SN- Lecture 10

of NetworksStrategic Formation

Lecture 10

We knowPeople make choices thinking about what others are doing


our social relationships have an important effect on

how we behave


our social relationships have an important effect on

how we behave

Our networks are valuable, they are the capital we use

to achieve our goals

Network FormationIn many social settings, our relationships are not given


We decide with whom to link


We decide with whom to link

In this class we will study how networks are formed by

strategic decision-makers

Aim Lecture 10To Understand

The basic of network strategic formation

Simple cases of how to model network formation

The relation between individual incentives & social welfare from a network perspective (emergence)

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Some basic intuition


Costs & Benefits

Actors choose links

Contrasts incentives & Social efficiency


Costs & Benefits

Actors choose links


Nodes as actors making choices


Costs & Benefits

There are costs and benefits associated with each network

Actors choose links




Costs & Benefits


Actors choose links



We can also call them agents, players, people, countries, etc...


Costs & Benefits


Actors choose links



We can also call them agents, players, people, countries, etc...

We will contrast the incentives individuals have to form relationships with what’s best for society

How to model choicesIncentives to form and severe links


Is consensus needed? (undirected/directed)

What happens on the network?

Are links adjustable in intensity?

Can they coordinate changes in the network?


Is consensus needed? (undirected/directed)Does somebody has to say yes to me to relate with her?









Citations networks vs. friendships or alliances







I’ll form a new alliance with you but only if we can form an alliance with someone else at the same time







I’ll from a new alliance with you but only if we can form an alliance with someone else at the same time

What’s generating value in the network? where? Costs & benefits?







I’ll from a new alliance with you but only if we can form an alliance with someone else at the same time

What’s generating value in the network? where? Costs & benefits?

Only whether a relationship exists or not? How intense is it?

Questions that ariseWe will address some of them

Which networks are likely to form?

Are some networks more stable than others?

We will address some of them

Which networks are likely to form?Are they the right ones from society’s perspective?


Questions that arise




if not, how different are they?





if not, how different are they?

How likely is it that changes can occur in certain networks?New people coming in? Creating new relationships?


Representationof strategic network formation

(Jackson & Wollinsky, 1996)



Payoffs:

The network g generates some payoffs to a given individual i


Payoff to i if the network is gui(g)


Payoffs:





Payoffs:


For simplicity:

We are going to look at undirected network formation




Payoffs:


For simplicity:

We are going to look at undirected network formation

We could look at directed, weighted, etc.

ModelingIncentives & Equilibrium


is modeled as a game where

The network:



The network:

Each agent announces who they wish to link with



The network:


& a link forms if and only if both agents name each other



The network:

What is a Nash equilibrium in this game?


& a link forms if and only if both agents name each other

Equilibrium NetworksNash equilibrium


A situation where nobody can gain by changing his/her actions (announcements)


Given the actions (announcements) of other individuals

A situation where nobody can gain by changing his/her actions (announcements)

for undirected networksProblem of Nash

mutual agreement to form a link

Problem of Nashfor undirected networks


If they are separated, each gets a value of 0 and if together 1.Example (i & j):

for undirected networks

Example (i & j):If they are separated, each gets a value of 0 and if together 1.

They play a game of simultaneous announcement


Problem of Nash


If they are separated, each gets a value of 0 and if together 1.


If i thinks j won’t announce, she might as well not announce

Example (i & j):


Problem of Nash


i

j

0

0

Case 1: Neither announces is a Nash equilibrium



Example (i & j):



Problem of Nash


ii

jj

1

1

0

0

Case 1: Neither announces is a Nash equilibrium



Case 2: Both announce is a Nash equilibrium

Example (i & j):



Problem of Nash

Pairwise StabilityModeling Incentives

A network (N,g) is pairwise stable if

ii

jj

1

1

0

0



ii

jj

1

1

0

0

No actor gains from severing a link (one person can do it)

No two actor gain from forming a link (at least one is better off)



ii

jj

1

1

0

0

Relationship must be beneficial to be maintained





ii

jj

1

1

0

0


Beneficial relationship are pursued when available



Some Notation


Pairwise Stability

Some Notation


No agent gains from severing a link (one person can do it)

ui(g) ≥ ui(g-ij) for i and ij in g

Pairwise Stability

Some Notation




No two agents gain from forming a link (at least one is better off)

ui(g+ij) ≥ ui(g) implies uj(g+ij) < uj(g) for ij not in g

Pairwise Stability

Some Notation





It’s still a very weak concept

Pairwise Stability


Some Notation






Only looks at pairs of individuals

Pairwise Stability


Some Notation






Only looks at pairs of individualsOne link at a time

Pairwise Stability


Some NotationPairwise Stability






Only looks at pairs of individualsOne link at a time

(It’s a minimum)


Back to our example

ii

jj

1

1

0

0

Back to our example

ii

jj

1

1

0

0



Back to our example

ii

jj

1

1

0

0



Which is the pairwise stable situation (network)?

Back to our example

ii

jj

1

1

0

0




Back to our example

ii

jj

1

1

0

0




Although both are Nash equilibria

RememberLave & March


ModelsAre a simplified version of the world


ModelsAre a simplified version of the world

Models are created by speculating about the process that could have produced

the observed outcomes

Collaboration NetworksPairwise Stability

0 0

0 0

Think of situations in which we collaborate with others (i.e.,

friendships, projects, inter-firm R&D)

The more collaborations you have the better for you

&

ButThe more collaborations your

collaborators have the worse for you

Negative Externalities


0 0

0 0

0 3

0 3

i f 2 actors are related, each gets 3 out of it.


0 0

0 0

0 3

0 3

Will this situation tend to change or is it likely to be stable?

i f 2 actors are related, each gets 3 out of it.


0 0

0 0

0 3

0 3

3 3

3 3

The two other actors can be better if they relate


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

NEGATIVE EXTERNALITIES

But now, the new connection benefits directly the linked actors & affects negatively & indirectly the existing neighbors


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

Think about the attention you get from your parents if you are alone or if you have to

share it with your brothers & sisters

NEGATIVE EXTERNALITIES


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

2.5 2.5

2.5 2.5

Everyone has incentives to create a new connection

without intending to harm their neighbors

Think about Schelling’s Segregation


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

2.5 2.5

2.5 2.5

2.78 2

2 2.78


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

2.5 2.5

2.5 2.5

2.78 2

2 2.78

2.33 2.33

2.33 2.33

Until the networks is

complete


0 0

0 0

0 3

0 3

3 3

3 3

3.25 3.25

2 2

2.5 2.5

2.5 2.5

2.78 2

2 2.78

2.33 2.33

2.33 2.33

PairwiseStable

Stability & EfficiencyThink about this two networks


3 3

3 3

Each could have gotten 3 & society 12


3 3

3 3

Each got 2.33 & society 9.32

2.33 2.33

2.33 2.33

PairwiseStable


3 3

3 3

Each got 2.33 & society 9.32

2.33 2.33

2.33 2.33

PairwiseStableEfficient

Not stable to individual incentives

EfficiencyA network is efficient if the total

sum of all the utilities from all the players is the maximal one can

achieve

The situation in which society as a whole is better offBut still some can be in very bad shape

Notation:A network g is efficient if:

g maximizes ∑ui(g’)

Emergence

3 3

3 3

Although individuals do not intend to harm their neighbors

2.33 2.33

2.33 2.33

Efficiency

Pursuing personal incentives leads to social inefficiency

Their actions have implications for other people they are not

necessarily taking into account

From our friends’ friends

Positive Externalities

Other Effects

In different settings

I get indirect benefits the more friends my friends have

Think about popularity, transmission of valuable information, access to jobs

We will cover some of this in other lectures

Some Notation

Positive Externalities

uk(g+ij)≥uk(g) if ij not in g, for every k≠i,j

Everyone is equal or better with the new link

Negative Externalities

uk(g+ij)≤uk(g) if ij not in g, for every k≠i,j

At least someone is worse with the new link

Checklist

Strategic network formation assumes actors are rational decision makers

Externalities from relationships influence the networks that are likely to be formed

Social efficiency can be affected by individual incentives

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Questions?

SN- Lecture 10

Technology

network actors

network formation

choices incentives

severe links

citations networks

network perspective

social relationships

incentives individuals