Evolutionary dynamics (sexual recombination as a learning rule)

Overview

The question is how a population's gene frequencies change when individuals reproduce sexually, recombining genes from two parents, and fitness selects which genotypes survive. The shared core is a generation step: selection weights each genotype by its fitness, and recombination mixes the genes of the selected genotypes. Each gene can then be read as a player that keeps a probability over its alleles, the versions it can take 1502.05056 §2.1.

The invariant the variants share is that the allele marginals, not the full joint population, follow a multiplicative-weights (polynomial-weights) update with a parameter-free or mixed utility 1502.05056 §3 1502.05056 §4. The earlier theory of this link needed weak selection and an assumption that the population stays a product of independent allele frequencies; the paper treats the general case 1502.05056 §1.

Variants

Selection before recombination (SR)

Each genotype is first weighted by fitness, then the population recombines. For two genes, the SR update mixes a recombinant term, weighted by the recombination rate r, with a selection-only term 1502.05056 §2.1 1502.05056 eq. 1. Its gene-level marginals equal a parameter-free polynomial-weights update, for any fitness matrix, any r and any population distribution 1502.05056 §3 1502.05056 prop. 1. Card: On Sex, Evolution, and the Multiplicative Weights Update Algorithm.

Recombination before selection (RS)

Recombination first produces the product of the two gene marginals, then selection acts on the mixed population 1502.05056 eq. 2. The gene marginals match a polynomial-weights update whose utility mixes an independent value (using the other gene's marginal) and a conditional value (using the joint), with mixing weight equal to r 1502.05056 §4 1502.05056 prop. 2. Card: On Sex, Evolution, and the Multiplicative Weights Update Algorithm.

k loci

For k genes with several alleles each, the paper gives an SR update as a sum over all subsets of genes inherited from the first parent, and an RS update as a product of marginals. The SR marginal lemma and the polynomial-weights correspondence hold for any k 1502.05056 app. A. Card: On Sex, Evolution, and the Multiplicative Weights Update Algorithm.

When to use which

The per-gene learning rule needs the full joint distribution to compute the conditional utility, so each gene would have to observe the whole population 1502.05056 §2.3. This is a model assumption, not a constraint the paper resolves.

Implementation sketch

The paper's update rules restated in our own notation. Not the authors' code; the paper releases no code. Shapes: P and W are (n, m).

# One SR generation  [arxiv:1502.05056 eq. 1]
# Input:  P (n, m) joint frequencies, sum 1; W (n, m) fitness; r in [0, 0.5]
# Output: next P (n, m)

def sr_step(P, W, r):
    # Average fitness of the current population  [arxiv:1502.05056 §2.1]
    w_bar = sum(P * W)
    # Selection-only term  [arxiv:1502.05056 eq. 1]
    selected = P * W / w_bar
    # Recombinant term; the sums factorise into row and column sums  [arxiv:1502.05056 §3]
    row = sum(P * W, axis=1)
    col = sum(P * W, axis=0)
    recombinant = outer(row, col) / w_bar**2
    # Mix with the recombination rate  [arxiv:1502.05056 eq. 1]
    return r * recombinant + (1 - r) * selected

# One RS generation  [arxiv:1502.05056 eq. 2]
# Input:  P (n, m); W (n, m); r
# Output: next P (n, m)

def rs_step(P, W, r):
    # Marginals of the current population  [arxiv:1502.05056 §2.1]
    x = sum(P, axis=1)
    y = sum(P, axis=0)
    # Recombined population is the product of the marginals  [arxiv:1502.05056 §2.1]
    recombined = outer(x, y)
    # Select the mixed population  [arxiv:1502.05056 eq. 2]
    mixed = W * (r * recombined + (1 - r) * P)
    # Normalise  [arxiv:1502.05056 eq. 2]
    return mixed / sum(mixed)

# Parameter-free polynomial-weights step for one gene  [arxiv:1502.05056 §2.3]
# Input:  x (n,) strategy; g (n,) expected utility per allele
# Output: next strategy (n,)

def pw_step(x, g):
    # Weight each allele by its expected utility, renormalise  [arxiv:1502.05056 §2.3]
    weighted = x * g
    return weighted / sum(weighted)

The utility for gene 1 is conditional, using P(j | i) = p_ij / x_i, or independent, using the other gene's marginal; the RS case mixes the two with weight r 1502.05056 §2.3 1502.05056 §4. Check the SR and RS steps against the paper's worked table before relying on them, because the RS column there does not add up (see Contradictions).

Contradictions

• Earlier theory versus this paper. The earlier result that gene-level multiplicative weights describe sexual dynamics needed weak selection and product-form populations 1502.05056 §1 1502.05056 §3. The paper argues the product assumption is inconsistent with the dynamics 1502.05056 app. B and drops both assumptions for the marginal equivalence 1502.05056 prop. 1 1502.05056 prop. 2.
• Convergence and fitness claims. The paper revises the convergence and fitness implications the earlier work drew for coordination games 1502.05056 §6. Its own result gives only diminishing regret per gene 1502.05056 §5.1 1502.05056 cor. 3, and it reports that no fitness or quality guarantee follows from that 1502.05056 §6. Convergence under weak selection rests on known results, and for general fitness matrices the paper reports that convergence is open 1502.05056 §5.2 1502.05056 §6.
• Internal inconsistencies in the paper. The text restricts r to [0, 0.5] while the main proposition is stated for r in [0, 1] 1502.05056 §2.1 1502.05056 §3. The RS column of one table does not sum to its printed row total 1502.05056 tab. 3. These are not resolved here; the values to test against come from the dynamics, not the table.

Papers

• On Sex, Evolution, and the Multiplicative Weights Update Algorithm: shows that gene marginals under SR and RS follow polynomial-weights updates for general fitness, recombination rate and population, and limits the regret implication to diminishing regret.