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 – Classical Game Theory – Axelrod’s (1984) Tournaments – ( Mainstream ) Evolutionary Game Theory • Our work: – Methodology: Agent-based modelling – Results and discussion • Conclusions
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
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.
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.
… and its simplest formalisation Player 2 The Prisoner’s Dilemma Cooperate Defect 3 0 Cooperate 3 4 Player 1 4 1 Defect 0 1 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 Player 2 The Prisoner’s Dilemma Cooperate Defect 3 0 Cooperate 3 4 Player 1 4 1 Defect 0 1 Examples of strategies or behavioural traits: ALL D: Always Defect ALL C: Always Cooperate TFT: C and then do what the other player did
The initial population ALLD ALLC Another strategy TFT
The pairing and the game ALLD ALLC Another strategy TFT
The selection Higher payoffs Lower payoffs Old population New entrants … (death) New population
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
CLASSICAL GAME THEORY: Player 2 The Prisoner’s Dilemma Cooperate Defect 3 0 Cooperate 3 4 Player 1 4 1 Defect 0 1 • Played only once: Rational players defect . • Played any finite number of times: Rational players ALWAYS defect! Crucial assumption: Common knowledge of rationality
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
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?
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
EVOLUTIONARY GAME THEORY What strategies (i.e. behavioural traits) are likely to emerge and be sustained under evolutionary pressures? ALLD ALLC Another strategy TFT
Mainstream EVOLUTIONARY GAME THEORY PROBLEM: Some assumptions made to achieve mathematical tractability: • Infinite populations • Only deterministic strategies • Pairing: Random • Selection: Proportional fitness rule • No mutation or random drift Even with many of these assumptions, we don’t really know what strategies are more plausible
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
Definition of the (unbiased) strategy space [ PC , PC/C , PC/D ] • 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]
The initial population (different sizes) [ 1, 1, 1]
The pairing (random, children together…) … and (different) number of rounds
The selection (roulette wheel, tournament…) Higher payoffs Lower payoffs Old population … and the mutation New entrants New population
ALLC TFT The (unbiased) strategy space ALLD
The modelling framework interface
EVO-2 x2 − A Modelling Framework to Study the Evolution of Strategies in 2x2 Symmetric Games under Various Competing Assumptions Izquierdo et al. PC PC/D PC/C
EVO-2 x2 − An Application to the Study of the Evolutionary Emergence of Cooperation Stochastic strategies Deterministic strategies TFT: 0.16% TFT: 58% ALLD: 60% ALLD: 8%
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
RESULTS AND DISCUSSION Stochastic strategies Deterministic strategies TFT: 0.16% TFT: 58% ALLD: 8% CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; ALLD: 60% num-players = 100; mutation-rate = 0.05; rounds-per-match = 10; selection-mechanism = roulette wheel ; pairing-settings = random pairings ;
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 ;
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 ;
RESULTS AND DISCUSSION Random pairings Children together TFT: 1% TFT: 22% ALLD: 1% CC-payoff = 3; CD-payoff = 0; DC-payoff = 5; DD-payoff = 1; ALLD: 72% num-players = 100; mutation-rate = 0.05; rounds-per-match = 5; selection-mechanism = roulette wheel ;
RESULTS AND DISCUSSION TFT-10 ALLD-30 1 0.8 0.6 0.4 0.2 0 2 3 4 5 6 7 8 9 10 20 30 40 50 100 num-strategies 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 ;
RESULTS AND DISCUSSION 5.E+08 Number of outcomes 4.E+08 CC 3.E+08 CD/DC 2.E+08 DD 1.E+08 0.E+00 2 3 4 5 6 7 8 9 10 20 30 40 50 100 num-strategies 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 ;
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
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.
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.
ACKNOWLEDGEMENTS • Edoardo Pignotti, University of Aberdeen • Bruce Edmonds, Manchester Metropolitan University • Nick Gotts The Macaulay Institute
End On the structural robustness of evolutionary models of cooperation Segismundo S. Izquierdo Luis R. Izquierdo
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