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Games on Graphs

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).

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Games on Graphs

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  1. Games on Graphs Rob Axtell

  2. 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)

  3. 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

  4. The Data

  5. The Data

  6. The Data

  7. Coordination Gamein Social Networks A agents, each has a social network, Ni

  8. Coordination Gamein Social Networks A agents, each has a social network, Ni x {working, retired}A is the state of the society

  9. Coordination Gamein Social Networks A agents, each has a social network, Ni x {working, retired}A is the state of the society

  10. Coordination Gamein Social Networks A agents, each has a social network, Ni x {working, retired}A is the state of the society

  11. Coordination Gamein Social Networks A agents, each has a social network, Ni x {working, retired}A is the state of the society

  12. Base Case Parameterization

  13. Typical Time Series:Rapid Establishment of Age 65 Norm

  14. Typical Time Series:Nonmonotonic Path to Age 65 Norm

  15. Establishment of Age 65 RetirementNorm as a Function of Population Types

  16. Establishment of Age 65 RetirementNorm as a Function of 

  17. Establishment of Age 65 RetirementNorm as a Function of Network Size

  18. Establishment of Age 65 Retirement Normas a Function of Variance in Network Size

  19. Establishment of Age 65 Retirement Normas a Function of S, |N| ~ U[10, S]

  20. Establishment of Age 65 Retirement Normas a Function of the Extent of Social Networks

  21. Establishment of Age 62 Retirement Normas a Function of the Extent of Social Networks

  22. Establishment of Age 65 Retirement Normas a Function of the Coupling Between Groups

  23. Effect of Interaction Topology • Random graphs

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

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

  26. New Parameterization

  27. Comparison of Random Graph, Latticeand Small World Social Networks(Network size = 24)

  28. An Empirical Agent Model of Smoking with Peer Effects • Population of Agents • Arranged in classrooms • Each agent has a socialnetwork • 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

  29. Agent Behavior • Agents update their behavior periodically • Smoking reduces threshold: • Decreases with amount smoked • Decreases with intensity of smoking t t0 amount of smoking

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

  31. < Run Model>

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

  33. Estimating the Peer Effects Real world

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

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

  36. Conventional (Mis-)Specification

  37. Typical Results • Nakajima (2003) • 2000 National Youth Tobacco Survey (NYTS) • 35K students • Grades 6-12 • 324 high schools • Peer effects estimated: • rff = 0.89 • rfm = rmf = 0.48 • rmm = 0.94 • Krauth…

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

  39. 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

  40. 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

  41. 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)

  42. 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

  43. 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

  44. 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

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

  46. Drug-Related Homicide • Goal: explain peaks and troughs in drug homicide rates (e.g., Watts: 30<->120/100K) • 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

  47. Typical Model Output Annual drug-related homicides Year

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

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