N, g) is complete if for every i, j ∈ N, ij ∈ g. If

(N′_{, g}′) _{is such that} _N′⊆_N_, _g′⊆_g _and (_N′_{, g}′) _{is complete, we say that} (_N′_{, g}′) _is

a clique of(N, g).

Definition 1.1.2. A path in a network (N, g) is a sequence of distinct nodes

i1, ..., iK ∈N such that ikik+1 ∈g for each k=1,2, ..., K−1. The length of a path is

the number of links in it.

Definition 1.1.3.Let(N, g)be a network. The distance between two nodesi, j∈N

is the length of the shortest path betweeniandj (if it exist). We denote it byd(i, j)

in case of existence. The average distance of(N, g) is the average among distances between two different nodes of the network. The diameter of the network is the maximum among distances between two different nodes in the network.

Definition 1.1.4. A cycle in a network (N, g) is a path i1, ..., iK ∈ N such that

i1 = iK. If the graph (N, g) doesn’t have some cycle, we say that this network is

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Definition 1.1.5. A network (N, g) is connected if for all i, j ∈N, there exists a path in the network between them.

Definition 1.1.6. A component of a network (N, g) is a connected subnetwork

(N′_{, g}′) _(i.e _N′ ⊆_{N, g}′ ⊆_g)_{, such that} _i∈_N′ _and _ij ∈_g _implies _j ∈_N′ _and _ij ∈_g′_).

Then a component of a network is a maximal connected subnetwork.

The idea behind using networks in game theory is to represent situations where utility of every player is influenced by the choices of their friends and acquaintances , and then payoffs that an individual receives from various choices depends on the behaviour of his neighbours. To do this representation of some strategic situations, we need some additional definitions.

Definition 1.1.7. The set of neighbours of a node i ∈ N, in a network (N, g), is defined byNi(g) ∶= {j ∈N ∶ij ∈g}. The degree of a node i∈N, in a network(N, g),

is defined byδg(i) ∶= ∣Ni(g)∣.

We use G(N) to denote the set of possible networks over nodes of a set N. Obviously G(N) is the set of subsets of all possible links. Then if N is finite,

∣G(N)∣ = 2∣N∣(∣N∣−1) _{if we consider directed graphs and} ∣_G(_N)∣ = ₂

∣N∣(∣N∣−1)

2 if we

con-sider undirected graphs. We have mentioned all definitions of deterministic graphs that we will need to the development of this thesis. Now, random graphs will be defined.

1.1.1 Random graphs

Complex graphs and networks are ubiquitous throughout science and engineering, and posses beautiful mathematical properties. These large and complex graphs and networks appear everywhere: the internet, social networks, strategic interaction, etc, and are often formed by some random process. Static random networks are graphs where the set of nodes is fixed, but edges are determined by some random process or probability distribution. These models of random networks find their origin in the studies of Solomonoff and Rapoport(1951), and Erdös and Rényi (1959), where the first authors proposed a random graph selected uniformly among all graphs of a fixed number of nodes and fixed number of edges, and Erdös and Rényi proposed a related random graph where every possible edge is formed independently with the same probability.

Definition 1.1.8. A random graph of Solomonoff and Rapoport, G(n, m), is a random graph obtained by sampling uniformly from all graphs withn nodes and m

edges.

It is easy to observe that an undirected graph with n nodes has (n

2) possible links, and then there are ((

n

2)

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m ≤ (n

2). For the case of directed graphs of n nodes, this quantity is equal to

(n(n−1)

m ) wherem≤n(n−1).

Definition 1.1.9. Forn∈_Nand 0<p<1, Erd¨os and R´enyi presented an undirected random graph, G(n, p), where the set of nodes satisfies ∣N∣ = n, and (gij)_i,j∈N,i<j

is a collection of independent and identically distributed (i.i.d) Bernoulli random variables with _P(gij =y) =py(1−p)1−y for y ∈ {0,1}. Then in an Erd¨os and R´enyi

random graph, every network is achieved with positive probability. When m is nearly p(n

2), since V ar(

∑i,j∈N,i<jgij

(n

2) ) =

(n

2)p(1−p)

(n

2) 2

n→∞

ÐÐ→0, Chebyshev’s inequality implies ∑i,j∈N,i<jgij

(n₂)

p

→ E[∑i,j∈N,i<j₍n gij

2) ] =

p ≈ m

(n₂), and then where n is very

large, the number of edges in G(n, m) and G(n, p) are very similar. We define two regimes of interest:

1. Fixed edge probability: Fix p ∈ (0,1) as a fixed probability to form a link. In this regime, every node has an expectation to being connected with (n−1)p

nodes.

2. Low complexity: Take p∼ _nc for some c>0. Here the probability of forming a link decreases with the number of nodes. In this regime, every node has a expectation of being connected with (n−1)p∼c nodes. Thus every node has a constant asymptotic expectation for its number of neighbours, when the number of players goes to infinity.

If G = (N, g) is an Erd¨os-R´enyi graph G = (n, p) and v ∈N, the random variable

Yv ∶= ∑i∈N,i≠vniv ∼ Bin((n−1), p) represents the number of neighbours of v. For

y∈ {0,1, ..., n−1}, we have _P(Yv =y) = (

n−1

y )p

y₍₁₋_p₎n−1−y

.

Theorem 1.1.1. Let G= (n, p) be an Erd¨os-R´enyi graph of low complexity regime

with limn→∞p(n−1) =c for some c>0. Then for every node v ∈N, we have Yv

d

→ P oi(c) (i.e limn→∞P(Yv =y) =

cye−c

y! for every y ∈N). Then under low complexity

regime, the number of neighbours of every nodev converges to a Poisson distribution with expectation equal to c.

Proof: For ally, n∈_N, we have the following:

P(Yv =y) = (

n−1

y )p

y₍₁₋_p₎n−1−y1

{0,1,...,n−1}(y)

= (n−_y1)[(n−1)p]y(n−1)−y(

1−p)n−1−y1

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= (n−1)!

y!(n−1−y)!(n−1)y[(n−1)p]

y(1−p) n−1

(1−p)y 1{0,1,...,n−1}(y)

For very largen,(n−1−y)!(n−1)y ∼ (_n−₁−_y)_!(_n−_y)(_n−_y+₁)_...(_n−₁) = (_n−₁)_!

and then limn→∞

(n−1)!

(n−1−y)!(n−1)y =1.

Also, for all n ≥ y+1 ∈ _N, we have y ≤ n−1 and then ∣1{0,1,...,n−1}(y) −1∣ = 0.

This implies limn→∞1{0,1,...,n−1}(y) =1.

On the other hand, we have limn→∞p = 0, which implies limn→∞(1−p)

y

= 1 and limn→∞(1−p)

n−1

=limn→∞e

−p(n−1)=_e−c_{. Then, we have the following:}

lim

n→∞P(

Yv =y) = lim n→∞

(n−1)!

y!(n−1−y)!(n−1)y[(n−1)p]

y(1−p) n−1

(1−p)y 1{0,1,...,n−1}(y)

= _y1_!⋅1⋅cy⋅e−c⋅

1= c

y_e−c

y!

The following theorem is useful to characterize the probability of having a com-ponent encompassing all nodes and the probability of having an isolated node in an Erd¨os-R´enyi graph.

Theorem 1.1.2. Let G = (n, p(n)) an Erd¨os-R´enyi graph. If limn→∞

p(n)n

ln(n) = 0,

then the probability of having isolated nodes tends to1, and if limn→∞

p(n)n

ln(n) = ∞, the

probability of having isolated nodes is tends to 0.

Proof: This theorem is Theorem 4.2.1 of [8], but I did an easier proof. We define

f(n) ∶= np(n) −ln(n). Then p(n) = ln(n)+f(n)

n . In any case, we use Xn to denote

the number of isolated nodes when there aren nodes. A given node is isolated with probability(1−p(n))n−1_{, so}

E[Xn] =n(1−p(n))n−1.

Suppose that limn→∞

p(n)n

ln(n) =0, then we have limn→∞−f(n) = ∞, and limn→∞p(n) =0

since we have limn→∞

p(n)

ln(n) n

=limn→∞

p(n)n

ln(n) =0 with limn→∞

ln(n)

n =0. AlsoXn(Xn−1)

represents the number of ordered pairs of isolated nodes, so we have_E[Xn(Xn−1)] =

n(1−p(n))n−1(_n−₁)(₁−_p(_n))n−2 =_n(_n−₁)(₁−_p(_n))2n−3_{. We have the following:}

V ar(Xn) =E[(Xn−E[Xn])2] =E[Xn2] −E[Xn]2

=E[Xn(Xn−1)] +E[Xn] −E[Xn]2

=n(n−1)(1−p(n))2n−3+

E[Xn] −n2(1−p(n))2n−2

≤n2(1−p(n))2n−3+

E[Xn] −n2(1−p(n))2n−3(1−p(n))

=n2_p(_n)(₁−_p(_n))2n−3+

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Fix n ≥3. We want to find the maximum of np(1−p)n−2 _for _p ∈ [₀_,₁]_{. We have}

0=n(0)(1−0)n−2 =_n(₁−₁)n−2_{. The first order condition is the following:}

0=n(1−p)n−2−_np(_n−₂)(₁−_p)n−3 =_n(₁−_p)n−3[₁−_p(_n−₁)]

The first order condition is satisfied by p=1 and p= _n1₋₁. In this case, the value of the objective function is equal to _nn₋₁(1−_n1₋₁)n−2= n

n−1(

n−2

n−1)

n−2≤ n

n−1 =

1 1−1

n ≤

1 1−1

3 = 3 2.

We have the following forn≥3:

V ar(Xn) ≤E[Xn](np(n)(1−p(n))n−2+1) ≤E[Xn] (

3

2+1) = 5 2E[Xn]

If we use µn to denote E[Xn] ,we have for n≥3 and >0, the following by

Cheby-shev’s inequality:

P⎛

⎝Xn−µn≤ −

√

5µn

2

⎞ ⎠≤P

⎛

⎝∣Xn−µn∣ ≥

√

5µn

2

⎞

⎠≤P(∣Xn−µn∣ ≥

√

V ar(Xn)) ≤

1

2

Also: lim

n→∞

µn= lim n→∞

n(1−p(n))n−1= _lim

n→∞

n(1−p(n))n

1−p(n) =nlim→∞

ne−np(n)= _lim

n→∞

ne−ln(n)−f(n)

= lim

n→∞

e−f(n)= ∞

Thus it is possible to construct a sequence (n)n∈N such that 0≤n≤

√

2µn

5 for n∈N

and nÐ→ ∞. there exists N ≥3, such that n>0 for n≥N. For n≥N, we have:

P(Xn=0) ≤P

⎛

⎝Xn≤µn−

√

5µn

2 n

⎞ ⎠≤

1

2 n

Since limn→∞

1 2

n = 0, we have limn→∞P(Xn = 0) = 0, and then the probability of having an isolated node tends to 1.

Now, assume limn→∞

p(n)n

ln(n) = ∞. We have P(Xn ≥ 1) ≤ E[Xn] = n(1−p(n))

n−1_,

by Markov’s inequality. If limn→∞p(n) = 0, we previously proved limn→∞n(1−

p(n))n−1=_lim

n→∞e

−f(n)_{, and since lim}

n→∞

p(n)n

ln(n) = ∞, limn→∞f(n) =limn→∞[np(n)−

ln(n)] = ∞. Then limn→∞n(1−p(n))n

−1 =_lim

n→∞e

−f(n)=_{0 in this case. If is not true}

that limn→∞p(n) =0, for all sufficiently largen∈N, we have n(1−p(n))n

−1 ∼_e−f(n)

or (1−p(n))n−1 ∼_{0 when} _p(_n) _{is not nearly to 0. Since} (₁−_p(_n))n−1 _{is an}

expo-nential function in n, we have n(1−p(n))n−1 ∼_{0 in the second case, and we have}

proved that limn→∞e

−f(n)=_{0. In any case, we have the following:}

lim

n→∞P(

Xn≥1) ≤ lim n→∞

n(1−p(n))n−1=₀

Thus the probability of having some isolated node tends to 0 if limn→∞

p(n)n

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Corollary 1.1.1. In an Erd¨os-R´enyi graph of the low complexity regime, the prob-ability of having an isolated node tends to 1 when the number of nodes tends to infinity, and in the fixed edge probability regime, the probability of having an isolated node tends to 0.

Although Erdös-Rényi graph is the simplest random graph model, it’s not a re-alistic model of most real-world networks. More complicated models capture certain features of real-world graphs that are not well described by the Erdös-Rényi model, and we will see some of these models.

1.1.2 Markov graphs and p* networks

Markov graphs are a generalization of Erd¨os-R´enyi random graphs, which have been useful in statistical analysis of observed networks and were introduced by Frank and Strauss (1986). These random graph models were later imported to the social net-work literature under the name ofp∗ _{networks. Markov graphs provide a model that}

allows for specific dependencies between the probabilities with which different links are formed.

To model the dependence in the link formation process, Frank and Strauss con-sider a graph (N, g) with n nodes, and use a graph D where all the possible (n

2) links of the original graph (or n(n−1)links if (N, g)is a directed graph) are nodes of the new graph D. An edge in D represents dependence among two links of the original graph, and then D allows to represent quite complicated combinations of dependence between links of the original graph. For example, in an Erd¨os-R´enyi graph,D is a graph without edges.

Let C(D) be the set of all cliques of D, and for A ∈ C(D), we define the func-tion IA ∶G(N) → {0,1} by IA(g) = 1 if A ⊆ g, and IA(g) = 0 otherwise, for every

possible graphg among the nnodes ofN. Frank and Strauss showed that the prob-ability of achieved a given networkg depends only on which cliques ofDit contains, and it can be written as:

log(_P(g)) = ∑

A∈C(D)

αAIA(g) −c (1.1)

for some constants{αA}A∈C(D), c.

1.1.3 Expected degree model of Chung and Lu

In the Erd¨os-R´enyi random graph,G(n, p), every node has an expected degree equal to(n−1)p. Chung and Lu [2] provide a random model that allows different expected degree for some nodes, and then they provide a random model that approximates a

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graph with a given degree sequence.

Chung and Lu consider a general model G(d) for random graphs with given ex-pected degree sequenced= (d1, d2, ..., dn) where the edge between i, j∈N is chosen

independently with probabilitypij proportional to the productdidj. Then the

prob-ability of having an edge betweeni, j∈N ispij =didjpwherep= _∑ 1

k∈Ndk, and authors assume (maxk∈Ndk)2 < ∑k∈Ndk so that the probabilities {pij}_i,j∈N are strictly

be-tween 0 and 1. The construction of this model allows links from a node to himself, and the expected degree of a node i∈N is the following:

∑

j∈N

didj

∑k∈Ndk

=di (1.2)

Then starting with a sequence of expected degrees, we can construct a random graph model with similar degrees to expected if the number of nodes is arbitrarily large, and then this model is very useful to generate a random graph with heterogeneous expected degree which allow us to study some phenomena in social networks as cen-trality, diffusion processes and contagion, etc.

Authors of this random graph model provide a lot of interesting probabilistic prop-erties of this model, but they are not necessary for the development of this thesis.

1.2 Non-cooperative games

In non-cooperative games, agents are acting in self-interested ways to maximize their payoffs and equilibrium notions are applied to predict outcomes, which are combi-nations of played strategies for every player. On the other hand, cooperative game theory examines coalitions (subsets of players) and how payoffs might be allocated within coalitions. However, non-cooperative games and cooperative games are not completely opposed concepts, because non-cooperative games may be analysed as a part of a cooperative game where the expected payoffs of the non-cooperative game for every player can be seen as the value for the coalition formed only by each player, in some cooperative game. Then, cooperative games allow us to analyse situations where agents are acting in self-interested ways to maximize their payoffs, but agents willing to form coalitions with other players. To analyze non-cooperative games, we need a set of players, strategies for every player, and an associated payoff for every player when every possible combination of strategies is played.

1.2.1 Games on normal form and Nash equilibrium

The normal form is a way to represent a game in the sense of the final part of the last paragraph. A strategic game in normal form is a triple G ∶= (N, S, u) whose elements are the following:

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• The set of players of the game is N = {1,2, .., n}

• For each i∈N, Si is a nonempty set of strategies of player i and S ∶= ∏ni=1Si

is the set of strategy profiles.

• For each i∈N, ui∶S→R is the payoff function of player iand u∶= ∏ni=1ui;ui

assigns to each strategy profiles∈S, the payoff that playerigets ifsis played, and then the payoff of every player may depend of the strategies played by other players, and not only by itself.

The next step is to propose solution concepts that aim to describe how rational agents should behave. The most important solution concept for strategic games is Nash equilibrium. It was introduced by Nash (1950). To define the concept of Nash equilibrium, given the strategic profile s ∈S, we use (s−i, s

∗

i) to denote the profile

(s1, ..., si−1, s ∗

i, si+1, ..., sn).

Definition 1.2.1. Let G = (N, S, u) be a strategic game. A Nash equilibrium in pure strategies of G is a strategy profile s∗ ∈_S _{such that, for each} _i∈_N _{and each}

si∈Si,

ui(s∗) ≥ui(s∗−i, si) (1.3)

Then a Nash equilibrium in pure strategies is a strategic profile where given the strategies of the other players, every player maximizes its payoff function, and this implies that every player has no incentives to change its strategy individually. The set of all Nash equilibrium in pure strategies may be empty, countable, or uncountable.

Example 1.2.1. (Rock, paper, and scissors) Consider the game of rock, paper, and scissors between two players, where both players simultaneously announce one thing among rock, paper, and scissors. Rock wins to scissors, scissors wins to paper, and paper wins to rock. This game can be represented as a strategic game G= (N, S, u)

where N ∶= {1,2}, S1 =S2 = {R, P, S}, and the payoff functions are the following:

• u1(R, P) = u1(P, S) = u1(S, R) = −1, u1(R, R) = u1(P, P) = u1(S, S) = 0,

u1(R, S) =u1(P, R) =u1(S, P) =1

• u2(R, S) = u2(P, R) = u2(S, P) = −1, u2(R, R) = u2(P, P) = u2(S, S) = 0,

u2(R, P) =u2(P, S) =u2(S, R) =1

We can see that this game doesn’t have Nash equilibrium in pure strategies. If the first player chooses R, then the second player maximizes its payoff when he chooses P, but given this strategy of the second player, the first player doesn’t maximizes its payoff. Analogously, we analyse cases when the first player chooses P or S.

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Example 1.2.2. Consider a strategic game G= (N, S, u) where N = {1,2, ..., n} for some n ∈_N, for all player i∈_N, Si = [0,∞) and ui ∶S = [0,∞)n →R is defined by

ui(s1, s2, ..., sn) =min{s1, ..., sn} −

min{s1,...,sn}2

n .

For i ∈ _N, we can rewrite ui = y− y 2

n where y =min{s1, ..., sn}, and given

strate-gies of other players, every player i ∈ _N wants to choose an strategy si, to define

y=min{s1, ..., sn} which maximizes its utility function, and this strategy exists

be-cause the utility function is concave. The condition of first order for every player

i∈_N is the following:

0=∂ui

∂y =1−

2y n

Then we have y = n₂ in equilibrium, and we conclude that every strategic profile

(s1, s2, ..., sn) ∈ [0,∞)n such that min{s1, ..., sn} = n₂ is a Nash equilibrium in pure

strategies, and ifn≥2, this set is clearly uncountable.

According to the previous examples, it is useful to establish some theorems to verify if a non-cooperative game has a Nash equilibrium in pure strategies.

Definition 1.2.2. LetX⊆_Rk_{, Y} ⊆

Rm for somek, m∈N. A correspondenceF from

X toY is a function F ∶X→_P(Y)(we can use the notationF ∶X⇉Y to denote a correspondence fromX toY). IfP is a property for some subsets of Y, we say that the correspondence F defined in the sense above, is P-valued if for allx∈X, F(x)

has theP property.

Definition 1.2.3. A correspondence F ∶ X → _P(Y) is upper hemicontinuous at

x∈X if for every pair of sequences (xn)n∞=1 ⊆X and (yn) ∞

n=1⊆Y, such that

• limn→∞xn=x

• yn∈F(xn)for all n∈N, there exists a subsequence(ynm)

∞

m=1of(yn) ∞

n=1andy

∗∈_F(_x)_{such that lim}

m→∞ynm =

y∗_.

The correspondence F is upper hemicontinuous if it is upper hemicontinuous at every x∈X.

Definition 1.2.4. LetG= (N, S, u)a strategic game . For eachi∈N,i’s best reply correspondence,Bi ∶S−i ⇉Si, is defined for each s−i∈S−i, by

Bi(s−i) =argmaxsi∈Siui(s−i, si) (1.4)

Also, we define the correspondence BR ∶ S ⇉ S by BR(s) ∶= ∏n_i₌₁BR(s−i) for

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For s ∈ S, we have s ∈ B(s), if and only if si ∈ BR(s−i) for all i ∈ N, and by

definition of BR(s−i), this occurs if and only if ui(s−i, si) ≥ui(s−i, s ∗

i) for all i ∈N

and s∗

i ∈Si. Then s∈B(s) if and only if s is a Nash equilibrium in pure strategies.

Then if we prove the existence on a strategic profile s ∈S such that s ∈B(s), we prove the existence of a Nash equilibrium in pure strategies.

Theorem 1.2.1. (Kakutani fixed-point theorem) For some n∈_N, let X ⊆_Rn _{be a}

nonempty, convex, and compact set, and letF ∶X⇉X be an upper hemicontinuous, nonempty, closed, and convex valued correspondence. Then, there is a x ∈X such thatx∈F(x), i.e, F has a fixed point.

To do the proof of the last theorem, we need some definitions and one lemma. Definition 1.2.5. LetX, Y be sets, F ∶X⇉Y be a correspondence, andf ∶X→Y

be a function. We define the graph ofF byGr F ∶= {(x, y) ∈X×Y ∶y∈F(x)}, and the graph of f byGr f ∶= {(x, f(x)) ∶x∈X}.

Lemma 1.2.1. (von Neumman’s Approximation Lemma) LetX ⊆_Rm_{be a nonempty}

compact subset,Y ⊆_Rn_{be a nonempty convex subset, and}_F ∶_X ⇉_Y _{an upper}

hemi-continuous correspondence with nonempty, compact, and convex values. Then for any>0, there exists a continuous functionf∶X →Y such that Gr f⊆B(Gr F),

where B(Gr F) ∶= {z∈X×Y ∶inf(x,y)∈Gr F d(z,(x, y)) <}, and d is a metric over

Rm+n_.

The proof of the last lemma can be found in Chapter 13 of [1]. Now we are ready to prove Kakutani fixed-point theorem.

Proof of Kakutani fixed-point theorem: Let X ⊆ _Rm _{for some} _m ∈

N be a nonempty, convex, and compact set, and let F ∶X ⇉X be an upper hemicontinu-ous, nonempty-valued, closed-valued, and convex-valued correspondence. We have all hypothesis of von Neumman’s approximation lemma, with exception of the fact thatF has compact values, but this is easy to verify. Forx∈X, F(x) ⊆X is closed by hypothesis, and sinceX is compact,F(x)is compact because it is a closed subset of a compact space.

Let be n ∈ _N. Since _n1 > 0, by von Neumman’s approximation lemma,there ex-ists a continuous function fn ∶X →X such that Gr fn⊆B1

n(Gr F). Since X is a nonempty, compact and convex subset of the Euclidean space, and every fn is

con-tinuous, then by the Brouwer fixed point theorem, everyfnhas a fixed pointx∗n∈X.

For everyn∈_N, we have(x∗

n, x∗n) = (x∗n, f(x∗n)) ∈Gr fn⊆B1

n(Gr F), and this implies that there exists(xn, yn) ∈Gr F such thatd((x∗n, x∗n),(xn, yn)) < _n1. Since (x∗n)∞n=1 ⊆

Xand Xis bounded, we have by Bolzano-Weierstrass that there exists a convergent subsequence(x∗

nm)

∞

m=1⊆ (x ∗

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is an element of X. Then x∗

nm Ð→x ∈X. For m ∈N, d((x

∗

nm, x

∗

nm),(xnm, ynm)) <

1

nm, and this implies that limm→∞(xnm, ynm) = limm→∞(x

∗

nm, x

∗

nm) = (x, x). Since limm→∞ynm =x, every subsequence of(ynm)

∞

m=1 converges to x, and by upper

hemi-continuity ofF, we have x∈F(x), and this proves thatF has a fixed point.

Theorem 1.2.2. (Glicksberg) Let G = (N, S, u) be a strategic game such that for each i∈N, we have the following:

• Si ⊆Rmi is nonempty, compact and convex.

• ui is continuous and ui(s−i,⋅) is quasiconcave in si.

Then, the game G has, at least, one Nash equilibrium.

Proof: S = ∏n_i=1Si is nonempty, compact and convex because it is a Cartesian

product of sets which are nonempty, compact and convex. For i ∈N, we have the following properties for the correspondence BRi:

• Nonempty-valuedness: Since every functionui is continuous, we have that

given s−i, ui(s−i,⋅) reaches a maximum over Si because this set is compact.

Then BRi is nonepty-valued.

• Closed-valuedness: Let bes−i ∈S−i, and(sn) ∞

n=1 ⊆BRi(s−i)such thatsnÐ→

s for some s∈_R. Then for n∈_N, ui(s−i, sn) ≥ui(s−i, s

∗) _{for all} _s∗∈_S

i. Since

the function ui is continuous, we haveui(s−i, s) ≥ui(s−i, s

∗)_{for all}_s∗∈_S

i, and

then s∈BRi(s−i). Then BRi(s−i) is closed.

• Convex-valuedness: Let be s−i ∈S−i,s1, s2∈BRi(s−i), and λ∈ [0,1]. Then

ui(s−i, s1) = ui(s−i, s2) ≥ ui(s−i, s) for all s ∈ Si. By the quasiconcavity of ui

in si, we have ui(s−i, λs1+ (1−λ)s2) ≥min{ui(s−i, s1), ui(s−i, s2)} ≥ui(s−i, s)

for all s ∈ Si. Then λs1+ (1−λ)s2 ∈ BRi(s−i), and BRi is a convex-valued

correspondence.

BR is a nonempty-valued, closed-valued, and convex-valued correspondence because it is a finite Cartesian product of sets with every one of these proper-ties. Now, we need to prove upper hemicontinuity of BR. Let be s ∈S, and

(sn)∞n=1,(yn) ∞

n=1 ⊆S be sequences such that:

– limn→∞sn=s

– yn∈BR(sn) ⊆S for all n∈N

Since (yn)∞n=1 ⊆S and S is compact, by Bolzano-Weierstrass, the exists a

sub-sequence (ynm)

∞

m=1 ⊆ (yn) ∞

=1 such that ynm Ð→y for some y ∈ S. We need to show that y ∈BR(s). For i ∈N, n ∈_N, we have ui(sn−i, yni) ≥ui(sn−i, si) for all si ∈Si. Taking limits and using the continuity of ui, we have ui(s−i, yi) ≥

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ui(s−i, si)for alli∈N, andsi ∈Si , and this impliesyi∈BRi(s−i)for alli∈N,

and by definition of BR, we have y∈BR(s). Then the correspondence BR is upper hemicontinuous.

All hypothesis of Kakutani fixed-point theorem were proven. By this theo-rem, there exists s∈S such thats∈BR(s), and then s is a Nash equilibrium in pure strategies.

The last theorem can help us verify if a given strategic game has a Nash equilibrium in pure strategies, but this theorem requires some strong assumptions over sets of strategies and utility functions. There is a weaker concept to Nash equilibrium in pure strategies, and it is Nash equilibrium in mixed strategies, which always exist in a strategic game with finite sets of strategies.

Definition 1.2.6. Let G= (N, S, u)be a strategic game. We say that G is a finite game if, for each i∈N, ∣Si∣ < ∞.

If Si ⊆Rmi is a finite set with ∣Si∣ ≥2, Si is not convex, and then Glicksberg’s

theorem can not be applied, but there is a ”theoretical trick” that allows to extend the game and guarantee the existence of Nash equilibrium in the extended version of every finite game.

Definition 1.2.7. Let G= (N, S, u) be a strategic game of n players. The mixed extension of G is the strategic game E(G) ∶= (N, M, U), whose elements are the following:

• Set of (mixed) strategies: For each i ∈ N, Mi ∶= ∆(Si) is the set of

strategies for player i, where ∆ denotes the set of probability distributions over a given set, and M ∶= ∏n_i=1Mi. We assume that strategies of different

players are independent.

• Payoff functions: For each i∈N, we define the function Ui ∶ M Ð→R by

Ui(m) ∶= ∫_Sui(s)m(ds)for all m∈M, and U ∶= ∏ni=1ui.

Theorem 1.2.3. (Nash theorem) LetG= (N, S, u)be a finite strategic game. Then, the mixed extension of E(G), has at least, one Nash equilibrium. We say that these equilibria are Nash equilibria in mixed strategies of the original game G. Obviously a Nash equilibrium in pure strategies of the game G is a Nash equilibrium in mixed strategies where players choose degenerated probabilities.

Proof: Since the gameG is finite, we have for all i∈N, the following:

Mi=∆(Si) =⎧⎪⎪⎨⎪⎪

⎩

m∈_R∣Si∣∶ ∣Si∣

∑

k=1

mk=1, mj ≥0f or j∈ {1,2, ...,∣Si∣}⎫⎪⎪⎬⎪⎪

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and Ui(m) = ∑s∈Sui(s) ∏

n

k=1mk(sk) for all m∈M.

Clearly every setMi is nonempty, convex and compact.

Also, it is clear that everyUiis a continuous function because∏nk=1mk(sk)is

continu-ous inm∈M since projections are continuous, and then fors∈S,ui(s) ∏nk=1mk(sk)

is continuous in m∈M, and this implies that Ui is continuous because it is a finite

sum of continuous functions. Also every Ui is quasiconcave in mi because for all

m−i ∈M−i, m1, m2 ∈Mi and λ∈ [0,1], we have the following:

Ui(m−i, λm1+ (1−λ)m2) = ∑

s∈S

ui(s)(λm1(si) + (1−λ)m2(si)) n

∏

k=1,k≠i

mk(sk)

=λ∑

s∈S

ui(s)m1(si) n

∏

k=1,k≠i

mk(sk) + (1−λ) ∑ s∈S

ui(s)m1(si) n

∏

k=1,k≠i

mk(sk)

=λUi(m−i, m1) + (1−λ)Ui(m−i, m2) ≥min{Ui(m−i, m1), Ui(m−i, m2)}

Then, by Glicksberg’s, the extended game E(G) has a Nash equilibrium.

Now we provide a simple and useful theorem to find Nash equilibria in mixed strate-gies for some games.

Theorem 1.2.4. (Indifference principle) Let Γ= (N, S, u) be a strategic game and

⃗ σ∗ ∈ ∏

i∈N∆(Si) a Nash equilibrium in mixed strategies for this game. For every

player i ∈ N and any strategies si, s_i′ ∈ Si such that σ⃗_i∗(si),σ⃗_i∗(s′_i) > 0, we have

Ui(si,σ⃗−∗i) =Ui(s ′

i,σ⃗

∗ −i).

Proof: Suppose that for some playeri∈N, there exists some strategiess′

i, s

′′

i ∈Si

with σ⃗∗

i(s

′

i),σ⃗

∗

i(s

′′

i) >0, but Ui(s

′

i,σ⃗

∗

−i) >Ui(s ′′

i,σ⃗

∗

−i). If we define a mixed strategy

⃗

σi∈∆(Si) byσ⃗i(s′′i) ∶=0,σ⃗i(s′i) = ⃗σ

∗

i(s

′

i) + ⃗σ

∗

i(s

′′

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s′

i≠sj ≠s′′i, we have the following:

Ui(⃗σi,σ⃗−∗i) = ∑

s∈S

⎛

⎝ui(s)⃗σi(si) ∏ j∈N∖{i}

⃗ σ∗

j(sj)

⎞

⎠= ∑s∈S

s′ i≠si≠s′′_i

⎛

⎝ui(s)⃗σi∗(si) ∏ j∈N∖{i}

⃗ σ∗

j(sj)

⎞ ⎠

+ ∑

s−i∈S−i

⎛

⎝ui(s′i, s−i)(⃗σ ∗

i(s

′

i) + ⃗σ

∗

i(s

′′

i)) ∏ j∈N∖{i}

⃗ σ∗

j(sj)⎞

⎠

= ∑

s∈S

⎛

⃗ σ∗

j(sj)

⎞ ⎠

+ (⃗σ∗

i(s

′

i) + ⃗σ

∗

i(s

′′

i))Ui(s′i,σ⃗

∗ −i)

> ∑

s∈S

⎛

⃗ σ∗

j(sj)⎞

⎠

+ ⃗σ∗

i(s

′

i)Ui(s′i,σ⃗

∗ −i) + ⃗σ

∗

i(s

′′

i)Ui(s′′i,σ⃗

∗ −i)

= ∑

s∈S

⎛

⃗ σ∗

j(sj)

⎞

⎠=Ui(⃗σi∗,σ⃗

∗ −i)

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Chapter 2 Strategic random network

formation

To define a game on a graph or a random graph, we need to specify the deterministic graph or the random process that generates the random graph. However, we can explain the network formation process, which is the process through agents build or choose the deterministic graph or the generating process of the random graph over which we define the game. Leaving aside the approach of game theory, we can say that many aspects of our lives are affected by social interactions with others, and then to understand many kinds of social phenomena, it is imperative to study the networks of these social relationships, rather than restricting the level of the analysis to any particular relationship. Thus it is important to understand and explain the structure of social networks, how and why they are formed, and how their proper-ties are related with interaction occurring through them, because some properproper-ties of the resulting graph as connectedness, degree distribution, etc, result and may be explained by economic fundamentals.

Almost all studies of network formation are based primarily on strategic motivations as opposed to those based primarily on random events. However, real networks are formed through a combination of randomness and self motivated behaviours. Now, we will analyse processes of network formation through a combination of strategic interaction and random events, and this will allow for situations where link prob-abilities are not viewed as exogenous parameters, but as the outcomes of strategic decisions. Nevertheless, some randomness of network formation process can be in-dependent from individual decisions. To this aim, we will define some stability properties of social networks, some measures of social welfare, with the final pur-pose of presenting some models of strategic random network formation processes. To model the process of network formation, we need to consider the benefits and costs for every player of maintaining a determined social network. IfN ∶= {1,2, ..., n}

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R. If we consider undirected graphs, the process of forming a link involves mutual consent, and for all type of graphs, severing a relationship only involves the consent of one player. In the basic model of network formation, presented in [8] by Jackson, the network formation process is based on a process where agents simultaneously announce whether or not they wish to be linked to each other. If they both announce that they wish to form the link, then it is formed, while if either wants to drop a relation, it is dropped. A natural idea to analyse the result of the network formation process is to use a Nash equilibria approach. However a Nash equilibrium requires stability for all the dynamic process from the beginning, and we only need stabil-ity from some stage of the process. A very simple stabilstabil-ity concept that captures mutual consent is pairwise stability, which is presented in [8] and was developed by Jackson and Wolinsky.

Definition 2.0.1. A network g∈G(N) is pairwise stable if: 1. For all ij∈g, ui(g) ≥ui(g−ij)and uj(g) ≥uj(g−ij)

2. For all ij∉g, if ui(g+ij) >ui(g), thenuj(g+ij) <uj(g)

Then a network is pairwise stable if no player wants to sever a link and no two players want to add a link. Pairwise stability is a weak notion in that it only considers deviations on a single link at a time, but it could be that a player would not benefit from severing or forming any single link but would benefit from severing or forming several links simultaneously, and yet the network could still be pairwise stable. Now, we will define some measures of societal welfare in network formation. Definition 2.0.2. A network g ∈ G(N) is efficient relative to a profile of utility functions (u1, ..., un) if ∑ni=1ui(g) ≥ ∑

n i=1ui(g

′) _{for all} _g′ ∈_G(_N)_{. Since} _N _{is finite,}

there are a finite number of possible networks inG(N), and then there exists a least one efficient network.

Definition 2.0.3. A network g ∈G(N) is Pareto efficient relative to (u1, ..., un) if

there is notg′∈_G(_N)_{such that}_u

i(g′) ≥ui(g)for alli∈N, and with strict inequality

for some agent i∗∈_N_.

Theorem 2.0.1. Let beg ∈G(N). Then g is efficient relative to (u1, ..., un) if and

only if is Pareto efficient relative to all payoff functions (uˆ1, ...,uˆn) with ∑ni=1ui =

∑n i=1uˆi

Proof:

Ô⇒ Suppose that g is efficient relative to (u1, ..., un) and (uˆ1, ...,uˆn) is a

pro-file of utility functions such that∑n_i=1ui= ∑

n

i=1uˆi. If g is not Pareto efficient relative

to(uˆ1, ...,uˆn), there existsg′∈G(N)with ˆui(g′) ≥uˆi(g)for alli∈N, and with strict

inequality for some agent. Then ∑n_i₌₁ui(g′) = ∑ni=1uˆi(g

′) > ∑n

i=1uˆi(g) = ∑

n

i=1ui(g)

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⇐Ô Suppose that g is not efficient relative to (u1, ..., un). Then there exists a

network g′ ∈ _G(_N) _{such that} ∑n

j=1uj(g

′) > ∑n

j=1uj(g). For i ∈ N, we define a

function ûi ∶ G(N) → R by ûi(g′′) ∶= ui(g′′) for all g′′ ∈ G∖ {g′}, and ûi(g′) ∶=

ui(g) +

∑n_j=₁(uj(g′)−uj(g))

∣N∣ >ui(g), which implies that g is not Pareto efficient relative

to(uˆ1, ...,uˆn). Furthermore, we have: n

∑

j=1

ˆ

uj(g′) = n

∑

j=1

uj(g) + n

∑

j=1

(uj(g′) −uj(g)) = n

∑

j=1

uj(g′)

Thus∑n_j=1uˆj = ∑

n j=1uj.

The last theorem implies that ifg∈G(N)is efficient relative to(u1, ..., un), theng is

Pareto efficient to the same utility profile, but the converse is not true. Furthermore, there is not a relation between pairwise stability and efficiency, as can be seen in the following example for four individuals taken from [8]:

Figure 2.1: Example of pairwise stability but not efficiency or efficiency but not pairwise stability

In this case, there are 24(42−1) = 26 = 64 undirected graphs for four individuals.

We also assume that u1(g) = ...= u4(g) = 0 for all possible networks that are not

in the figure. The third network of the first row is efficient relative to (u1, ..., u4),

and by the last theorem, this network is Pareto efficient relative to the same utility profile. Clearly the last network of the first row is not efficient because the sum of the utilities is equal to 10.5 which is less than 12, but this network is Pareto efficient, because if we improve the utility of third or fourth player, we need to decrease the

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utility of first and second player. Neither of these networks is pairwise stable, be-cause in the first, the first and second player have incentives to form a link between them, and in the second network, third and fourth player have incentives to form a link. The last network of the second row is pairwise stable because if the players 1 and 4 or 2 and 3 form a link, then the utility of the players that formed the link, decrease, and if a player wants to drop some link, its utility decrease to 0 or 2. Also, the last network obviously is not efficient neither Pareto efficient since it is possible to increase utility of all players in the efficient network.

In general, if we use the concept of pairwise stability to characterize the process of network formation, the resulting network can be very inefficient in terms of soci-etal welfare, and this fact is consequence of the presence of externalities (decisions of one individual that affect the utility of some other individual). To illustrate this result, we will present a particular case of the distance-based utility model, which is a model of network formation where the utility of every individual depends on its direct connections and its indirect connections via paths between the individual and other individuals.

Let b ∶ {1,2, ..., n−1} → _R denote the net benefit that a player gets from con-nections as a function of the distance between the individuals. If an individual forms a link incurs in a cost of c ≥0, and we assume that b(k) > b(k+1) > 0 for all k = 1,2, ..., n−2, so that the utility deteriorates with the distance between in-dividuals. For every player i ∈N, and every network g ∈ G(N), we use Ni(_g) _to

denote the set of individuals for which there exists a path between them andiunder the network g, and for j ∈ Ni(_g)_, _l

ij(g) denotes the length of the shortest path

between i and j under the network g. In the distance-based utility model, every utility functionui∶G(N) →R is given by the following:

ui(g) ∶= ∑ j∈Ni(g)

b(lij(g)) −di(g)c

It’s obvious that in this network formation model, a player would benefit from a link formed or deleted by other player though new or deleted indirect connections, and thus there are externalities. To illustrate the tension between stability and efficiency, we assume n > 4 and b(1) < c <b(1) +b(2), where the upper bound of

c is assumed less than the upper bound presented by Jackson in [8], because the following statement is not necessarily true for the bound of Jackson.

Proposition 2.0.1. If b(1) < c < b(1) +b(2) and n > 4, the unique efficient net-work structure in the distance-based utility model is a star encompassing all nodes (i.e there exists k ∈ N such that g ∶= {ik∶i∈N}) is the unique efficient network. However, this network is not pairwise stable.

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Proof: A star network component of k nodes leads to a total utility equal to:

(k−1)(b(1) −c) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

utility of center node

+(k−1)(b(1) + (k−2)b(2) −c) ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

utility of every other node

=2(k−1)(b(1)−c)+(k−1)(k−2)b(2)

(2.1) A component ofk nodes has at least k−1 links. If we have a component of k nodes and m≥k−1 links, the total utility due to direct connections is 2m(b(1) −c)since every individual that forms a direct connection has a net gain ofb(1) −cdue to this direct connection. Also, there are k(k−1)

2 −m pairs of players which are at a distance

of at least 2. Then the utility of these component is at most the following quantity: 2m(b(1) −c) + (k(k−1) −2m)b(2) (2.2) The difference between (2.1) and (2.2) is equal to 2(k−1−m)(b(1) −c−b(2)). We have b(1) −c−b(2) <b(1) −c<0 and k−1−m ≤0. If k−1−m<0, then the last difference is greater than 0, and then every component of k nodes in the efficient network has exactlyk−1 links.

Any component of k nodes and k−1 links which is not a star, has a total util-ity less than 2(k−1)(b(1) −c) + (k−1)(k−2)b(2), since there are some nodes of the component at a distance of more than 2, and every direct connection helps to create some indirect connection. By (2.1), all components of efficient networks are star components.

In a star encompassing all nodes, the utility is the following:

2(n−1)(b(1) −c) + (n−1)(n−2)b(2) >2(n−1)(b(1) −c) + (n−1)(n−2)(c−b(1))

= (n−1)(2+2−n)(b(1) −c) = (n−1)(4−n)(b(1) −c) >0

Then the empty network is not an efficient network and then there exists a com-ponent with at least 2 nodes in every efficient network in the distance-based utility model. If we have two star components involvingk1≥1 and k2 ≥2 nodes, the total

utility associated to these components is the following by (2.1):

2(k1−1)(b(1) −c) + (k1−1)(k1−2)b(2) +2(k2−1)(b(1) −c) + (k2−1)(k2−2)b(2)

If we have a star component involvingk1+k2 nodes, the utility of these component

is the following:

2(k1+k2−1)(b(1) −c) + (k1+k2−1)(k1+k2−2)b(2)

The difference between the last expressions is the following:

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= −2(b(1) −c) +b(2)[(k1−1)(k1−2−k1−k2+2) + (k2−1)(k2−2) −k2(k1+k2−2)]

= −2(b(1) −c) +b(2)[−(k1−1)k2+ (k2−2)(k2−1−k2) −k2k1]

= −2(b(1) −c) +b(2)(−2k1k2+k2− (k2−2)) = −2[b(1) −c+b(2)(k1k2−1)]

Also, we haveb(1)−c+b(2)(k1k2−1) ≥b(1)−c+b(2)(2−1) =b(1)+b(2)−c>0, and

then the total utility of disjoint star components is less than the total utility of a star component encompassing nodes of these disjoint components. Then the unique efficient network is a star network encompassing all nodes of the network.

However these network is not pairwise stable, because if the center node drops a link, its utility varies inc−b(1) >0.

2.1 Stochastically stable networks

Consider the following process for growing a network: at each point in time a link is randomly chosen, with equal weight on all links. If the link is not in the network, the two players involved in the link have the choice to add it to the network if it makes each of them weakly better in terms of payoffs and makes at least one of them strictly better off. If the link is already in the network, then either of the players involved in the link can choose to delete it if it would increase the payoff for some player. If this process continues, then it will come to rest at a pairwise stable network. If there do not exist some pairwise stable network, then it will keep cycling. We can add some exogenous randomness in the network formation process, so that links occasionally are added or deleted even though the benefits do not outweigh the costs. The following variation to the previous process of network formation was introduced by Jackson and Watts [9]: in every period of the process of network formation, the intent of the players (to add a link, to delete a link, or to leave the network as it is) is carried out with probability 1−, and with probability 0<<1, there is an error or perturbation in the process of network formation, and the reverse occurs.

If we represent the process of network formation by a Markov chain where the states are the possible networks that may emerge, then clearly the Markov chain is irreducible, and by the fundamental theorem of Markov chains (see theoremA.1.1), the Markov chain associated to this process has a stationary distribution. For ex-ample, consider N ∶= {1,2,3}. In the empty network, every player has an utility equal to 0, in all networks with exactly one link, every player has an utility equal to −a for some a > 0, in the networks with two links, every player has an utility equal to a, and in the complete network every player has an utility equal to 2a. If the number of links of the network in every period determined the states of the

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random process, the following matrix is the Markov chain associated to the process of network formation:

P() ∶= ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣

1− 0 0

1− 3 3+ 2 3

2(1−)

3 0

0 2

3

2(1−)

3 +

3

1−

3

0 0 1−

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ = ⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣

1− 0 0

1−

3

2(1−)

3 0

0 2₃ 2−

3

1−

3

0 0 1−

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦

To find the stationary distribution ofP(), we need to find a probability row vector

π() such thatπ()P() =π(), or in the appendix we prove in theorem A.1.1 that

π() = [⃗0T_,₁]_B()−1_{, where} _B()_{is the matrix} _A() ∶= [_P() −_I,⃗₁]_{without the first}

column. Then we have the following:

π() = (0,0,0,1)

⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣

0 0 1

−1 2(1−)

3 0 1

2 3

−−1

3

1−

3 1

0 − 1

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ −1

= (0,0,0,1)

⎡⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣

1++2

(1+2)

−3

1+2

9 1+2

−(−1)2

(1+2)

3(+1)

2(1+2)

3 2(1+2)

−9

2(1+2)

3(−1)

2(1+2)

3− 2 3 2 −3 2

3−5

2 −2

1+2

32 1+2

3−32

1+2

(1−)2

1+2

⎤⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦

Then we haveπ() = ((1−)

1+2 ,

32 1+2,

3(1−)

1+2 , (1−)2

1+2 ). Clearly lim→0π() = (0,0,0,1), and

this implies that the process of network formation converges with positive probability to the complete network when the probability of error or perturbation converges to 0, and in this case we say that the complete network is stochastically stable. We use µ(g, ) to represent the stationary probability of the network g ∈ G(N) when the error probability is.

Definition 2.1.1. A network g∈G(N) is stochastically stable if lim→0µ(g, ) >0.

To characterize the set of stochastically stable networks, we need some defini-tions.

Definition 2.1.2. Two networks g, g′ ∈ _G(_N) _{are called adjacent if there exists}

some linkij such that g′=_g−_ij _or _g′=_g+_ij_.

Definition 2.1.3. A path of networks p ∶= (g1, ..., gK) is a sequence of networks

such that for i=1,2, ..., K−1, the networks gi and gi+1 are adjacent. This path is

called an improving path ifgi+1=gi−kl for some linkkl implies uk(gi−kl) >uk(gi)

andgi+1=gi+kl for some linkkl impliesuk(gi+kl) >uk(gi) andul(gi+kl) ≥ul(gi).

Then an improving path is a sequence of adjacent networks which result of the pure rational process of network formation, without error probabilities.

Definition 2.1.4. The resistance of a pathp= (g1, ..., gK)is the minimum number of

errors needed in order to go fromg1togK, and is equal tor(p) ∶= ∑iK=−111(gi ∉im(gi+1)),

where for all networks g ∈ G(N), im(g) is the set of networks for which there exists an improving path from them to g. The resistance between two networks

g, g′ ∈ _G(_N) _{is denoted by} _r(_{g, g}′) _{and is equal to the minimum} _r(_p) _{among all}

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Definition 2.1.5. Given a network g ∈ G(N), a g-tree is a directed graph which has as vertices all networks and has an unique directed path leading from each

g′∈_G(_N) ∖ {_g} _to_g_{, and we use} _T(_g) _{to denote the set of}_g_-trees.

Definition 2.1.6. Let be g ∈G(N). The resistance of a g-tree t ∈T(g) is defined byr(t) ∶= ∑_g′_g′′_∈_tr(g′, g′′)

Definition 2.1.7. The resistance of a network g∈G(N)is defined by the quantity

r(g) ∶=mint∈T(g)r(t).

Now, we can state one of the principal results of this section.

Theorem 2.1.1. The set{g∈G(N) ∶r(g) ≤r(g′) ∀_g′∈_G(_N)}_{is the set of}

stochas-tically stable networks, and since there is a finite number of networks, there exists some stochastically stable network.

Proof: In this case, the set of states is the set of all possible networks. For each network g ∈ G(N), we define P

g ∶= ∑t∈T(g)∏g′_g′′_∈_tP()_g′_g′′. Since the per-turbed Markov chain is irreducible for some>0 near to 0, there exists a sequence

(gn)

∣G(N)∣

n=1 =G(N)of all networks such thatP()gngn+1 >0 for alln=1, ...,∣G(N)∣−1,

and there is a unique path from any other network to g, and thus this sequence is associated to a g-tree, and this implies P

g > 0 for all g ∈ G(N). We define

µ g ∶=

P g

∑g′ ∈G(N)P_g′ >

0 for every network g ∈G(N). Obviously µ _{is a probability}

dis-tribution for the states of the stochastic process. We define r∗ ∶= _min

g∈G(N)r(g). Let be g ∈ G(N), and T is a g-tree such that

r(g) =r(T). We have the following identity:

r(T)−r∗

∏

g′_g′′_∈_T

−r(g′,g′′)

P()g′_g′′ =r

(T)−r∗

− ∑g′_{g′′ ∈T}r(g′,g′′)

∏

g′_g′′_∈_T

P()g′_g′′ =

−r∗

∏

g′_g′′_∈_T

P()g′_g′′

(2.3)

In the g-tree constructed at the beginning of the proof, we have P()g′_g′′ > 0 for all networks connected in this g-tree. Since in T, we minimize ∑_g′_g′′

∈tr(g

′_{, g}′′) _over

all t∈T(g), we have P()g′_g′′ >0 for all g′g′′ ∈T, because if this quantity is equal to 0 for two networks connected inT, it is impossible to pass from one of these two networks to the other in the perturbed process, and then the minimum number of errors for pass from one of these two networks to the other is at least 2, which is greater than 1 that is the maximum possible resistance between these two networks whenP()g′_g′′ >0 for all g′g′′∈T.

We have 0 < ∏_g′_g′′_∈_T −r(g ′_,g′′

)_P()

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im-plies lim→0 −r∗∏

g′_g′′_∈_TP()_g′_g′′ =0, so we have the following:

lim

→0

−r∗_P

g =lim →0 ∑

t∈T(g)

−r∗ ∏

g′_g′′_∈_t

P()g′_g′′ = ∑

t∈T(g),r(t)=r(g)

lim

→0(

−r∗ ∏

g′_g′′_∈_t

P()g′_g′′) =0 (2.4) where the penultimate equality holds because if for somet∈T(g), we haveP()g′_g′′ = 0 for someg′_g′′∈_t_{, then} _t _{is not a}_g_{-tree of minimum resistance.}

On the other hand, if for some networks g′_{, g}′′ ∈ _G_{, we have} _P

g′_g′′ > 0, it is pos-sible to pass from g′ _to _g′′ _{in the stationary state, so} _r(_g′_{, g}′′) =_{0, and this implies}

lim→0

−r(g′,g′′)_P()

g′_g′′ = lim_→₀P()_g′_g′′ = P_g′_g′′ > 0. If g′, g′′ ∈ G are such that

P()g′_g′′ >0 and P_g′_g′′ =0, it is possible to pass from g′ to g′′ in the perturbed pro-cess but not in unperturbed propro-cess, and then the minimum number of errors to do this is 1, sor(g′_{, g}′′) =_{1 and}_P()

g′_g′′ is linear in, and then lim_→₀−r(g ′_,g′′

)_P()

g′_g′′ = lim→0

−1_P()

g′_g′′ >0. By the two previous cases, if P()_g′_g′′ > 0 ,we always have lim→0

−r(g′,g′′)_P()

g′_g′′ >0. If we haver(T) =r(g) =r∗_{, lim}

→0

−r(g′,g′′)_P()

g′_g′′ >0, (2.3) imply the following:

lim

→0

−r∗

P_g= ∑

t∈T(g),r(t)=r(g)

lim

→0(

−r∗

∏

g′_g′′_∈_t

P()g′_g′′) >0

. We have µ g =

Pg

∑g′ ∈G(N)P_g′ =

e−r∗Pg

∑g′ ∈G(N)e−r ∗

P_g′

. Since there exists a network g∗∈_G(_N)

such that r(g∗) = _r∗_{, we have lim}

→0∑g′_∈_G₍_N₎e−r ∗

P

g′ ≥ lim→0(e −r∗_P

g∗) > 0. The previous paragraph implies lim→0µ=0 if r(g) >r

∗ _{and lim}

→0µ>0 if r(g) =r ∗_.

Freidlin and Wentzell in chapter 6 of [5], prove a lemma which allow us to affirm that

µ _{is the stationary distribution of the Markov chain of the pertubed process,} _P()_.

Since lim→0µ

>_{0 if and only if}_r(_g) =_r∗_{, we have that}{_g∈_G(_N) ∶_r(_g) ≤_r(_g′) ∀_g′∈_G(_N)}

is the set of stochastically stable networks.

Theorem 2.1.2. If there exists some pairwise stable network, then every stochasti-cally stable network is pairwise stable

Proof: Let g ∈G(N) be a stochastically stable network, and suppose that g is not pairwise stable. If the set of pairwise stable networks is not empty, there exists some g∗ ∈ _G(_N) _{such that} _g ∈_im(_g∗)_{. Consider a} _g_{-tree with resistance equal to}

r(g), and construct a new graph of the following form: create a direct edge from

g to g∗_{, and delete the directed edges from} _g∗_{, and join the resulting components}

without altering the structure of graph. An example of this process is provide in the following figure:

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Figure 2.2: Relation between stochastically stability and pairwise stability

Since in the first graph, there is a unique path from other networks tog, in the final path there is a unique path from other networks tog∗_{, and then the resulting}

directed graph is ag∗_{-tree. Also, since we add a link from}_g _to_g∗_{, which is a possible}

transition without errors, and delete some links from g∗ _{to other networks, which}

are transition only possible with errors, the resistance of the resulting graph is less than the resistance of the initial graph, and by definition of resistance of a network, we have r(g∗) <_r(_g)_{, which by the previous theorem contradicts the fact that} _g _is

stochastically stable.

By the last two theorems, we can finding stochastically stable networks by find-ing networks of minimum resistance among pairwise stable networks if the set of pairwise stable networks is non empty or finding the network of minimum resistance among all networks in opposite case. However, with these tools, find the resistance of a network may be complicated if the set networks is big.

Tercieux and Vannetelbosch proposed in [13], a refinement of pairwise stability, which also allow us to calculate the set of stochastically stable networks. This re-finement is called p-pairwise stability, and we need to define some concepts before presenting this refinement.

Definition 2.1.8. IfgN ∈_G(_N)_{denotes the complete network between individuals}

inN, we define the distance between two networks g, g′∈_G(_N)_{by the following:}

d(g, g′) ∶= ∣{ig∈g

N ∶ (_ij∈_g∧_ij∉_g′) ∨ (_ij ∈_g′∧_ij∉_g)}∣

∣gN∣ (2.5)

Then the distance between two networks is the proportion of links in which the two networks differ. For subsets of networks G, G′ ⊆_G(_N)_{, we define the distance}

between them as d(G, G′) ∶=_min

g∈G,g′_∈_G′d(g, g′).

Definition 2.1.9. We say that an improving path p= (g1, ..., gK)goes directly from

g′ ∈_G(_N) _to _G ⊆_G(_N) _if _g

On the asymptotic behaviour of strategic interaction on random undirected graphs