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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
00
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?