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Evolution: Games, dynamics and algorithms Karen Page Bioinformatics Unit Dept. of Computer Science, UCL Evolution Darwinian evolution is based on three fundamental principles: reproduction, mutation and selection Concepts like fitness and natural selection are best defined in terms of mathematical equations We show how many of the existing frameworks for the mathematical description of evolution may be derived from a single unifying framework Summary of what will be discussed Games, evolutionary game theory Key frameworks of evolutionary dynamics Deriving a unifying framework An application to Fisher’s Fundamental Theorem Relationship with Genetic Algorithms What is game theory? Formal way to analyse interactions between agents who behave strategically Mathematics of decision making in conflict situations Usual to assume players are “rational” Widely applied to the study of economics, warfare, politics, animal behaviour, sociology, business, ecology and evolutionary biology Assumptions of game theory The game consists of an interaction between two or more players Each player can decide between two or more welldefined strategies For each set of specified choices, each player gets a given score (payoff) The Prisoners’ Dilemma Probably most studied of all games Not enough evidence to convict two suspects of armed robbery, enough for theft of getaway car Both confess (4 years each), both stay quiet (2 years each), one tells (0 years) the other doesn’t (5 years) Stay quiet= cooperate (C) ; confess = defect (D) Payoff to player 1: player2 C D C R S player1 D T P R is REWARD for mutual cooperation =3 S SUCKER’s payoff =0 T TEMPTATION to defect =5 P PUNISHMENT for mutual defection=1 with T>R>P>S The problem of cooperation What ever player 2 does, player 1 does better by defecting: C pl.1 D pl. 2 C 3 5 D 0 1 Classical game theory both players D Shame because they’d do better by both cooperating Cooperation is a very general problem in biology Everyone benefits from being in cooperative group, but each can do better by exploiting cooperative efforts of others Trade wars and cartels Import tariffs - Should countries remove them? Price fixing- why not cheat? Repeated games In many situations, typically players interact repeatedly- repeated Prisoners Dilemma Strategies can involve memory, use reciprocity Tit-for-tat Pavlov Game theory and a computer tournament Game theory says it is rational to defect in single game or fixed number of rounds Axelrod’s tournament- double victory for Titfor-Tat Evolutionary Game Theory So how can cooperation be explained? Evolutionary games John Maynard Smith- evolution of animal behaviour Behaviour shaped by trial and error- adaptation through natural selection or individual learning Players no longer have to be ‘rational’: follow instincts, procedures, habits rather than computing best strategy. Games played in a population. Scores are summed. Strategies which do well against the population on average propagate. Phenotypic approach to evolution Frequency-dependent selection Simple evolutionary game simulations Everyone starts with a random strategy Everyone population plays game against everyone else The payoffs are added up The total payoff determines the number of offspring (Selection) Offspring inherit approximately the strategy of their parents (Mutation) [Note similarity to genetic algorithms.] [Nash equilibrium in a population setting- no other strategy can invade] Evolution in the Prisoners’ Dilemma Standard evolutionary game (random interactions) all Defect Modifications- spatial games: Interactions no longer random, but with spatial neighbours: Sum scores. Player with highest score of 9 shaded takes square (territory, food, mates) in next generation Some degree of cooperation evolves! Simulations of the spatial Prisoners Dilemma 75 generations Winner-takes-all selection No mutation Red=d(d last) Blue=c(c last) Yellow=d(c last) Green=c(d last) Conclusions on Evolutionary Games Game theory can be applied to studying animal and human behaviour (economics - evolutionary biology). Often traditional game theory’s assumption of ‘rationality’ fails to describe human/ animal behaviour Instead of working out the optimal strategy, assume that strategies are shaped by trial and error by a process of natural selection or learning. This can be modelled by evolutionary game theory. Space can matter Evolutionary dynamics General framework Quasispecies equation Replicator-mutator equation Lotka-Volterra equation replicator-mutator Price equation Game dynamical equation Price equation replicator Price equation Adaptive dynamics The replicator equation Replicator equation describes evolution of frequencies of phenotypes within a population with fitnessproportionate selection Eg. game theory, replicators like “Game of Life” Frequency of type i is y and fitness of type i is f i then i yi yi ( f i ( y ) f ( y )), where f ( y ) i yi f i ( y ) The equivalence with Lotka Volterra equations Lotka Volterra systems of ecology describe the numbers of animals (eg. fish) of different species and are of the form: xi xi f i ( x ) where xi is the abundance of species i, fi its fitness and there are n species in total. Often these interacting species oscillate in abundance. There is a precise equivalence with the replicator system for (n+1) types given by the substitution yi xi ( X 1) i 1,, n yn 1 1 ( X 1) . Replicator equation with mutation and quasispecies Suppose there are errors in replicating. The probability of type j mutating to type i is q ji . We obtain a replicator equation with mutation: yi j y j f j ( y )q ji f ( y ) yi The equivalent with numbers rather than frequencies of types is xi j x j f j q ji When the fitnesses do not depend on frequencies, this is the quasispecies eqn. (Probably the case in most GAs?) Quasispecies equation Describes molecular evolution (Eigen) yi j y j f j q ji f yi N biochemical sequences Biochemical species i has frequency yi Replication at rate fi is error-prone - mutation to type j at rate qij Adaptive dynamics framework Game consists of a continuous space of strategies (eg.) Population is assumed to be homogeneous- all players adopt same strategy Mutation generates variant strategies very close to the resident strategy If a mutant beats the resident players it takes over otherwise it is rejected Adaptive dynamics illustrates the nature of evolutionary stable strategies Adaptive dynamics equations Strategies are described by continuous parameters : S ( p1 , p2 ,...., pn ) ' ' ' ' S ( p , p ,...., p Expected score of mutant 1 2 n ) against S is given by E(S’,S) The adaptive dynamics flow in the direction which maximises the score: E S ' , S p i , i 1,, n ' pi S ' S We can derive Price’s equation from replicator-mutator equation Price’s equation from population genetics describes any type of selection. Suppose an individual of type i, frequency yi , has some trait p of value ip p i yi pi , so using the replicator equation with mutation we obtain p Cov( f , p ) E ( fp ) This applies when the values of pi are const. [p is the expected mutational change in p.] Price’s equation p Cov( f , p ) E ( fp ) selection mutation Price’s equation gives rise to adaptive dynamics If we assume that the mutation is localised and symmetrical then we can neglect the second term in Price’s eqn. Assume population is almost homogeneous and fitness is differentiable then we can Taylor expand the fitness, obtaining f p ( p )Var( p ) p cf. adaptive dynamics: E S ' , S p i ' pi S ' S General framework Quasispecies equation Replicator-mutator equation Lotka-Volterra equation replicator-mutator Price equation Game dynamical equation Price equation replicator Price equation Adaptive dynamics Fisher’s fundamental theorem Suppose fitnesses of genotypes constant. Can consider f as the trait p and obtain (for symmetric mutation): f Cov( f , f ) Var( f ) Fisher’s fundamental theorem of NS In general, fitnesses of genotypes depend on environment. In game theory context, depend on the frequencies of other genotypes. Fisher’s theorem doesn’t apply- eg. PD Generalized version We can use Price’s equation to obtain a generalized version of Fisher’s fundamental theorem: f Var ( f ) Cov( f , g ) where g i j f j xi xj This applies when the f i s depend linearly on the frequencies of genotypes- normally the case in evolutionary game theory. Fisher’s theorem and GAs In most GAs, fitnesses of particular solutions (chromosomes) probably fixed and so (except for the complication of recombination) Fisher’s theorem should hold: f Var ( f ) So for a GA with fitness-proportionate selection, no recombination and fixed fitness for a given solution, the average fitness of the population of solutions increases until there is no diversity left in the fitnesses. Conclusions on unifying evolutionary dynamics Unifying framework Different frameworks for different problems. We derive from Price’s equation a generalized version of Fisher’s Fundamental Theorem of Natural Selection. The Price – replicator framework can also be applied to discrete time formulations and to formulations with sexual reproduction. Relationship to GAs Evolutionary games and genetic algorithms Two-way interaction: 1) So far discussed computer simulations of evolutionary processes, eg. evolution of animal behaviour 2) Evolutionary computation, eg. genetic algorithms = computer science based on theory of biological evolution Evolutionary games very like genetic algorithms- but 1) Population size is usually quite large and may be few phenotypes: space well searched but not v. efficient. 2) Usually no recombination 3) Fitnesses depend on interactions Genetic Algorithms Evolutionary models are computer algorithms which use evolutionary methods of optimisation to solve practical problems (cf. finding stable strategies in games rather than working out ‘rational’ solution)- eg. Evolutionary programming, genetic algorithms Evolutionary operations involved in genetic algorithms: selection, mutation, recombination: How evolutionary dynamics relates to GAs GAs evolve by selection and mutation their dynamics can be (to some extent) described by the replicator equation with mutation (cf. unifying framework). The replicator equation describes fitness-proportionate selection. Ficici, Melnik and Pollack (2000) - effects of different types of selection (eg. truncation) on the dynamics of the Hawk-Dove game + relevance for evolutionary algorithms. Can lead to different dynamics. Must also consider the effects of recombination. Incorporating recombination into the replicator framework Do this by assuming that rjk;i = probability that when parent chromosome of type j combines with parent chromosome of type k, an offspring of type i is formed. No mutation, recombination after replication: yi ' j ,k r jk ;i [NB discrete-time version] f j fk y j yk f f Adding in mutation Add in mutation. Assume, as before, q ij is probability type i mutates to form type j ( qiilarge). Assume this happens after recombination. What we had before was f j fk yi ' j ,k r jk ;i y j yk What we have now is yi ' ' l qli yl ' j ,k ,l qli rjk ;l f f f j fk y j yk f f The diversity of the population and adaptive dynamics From Fisher’s theorem, see that no diversity of fitness in population no further increase in average fitness. However, because the variation in the parameters of the your system has become very small (population convergence), does not mean no further evolution. In the case of small variation, we can apply the adaptive dynamics framework which shows how the average values of traits (parameters) will change in time f p ( p)Var ( p) p Relationship: evolutionary games & GAs - Conclusions Often evolution leads in the long run to ‘optimal’ solutions, like Nash equilibria. Ability of evolutionary processes to seek out optimal strategies has been exploited in computer science by the development of genetic algorithms and evolutionary computation for problem solving. Comparing with the use of computer simulations to study biological evolution, we see that there is a twoway interaction between biological evolutionary theory and computer science. Relationship to GAsConclusions Frameworks of evolutionary dynamics can be applied to GAs by modifying them to include recombination. Which framework is most informative depends on the individual problem, but we have shown they are equivalent. Eg. can look at detailed dynamics using the replicatormutator framework Or we can look at a “converged” population using the adaptive dynamics framework. Looking further at the relationship between GAs and evolutionary dynamics could yield new solutions/ techniques for both. Acknowledgements Martin Nowak (IAS, Princeton) Terry Leaves (BNP Paribas, London) Karl Sigmund (Univ. Vienna) Steven Frank (Univ. California, Irvine) Peter Bentley (UCL) Christoph Hauert (Univ. British Columbia) http://www.univie.ac.at/virtuallabs/Spatial2x2Ga mes/ Anargyros Sarafopoulos (Univ. Bournemouth) Bernard Buxton (UCL)
