Games On NetworksGames On NetworksLecture 12

Aim Lecture 12To understand

how networks & games are related

Games with strategic complements

Games with heterogeneous players

Games with endogenous link formation

An experiment on network games

So far...

We are interested in understanding networks & behavior

Now by

bringing strategic interaction into play

Network Games

In network gamesDecisions to be madeNot just a simple diffusion or contagion process

People care about what other people are doing

Complementarities

I want to buy/use a certain software only if other people are buying/using that same software

Languages, attending to social events...

Interdependencies between individuals

For this purpose

We use game theory as a tool to try and understand how behavior relates to network

structure

Players on a network

Different individuals each making decisions

Games on Networks

Care about the actions of neighbors

Interdependencies in payoffs (i.e., their utility function)

Our Focus

What can we say about behavior &

how it relates to network structure

Network Games

Strategic Complementarities

With

Each player chooses an action

xi in {0,1}

Network game

Either use the software or not

Payoff will depend on

my choice the choice of my neighbors

ui(xi, xNi)

How many choose 1 or 0Whether I choose 1 or 0

Either take an action or not

simplifying assumptions

1

2 Care about actions of neighbors but not who they are (Identities)

3 Fixed networks

1 or 0, but not a wide range of choices

I don’t have best friends or closer neighbors

I cannot choose my neighbors but they are given

It will become richer soon...

Complements vs. Substitutes

Types of games

As more of my friends take an action, it is more attractive to me

ui(1,m)-ui(0,m)≥ ui(1,m’)-ui(0,m’) for all m≥m’

Strategic complements - Increasing differences

Strategic substitutes - Decreasing differencesAs more of my friends take an action, it is less attractive to me

ui(1,m)-ui(0,m)≤ ui(1,m’)-ui(0,m’) for all m≥m’

Coordination

Anti-Coordination

Example 1An agent chooses action 1 if at least

two neighbors do: threshold of one.

Payoff action 0: ui=0

Payoff action 1: ui=-1, if less than two neighbors choose 1

ui=1 per coordination, if at least two neighbors choose 1

Remember: payoffs are ordinal. The number doesn’t matter

Example 1

For example:

I only choose to use a technology (choose 1) if at least 2 of my friends are using it, otherwise I rather not do it.

An agent chooses action 1 if at least two neighbors do: threshold of one.

0

0

0

0

00

0

0

0

0

0

Only one neighbor, so they can’t choose 1

Example 1

For example:

0

0

0

0

00

0

0

0

0

0But, how about this guy?

I only choose to use a technology (choose 1) if at least 2 of my friends are using it, otherwise I rather not do it.

An agent chooses action 1 if at least two neighbors do: threshold of one.

Example 1

For example:

I only choose to learn to play bridge if at least 2 of my friends know how to play it. otherwise I rather not do it.

0

0

0

0

01

0

0

1

1

0Each has at least

two friends choosing 1

An agent chooses action 1 if at least two neighbors do: threshold of one.

Nash Equilibrium

No player wants to change her behavior alone, fixing what her neighbors are doing

0

0

0

0

00

0

0

0

0

0

Case 1: The technology is never used

Nash Equilibrium

Case 2: These three people adopted the technology because each of them has 2 neighbors using it

0

0

0

0

01

0

0

1

1

0

In this case no one else would want to do it

Network Games

Heterogeneous Preferences

With

&

Strategic Complementarities

Either take an action or not

simplifying assumptions

1

2 Care about actions of neighbors but not who they are (Identities)

3 Fixed networks

1 or 0, but not a wide range of choices

I don’t have best friends or closer neighbors

I cannot choose my neighbors but they are given

Strategic complementarities

Games with complements

The choice to take an action by my friends increases my relative payoffs to taking that action (tipping point)

Tipping point: Threshold (ti)Number of my neighbors adopting the technology (choosing action 1)

0 niti

------- I choose 0 ------- ------- I choose 1 -------

if ni(1)<ti I choose 0 & if ni(1)≥ti I choose 1

Education decisions: university, human capital?

Some examples of SC

Increases access to jobs (the more people you know who are well educated the higher the chances)

Invest if at least k neighbors do the same

Technology adoption/ learn a language

Peer influence were the pressure grows with the size of the crowd

Cheating, doping

Anti-social behavior: smoking among teens

If you are in sports and others are taking “performance enhancing drugs”The relative payoff goes up the more others are doing it... even though the more people do it doesn’t make it better for an athetle

Example 2

Players have preferences for the different options: like one more than the other

any two players are identicalSo far:

Heterogeneous Network Games: Conflicting Preferences with P. Hernández & A. Sánchez (GEB, 2013)

Types:ui= a for 1 & b for 0

ui= b for 1 & a for 0a>b

Example 2

1 > 0 0>1

2a,2a a,b

b,a 2b,2b

1

0

1 0

2a,2b a,a

b,b 2b,2a

1

0

1 0

2b,2b b,a

a,b 2a,2a

1

0

1 0

Conflicting preferences: Players want to be together, but have different opinions about which is the most desirable outcome

Example 2An agent chooses her favorite action

if ti=1/3 of her neighbors choose it

Neighbors choosing 1

0 niti ti

0

0

1

0

1

1

There are two thresholds: One to choose

what I like (ti) & one to choose what I dislike (ti)

With heterogeneity

We can understand & model the strength of social influence needed to make players adopt

certain behaviors

Education decisions: The effect is different for people who would prefer to study than for those who would prefer to jump into the job market

Anti-social behavior: Kids who would like smoking need less influence to start than others

Example 2We know need to consider:

preference (type) & behavior (action chosen)

00

00

00

00

0000

00

00

00

00

00

The first digit is the preference & the second the behavior

Example 2We know need to consider:

preference (type) & behavior (action chosen)

Case 1: This is a Specialized satisfactory equilibrium.

00

00

00

00

0000

00

00

00

00

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All choose the same action &

are happy

Example 2We know need to consider:

preference (type) & behavior (action chosen)

11

11

11

11

1111

11

11

11

11

11

All choose the same action &

are happy

Case 1: This is a Specialized satisfactory equilibrium.

Example 2We know need to consider:

preference (type) & behavior (action chosen)

Case 2: This is a Specialized frustrated equilibrium.

00

00

00

00

0010

10

10

10

10

00

All choose the same action but

the 1’s are frustrated

Example 2We know need to consider:

preference (type) & behavior (action chosen)

Case 2: This is a Specialized frustrated equilibrium.

01

01

01

01

0111

11

11

11

11

01

All choose the same action but

the 0’s are frustrated

Example 2We know need to consider:

preference (type) & behavior (action chosen)

Case 3: This is a Hybrid satisfactory equilibrium.

00

00

00

00

0011

11

11

11

11

11

All choose the preferred action (happy) & both actions coexist

Example 2We know need to consider:

preference (type) & behavior (action chosen)

Case 4: This is a Hybrid frustrated equilibrium.

00

00

10

00

0001

11

11

11

11

01

Some choose the non-preferred action & both actions coexist

What does this mean?People’s relationships with others determine the benefits they can get and the goals they can achieve

1

Marsden & Gorman, 2001

About 2/3 of the working population in western industrialized societies (informal social ties)

Participation in political protestAffected by friendship and family networks

Opp & Gern, 1993

Finding jobs

But...

2 People’s characteristics determine the social relationships they form

Opportunity (Contact theory)

More of a chance of meeting your own type

The possibility that you meet people could be biased by attributes (i.e, race)

Costs & benefits

Social pressure or social competition

Common attributes (i.e., language, culture, knowledge) make it easier

Network Games

Endogenous link formation

With

&

Strategic Complementarities

Heterogeneous Preferences

&

Either take an action or not

simplifying assumptions

1

2 Care about actions of neighbors but not who they are (Identities)

3 Fixed networks

1 or 0, but not a wide range of choices

I don’t have best friends or closer neighbors

I cannot choose my neighbors but they are given

A person’s social network promotes her goal achievement

&

1

2A person will invest in her social network (i.e., form relationships) depending on its instrument value

Combined arguments

What if people’s preferences on the possible outcomes are in conflict?

Heterogeneity in preferencesHeterogeneity in preferences

The Model

2 stage network game

Nodes: Purposive rational actors

Players

Type ✓i 2 {0, 1}

N = {1, . . . , n}

Affiliation

Each player announces who they wish to link with

pi = (p1, . . . , pn)

A link forms iff both players propose to each other

Behavior Adoption

Players observe the network and choose an action

X = {0, 1}

A player of type 1 prefers action 1 over 0

The ModelThe network generates payoffs to the players

u

i

(✓i

, (p1, . . . , pn), xi

, x

ki(g)) = �

✓ixi(1 +

Pki

j=1 I{xj=xi})� cp

i

Indicator functioncoordination

Type parametera if happy b if frustrated

Linking costfor every proposal

a>b>c

Coordination game with strategic complementarities

The ModelEquilibrium examples

Subgame perfect Nash equilibrium

No links with uncoordinated neighbors are kept

Only links with coordinating neighbors are part of an equilibrium

No unreciprocated links are proposed

There are multiple networks that satisfy these properties

The more neighbors coordinating with the better

Threshold model (tipping point to choose favorite action)

The effect of different levels of conflict in the preferences (microlevel) on the emerging network configurations (macrolevel)

Experimental Design

3 treatments

No conflict (15-0) - 30 subjects+

+ Low conflict (12-3) - 45 subjects

+ High conflict (8-7) - 45 subjects

One-shot + 15 players + 20 rounds + z-tree

Undergraduate students Universitat de Valencia (Spain)

Experimental Design

Multiple equilibria

Link Proposal

Action

1 socially optimal

Complete network

Favorite behavior of the majority

ResultsWho is proposing connections to whom?

Subjects segregate between types (preferences) in the relationships

they form1

Maj proposes & connects most often in No and least often in High+

Min proposes & connects most often in High and least often in Low

In High, Min sends more proposals between groups than Maj

+

+

2 components in heterogeneity

ResultsWho is proposing connections to whom?

Box plots of proposals & connections within & between groups

Results

Subjects attempt to maximize payoffs within the equilibrium they choose2

How is the level of connectivity?

Differences between treatments and groups are systematic+

ANOVA: (Ftreatment=5.00, p<0.01; Fgroup=79.92, p<0.001)

+ Min networks are denser than the Maj networks

+ Density is lower in Low than in No

+ Max. density is reached for Min in High

Results

Box plots of density by treatment and groups

How is the level of connectivity?

Results

Subjects reach stable configuration in homogeneity or minorities in heterogeneity3

Is there any stable subgroup & when?

NO: complete density from round 12 on

LOW & HIGH: Do not reach complete density + lower density than Min

MinorityLOW: complete density from round 4 on.

HIGH: start off with higher densities + complete density from round 6

Once complete density is reached it remains very stable.

Majority

ResultsIs there any stable subgroup & when?

Is there any stable subgroup & when?

Results

Is there any stable subgroup & when?

Results

ImplicationsConflicting preferences

social optimality is not reached (risk-dominant equilibrium)

Subjects aim for the payoff dominant equilibrium within their segregated Eq.

Individual preferences are more focal than payoffs Game theoretic Lit: exogenous networks (risk-dominant equilibrium)endogenous networks (payoff dominant equilibrium)

Existence of conflict (not level) leads to segregationSocial identity theory, homophily, etc.

Further: Less individualistic societies?

Checklist

We have learned thatnetwork games can represent multiple social & economic problems

Actors relations in the network affect their behavior

Actors preferences influence the relationships they form

Experimentally, individual characteristics are more salient than payoffs

Questions?