= Evolutionary game theory
<Evolutionary game theory> studies how a population's composition changes when the success of a <pure strategy> depends on which other <pure strategies> it encounters. The central feedback is a <frequency-dependent payoff>: becoming common changes the opponents a strategy meets, which can change whether it does well. In a <coordination game>, common behaviour can reinforce itself; in the coexistence example below, being rare is an advantage. This makes <evolutionary game theory> useful for asking which behaviours can persist and which initial compositions lead to different outcomes, without assuming that each participant solves an <optimization> problem.
In the basic model, a large, well-mixed population has finitely many inherited <pure strategies>; interactions sample opponents according to their population frequencies. Assume that a strategy's <expected payoff> determines its per capita net reproductive growth rate, with a common baseline allowed. These are modelling assumptions: reproduction must transmit the strategy, encounters must match the proposed sampling rule, and the conversion from <payoff> to growth must be justified for an application. The <replicator equation> below describes the resulting continuous-time frequencies. Taylor and Jonker's https://static1.squarespace.com/static/5dc9917c4cdb1467e1e54c4c/t/6526a5647942e33b89ab7af1/1697031525653/1978%2Bevolutionarily%2Bstable%2Bstrategies%2Band%2Bgame%2Bdynamics.pdf[Evolutionarily Stable Strategies and Game Dynamics (1978), section 3, pp. 148–149]{ref} explicitly derive this growth model.
For learned behaviour, reproduction can instead be replaced by an assumed copying rule: an individual samples another individual uniformly and copies a better-performing <pure strategy> at a rate proportional to the positive <payoff> difference. At frequencies $x_i,x_j$, the net flow from $j$ to $i$ is $x_ix_j(f_i-f_j)$, because the two directional positive parts subtract to $f_i-f_j$. Summing over $j$ gives $x_i(f_i-\bar f)$, the <replicator equation>. This derivation states a particular mechanism; an arbitrary rule for learning or imitation need not give the same <dynamical system>.
Three questions must be distinguished. A <symmetric Nash equilibrium> tests whether an individual can improve its <expected payoff> against the resident composition. An <evolutionarily stable strategy> tests whether each distinct rare mutant is outperformed after it is introduced. <Dynamical stability> tests the response of an actual <dynamical system> to small changes of composition, and <asymptotic stability> additionally requires nearby trajectories to converge. For a finite <symmetric bimatrix game> under the continuous-time <replicator equation>, every <evolutionarily stable strategy> is a <symmetric Nash equilibrium> and a <locally asymptotically stable> rest point; the converse from <asymptotic stability> to an <evolutionarily stable strategy> fails in general. See https://pmc.ncbi.nlm.nih.gov/articles/PMC4113915/[Ross Cressman and Yi Tao, The replicator equation and other game dynamics (2014), Theorems 1–2 and Example 1]. These statements refer to stability on the full <probability simplex>, including perturbations introducing currently absent <pure strategies>.
A concrete experiment makes the frequency feedback visible. In 2009, Gore, Youk and van Oudenaarden competed two <yeast> strains: a producer of the <enzyme> invertase, which enables <sucrose> <hydrolysis>, and a mutant lacking the corresponding SUC2 <gene>. Released sugars can feed neighbours, so avoiding production can be advantageous. They used shaken 5 mL cultures with 5% <sucrose> at $30\,{}^\circ\mathrm C$, diluted daily after 23 hours of growth. Different labelled <proteins> allowed the two strains to be counted separately by flow cytometry; selective plating periodically checked the counts. The cooperator required externally supplied histidine, so restricting that <amino acid> imposed an adjustable growth disadvantage. See https://www.youklab.org/papers/Nature2009_GoreAndYouk.pdf[Snowdrift game dynamics and facultative cheating in yeast, pp. 253 and 255–256, Methods Summary].
Starting with rare producers and, separately, rare nonproducers tested both invasion directions. Under the reported coexistence conditions each rare strain increased; changing histidine or added <glucose> shifted the eventual composition. Their explanation used nonlinear growth benefits, rather than a fixed pairwise <payoff matrix>. This motivates testing both rarity regimes before inferring an <evolutionarily stable strategy> or fitting a <replicator equation>; it does not identify the illustrative tables below with measured <payoffs>. See https://www.youklab.org/papers/Nature2009_GoreAndYouk.pdf[Gore, Youk and van Oudenaarden, Figures 1–3].
An early motivation was understanding why animal contests can involve restraint even when escalation might win an individual encounter. Maynard Smith and Price's https://public.ek-cer.hu/~szabo/EGT20/library/maynard_n73.pdf[The Logic of Animal Conflict (1973), Nature 246, 15–18]{ref} introduced the <evolutionarily stable strategy> in this setting. Taylor and Jonker's https://static1.squarespace.com/static/5dc9917c4cdb1467e1e54c4c/t/6526a5647942e33b89ab7af1/1697031525653/1978%2Bevolutionarily%2Bstable%2Bstrategies%2Band%2Bgame%2Bdynamics.pdf[1978 paper]{ref} then studied explicit continuous and discrete dynamics. Static resistance to invasion and the paths populations follow are related questions, but they require separate definitions.
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