Source
Peters & Adamou (2022), LIT-035, “The ergodicity solution of the cooperation puzzle”, Phil. Trans. R. Soc. A 380(2227), read in full in NOTE-059.
What was actually shown
The drag (§2). A geometric Brownian motion dx = x(μ dt + σ dW) has ensemble growth rate μ (Eq. 2.7), but a single trajectory grows at μ − σ²/2 almost surely (Eq. 2.8). Fluctuations cost growth.
Pooling (§§3–4). N members grow, pool and split equally each step. The shared process is again a GBM with the same drift and volatility σ/√N (Eq. 3.9). Each member’s time-average rate is μ − σ²/(2N) (Eq. 4.1), a premium of (σ²/2)(1 − 1/N) (Eq. 4.2) that is largest per added member at small N and tends to μ as N → ∞. Members out-grow even the arithmetic mean of the same number of non-sharers (Fig. 2; μ = 0.15, σ² = 0.2, giving 0.05 for a loner and 0.10 for a pair member by NOTE-059’s arithmetic).
Correlation (§5b). With equal pairwise correlation ρ, the effective volatility is σ√((1 + (N−1)ρ)/N) (Eq. 5.12). There is a benefit iff ρ < 1, and with many members it caps at μ − σ²ρ/2 (Eq. 5.13). Perfect anticorrelation is possible only for N = 2, which gives the full rate μ.
Unequal members (§5a). The cooperative’s rate is (1/N)Σ(μ_i − σ_i²/(2N)) (Eq. 5.5). Member j gains by joining iff that exceeds μ_j − σ_j²/2 (Eq. 5.6), and the group gains by admitting j iff it exceeds the (N−1)-member rate (Eq. 5.7). A member skilled enough does better alone.
What could have come out otherwise. Nothing, given the model: it is a derivation. What could fail is the model’s fit to any system.
What this does not say
- That this solves the cooperation puzzle. The puzzle is why cooperators are not exploited. Defection, walk-away and invasion are set aside (§4a), and the “ignore-for-tat” remedy is sketched with no stability analysis.
- That cooperation is the behavioural default. That follows only with zero sharing cost, perfect continuous pooling, no defection and unconstrained growth. The authors grant that coordination costs “may … make cooperation disadvantageous” (§3), and no cost is analysed.
- That it is new. The paper itself says McCloskey and Winterhalder obtain “the same expression for the variance”, and that Kennedy et al. treat it as a special case. The novelty is the time-average framing.
- That it explains the move to multicellularity. That is a suggestion with no model (§1, §6).
- That growth-limited populations behave the same. GBM is unconstrained self-reproduction, and the authors say resource-limited growth “would be poorly described” by it (§2).
Connections
- Morality as cooperation (THEORY-468) and Nowak’s five mechanisms (THEORY-567) explain cooperation by the structure of the game: kinship, reciprocity, partner choice. This needs none of them. The two are not rivals: variance pooling is a payoff to sharing, and those mechanisms are about keeping sharers from being exploited, which this model assumes away.
- Major transitions (THEORY-141). The paper’s unicellular-to-multicellular remark points at the stability of collectives. It supplies a reason collectives would pay, not the policing or alignment that THEORY-141 holds they need.