THEORY-899: In an infinite, well-mixed population whose members each sample the quality of their own type, imitation of success has the expected update of a single Cross Learning bandit agent (the Taylor replicator dynamic), and the weighted voter model, in which members broadcast their type in proportion to their sample and adopt the first type they perceive, has the expected update of Maynard-Cross Learning, π_a ← π_a + (r_k / v^π)(1[a = k] − π_a), the Maynard-Smith replicator dynamic; the population vector is the policy and no member learns

Source

Soma, Bouteiller, Hamann & Beltrame, LIT-1484, read in NOTE-1132 from arXiv:2410.17517v6: Propositions 1 and 2 (pp. 7–11), Lemmas 1 and 2 with their proofs (pp. 20–21).

What was actually shown

• Imitation of success. Inflow to type a from b is π_b π_a E[r_a], and the net change is π_a(q_a − v), the Taylor replicator dynamic. The expected Cross Learning update, Σ_k π_k dπ_a(k), is the same (Lemmas 1 and 2).
• Weighted voter. Bee i of type k, with sample r_k⁽ⁱ⁾, draws from type a an expected flow (1/N)(r_k⁽ⁱ⁾ / v) π_a. Summed over bees this is (π_a / v)(q_a − v), the Maynard-Smith dynamic, and it equals the expected 1-MCL update (Eqs. 10–14).
• The reading. The bee “only “blindly” imitates its peers, irrespective of their payoffs!” (p. 10), so the reward-driven policy improvement exists only at the level of the population.

The derivation was re-checked step by step when filed.

What it does not show

That honey-bee swarms follow this dynamic: the weighted voter model omits spontaneous commitment, stop signals, direct switches and quorum sensing. That the swarm is an agent in any sense beyond this identity. Or anything for finite populations, where the authors’ simulations show options lost for good (THEORY-905).

Note of 2026-10-30: the second check

The second check at the status review (ADR-065 §5) followed each step.

Imitation of success. A revising member of type b meets a member of type a with probability π_a and adopts a with probability equal to that member’s sample, whose mean is q_a. The inflow to a from b is π_b π_a q_a, so the inflow to a is π_a q_a. The outflow from a is π_a Σ_b π_b q_b = π_a v. The net change per revision is π_a(q_a − v), the Taylor replicator dynamic. Cross Learning with action k drawn from π and reward r has expected update E[r(1[a = k] − π_a)] = Σ_k π_k q_k (1[a = k] − π_a) = π_a q_a − π_a v, the same.

Weighted voter. Each member broadcasts its type in proportion to its sample, so in a large well-mixed neighbourhood the first type perceived is a with probability π_a q_a / v. The reviser’s expected change of share is π_a q_a / v − π_a = (π_a / v)(q_a − v), the Maynard-Smith replicator dynamic. The expected Maynard-Cross Learning update, Σ_k π_k (q_k / v)(1[a = k] − π_a), is the same. With one of N members revising per step, each is scaled by α = 1/N.

A script (numpy, in the session’s scratchpad) checked these. In 1000 random cases of 2 to 6 types, the four expected updates agreed exactly as stated. A member-level simulation of the weighted voter (N = 4000, 4000 revisions, π = (.5, .3, .2), q = (.3, .6, .9)) gave a drift, in units of 1/N, of (−.217, .064, .153) against the predicted (−.206, .053, .153), within one standard error. As a control, an unweighted voter gave (−.003, −.007, .010), zero within noise.