Algorithms, Games, and Evolution Erick Chastain ∗, Adi Livnat y, Christos Papadimitriou z and Umesh Vazirani z ∗Computer Science Department, Rutgers University, Piscataway, NJ 08854 USA, [email protected].,yDepartment of Biological Sciences, Virginia Tech, Blacks• burg, VA 24061 USA, [email protected]., and zComputer Science Division, University of California at Berkeley, CA, 94720 USA, [email protected], vazi• [email protected]. Contributed by Christos Papadimitriou, April 2, 2014 (sent for review November 16, 2013) Even the most seasoned students of evolution, starting with Dar• so as to maximize a certain convex combination of (a)cumulative ex- win himself, have occasionally expressed amazement that the pected fitness and (b) the entropy of its distribution on alleles. This mechanism of natural selection has produced the whole of Life connection between evolution, game theory, and algorithms seems to as we see it around us. There is a computational way to articulate us rife with productive insights; for example, the dual view just men- the same amazement: “What algorithm could possibly achieve all tioned sheds new light on the maintenance of diversity in evolution. this in a mere three and a half billion years?” In this paper we propose an answer: We demonstrate that in the regime of weak Game theory has been applied to evolutionary theory before, to selection, the standard equations of population genetics describ• study the evolution of strategic individual behavior (see, e.g., [6, 7]). ing natural selection in the presence of sex become identical to The connection between game theory and evolution that we point out those of a repeated game between genes played according to mul• here is at a different level, and arises not in the analysis of strategic tiplicative weight updates (MWUA), an algorithm known in com• individual behavior, but rather in the analysis of the basic population puter science to be surprisingly powerful and versatile. MWUA genetic dynamics in the presence of sexual reproduction. The main maximizes a trade•off between cumulative performance and en• ingredients of evolutionary game theory, namely strategic individual tropy, which suggests a new view on the maintenance of diversity behavior and conflict between individuals, are extraneous to our anal- in evolution. ysis. We now state our assumptions and results. We consider an in- Significance statement: Theoretical biology was founded on the finite panmictic population of haplotypes involving several unlinked mathematical tools of statistics and physics. We believe there are pro- (i.e., fully recombining) loci, where each locus has several alleles. ductive connections to be made with the younger field of theoretical These assumptions are rather standard in the literature. They are computer science, which shares with it an interest in complexity and made here in order to simplify exposition and algebra, and there is functionality. In this paper, we find that the mathematical descrip- no a priori reason to believe that they are essential for the results, tion of evolution in the presence of sexual recombination and weak beyond making them easily accessible. For example, Nagylaki’s the- selection is equivalent to a repeated game between genes played ac- orem [4], which is the main analytical ingredient of our results, holds cording to the multiplicative weight updates algorithm (MWUA), an even in the presence of diploidy and partial recombination. algorithm that has surprised computer scientists time and again in its Nagylaki’s theorem states that weak selection in the presence of usefulness. This equivalence is informative for the pursuit of two key sex proceeds near the Wright manifold, where the population genetic problems in evolution: the role of sex, and the maintenance of varia- dynamics become (SI) tion. ( ) t+1 1 t t xi (j) = xi(j) Fi (j) ; Weak selection j sex and recombination j coordination games j learning Xt algorithms j multiplicative weights update t where xi(j) is the frequency of allele j of locus i in the population at generation t, X is a normalizing constant to keep the frequencies t recisely how does selection change the composition of the gene summing to one, and Fi (j) is the mean fitness at time t amongst Ppool from generation to generation? The field of population ge- genotypes that contain allele j at locus i (see [4] and the SI). Un- netics has developed a comprehensive mathematical framework for der the assumption of weak selection, the fitness of all genotypes are answering this and related questions [1]. Our analysis in this paper close to one another, say within the interval [1 − ϵ, 1 + ϵ], and so the focuses particularly on the regime of weak selection, now a widely fitness of genotype g can be written as Fg = 1 + ϵ∆g, where ϵ is the used assumption [2, 3]. Weak selection assumes that the differences selection strength, assumed here to be small, and ∆g 2 [−1; 1] can in fitness between genotypes are small relative to the recombination be called the differential fitness of the genotype. With this in mind, rate, and consequently, through a result due to Nagylaki et al. [4], the equation above can be written (see also [1] Section II.6.2), evolution proceeds near linkage equi- ( ) librium, a regime where the probability of occurrence of a certain t+1 1 t t xi (j) = xi(j) 1 + ϵ∆i(j) ; (1) genotype involving various alleles is simply the product of the prob- Xt t abilities of each of its alleles. Based on this result, we show that where ∆i(j) is the expected differential fitness amongst genotypes evolution in the regime of weak selection can be formulated as a re- that contain allele j at locus i (see Figure 1 for an illustration of pop- peated game, where the recombining loci are the players, the alleles ulation genetics at linkage equilibrium). in those loci are the possible actions or strategies available to each player, and the expected payoff at each generation is the expected fit- ness of an organism across the genotypes that are present in the pop- Reserved for Publication Footnotes ulation. Moreover, and perhaps most importantly, we show that the equations of population genetic dynamics are mathematically equiva- lent to positing that each locus selects a probability distribution on al- leles according to a particular rule which, in the context of the theory of algorithms, game theory and machine learning, is known as mul- tiplicative weight updates (MWUA). MWUA is known in computer science as a simple but surprisingly powerful algorithm (see [5] for a survey). Moreover, there is a dual view of this algorithm: each locus may be seen as selecting its new allele distribution at each generation www.pnas.org/cgi/doi/10.1073/pnas.0709640104 PNAS Issue Date Volume Issue Number 1–5 We now introduce the framework of game theory (see Figure 2 nection between three theoretical fields: evolutionary theory, game for an illustration) and the “multiplicative weight updates” algorithm theory, and the theory of algorithms/machine learning. (MWUA, see SI), studied in computer science and machine learning, So far we have presented the MWUA by “how it works” (infor- and rediscovered many times over the past half-century; as a result mally, it boosts alleles proportionally to how well they do in the cur- of these multiple rediscoveries, the algorithm is known with various rent mix). There is an alternative way of understanding the MWUA names across subfields: “the experts algorithm” in the theory of algo- in terms of “what it is optimizing.” That is, we imagine that the al- rithms, “Hannan consistency” in economics, “regret minimization” in lele frequencies of each locus in each generation is the result of a game theory, “boosting” and “winnow” in artificial intelligence, etc. deliberate optimization by the locus of some quantity, and we wish Here we state it in connection to games, which is only a small part to determine that quantity. P t t τ of its applicability (see the SI for an introduction to the MWUA in Returning to the game formulation, define Ui (j) = τ=0 ui (j) connection to the so-called experts problem in Computer Science). to be the cumulative utility obtained by player i by playing strategy j A game has several players, and each player i has a set Ai of over all t first repetitions of the game, and consider the quantity possible actions. Each player also has a utility, capturing the way X 1 X whereby her actions and the actions of the other players affect this xt(j)U t(j) − xt(j) ln xt(j): (3) i i ϵ i i player’s well being. Formally the utility of a player is a function j j that maps each combination of actions by the players to a real num- The first term is the current (at time t) expected cumulative utility. ber (intuitively denoting the player’s gain, in some monetary unit, if The second term of (3) is the entropy (expected negative logarithm) all players choose these particular actions). In general, rather than of the probability distribution fxi(j); j = 1;::: jAijg, multiplied by choosing a single action, a player may instead choose a mixed or ran- a large constant 1 . Suppose now that player i wished to choose the domized action, that is, a probabilistic distribution over her action ϵ probabilities of actions xt(j)’s with the sole goal of maximizing the set. Here we only need to consider coordination games, in which all i quantity (3). This is a relatively easy optimization problem, because players have the same utility function — that is, the interests of the the quantity (3) to be maximized is strictly concave, and therefore players are perfectly aligned, and their only challenge is to coordinate it has a unique maximum, obtained through the KKT conditions [8] their choices effectively.