[PPT] - ( CONT D ) Each can either say that his/her partner committed the PowerPoint Presentation

SLIDE 1

ASSORTATIVITY IN SOCIAL DILEMMAS

Alexandros Rigos University of Leicester → Lund University (with Heinrich Nax, ETH Zürich) Controversies in Game Theory III 2016-06-01

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS TITLE

SLIDE 2

INTRODUCTION

THE TRAGEDY OF THE COMMONS

TRAGEDY OF THE COMMONS

A situation where individuals acting independently and rationally according to each’s self-interest behave contrary to the best interests of the whole group by depleting some common resource.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 1 / 30

SLIDE 3

INTRODUCTION

THE PRISONER’S DILEMMA AND THE TRAGEDY OF THE COMMONS

Arguably the best-known game in game theory is the Prisoner’s Dilemma (PD). It is the story of two “partners in crime” who the police has caught. They are being interrogated separately by the police.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 2 / 30

SLIDE 4

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 5

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). Prisoner B Prisoner A

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 6

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). Prisoner B C D Prisoner A C D

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 7

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. Prisoner B C D Prisoner A C D

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 8

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. Prisoner B C D Prisoner A C

1,-1

D

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 9

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. If one partner Defects, the police let him to go free while his partner gets 5 years in prison. Prisoner B C D Prisoner A C

1,-1

D

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 10

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. If one partner Defects, the police let him to go free while his partner gets 5 years in prison. Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 11

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. If one partner Defects, the police let him to go free while his partner gets 5 years in prison. If both partners Defect, they get 4 years in prison each. Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 12

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Each can either say that his/her partner committed the crime (Defect) or to remain silent (Cooperate). If both partners Cooperate, they get charged for some minor offence and they get 1 year in prison each. If one partner Defects, the police let him to go free while his partner gets 5 years in prison. If both partners Defect, they get 4 years in prison each. Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 3 / 30

SLIDE 13

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 14

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 15

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 16

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 17

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 18

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D. If Prisoner B plays D, Prisoner A would play D.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 19

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D. If Prisoner B plays D, Prisoner A would play D. The same goes for Prisoner B.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 20

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D. If Prisoner B plays D, Prisoner A would play D. The same goes for Prisoner B.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 21

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D. If Prisoner B plays D, Prisoner A would play D. The same goes for Prisoner B.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 22

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

How would they choose if they

Act independenty of each other. Try to maximise the amount of time out of jail?

If Prisoner B Cooperates (plays C), Prisoner A would play D. If Prisoner B plays D, Prisoner A would play D. The same goes for Prisoner B. If both players best-respond to each other, the outcome is (D,D).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 4 / 30

SLIDE 23

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

What is the problem?

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 5 / 30

SLIDE 24

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

What is the problem? In the outcome (Nash Equilibrium), they get 8 years in prison together.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 5 / 30

SLIDE 25

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

What is the problem? In the outcome (Nash Equilibrium), they get 8 years in prison together. By playing C, they could have been getting 2 years in prison together.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 5 / 30

SLIDE 26

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

What is the problem? In the outcome (Nash Equilibrium), they get 8 years in prison together. By playing C, they could have been getting 2 years in prison together. The incentives are not there for the “efficient” outcome to occur.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 5 / 30

SLIDE 27

INTRODUCTION

THE PD AND THE TRAGEDY OF THE COMMONS (CONT’D)

Prisoner B C D Prisoner A C

1,-1
5,0

D 0,-5

4,-4

What is the problem? In the outcome (Nash Equilibrium), they get 8 years in prison together. By playing C, they could have been getting 2 years in prison together. The incentives are not there for the “efficient” outcome to occur. ⇒ Tragedy of the Commons.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 5 / 30

SLIDE 28

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 29

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0 They follow different strategies and meet at random.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 30

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication

r imitation).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 31

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication

r imitation).

The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 32

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication

r imitation).

The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS). A population full of D-individuals cannot be invaded by C-individuals.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 33

INTRODUCTION

PD WITH RANDOM INTERACTIONS LEADS TO TRAGEDY

OF THE COMMONS

Consider a large population playing a Prisoners’ Dilemma. C D C 9,9

2,11

D 11,-2 0,0 They follow different strategies and meet at random. More successful strategies evolve more quickly (because of replication

r imitation).

The unique Nash equilibrium is D-D and all individuals following D is the unique Evolutionarily Stable Strategy (ESS). A population full of D-individuals cannot be invaded by C-individuals. ⇒ Tragedy of the Commons.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 6 / 30

SLIDE 34

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 35

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 36

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 37

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs!

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 38

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 39

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached. ⇒ Assortativity can help overcome the tragedy of the commons.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 40

INTRODUCTION

ASSORTATIVITY AND THE TRAGEDY OF THE COMMONS

In the same PD as before.. . C D C 9,9

2,11

D 11,-2 0,0 . . . let individuals meet assortatively. Cs have higher chance (when compared to Ds) to meet Cs. Extreme case: Cs always meet Cs! Then efficient outcomes can be reached. ⇒ Assortativity can help overcome the tragedy of the commons. Assortativity can be the outcome of different processes.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 7 / 30

SLIDE 41

INTRODUCTION

LOCAL INTERACTIONS

Interact mostly with your neighbours. (Boyd and Richerson, 2002; Grund, Waloszek, and Helbing, 2013)⇒ Spatial assortativity.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 8 / 30

SLIDE 42

INTRODUCTION

KIN SELECTION

Interact mostly with your relatives. (Hamilton, 1964)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 9 / 30

SLIDE 43

INTRODUCTION

GREENBEARD EFFECT

Recognise others’ types. (Dawkins, 1976)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 10 / 30

SLIDE 44

INTRODUCTION

HOMOPHILY

You mostly interact with people that are like you in some characteristic. Alger and Weibull (2012) (social preference assortativity)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 11 / 30

SLIDE 45

QUESTIONS

We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 12 / 30

SLIDE 46

QUESTIONS

We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 12 / 30

SLIDE 47

QUESTIONS

We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes?

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 12 / 30

SLIDE 48

QUESTIONS

We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes? To what extent can populations endogenise the solution to a tragedy-of-the-commons problem?

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 12 / 30

SLIDE 49

QUESTIONS

We know that assortativity (of any kind) can theoretically help resolve the Tragedy of the Commons. (Jensen and Rigos, 2014, among others) The assortativity level is usually taken as given. What levels of assortativity would be the outcomes of natural processes? To what extent can populations endogenise the solution to a tragedy-of-the-commons problem? ⇒ Endogenise assortativity

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 12 / 30

SLIDE 50

MAIN IDEA

Study Social Dilemmas that differ in their

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 13 / 30

SLIDE 51

MAIN IDEA

Study Social Dilemmas that differ in their

strategic structure

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 13 / 30

SLIDE 52

MAIN IDEA

Study Social Dilemmas that differ in their

strategic structure efficiency structure

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 13 / 30

SLIDE 53

MAIN IDEA

Study Social Dilemmas that differ in their

strategic structure efficiency structure

Let the population decide upon their own level of assortativity based on a “voting” rule.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 13 / 30

SLIDE 54

MAIN IDEA

Study Social Dilemmas that differ in their

strategic structure efficiency structure

Let the population decide upon their own level of assortativity based on a “voting” rule. Think about it as a system with agents who interact, get results and give feedback to a central planner.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 13 / 30

SLIDE 55

SOCIAL DILEMMAS

C D C r, r a, 1 D 1, a 0,0 We keep r ∈ (0, 1) and a ∈ (−1, r) so that C always increases the payoff of the other player (“public goods” character) D is always a best response to C

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 14 / 30

SLIDE 56

SOCIAL DILEMMAS

SOCIAL DILEMMA CLASSIFICATION

C D C r, r a, 1 D 1, a 0,0 C D C 0.8, 0.8 − 0.2, 1 D 1, − 0.2 0,0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD SD UD

1. Prisoners’ Dilemma (PD): 2r > 1 + a and a < 0.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 15 / 30

SLIDE 57

SOCIAL DILEMMAS

SOCIAL DILEMMA CLASSIFICATION

C D C r, r a, 1 D 1, a 0,0 C D C 0.3, 0.3 − 0.2, 1 D 1, − 0.2 0,0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD SD UD

1. Prisoners’ Dilemma (PD): 2r > 1 + a and a < 0.
2. Missing Hero Dilemma (MHD): 2r < 1 + a and a < 0.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 15 / 30

SLIDE 58

SOCIAL DILEMMAS

SOCIAL DILEMMA CLASSIFICATION

C D C r, r a, 1 D 1, a 0,0 C D C 0.3, 0.3 0.2, 1 D 1, 0.2 0,0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD SD UD

1. Prisoners’ Dilemma (PD): 2r > 1 + a and a < 0.
2. Missing Hero Dilemma (MHD): 2r < 1 + a and a < 0.
3. SnowDrift Game (SD): 2r < 1 + a and a > 0.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 15 / 30

SLIDE 59

SOCIAL DILEMMAS

SOCIAL DILEMMA CLASSIFICATION

C D C r, r a, 1 D 1, a 0,0 C D C 0.8, 0.8 0.2, 1 D 1, 0.2 0,0

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD SD UD

1. Prisoners’ Dilemma (PD): 2r > 1 + a and a < 0.
2. Missing Hero Dilemma (MHD): 2r < 1 + a and a < 0.
3. SnowDrift Game (SD): 2r < 1 + a and a > 0.
4. Underprovision Dilemma (UD): 2r > 1 + a and a > 0.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 15 / 30

SLIDE 60

SOCIAL DILEMMAS

SOCIAL DILEMMA CLASSIFICATION

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a r a = 2r − 1 a = 0 a = r PD MHD SD UD

Efficient Outcome C-D (a > 2r − 1) C-C (a < 2r − 1) Best Reply C (a > 0) SD UD vs D D (a < 0) MHD PD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 16 / 30

SLIDE 61

MODEL

THE MODEL

Continuum of individuals in continuous time

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 62

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 63

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 64

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics ˙ x = x(1 − x)(πC − πD)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 65

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics ˙ x = x(1 − x)(πC − πD) πC (πD) is the average payoff of Cooperators (Defectors) in the population.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 66

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics ˙ x = x(1 − x)(πC − πD) πC (πD) is the average payoff of Cooperators (Defectors) in the population. We are looking for asymptotically stable states of the above dynamics.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 67

MODEL

THE MODEL

Continuum of individuals in continuous time x is the proportion of cooperators (C-individuals) At each time t they get matched in pairs (how? → next slide) More successful behaviour is imitated by more individuals ⇒ Replicator Dynamics ˙ x = x(1 − x)(πC − πD) πC (πD) is the average payoff of Cooperators (Defectors) in the population. We are looking for asymptotically stable states of the above dynamics. These states generically exist.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 17 / 30

SLIDE 68

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 69

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 70

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 71

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure) More generally: α = Pr(C meets C) − Pr(D meets C)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 72

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure) More generally: α = Pr(C meets C) − Pr(D meets C)

Proportions of pair types under assortative matching with assortativity α:

C-C: f1(x, α) = αx + (1 − α)x2 C-D: f2(x, α) = 2x(1 − x)(1 − α) D-D: f3(x, α) = α(1 − x) + (1 − α)(1 − x)2

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 73

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure) More generally: α = Pr(C meets C) − Pr(D meets C)

Proportions of pair types under assortative matching with assortativity α:

C-C: f1(x, α) = αx + (1 − α)x2 C-D: f2(x, α) = 2x(1 − x)(1 − α) D-D: f3(x, α) = α(1 − x) + (1 − α)(1 − x)2

πC(x, α) = (r · f1(x, α) + a · f2(x, α)/2)/x πD(x, α) = (0 · f3(x, α) + 1 · f2(x, α)/2)/(1 − x)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 74

MODEL

MATCHING

The matching process follows a rule with a constant index of assortativity α (∈ [0, 1]).

α = 0 → uniformly random matching (C’s and D’s have the same probability to meet C’s) α = 1 → full assortativity (C’s meet C’s and D’s meet D’s for sure) More generally: α = Pr(C meets C) − Pr(D meets C)

Proportions of pair types under assortative matching with assortativity α:

C-C: f1(x, α) = αx + (1 − α)x2 C-D: f2(x, α) = 2x(1 − x)(1 − α) D-D: f3(x, α) = α(1 − x) + (1 − α)(1 − x)2

πC(x, α) = (r · f1(x, α) + a · f2(x, α)/2)/x πD(x, α) = (0 · f3(x, α) + 1 · f2(x, α)/2)/(1 − x) In asymptotically stable states (ESS) of the dynamics: Higher assortativity ⇒ More Cooperators Higher efficiency (avg. payoff)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 18 / 30

SLIDE 75

MODEL

VOTING FOR ASSORTATIVITY

Now the population will get to (dynamically) select how assortativity itself evolves.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 19 / 30

SLIDE 76

MODEL

VOTING FOR ASSORTATIVITY

Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 19 / 30

SLIDE 77

MODEL

VOTING FOR ASSORTATIVITY

Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity. They compare their payoff to the highest and the lowest payoff they can receive in the Social Dilemma.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 19 / 30

SLIDE 78

MODEL

VOTING FOR ASSORTATIVITY

Now the population will get to (dynamically) select how assortativity itself evolves. Each individual gets one vote and can vote either for an increase or a decrease of assortativity. They compare their payoff to the highest and the lowest payoff they can receive in the Social Dilemma. They vote using a logit choice rule based on the payoffs they receive.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 19 / 30

SLIDE 79

MODEL

VOTING (FEEDBACK)

Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 20 / 30

SLIDE 80

MODEL

VOTING (FEEDBACK)

Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff. Heterogeneous pairs (C-D): probability to vote for an increase in assortativity decreasing in payoff.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 20 / 30

SLIDE 81

MODEL

VOTING (FEEDBACK)

Agents matched in Homogeneous pairs (either C-C or D-D): probability to vote for an increase in assortativity increasing in payoff. Heterogeneous pairs (C-D): probability to vote for an increase in assortativity decreasing in payoff. Example: A Defector gets matched to a Cooperator and receives the highest payoff in the game. Then, he/she votes for a decrease in assortativity for sure.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 20 / 30

SLIDE 82

MODEL

COUNTING THE VOTES

v+: Number of votes for more assortativity v−: Number of votes for less assortativity

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 21 / 30

SLIDE 83

MODEL

COUNTING THE VOTES

v+: Number of votes for more assortativity v−: Number of votes for less assortativity Assortativity follows a replicator-like dynamic based on the number of votes received: ˙ α = α(1 − α)(v+ − v−)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 21 / 30

SLIDE 84

MODEL

CO-EVOLUTION

˙ x = x(1 − x)(πC − πD) (1) ˙ α = α(1 − α)(v+ − v−) (2)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 21 / 30

SLIDE 85

RESULTS

FULL EQUILIBRIA

DEFINITION (FULL EQUILIBRIUM)

For a Social Dilemma G, a full equilibrium is a pair (x∗, α∗) that is an asymptotically stable solution of the system (1) – (2). In Social Dilemmas full equilibria always exist.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 22 / 30

SLIDE 86

RESULTS

FULL EQUILIBRIA

DEFINITION (FULL EQUILIBRIUM)

For a Social Dilemma G, a full equilibrium is a pair (x∗, α∗) that is an asymptotically stable solution of the system (1) – (2). In Social Dilemmas full equilibria always exist.

OBSERVATION

For any Social Dilemma G, all full equilibria have either α∗ = 0 (x0) or α∗ = 1 (x1).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 22 / 30

SLIDE 87

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 88

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α MHD (r=0.20, a=-0.40)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 89

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α SD (r=0.40, a=0.10)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 90

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α SD (r=0.55, a=0.40)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 91

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α SD (r=0.80, a=0.70)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 92

RESULTS

DYNAMICS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α UD (r=0.80, a=0.40)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 23 / 30

SLIDE 93

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

We want to measure how stable are the possible outcomes (full equilibria).

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 24 / 30

SLIDE 94

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

We want to measure how stable are the possible outcomes (full equilibria). Efficiency is increasing in equilibrium assortativity ⇒ Robustness of the equilibrium with α = 1 is a measure of efficiency.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 24 / 30

SLIDE 95

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

We want to measure how stable are the possible outcomes (full equilibria). Efficiency is increasing in equilibrium assortativity ⇒ Robustness of the equilibrium with α = 1 is a measure of efficiency. We use an invasion barrier approach.

FULL ASSORTATIVITY ROBUSTNESS

“What is the largest invasion a population at x1 can sustain?”

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 24 / 30

SLIDE 96

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 25 / 30

SLIDE 97

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 25 / 30

SLIDE 98

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 25 / 30

SLIDE 99

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 25 / 30

SLIDE 100

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – EXAMPLE

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x α PD (r=0.30, a=-0.60) ̺G

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 25 / 30

SLIDE 101

RESULTS

FULL ASSORTATIVITY ROBUSTNESS – RESULTS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 1 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 26 / 30

SLIDE 102

RESULTS

DIFFERENT SPEEDS

What if behaviour (strategy) evolved at a different pace than assortativity?

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 27 / 30

SLIDE 103

RESULTS

DIFFERENT SPEEDS

What if behaviour (strategy) evolved at a different pace than assortativity? We introduce a parameter s that shows this relative speed:

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 27 / 30

SLIDE 104

RESULTS

DIFFERENT SPEEDS

What if behaviour (strategy) evolved at a different pace than assortativity? We introduce a parameter s that shows this relative speed: ˙ x = x(1 − x)(πC − πD) ˙ α = s · α(1 − α)(v+ − v−)

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 27 / 30

SLIDE 105

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 0.01 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 28 / 30

SLIDE 106

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 0.1 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 28 / 30

SLIDE 107

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 1 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 28 / 30

SLIDE 108

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 10 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 28 / 30

SLIDE 109

RESULTS

FULL ASSORTATIVITY ROBUSTNESS

−1 −0.5 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Payoff to a Cooperator facing a Defector (a) Payoff to a Cooperator facing a Cooperator (r) s = 100 0.2 0.4 0.6 0.8 1 Full assortativity robustness (̺) PD MHD SD UD

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 28 / 30

SLIDE 110

CONCLUSION

We endogenised the level of assortativity for a matching process in democratic regimes.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 29 / 30

SLIDE 111

CONCLUSION

We endogenised the level of assortativity for a matching process in democratic regimes. For our class of social dilemmas only full or null assortativity can be stable.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 29 / 30

SLIDE 112

CONCLUSION

We endogenised the level of assortativity for a matching process in democratic regimes. For our class of social dilemmas only full or null assortativity can be stable. The endogenisation of assortativity can help mitigate tragedies of the commons

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 29 / 30

SLIDE 113

CONCLUSION

We endogenised the level of assortativity for a matching process in democratic regimes. For our class of social dilemmas only full or null assortativity can be stable. The endogenisation of assortativity can help mitigate tragedies of the commons The situations it helps the most are UD and some SD. Partial help for

PD. No help for MHD.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 29 / 30

SLIDE 114

CONCLUSION

We endogenised the level of assortativity for a matching process in democratic regimes. For our class of social dilemmas only full or null assortativity can be stable. The endogenisation of assortativity can help mitigate tragedies of the commons The situations it helps the most are UD and some SD. Partial help for

PD. No help for MHD.

More often voting (feedback) helps mostly PDs with a < r − 1.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 29 / 30

SLIDE 115

FUTURE WORK

FUTURE

Theory: Find other ways to endogenise assortativity.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 30 / 30

SLIDE 116

FUTURE WORK

FUTURE

Theory: Find other ways to endogenise assortativity.

e.g. networks, matching/search models.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 30 / 30

SLIDE 117

FUTURE WORK

FUTURE

Theory: Find other ways to endogenise assortativity.

e.g. networks, matching/search models.

Experiments: Test predictions of our endogenous assortativity model.

HH NAX AND A RIGOS ASSORTATIVITY IN SOCIAL DILEMMAS 30 / 30