Games on Graphs
description
Transcript of Games on Graphs
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Games on Graphs
Rob Axtell
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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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Coordination in Transient Social Networks:A Model of the Timing of Retirement
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
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The DataThe Data
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The DataThe Data
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The DataThe Data
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Coordination Gamein Social Networks
Coordination Gamein Social Networks
A agents, each has a social network, Ni
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Coordination Gamein Social Networks
Coordination Gamein Social Networks
A agents, each has a social network, Ni
x {working, retired}A is the state of the society
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Coordination Gamein Social Networks
Coordination Gamein Social Networks
Ui x( ) = u xi ,xj( )j∈Ni
∑
A agents, each has a social network, Ni
x {working, retired}A is the state of the society
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work retire
work w , w 0, 0
retire 0, 0 r, r
Coordination Gamein Social Networks
Coordination Gamein Social Networks
Ui x( ) = u xi ,xj( )j∈Ni
∑
A agents, each has a social network, Ni
x {working, retired}A is the state of the society
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work retire
work w , w 0, 0
retire 0, 0 r, r
Coordination Gamein Social Networks
Coordination Gamein Social Networks
Ui x( ) = u xi ,xj( )j∈Ni
∑
A agents, each has a social network, Ni
x {working, retired}A is the state of the society
τ =w
w + r
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Base Case ParameterizationBase Case Parameterization
Parameter Value
Agents/cohort, C 100
rational agents 10%
imitative agents 85%
imitation threshold, τ 0.50
, social network size S [10, 25]U
, network age extent E [-5, 5]U
random agents 5%
p 0.50
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Typical Time Series:Rapid Establishment of Age 65 Norm
Typical Time Series:Rapid Establishment of Age 65 Norm
5 10 15 20 Time
0.2
0.4
0.6
0.8
1FractionRetired
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Typical Time Series:Nonmonotonic Path to Age 65 Norm
Typical Time Series:Nonmonotonic Path to Age 65 Norm
100 200 300 400 500Time
0.2
0.4
0.6
0.8
1FractionRetired
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Establishment of Age 65 RetirementNorm as a Function of Population Types
Establishment of Age 65 RetirementNorm as a Function of Population Types
0 5 10 15 20 25% Rational
10
20
50
100
200
500
1000
2000
Transition Time
0% Random5%
10%
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Establishment of Age 65 RetirementNorm as a Function of τ
Establishment of Age 65 RetirementNorm as a Function of τ
0.05 0.1 0.15 0.2 0.25ThresholdStd. Dev.
20
40
60
80
100TransitionTime
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Establishment of Age 65 RetirementNorm as a Function of Network Size
Establishment of Age 65 RetirementNorm as a Function of Network Size
5 10 15 20 25 30 35 40MeanNetworkSize
1020
50
100
200
5001000
2000
TransitionTime
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Establishment of Age 65 Retirement Normas a Function of Variance in Network Size
Establishment of Age 65 Retirement Normas a Function of Variance in Network Size
2 4 6 8 10NetworkSizeStd. Dev.
10
15
20
30
TransitionTime
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10 20 30 40 50 60 70 80Max. NetworkSize
10
100
1000
TransitionTime
Establishment of Age 65 Retirement Normas a Function of S, |N| ~ U[10, S]
Establishment of Age 65 Retirement Normas a Function of S, |N| ~ U[10, S]
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2 4 6 8 10Extent
10
152030
5070100150200
300Transition Time
5% Rational10% Rational
Establishment of Age 65 Retirement Normas a Function of the Extent of Social Networks
Establishment of Age 65 Retirement Normas a Function of the Extent of Social Networks
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0 2 4 6 8 10 12 14% Rational
10
1520
30
5070100
150200
Transition Time
Establishment of Age 62 Retirement Normas a Function of the Extent of Social Networks
Establishment of Age 62 Retirement Normas a Function of the Extent of Social Networks
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0.05 0.1 0.15 0.2 0.25Coupling
20
50
100
200
500
1000Transition Time
Sub- population withrational agents
Sub- population withoutrational agents
Establishment of Age 65 Retirement Normas a Function of the Coupling Between Groups
Establishment of Age 65 Retirement Normas a Function of the Coupling Between Groups
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Effect of Interaction TopologyEffect of Interaction Topology
• Random graphs
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Effect of Interaction TopologyEffect of Interaction Topology
• Random graphs
• Regular graphs (e.g., lattices)
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Effect of Interaction TopologyEffect of Interaction Topology
• Random graphs
• Regular graphs (e.g., lattices)
• ‘Small-world’ graphs
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New ParameterizationNew Parameterization
Parameter Value
Agents/cohort, C 100
rational agents 10%
imitative agents 85%
imitation threshold, τ 0.50
, social network size S 24 , network age extent E [-5, 5]U
random agents 5%
p 0.50
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0 5 10 15 20% Rational
10
20
50
100
200
500
1000
2000
Transition Time
Random graph
Small world
p = 10% p = 25%
Lattice
Comparison of Random Graph, Latticeand Small World Social Networks
(Network size = 24)
Comparison of Random Graph, Latticeand Small World Social Networks
(Network size = 24)
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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, τ: fraction (f) of an
agent’s social network who must smoke before an agent adopts smoking
– Behavioral rule: If f > τ then smoke, else don’t (or quit)
– Threshold of 1 means non-smoker, 0 first adopter
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Agent Behavior• Agents update their behavior periodically
• Smoking reduces threshold:– Decreases with amount smoked– Decreases with intensity of smoking
τ
amount of smoking
τ0
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Visualization
Cohorts
Threshold 1 agent (never smokes)
Intermediate threshold agent
Threshold 0 agent (always smokes)
Non social network agent
Smoker
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< Run Model>
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Typical Output: Smoking Time Series
Lesson: Significant temporal variations in aggregate data;non-equilibrium, non-monotonic
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Estimating the Peer Effects
Real world
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Estimating the Peer Effects
Real worldStandard specification
Extent of peer effects
Estimation ofmis-specifiedmodel
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Estimating the Peer Effects
Real worldStandard specification
Extent of peer effects
Estimation ofmis-specifiedmodel
Agent-Based Model
Estimation ofmis-specified modelwith ‘synthetic’ data
Estimation ofagent model
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Conventional (Mis-)Specification
€
yi =+1−1
⎧ ⎨ ⎩
€
yi* =xib + ρijy j +εi
j≠i∑
€
yi*
( )t+1
=xitb + ρijy j
t +εit
j≠i∑
€
θ ≡ b, ρ( )) θ =aρgmax
θlθ( )
€
lθ( ) = Q y j x j ,θ( )−log εxp Q η j x j ,θ( )∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥j=1
N
∑
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Typical Results
• Nakajima (2003)– 2000 National Youth Tobacco Survey (NYTS)
• 35K students
• Grades 6-12
• 324 high schools
– Peer effects estimated:• ρff = 0.89
• ρfm = ρmf = 0.48
• ρmm = 0.94
• Krauth…
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Crack, Gangs, Guns and Homicide:A Computational Agent Model
George TitaUC Irvine
Rob AxtellBrookings
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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
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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
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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)
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Payoffs to Selling Drugs
No guns GunsNo guns 3, 3 0, 4-G
Guns 4-G, 0 1, 1
where G is the price of buying+owning+using a gun
If G is large, this is the assurance (stag hunt) gameIf G is small, this is prisoner’s dilemma
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Pre-Crack Era
No gun GunNo gun 3, 3 0, 2
Gun 2, 0 1, 1
Payoffs low (relatively), price of guns (relatively) high
Two Nash equilibria in the assurance game, much like acoordination game; ‘no gun’ equilibrium is Pareto efficient
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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
No gun GunNo gun 3, 3 0, 4
Gun 4, 0 1, 1
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Economic Emergence of Gangs
‘Gun toting’ is dominant strategy for a gang of size N > G
No guns GunsNo guns 3N, 3N 0, 4N-G
Guns 4N-G , 0 N, N
Widespread ‘gun toting’ leads to drug-related homicide
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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
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Typical Model Output
Annual drug-related homicides
Year
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Summary
• Simple model:– Adaptive agents– Social networks
• Preliminary results:– Multiple regimes, sensitive to network structure– Qualitative plausibility
• Much future work to do– Comments welcome