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Craigiebuckler, Aberdeen, AB15 8QH, UK
APPEARANCES CAN BE DECEIVING:Lessons Learned Re-implementing Axelrod’s “Evolutionary Approach to Norms”
Luis R. Izquierdo1 & José M. Galán2
1 The Macaulay Institute, Aberdeen, UK2 University of Burgos, Spain & INSISOC Group
PRESENTATION OUTLINE
• Axelrod’s models• Method• Results and discussion• Conclusions
PRESENTATION OUTLINE
• Axelrod’s models• Method• Results and discussion• Conclusions
AXELROD’S MODELS: The Norms model
i defects
Boldness i > S
Temptation = 3Hurt = −1
i cooperates
each i
Enforcement = −2Punishment = −9
j punishes i
j does not punish i
Vengefulness j20-player PD
j sees i
j does not see i
S
each j≠i
AXELROD’S MODELS: The Norms model
• 20 agents; Random initial strategies• 1 round = 1 opp. to defect for everyone• 4 rounds = 1 generation
0 1 1 = 3 / 7
Avg +/— σReplicated
once
> Avg + σReplicated
twice
Payoffs
< Avg — σEliminated
• Evolutionary pressures
–MutationRate = 0.01– Selection mechanism
AXELROD’S MODELS: The Norms model
AverageVengefulness
AverageBoldness
1
10
5 runs
100 generations each
Norm collapse
Norm establishment
AXELROD’S MODELS: The MetaNorms model
j does not punish i
The Norms model
Vengefulness k
MEnforcement = −2MPunishment = −9
k meta-punishes j
k does not punish j
k sees j
k does not see j
S
each k ≠ i, j
i defects
Boldnessi > S
Temptation = 3Hurt = −1
i cooperates
j sees i
j does not see ieach i
S Enforcement = −2Punishment = −9
j punishes i
Vengefulness j
each j≠i
20-player PD
j does not punish i
AXELROD’S MODELS: The MetaNorms model
• 20 agents; Random initial strategies• 1 round = 1 opp. to defect for everyone• 4 rounds = 1 generation
0 1 1 = 3 / 7
Avg +/— σReplicated
once
> Avg + σReplicated
twice
Payoffs
< Avg — σEliminated
• Evolutionary pressures
–MutationRate = 0.01– Selection mechanism
AXELROD’S MODELS: The MetaNorms model
AverageVengefulness
AverageBoldness
1
10
5 runs
100 generations each
Norm establishment
Norm collapse
PRESENTATION OUTLINE
• Axelrod’s models• Method• Results and discussion• Conclusions
METHOD
• Computer models (Java & RePast)• Mathematical analysis – Markov chain
• Mathematical Abstractions
METHOD
• Mathematical Abstractions
ååå¹=
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¹=
++×+×=n
ijj
ji
n
ijj
ji
n
ijj
jii vbPbvEbHbTPayoff1
2
1
2
1 22)(Exp
Definition of Evolutionary Stable States
Maps of the dynamics
+ Continuity
+ Homogeneity
MAPS OF THE DYNAMICS
0 0.2 0.4 0.6 0.8 1Boldness0
0.2
0.4
0.6
0.8
1
Vengefulness
Norm establishment
Norm collapse
Expected PayoffsContinuityHomogeneity
Unique ESS
METHOD
• Computer models (Java & RePast)• Mathematical analysis – Markov chain
• Mathematical Abstractions
PRESENTATION OUTLINE
• Axelrod’s models• Method• Results and discussion• Conclusions
RESULTS AND DISCUSSION
• The Norms model• The MetaNorms model–Replication of the original
experiments–Exploration of parameter space–Other instantiations of the same
conceptual model
RESULTS AND DISCUSSION
• The Norms model• The MetaNorms model–Replication of the original
experiments–Exploration of parameter space–Other instantiations of the same
conceptual model
THE NORMS MODEL: Axelrod’s resultsAverage
Vengefulness
AverageBoldness
1
10
5 runs
100 generations each
Norm collapse
Norm establishment
THE NORMS MODEL: Dynamics
0 0.2 0.4 0.6 0.8 1Boldness0
0.2
0.4
0.6
0.8
1
Vengefulness
Norm establishment
Norm collapse
Expected PayoffsContinuityHomogeneity
Unique ESS
THE NORMS MODEL: Our results
500 runs; 1,000,000 generations each
RESULTS AND DISCUSSION
• The Norms model• The MetaNorms model–Replication of the original
experiments–Exploration of parameter space–Other instantiations of the same
conceptual model
THE METANORMS MODELAxelrod’s results
AverageVengefulness
AverageBoldness
1
10
5 runs
100 generations each
Norm establishment
Norm collapse
THE METANORMS MODEL: Our results
1,000 runs; 1,000,000 generations each
0 0.2 0.4 0.6 0.8 1Boldness0
0.2
0.4
0.6
0.8
1
Vengefulness
THE METANORMS MODEL: Dynamics
Norm establishment
Norm collapse
ESS
ESSExpected PayoffsContinuityHomogeneity
0.01 0.04Boldness
0.9
0.95
Vengefulness
RESULTS AND DISCUSSION
• The Norms model• The MetaNorms model–Replication of the original
experiments–Exploration of parameter space–Other instantiations of the same
conceptual model
THE METANORMS MODEL MutationRate = 0.001 (as opposed to 0.01)
300 runs; 200,000 generations each
AXELROD’S MODELS: The MetaNorms model
j does not punish i
The Norms model
Vengefulness k
MEnforcement = −2MPunishment = −9
k meta-punishes j
k does not punish j
k sees j
k does not see j
S
each k ≠ i, j
i defects
Boldnessi > S
Temptation = 3Hurt = −1
i cooperates
j sees i
j does not see ieach i
S Enforcement = −2Punishment = −9
j punishes i
Vengefulness j
each j≠i
20-player PD
j does not punish i
0 0.2 0.4 0.6 0.8 1Boldness0
0.2
0.4
0.6
0.8
1
Vengefulness
THE METANORMS MODELME = –0.2 ; MP = –0.9
Norm establishment
Norm collapse
ESS
Expected PayoffsContinuityHomogeneity
THE METANORMS MODELME = –0.2 ; MP = –0.9
300 runs; 200,000 generations each
AXELROD’S MODELS: The MetaNorms model
j does not punish i
The Norms model
Vengefulness k
MEnforcement = −2MPunishment = −9
k meta-punishes j
k does not punish j
k sees j
k does not see j
S
each k ≠ i, j
i defects
Boldnessi > S
Temptation = 3Hurt = −1
i cooperates
j sees i
j does not see ieach i
S Enforcement = −2Punishment = −9
j punishes i
Vengefulness j
each j≠i
20-player PD
j does not punish i
0 0.2 0.4 0.6 0.8 1Boldness0
0.2
0.4
0.6
0.8
1
Vengefulness
THE METANORMS MODEL Temptation = 10 (as opposed to T = 3)
THE METANORMS MODEL Temptation = 10 (as opposed to T = 3)
1,000 runs; 200,000 generations each
RESULTS AND DISCUSSION
• The Norms model• The MetaNorms model–Replication of the original
experiments–Exploration of parameter space–Other instantiations of the same
conceptual model
AXELROD’S MODELS: The MetaNorms model
• 20 agents; Random initial strategies• 1 round = 1 opp. to defect for everyone• 4 rounds = 1 generation
0 1 1 = 3 / 7
Avg +/— σReplicated
once
> Avg + σReplicated
twice
Payoffs
< Avg — σEliminated
• Evolutionary pressures
–MutationRate = 0.01– Selection mechanism
OTHER SELECTION MECHANISMS
• Random Tournament
vs.
OTHER SELECTION MECHANISMS
• Random Tournament
• Roulette wheel
OTHER SELECTION MECHANISMS
• Random Tournament
• Roulette wheel
• Average selection
≥ AvgReplicated
twice
Payoffs
< AvgEliminated
OTHER SELECTION MECHANISMS
• Random Tournament
• Roulette wheel
• Average selection
• Axelrod
Avg +/- σReplicated
once
> Avg + σReplicated
twice
Payoffs
< Avg + σEliminated
OTHER SELECTION MECHANISMS
300 runs; 20,000 generations each
PRESENTATION OUTLINE
• Axelrod’s models• Method• Results and discussion• Conclusions
CONCLUSIONS
• Run our models several times for many periods
• Exploration of the parameter space• Usefulness of complementary analytical
work• We should try not to conclude anything
beyond the scope of our models
Craigiebuckler, Aberdeen, AB15 8QH, UK
APPEARANCES CAN BE DECEIVING:Lessons Learned Re-implementing Axelrod’s “Evolutionary Approach to Norms”
Luis R. Izquierdo1 & José M. Galán2
1 The Macaulay Institute, Aberdeen, UK2 University of Burgos, Spain & INSISOC Group
