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arxiv: 2503.21849 · v2 · submitted 2025-03-27 · 🧬 q-bio.PE · math.PR

Selection of the fittest or selection of the luckiest: the emergence of Goodhart's law in evolution

Bastien Mallein , Francesco Paparella , Emmanuel Schertzer , Zs\'ofia Talyig\'as This is my paper

Pith reviewed 2026-05-22 23:08 UTC · model grok-4.3

classification 🧬 q-bio.PE math.PR
keywords evolutionselection pressureGoodhart's lawadaptation speedtraveling wavesrandomnessgene pool diversityphase transition
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The pith

In an evolutionary model, adaptation speed rises with selection pressure only up to a threshold, after which randomness dominates and slows adaptation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how randomness in selection affects long-term evolution using an idealized model. It finds that selection pressure boosts adaptation speed only until a certain point. Past that threshold, stronger selection gives more weight to luck over genetic fitness, reducing both adaptation speed and gene pool diversity. This is presented as a biological version of Goodhart's law. The behavior is explained mathematically by a phase transition in pulled traveling waves.

Core claim

The speed of adaptation increases with increasing selection pressure only up to a threshold. Beyond the threshold, any increase of the selection pressure results in more weight given to random effects rather than on genetic fitness in determining which individuals will successfully reproduce. This severely reduces the speed of adaptation and the diversity in the gene pool. The intricate response can be explained by a novel phase transition for pulled traveling waves.

What carries the argument

Pulled traveling wave dynamics that govern the long-term patterns of evolution under selection on phenotypes affected by random contingencies.

Load-bearing premise

The model assumes an idealized evolution where selection acts on phenotypes affected by random contingencies and long-term patterns follow pulled traveling wave dynamics.

What would settle it

Simulations or observations showing that adaptation speed continues to increase or does not decrease after a certain selection pressure threshold would falsify the claim.

Figures

Figures reproduced from arXiv: 2503.21849 by Bastien Mallein, Emmanuel Schertzer, Francesco Paparella, Zs\'ofia Talyig\'as.

Figure 1
Figure 1. Figure 1: Rates adaptations and effective population sizes in a branching-selection particle system [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Top left panel. Distribution of genotypes in a large population, plotted on a logarithmic scale. We observe for large populations the apparition of a deterministic genotypic profile. The profile is well approximated by a quadratic profile (in red). Top right panel. Convergence to the traveling wave solution of the deterministic dynamic, started from an initial Gaussian distribution (i.e. a parabolic log-pr… view at source ↗
Figure 3
Figure 3. Figure 3: Top panels. Rate of adaptation for the deterministic limiting model and its finite-size population correction, for the original stochastic model on the left pannel, and the noisy BRW right pannel (believed to be its scaling limit). In both cases, the rate of convergence is notably slow. Bottom left panel. Comparison between the critical value γc estimated for each µ as the value of γ maximizing γ 7→ v (N) … view at source ↗
Figure 4
Figure 4. Figure 4: Left panel. Speed for the asexual model as predicted by the explicit solution of (4.3). Right panel. Speed for the sexual model as predicted by the iterating the modified equation (4.9). For both models, if µ ∈ (.5, 1), the speed is maximized at intermediary values. However, in the sexual case, the speed is always 0 when the phenotypic noise is too large µ < 1/2. Bottom panel. Comparison of the speed of ev… view at source ↗
Figure 5
Figure 5. Figure 5: Fitness wave for the sexual model. In blue : generation 0 to 8. In red: generation 16 to 24. [PITH_FULL_IMAGE:figures/full_fig_p037_5.png] view at source ↗
read the original abstract

Biological evolution depends on the passing down to subsequent generations of genetic information encoding beneficial traits, and on the removal of unfit individuals by a selection mechanism. However, selection acts on phenotypes, and is affected by random contingencies. Thus, a combination of fitness and luck determines which individuals will successfully reproduce and give rise to the next generation. To understand how randomness in the selection mechanism affects the long-term patterns of evolution, we studied an idealized evolution model. We show through simulations and mathematical analysis, that the speed of adaptation increases with increasing selection pressure only up to a threshold. Beyond the threshold, any increase of the selection pressure results in more weight given to random effects rather than on genetic fitness in determining which individuals will successfully reproduce. This severely reduces the speed of adaptation and the diversity in the gene pool. Our findings may be considered as a biological instance of Goodhart's law: "When a measure becomes a target, it ceases to be a good measure". Finally, we show that this intricate response of evolution to natural selection can be mathematically explained by a novel phase transition for pulled traveling waves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies an idealized evolutionary model in which selection acts on phenotypes subject to random contingencies. Using simulations and mathematical analysis based on pulled traveling-wave dynamics, it claims that adaptation speed increases with selection pressure only up to a threshold; beyond this point, stronger selection amplifies the relative weight of stochastic effects over genetic fitness, thereby decreasing adaptation speed and reducing genetic diversity. The non-monotonic response is presented as a biological instance of Goodhart's law and is attributed to a novel phase transition in the pulled traveling-wave regime.

Significance. If the central threshold and phase-transition claims hold, the work supplies a mechanistic, traveling-wave account of why excessively strong selection can become counterproductive, with potential implications for both theoretical evolutionary dynamics and empirical studies of selection intensity. The combination of simulation evidence with an analytic traveling-wave explanation, together with the apparent absence of free parameters in the model, constitutes a clear strength that distinguishes the contribution from purely phenomenological descriptions.

major comments (1)
  1. [Mathematical analysis of traveling waves] The non-monotonic speed and diversity loss are derived under the maintained assumption that the population remains in the pulled traveling-wave regime even for selection pressures above the reported threshold. The abstract states that the effect is 'mathematically explained by a novel phase transition for pulled traveling waves,' yet no explicit verification (e.g., comparison of observed wave speed against the pulled-wave formula or inspection of leading-edge dynamics) is described that confirms the wave type does not switch to pushed or become unstable once the threshold is crossed. This verification is load-bearing for the claimed mechanism.
minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief, explicit statement of the precise definition of 'selection pressure' used in the model (e.g., the functional form relating fitness to reproductive success).
  2. [Figure captions] Figure captions should state the number of independent simulation replicates and the precise parameter values at which the threshold is reported to occur.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the potential significance of our work. We address the single major comment below.

read point-by-point responses

  1. Referee: [Mathematical analysis of traveling waves] The non-monotonic speed and diversity loss are derived under the maintained assumption that the population remains in the pulled traveling-wave regime even for selection pressures above the reported threshold. The abstract states that the effect is 'mathematically explained by a novel phase transition for pulled traveling waves,' yet no explicit verification (e.g., comparison of observed wave speed against the pulled-wave formula or inspection of leading-edge dynamics) is described that confirms the wave type does not switch to pushed or become unstable once the threshold is crossed. This verification is load-bearing for the claimed mechanism.

    Authors: We agree that explicit verification is necessary to substantiate that the reported phase transition occurs within the pulled regime. In the revised manuscript we will add a dedicated verification section (or appendix) that (i) directly compares simulated front speeds against the analytic pulled-wave speed formula across the full range of selection pressures, including values above the threshold, and (ii) examines the leading-edge density profiles to confirm consistency with pulled-front dynamics and the absence of a switch to pushed or unstable behavior. These additions will make the load-bearing assumption explicit and testable. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent traveling-wave analysis and simulations without reduction to fitted inputs or self-citations.

full rationale

The paper derives the non-monotonic adaptation speed and Goodhart-like effect from an idealized model whose long-term behavior is governed by pulled traveling waves, with the threshold arising from a claimed novel phase transition. No equations or text in the provided abstract or description reduce the central prediction to a self-definitional parameter, a fitted quantity renamed as output, or a load-bearing self-citation chain. The traveling-wave regime is invoked as an external modeling assumption whose consequences are then analyzed, rather than being defined in terms of the target result. This is the common case of a self-contained mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the model is described as idealized but no explicit free parameters, axioms, or invented entities are listed.

axioms (1)
  • domain assumption Idealized evolution model in which selection acts on phenotypes affected by random contingencies
    Stated directly in the abstract as the foundation of the study.

pith-pipeline@v0.9.0 · 5742 in / 1255 out tokens · 43671 ms · 2026-05-22T23:08:07.484003+00:00 · methodology

discussion (0)

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Reference graph

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    Let z ∈ T an individual at position u log(N) (u ∈ (−1, 0)). For the sake of simplicity, let us first assume that individuals are sampled with replacement and let pN(u) := N γ × eµu log(N) PN i=1 eµzi 0 . be the expected number of times our focal individual is selected. From (S.4.3), an easy computation shows that log(PN i=1 eµzi 0) log(N) ≈ 1 − µ. so that...

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