Segregation and Neighborhood Segregation and Neighborhood InteractionInteraction

Work in progressWork in progress

Jason Barr, Rutgers NewarkJason Barr, Rutgers Newark

Troy Tassier, FordhamTroy Tassier, Fordham

October 31, 2006October 31, 2006

MotivationMotivation

Schelling tipping model quite “pessimistic.”Schelling tipping model quite “pessimistic.” But says nothing about neighborhood But says nothing about neighborhood

interaction.interaction. Only says: given preferences, you move if # Only says: given preferences, you move if #

different than you hits a certain threshold.different than you hits a certain threshold. What role does neighborhood interaction What role does neighborhood interaction

play in “counteracting” the tipping play in “counteracting” the tipping phenomena?phenomena?

Schelling Model w. UtilitySchelling Model w. Utility NxNNxN lattice (we fix at lattice (we fix at 12x1212x12).). 130 agents; 10 % free slots; randomly placed at 130 agents; 10 % free slots; randomly placed at t=0t=0.. Moore neighborhood – 8 surrounding neighbors (except on edges).Moore neighborhood – 8 surrounding neighbors (except on edges). Calculate % of neighbors who are alike.Calculate % of neighbors who are alike. Calculate utility.Calculate utility. Choose randomly open location and calculate new utility.Choose randomly open location and calculate new utility. If If new utility>old utilitynew utility>old utility move, otherwise stay. move, otherwise stay. Asynchronous movement: begin with agent 1 and go through each Asynchronous movement: begin with agent 1 and go through each

agent.agent. Run the system 10,000 periods (each agent has option to move about Run the system 10,000 periods (each agent has option to move about

79 times).79 times). Segregation Measure: Avg. of % of agents with like neighbors.Segregation Measure: Avg. of % of agents with like neighbors.

1,0,

11

1 1

ji

N

i

n

jji

i

xx

xxnN

Segi

Utility FunctionsUtility Functions

5.,2

5.,2

xx

xxU(x)

is our measure of preference for integration

% Same

α

β

γ=1

γ=0

γ=.5

0 1

Utility Functions

1,0

.5

Some ExamplesSome Examples

0 2 4 6 8 10 12

0

2

4

6

8

10

12

0 2 4 6 8 10 12

0

2

4

6

8

10

12

t=0

Seg.=.5

t=10,000

Seg.=.88

0

0 2 4 6 8 10 12

0

2

4

6

8

10

12

5.0

0 2 4 6 8 10 12

0

2

4

6

8

10

12

t=0

Seg.=.5t=10,000

Seg.=. 7

Experiment 1: Segregation and UtilityExperiment 1: Segregation and Utility

88% 85%80%

75%68%

62% 63%56%

49%

74%

53%

0%

20%

40%

60%

80%

100%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Steepness of Utility Function

Segregation vs. Utility Function Type

Avg. % Sam

e in each nd.

Full Integration

Prisoner’s DilemmaPrisoner’s Dilemma

RivalRival

Coop.Coop. DefectDefect

AgentAgent

Coop.Coop. AA (agree) (agree) BB (bested) (bested)

DefectDefect CC (cheat) (cheat) DD (defect) (defect)

where C>A>D>B

Fix A and B: C=A+ε, D=B+μ

The Prisoner’s Dilemma on the LatticeThe Prisoner’s Dilemma on the Lattice

Agents play the PD against all their Agents play the PD against all their neighbors (in the Moore neighborhood).neighbors (in the Moore neighborhood).

Payoff to agent is average payoff from the Payoff to agent is average payoff from the play with neighbors.play with neighbors.

Each agent has a probability of playing Each agent has a probability of playing Coop. (Coop. (ppii), and Defect (), and Defect (1-p1-pii). ).

PD cont.PD cont.

Notice every agent is also a neighbor to one or Notice every agent is also a neighbor to one or more other agents.more other agents.

Strategy Approach I:Strategy Approach I:– When agent plays with neighbors, randomly choose When agent plays with neighbors, randomly choose

action (according to action (according to ppii) for each neighbor interaction.) for each neighbor interaction.– Neighbors also choose action according to Neighbors also choose action according to ppjj’s.’s.

Strategy Approach II:Strategy Approach II:– Each round, every agent chooses an action (according Each round, every agent chooses an action (according

to their probability).to their probability).– Plays same action for entire round (both as agent and Plays same action for entire round (both as agent and

rival).rival).

Probability Updating RuleProbability Updating Rule

p t

a a x

a a

a a Coop Def x

i

t j i jj

n

t j ij

n

i j j

1

1

1

1

,

,,

/ , ,where if Coop., 0 otherwise.

Agent’s probability is based on how well the neighbors do.

Notice that only the neighbors’ performance matters.

Performance and ProbabilityPerformance and Probability

j j j j j j

i

x y Ax y Bx y C x y D x y

pA y B y

Ay B y y y

,

;

1 1 1 1

1

1 1 1

where =1

nx

ij

j=1

ni

Denote:

xj=1 if rival Cooperates, 0 otherwise.

y=1 if agent Cooperates, 0 otherwise.

ExampleExample

p

E p tA B

A B

i t

i

, .

|..

5

1 51

2 5

0.4970

0.4975

0.4980

0.4985

0.4990

0.4995

0.5000

0 10 20 30 40 50

ε+μ=0.1

ε+μ=0.01

ε+μ=0.001

A+B

EquilibriaEquilibria

pA y B y

Ay B y y y

p yA

A

p yB

B

A

A

B

B

A B

i

i

i

1

1 1 1

11

01

1 1

1 1 0

|

|

Assuming A/B ≠ ε/μ then all play cooperate or all play defect are the only Nash Equilibria.

Experiment 2Experiment 2 PD on the Lattice: No Movement PD on the Lattice: No Movement

Coop.Coop. Def.Def.

Coop.Coop. A=2A=2 B=1.99B=1.99

Def.Def. C=2+C=2+εε D=1.991D=1.991

--Run the system for 100,000 iterations or hit absorbing state, which ever comes first.

--Take averages of 250 runs.

Experiment 2 ResultsExperiment 2 Results

0.00

0.25

0.50

0.75

1.00

2.0000 2.0025 2.0050 2.0075 2.0100 2.0125 2.0150 2.0175 2.0200 2.0225 2.0250 2.0275 2.0300 2.0325 2.0350

Random Action

Fixed Action

% D

efection

% Defection versus Cheating Payoff

Experiment 3:Experiment 3:“Conscious Movement”“Conscious Movement”

Play game with neighbors.Play game with neighbors. Receive average payoff.Receive average payoff. Pick random open location.Pick random open location. ““Play” game with neighbors there.Play” game with neighbors there. If agent's payoff is higher, move to new If agent's payoff is higher, move to new

neighborhood.neighborhood. Update probabilities based on chosen Update probabilities based on chosen

location (i.e., new or old).location (i.e., new or old).

Experiment 3:Experiment 3:“Conscious Movement”“Conscious Movement”

0.00

0.25

0.50

0.75

1.00

2 4 6 8 10 12 14

Fixed Action

Random Action

Cheating Payoff

% Defection vs. Cheating Payoff

Movement Increases Probability of Movement Increases Probability of CooperationCooperation

Agent’s probability adjustment is determined Agent’s probability adjustment is determined by neighbors’ actions.by neighbors’ actions.

If new location gives higher payoff it means If new location gives higher payoff it means that there is more cooperation by the new that there is more cooperation by the new neighbors.neighbors.

This, therefore, will increase agent’s This, therefore, will increase agent’s probability of cooperating in the future probability of cooperating in the future rounds.rounds.

Movement vs. No MovementMovement vs. No Movement

MovementMovement No MovementNo Movement

Fixed ActionFixed Action 2.5252.525 2.0252.025

Random ActionRandom Action 5.05.0 2.012.01

C payoffs that gives 50% prob. of all cooperate

Combined Schelling Plus GameCombined Schelling Plus Game

What does the interaction of the PD and Schelling What does the interaction of the PD and Schelling games do to cooperation and segregation?games do to cooperation and segregation?

Two experiments:Two experiments:– Move if , stay otherwiseMove if , stay otherwise– Find randomly chosen open spot, compare utilities and Find randomly chosen open spot, compare utilities and

move if new utility > old utilitymove if new utility > old utility

in

jji

i

i

i

aan

likeUUtility

1

,1

,%

where

0 iU

Utility Functions: Move if Utility<0Utility Functions: Move if Utility<0

“Low” Cheat payoff“Low” Cheat payoff

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0% 13% 25% 38% 50% 63% 75% 88% 100%

Combined Utility vs. % like for different PD outcomes , C=2.0025

D vs. all C

C vs. all D D vs. all D

D vs. 4 D C vs. 4 D

Utility Functions: Move if Utility<0Utility Functions: Move if Utility<0“High” Cheat payoff“High” Cheat payoff

-0.020

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0% 13% 25% 38% 50% 63% 75% 88% 100%

Combined Utility vs. % like for different PD outcomes, C=2.05

D vs. all D

D vs. 4 D

D vs. all C

Move if Utility<0Move if Utility<0 Interaction between nd. composition and game Interaction between nd. composition and game

outcome.outcome. If neighbors all defect, you move, regardless of types.If neighbors all defect, you move, regardless of types. If neighbors all cooperate, agent stays regardless of If neighbors all cooperate, agent stays regardless of

types.types. Movement depends on intermediate game results & Movement depends on intermediate game results &

% like you.% like you. Increasing cheating payoff:Increasing cheating payoff:

– increases likelihood that an agent will stay, since agent increases likelihood that an agent will stay, since agent “earns” more against rivals who cooperate.“earns” more against rivals who cooperate.

– But as “city” moves toward everybody defect then But as “city” moves toward everybody defect then movement will increase.movement will increase.

Experiment Results: Move if Utility<0Experiment Results: Move if Utility<0Coop. vs. Cheat payoffCoop. vs. Cheat payoff

0%

25%

50%

75%

100%

2 2.01 2.02 2.03

Move if Utility<0, Random Action, Flat Utility

Segregation

% Defection

C

Cooperation in RPD vs. Combined GameCooperation in RPD vs. Combined Game

0%

25%

50%

75%

100%

2.00 2.01 2.02 2.03

Game Only

Combined

Degree of Cooperation in RPD (no movement or random movement) and Combined Game

C

Random Action

Compare Utilities and Move if New Compare Utilities and Move if New Payoff is LargerPayoff is Larger

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

2 2.5 3 3.5 4 4.5 5C

Random Action, "Flat" Utility

Segregation

Defection

Remaining IssuesRemaining Issues

Random vs. Fixed actionRandom vs. Fixed action Initial Conditions and probabilities dynamics.Initial Conditions and probabilities dynamics. Conditions where increase coop. decreases Conditions where increase coop. decreases

segregation?segregation?