Games on Graphs Rob Axtell

Examples Abstract graphs: Coordination in fixed social nets (w/ J Epstein) Empirical graphs: Peer effects in fixed social networks w/addiction Dynamic graphs: Crime waves in endogenously changing networks (w/ George Tita)

Coordination in Transient Social Networks: A Model of the Timing of Retirement Joint work with J. Epstein In Behavioral Dimensions of Retirement Economics, H. Aaron, editor, Brookings Institution Press and Russell Sage Foundation

The Data

Coordination Game in Social Networks A agents, each has a social network, N i

Coordination Game in Social Networks A agents, each has a social network, N i x  {working, retired} A is the state of the society

Coordination Game in Social Networks A agents, each has a social network, N i x  {working, retired} A is the state of the society

Coordination Game in Social Networks A agents, each has a social network, N i x  {working, retired} A is the state of the society

Coordination Game in Social Networks A agents, each has a social network, N i x  {working, retired} A is the state of the society

Base Case Parameterization

Typical Time Series: Rapid Establishment of Age 65 Norm

Typical Time Series: Nonmonotonic Path to Age 65 Norm

Establishment of Age 65 Retirement Norm as a Function of Population Types

Establishment of Age 65 Retirement Norm as a Function of  

Establishment of Age 65 Retirement Norm as a Function of Network Size

Establishment of Age 65 Retirement Norm as a Function of Variance in Network Size

Establishment of Age 65 Retirement Norm as a Function of S, |N| ~ U[10, S]

Establishment of Age 65 Retirement Norm as a Function of the Extent of Social Networks

Establishment of Age 62 Retirement Norm as a Function of the Extent of Social Networks

Establishment of Age 65 Retirement Norm as a Function of the Coupling Between Groups

Effect of Interaction Topology Random graphs

Effect of Interaction Topology Random graphs Regular graphs (e.g., lattices)

Effect of Interaction Topology Random graphs Regular graphs (e.g., lattices) ‘Small-world’ graphs

New Parameterization

Comparison of Random Graph, Lattice and Small World Social Networks (Network size = 24)

An Empirical Agent Model of Smoking with Peer Effects Population of Agents –Arranged in classrooms –Each agent has a social network Agents are Heterogeneous –Distribution of initial thresholds,  : fraction (f) of an agent’s social network who must smoke before an agent adopts smoking –Behavioral rule: If f >  then smoke, else don’t (or quit) –Threshold of 1 means non-smoker, 0 first adopter

Agent Behavior Agents update their behavior periodically Smoking reduces threshold: –Decreases with amount smoked –Decreases with intensity of smoking  amount of smoking 

Visualization Cohorts  Threshold 1 agent (never smokes) Intermediate threshold agent Threshold 0 agent (always smokes) Non social network agent Smoker

Typical Output: Smoking Time Series Lesson: Significant temporal variations in aggregate data; non-equilibrium, non-monotonic

Estimating the Peer Effects Real world

Estimating the Peer Effects Real world Standard specification Extent of peer effects Estimation of mis-specified model

Estimating the Peer Effects Real world Standard specification Extent of peer effects Estimation of mis-specified model Agent-Based Model Estimation of mis-specified model with ‘synthetic’ data Estimation of agent model

Conventional (Mis-)Specification

Typical Results Nakajima (2003) –2000 National Youth Tobacco Survey (NYTS) 35K students Grades high schools –Peer effects estimated:  ff = 0.89  fm =  mf = 0.48  mm = 0.94 Krauth…

Crack, Gangs, Guns and Homicide: A Computational Agent Model George Tita UC Irvine Rob Axtell Brookings

Drug-Related Homicide in Largest 237 U.S. Cities, mid 1980s to Present (Blumstein, Cork, Cohen and Tita) Innovation in narcotics: crack cocaine Emergence of gangs Adoption of guns Rise of gun violence and homicide Diffusion of non-drug gun homicide

An Agent Model The problem domain well-suited to agent modeling because: –Heterogeneous actors –Social interactions –Purposive but not hyper-rational behavior –Non-equilibrium dynamics Preliminary results to be shown

Basic Features of Model Payoffs depend on context (to be described) Population of drug sellers who interact with one another through social networks (random graph, lattice and small world) Agents heterogeneous wrt age, network Agents removed by incarceration (fixed rate), becoming too old (age 40), or death (proportional to amount of gun toting)

Payoffs to Selling Drugs where G is the price of buying+owning+using a gun If G is large, this is the assurance (stag hunt) game If G is small, this is prisoner’s dilemma

Pre-Crack Era Payoffs low (relatively), price of guns (relatively) high Two Nash equilibria in the assurance game, much like a coordination game; ‘no gun’ equilibrium is Pareto efficient

Crack Era Payoffs high (relatively), price of guns (relatively) low ‘Gun toting’ is dominant strategy in prisoner’s dilemma, although ‘no gun’ outcome Pareto dominates the Nash outcome

Economic Emergence of Gangs ‘Gun toting’ is dominant strategy for a gang of size N > G Widespread ‘gun toting’ leads to drug-related homicide

Drug-Related Homicide Goal: explain peaks and troughs in drug homicide rates (e.g., Watts: /100K) Postulate homicide rate proportional to rate of gun ownership Homicide is one more way an agent can be removed from the population (in addition to being incarcerated and becoming too old) This can lead to oscillatory homicide rate dynamics

Typical Model Output Annual drug-related homicides Year

Summary Simple model: –Adaptive agents –Social networks Preliminary results: –Multiple regimes, sensitive to network structure –Qualitative plausibility Much future work to do –Comments welcome