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=

¹=

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