On the structural robustness of evolutionary models of cooperation - - PowerPoint PPT Presentation

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On the structural robustness of evolutionary models of cooperation - - PowerPoint PPT Presentation

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On the structural robustness of evolutionary models of cooperation Segismundo S. Izquierdo Luis R. Izquierdo IDEAL 2006 Burgos, 20-9-2006 PRESENTATION OUTLINE Aim and necessary background Previous work and problems with it

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On the structural robustness of evolutionary models of cooperation Segismundo S. Izquierdo Luis R. Izquierdo

IDEAL 2006 Burgos, 20-9-2006

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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AIM

To advance our formal understanding of the evolution of cooperation by determining in the context of social dilemmas what behavioural traits are likely to emerge and be sustained under evolutionary pressures.

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BACKGROUND: Social dilemmas …

  • Social Dilemmas:

– Each individual receives a higher payoff for a socially defecting choice than for a socially cooperative choice, no matter what the other individuals in society do, but – All individuals are better off if all cooperate than if all defect.

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Player 2 Cooperate Defect Cooperate 3 3 4 Defect 4 1 1 Player 1 The Prisoner’s Dilemma

Both players prefer defecting no matter what the other one does Both players are better off if they both cooperate than if they both defect

… and its simplest formalisation

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Player 2 Cooperate Defect Cooperate 3 3 4 Defect 4 1 1 Player 1 The Prisoner’s Dilemma

… and its simplest formalisation

Examples of strategies or behavioural traits:

ALL D: Always Defect ALL C: Always Cooperate TFT: C and then do what the other player did

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The initial population

ALLD ALLC Another strategy TFT

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The pairing and the game

ALLC ALLD Another strategy TFT

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The selection

Old population

New population

Higher payoffs Lower payoffs New entrants

(death) …

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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CLASSICAL GAME THEORY:

Player 2 Cooperate Defect Cooperate 3 3 4 Defect 4 1 1 Player 1 The Prisoner’s Dilemma

  • Played only once:

Rational players defect.

Crucial assumption: Common knowledge of rationality

  • Played any finite number of times:

Rational players ALWAYS defect!

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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AXELROD’S TOURNAMENTS

  • Finitely repeated PD (200 rounds)
  • Round robin (and vs. random strategy)
  • Under common knowledge of

rationality, everyone should play ALLD... ... but the winner was TFT !!! Would TFT be the winner under other (more general) conditions?

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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EVOLUTIONARY GAME THEORY

What strategies (i.e. behavioural traits) are likely to emerge and be sustained under evolutionary pressures?

ALLD ALLC Another strategy TFT

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Mainstream EVOLUTIONARY GAME THEORY

  • Infinite populations
  • Only deterministic strategies
  • Pairing: Random
  • Selection: Proportional fitness rule
  • No mutation or random drift

PROBLEM: Some assumptions made to achieve mathematical tractability:

Even with many of these assumptions, we don’t really know what strategies are more plausible

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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Definition of the (unbiased) strategy space

  • PC:

Probability to cooperate in the first round

  • PC/C: Probability to cooperate in round n (n > 1)

given that the other player has cooperated.

  • PC/D: Probability to cooperate in round n (n > 1)

given that the other player has defected. Example: [ 0.13, 0.34, 0.93] ALLC: [ 1, 1, 1] ALLD: [ 0, 0, 0] TFT: [ 1, 1, 0]

[ PC, PC/C, PC/D ]

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The initial population (different sizes)

[ 1, 1, 1]

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The pairing (random, children together…)

… and (different) number of rounds

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The selection (roulette wheel, tournament…)

Old population

New population

Higher payoffs Lower payoffs New entrants

… and the mutation

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The (unbiased) strategy space

ALLD TFT ALLC

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The modelling framework interface

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PC PC/C PC/D

EVO-2 x2 − A Modelling Framework to Study

the Evolution of Strategies in 2x2 Symmetric Games under Various Competing Assumptions

Izquierdo et al.

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EVO-2 x2 − An Application to the Study of

the Evolutionary Emergence of Cooperation

Stochastic strategies Deterministic strategies

TFT: 58% ALLD: 8% TFT: 0.16% ALLD: 60%

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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RESULTS AND DISCUSSION

Stochastic strategies Deterministic strategies

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; num-players = 100; mutation-rate = 0.05; rounds-per-match = 10; selection-mechanism = roulette wheel; pairing-settings = random pairings;

TFT: 0.16% TFT: 58% ALLD: 8% ALLD: 60%

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RESULTS AND DISCUSSION

Mutation rate = 0.05 Mutation rate = 0.01 TFT: 0.2% TFT: 3.2%

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; num-players = 100; rounds-per-match = 50; selection-mechanism = roulette wheel; pairing-settings = random pairings;

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RESULTS AND DISCUSSION

  • Pop. size = 100
  • Pop. size = 10

3.2 % 0.3 %

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; mutation-rate = 0.01; rounds-per-match = 50; selection-mechanism = roulette wheel; pairing-settings = random pairings;

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RESULTS AND DISCUSSION

Random pairings Children together TFT: 1% TFT: 22%

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; num-players = 100; mutation-rate = 0.05; rounds-per-match = 5; selection-mechanism = roulette wheel;

ALLD: 1% ALLD: 72%

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RESULTS AND DISCUSSION

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; num-players = 100; mutation-rate = 0.01; rounds-per-match = 10; selection-mechanism = roulette wheel; pairing-settings = random pairings;

0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 20 30 40 50 100 num-strategies

TFT-10 ALLD-30

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RESULTS AND DISCUSSION

CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; num-players = 100; mutation-rate = 0.01; rounds-per-match = 10; selection-mechanism = roulette wheel; pairing-settings = random pairings;

0.E+00 1.E+08 2.E+08 3.E+08 4.E+08 5.E+08

2 3 4 5 6 7 8 9 10 20 30 40 50 100 num-strategies Number of outcomes CC CD/DC DD

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PRESENTATION OUTLINE

  • Aim and necessary background
  • Previous work and problems with it

– Classical Game Theory – Axelrod’s (1984) Tournaments – (Mainstream) Evolutionary Game Theory

  • Our work:

– Methodology: Agent-based modelling – Results and discussion

  • Conclusions
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CONCLUSIONS (1/2)

  • What type of strategies are likely to

emerge and be sustained in evolutionary contexts is strongly dependent on assumptions that traditionally have been thought to be unimportant.

  • Strategies similar to ALLD and TFT are

the two most successful strategies in most contexts.

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CONCLUSIONS (2/2)

  • Strategies similar to ALLD tend to be the

most successful in most environments.

  • Strategies similar to TFT tend to spread

best:

– In large populations – where the individuals with similar strategies interact frequently – for many rounds – with low mutation rates – and only deterministic strategies are allowed.

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ACKNOWLEDGEMENTS

  • Edoardo Pignotti,

University of Aberdeen

  • Bruce Edmonds,

Manchester Metropolitan University

  • Nick Gotts

The Macaulay Institute

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End

On the structural robustness of evolutionary models of cooperation Segismundo S. Izquierdo Luis R. Izquierdo