Segregation and Neighborhood Interaction Work in progress Jason Barr, Rutgers Newark Troy Tassier, Fordham October 31, 2006

Motivation  Schelling tipping model quite “pessimistic.”  But says nothing about neighborhood interaction.  Only says: given preferences, you move if # different than you hits a certain threshold.  What role does neighborhood interaction play in “counteracting” the tipping phenomena?

Schelling Model w. Utility  NxN lattice (we fix at 12x12).  130 agents; 10 % free slots; randomly placed at t=0.  Moore neighborhood – 8 surrounding neighbors (except on edges).  Calculate % of neighbors who are alike.  Calculate utility.  Choose randomly open location and calculate new utility.  If new utility>old utility move, otherwise stay.  Asynchronous movement: begin with agent 1 and go through each agent.  Run the system 10,000 periods (each agent has option to move about 79 times).  Segregation Measure: Avg. of % of agents with like neighbors.

Utility Functions is our measure of preference for integration % Same α β γ =1 γ =0 γ =.5 01 Utility Functions.5

Some Examples t=0 Seg.=.5 t=10,000 Seg.=.88

t=0 Seg.=.5 t=10,000 Seg.=. 7

Experiment 1: Segregation and Utility

Prisoner’s Dilemma Rival Coop.Defect AgentCoop. A (agree) B (bested) Defect C (cheat) D (defect) where C>A>D>B Fix A and B: C=A+ε, D=B+μ

The Prisoner’s Dilemma on the Lattice  Agents play the PD against all their neighbors (in the Moore neighborhood).  Payoff to agent is average payoff from the play with neighbors.  Each agent has a probability of playing Coop. (p i ), and Defect (1-p i ).

PD cont.  Notice every agent is also a neighbor to one or more other agents.  Strategy Approach I: –When agent plays with neighbors, randomly choose action (according to p i ) for each neighbor interaction. –Neighbors also choose action according to p j ’s.  Strategy Approach II: –Each round, every agent chooses an action (according to their probability). –Plays same action for entire round (both as agent and rival).

Probability Updating Rule Agent’s probability is based on how well the neighbors do. Notice that only the neighbors’ performance matters.

Performance and Probability Denote: x j =1 if rival Cooperates, 0 otherwise. y=1 if agent Cooperates, 0 otherwise.

Example

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

Experiment 2 PD on the Lattice: No Movement Coop.Def. Coop.A=2B=1.99 Def. C=2+ε D= Run the system for 100,000 iterations or hit absorbing state, which ever comes first. --Take averages of 250 runs.

Experiment 2 Results

Experiment 3: “Conscious Movement”  Play game with neighbors.  Receive average payoff.  Pick random open location.  “Play” game with neighbors there.  If agent's payoff is higher, move to new neighborhood.  Update probabilities based on chosen location (i.e., new or old).

Experiment 3: “Conscious Movement”

Movement Increases Probability of Cooperation  Agent’s probability adjustment is determined by neighbors’ actions.  If new location gives higher payoff it means that there is more cooperation by the new neighbors.  This, therefore, will increase agent’s probability of cooperating in the future rounds.

Movement vs. No Movement Movement No Movement Fixed Action Random Action C payoffs that gives 50% prob. of all cooperate

Combined Schelling Plus Game  What does the interaction of the PD and Schelling games do to cooperation and segregation?  Two experiments: –Move if, stay otherwise –Find randomly chosen open spot, compare utilities and move if new utility > old utility

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

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

Move if Utility<0  Interaction between nd. composition and game outcome.  If neighbors all defect, you move, regardless of types.  If neighbors all cooperate, agent stays regardless of types.  Movement depends on intermediate game results & % like you.  Increasing cheating payoff: – increases likelihood that an agent will stay, since agent “earns” more against rivals who cooperate. –But as “city” moves toward everybody defect then movement will increase.

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

Cooperation in RPD vs. Combined Game Random Action

Compare Utilities and Move if New Payoff is Larger

Remaining Issues  Random vs. Fixed action  Initial Conditions and probabilities dynamics.  Conditions where increase coop. decreases segregation?