An interpretation and proof of the fundamental theorem of natural selection

Ewens, Warren J.

doi:10.1016/0040-5809(89)90028-2

Cited by 188 publications

(158 citation statements)

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“…[To be precise, Fisher's theorem has been interpreted (28)(29)(30) as stating that in an infinite population, the fitness component associated with a constant environment, both ecologically and genetically, increases at a rate proportional to the additive genetic variance.] Unlike the models considered here, Fisher's theorem applies under conditions of a changing environment and includes an explicit representation of a population's standing genetic variance.…”

Section: Models and Analysismentioning

confidence: 99%

The application of statistical physics to evolutionary biology

Sella

Hirsh

2005

Proc. Natl. Acad. Sci. U.S.A.

401

477

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A number of fundamental mathematical models of the evolutionary process exhibit dynamics that can be difficult to understand analytically. Here we show that a precise mathematical analogy can be drawn between certain evolutionary and thermodynamic systems, allowing application of the powerful machinery of statistical physics to analysis of a family of evolutionary models. Analytical results that follow directly from this approach include the steady-state distribution of fixed genotypes and the load in finite populations. The analogy with statistical physics also reveals that, contrary to a basic tenet of the nearly neutral theory of molecular evolution, the frequencies of adaptive and deleterious substitutions at steady state are equal. Finally, just as the free energy function quantitatively characterizes the balance between energy and entropy, a free fitness function provides an analytical expression for the balance between natural selection and stochastic drift

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Section: Models and Analysismentioning

confidence: 99%

The application of statistical physics to evolutionary biology

Sella

Hirsh

2005

Proc. Natl. Acad. Sci. U.S.A.

401

477

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“…Against X's fixed extortionate ZD strategy, a particularly simple evolutionary strategy for Y, close to if not exactly Darwinian, is for him to make successive small adjustments in q and thus climb the gradient in s Y . [We note that true Darwinian evolution of a trait with multiple loci is, in a population, not strictly "evolutionary" in our loose sense (14)]. …”

Section: Methods and Resultsmentioning

confidence: 99%

Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent

Press

Dyson

2012

Proc. Natl. Acad. Sci. U.S.A.

586

698

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The two-player Iterated Prisoner's Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y's score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores. Against such a player, an evolutionary player's best response is to accede to the extortion. Only a player with a theory of mind about his opponent can do better, in which case Iterated Prisoner's Dilemma is an Ultimatum Game.evolution of cooperation | game theory | tit for tat I terated 2 × 2 games, with Iterated Prisoner's Dilemma (IPD) as the notable example, have long been touchstone models for elucidating both sentient human behaviors, such as cartel pricing, and Darwinian phenomena, such as the evolution of cooperation (1-6). Well-known popular treatments (7-9) have further established IPD as foundational lore in fields as diverse as political science and evolutionary biology. It would be surprising if any significant mathematical feature of IPD has remained undescribed, but that appears to be the case, as we show in this paper.Fig . 1A shows the setup for a single play of Prisoner's Dilemma (PD). If X and Y cooperate (c), then each earns a reward R. If one defects (d), the defector gets an even larger payment T, and the naive cooperator gets S, usually zero. However, if both defect, then both get a meager payment P. To be interesting, the game must satisfy two inequalities: T > R > P > S guarantees that the Nash equilibrium of the game is mutual defection, whereas 2R > T þ S makes mutual cooperation the globally best outcome. The "conventional values" ðT; R; P; SÞ ¼ ð5; 3; 1; 0Þ occur most often in the literature. We derive most results in the general case, and indicate when there is a specialization to the conventional values. Fig. 1B shows an iterated IPD game consisting of multiple, successive plays by the same opponents. Opponents may now condition their play on their opponent's strategy insofar as each can deduce it from the previous play. However, we give each player only a finite memory of previous play (10). One might have thought that a player with longer memory always has the advantage over a more forgetful player. In the game of bridge, for example, a player who remembers all of the cards played has the advantage over a player who remembers only the last trick; however, that is not the case when the same game (same allowed moves and same payoff matrices) is indefinitely repeated. In fact, it is easy to prove (Appendix A) that, for any strategy of the longer-memory player Y, shorter-memory X's score is exactly the same as if Y had played a certain shorter-memory strategy (roughly, the marginalization of Y's...

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“…This is the change in mean fitness resulting only from natural selection, and holds regardless of dominance, epistasis, link age disequilibrium, frequency dependence, or nonrandom mating. It does not include changes in fitness resulting from changes in the average effect or excess of genes, which in turn may be caused by mutation, changes in the external environment, and changes in gene and genotype frequencies (Fisher 1958; for a more detailed discussion, see Price 1972;Ewens 1989; and Frank and Slatkin 1992; for relevant ex- perimentation, see Paquin and Adams 1983;Lenski et al 1991;and Spitze 1991). Thus, one way to estimate the change in fitness resulting from selection is to measure the additive variance of fitness What do we know of this quantity?…”

Section: E T H O D I: E S T Im a T In G T H E V A R Ia N C E O F F mentioning

confidence: 99%

The Evolution of Fitness

1995

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Abstract.-In every generation, the mean fitness of populations increases because of natural selection and decreases because of mutations and changes in the environment. The magnitudes of these effects can be measured in two ways: either directly, by comparing the fitnesses of selected and unselected populations, or indirectly, by measuring the additive variance of fitness and making use of the fundamental theorem of natural selection. The available data suggest that the amount by which natural selection increases mean fitness each generation (or degradation decreases mean fitness) will usually be between 0.1% and 30%; more tentatively, it is suggested that values will typically fall between 1% and 10%. These values can be used to set an upper limit of 5%-10% on the genetic advantage of mate choice.Key words.-Fitness variation, fundamental theorem, mate choice, natural selection, quantitative genetics, sexual selection.Received March 17, 1993. Accepted March 30, 1994 Fisher (1958) argued that the adaptedness or fitness of organisms tends to an equilibrium between opposing forces, increased on the one hand by natural selection, and decreased on the other by mutation and changes in the environment. Fitness distributions are thus seen as continually shifting back and forth every generation ( fig. 1), and, over the long term, increases resulting from selection will be approximately equal to decreases resulting from degradation. But just how large are these shifts-does selection typically increase mean fitness by a factor of 2 every generation, or 0.0002? Apart from its intrinsic interest, the answer would provide a way of quantifying natural selection that could be used, for ex ample, in setting upper limits on rates of phenotypic and molecular evolution, in setting an upper limit on the genetic advantage of mate choice, and in constructing realistic mod els for the evolution of sex and recombination. Estimates can be obtained in two ways: by comparing the mean fitness of selected and unselected populations directly, or by measuring the variance of fitness and making use of Fisher's (1958) fundamental theorem of natural selection. These methods are complementary, because the former is generally most useful in setting a lower limit on the change in mean fitness resulting from selection, the latter an upper limit. Here I develop these methods in more detail, review the available data, and discuss some implications for the evolution of mate choice. I begin with the indirect method of estimation, because it is the more familiar. M e t h o d I: E s t im a t in g t h e V a r ia n c e o f F it n e s sThe fundamental theorem of natural selection states that the relative increase in mean fitness resulting from selection is equal to the standardized additive (i.e., heritable) variance of fitness among genotypes:' Present address: Imperial College, Silwood Park, Ascot, Berk shire SL5 7PY UK. H-mail: aburt@ic.ac.uk where w is mean fitness before selection, and w' is mean fitness after selection. This simple principle, more stat...

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An interpretation and proof of the fundamental theorem of natural selection

Cited by 188 publications

References 10 publications

The application of statistical physics to evolutionary biology

The application of statistical physics to evolutionary biology

Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent

The Evolution of Fitness

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