The determinants of proxy treadmilling in evolutionary models of reliable signals

Abstract

Identifying the conditions of proxy treadmilling is crucial for determining whether reliable signals can persist over time. I present a framework that maps evolutionary models of reliable signals according to their assumptions regarding the effects of Goodhart's law. This framework can explain the contrasting outcomes of different modelling approaches, and identify in which models proxy treadmilling is expected to occur.

The target article raises the possibility that costly signals (i.e., signals whose level is regulated by their cost of production) are liable to a “proxy treadmill” process (McCoy & Haig, 2020). This is at odds with the conclusions of previous reliable signalling models, specifically evolutionary models that have suggested that costly signals can persist stably (Grafen, 1990; Johnstone, 1996; Zahavi, 1975). However, it is difficult to place the assumptions of these models within the scheme presented in Section 3 of the target article, partly owing to a terminological gap. To reevaluate the conclusions of evolutionary models of reliable signals in light of the target article's synthesis, I present a framework that delineates how proxy treadmilling is constrained by the assumptions of these models.

As model parameters, we include components that are commonly included in biological signalling models, and were already introduced in Fisher's runaway selection model (1930): (i) The degree to which the observer has a preference for the signal (e.g., the peahen's preference of long tails), p; (ii) the investment in the signal (e.g., elaboration of the peacock's tail), s; and (iii) the marginal cost of the signal (e.g., the survival cost associated with longer tails), m. As opposed to signal preference and investment, marginal cost often only appears implicitly (as in Fisher's model) or as a fixed property of the signal (Biernaskie, Perry, & Grafen, 2018; Harris, Daon, & Nanjundiah, 2020). By contrast, Goodhart's law subsumes the marginal cost into what determines the signal level; “proxy optimisation” in the target article refers both to an increase in signal level and a reduction in production cost. Although it has been recognised that marginal cost could change over time, resulting in the inflation and subsequent replacement of signals – as in the “inflation hypothesis” in Zahavi and Zahavi (1999) – these dynamics have not been modelled.

The interaction between the components is based on a previous model that tracks the inflation of signals (Harris et al., 2020). We assume that preference of a signal (e.g., mate selection based on signal level) can increase signal investment, s, at rate Δs, and that this investment is constrained by the marginal cost, m. If m is sufficiently low, this runaway process leads to signal inflation (Harris et al., 2020). We also allow m to decrease at rate Δm, owing to the same pressure that increases s. We assume that preference of the signal, p, will increase or decrease, depending on the benefit of observing reliable signals, at rate Δp; importantly, preference of a partially reliable signal can also be stable (Harris et al., 2020; Johnstone & Grafen, 1993). When Δs < Δm, we expect signal inflation to be constrained only by Δs.

We can now map outcomes of different models using two expressions, Δp/Δs and Δm/Δs (Fig. 1). These expressions define two axes: (i) the position on the x-axis determines the rate at which signal preference is “optimised” relative to the inflation process; and (ii) the position on the y-axis determines the degree to which marginal cost constrains signal inflation. Figure 1 also illustrates how models can be mapped in this parameter space. At position (A), where Δp ≫ Δs ≫ Δm, are models (such as the handicap principle) that assume a fixed marginal cost, and that preference change can eliminate inflated signals. However, even with these assumptions, proxy treadmilling could occur when considering perceptive error (Harris et al., 2020; Johnstone, 1994). Moving to (B) entails increasing Δm relative to Δs; this means that signal production can be optimised because of Goodhart's law in the same timescale as the change in signal investment. This also shifts the constraint on inflation to Δs. When Δm > Δs, signal inflation will depend only on Δs and not on m. This means that even costly signals will eventually inflate (leading to increased proxy treadmilling), whereas only signals with a sufficiently low Δs will be stable, consistent with “non-handicap” models (Számadó, 2011). Moving to (C) from (A) entails reducing Δp relative to Δs. This decreases the rate of proxy treadmilling, because inflated signals can persist for longer; indeed, it has been suggested that constraints on preference will reduce reliability (Johnstone & Grafen, 1993).

Figure 1. Mapping model outcomes based on the relationships between the rates Δp, Δs, and Δm. Arrows indicate the expected change of outcome resulting from moving along the respective axis. (A) When Δp ≫ Δs ≫ Δm, preference change can ignore inflated signals, while costly signals can be stable. (B) When Δm > Δs, signal inflation depends on Δs, and costly signals can inflate leading to a proxy treadmill, unless Δs is sufficiently low. (C) Reducing Δp relative to Δs, inflated signals can persist.

This framework demonstrates that there is a nontrivial relationship between the conclusions of evolutionary models of reliable signals and whether proxy treadmilling is expected to occur under the respective model assumptions. A fully developed model of inflation dynamics (such as Harris et al., 2020) that also allows Δm > 0 could indicate more precisely the conditions of proxy treadmilling, in terms of the axes depicted in Figure 1.

The terms introduced here can be used to differentiate between models that have reached contrasting conclusions regarding what maintains signal reliability. For instance, using this framework we find two possible explanations for “index signals”: the first is that Δm is negligible, i.e., index signals have high marginal costs that are fixed, and this prevents their increase; the second is that Δs is negligible, i.e., index signals have a fixed level that cannot be manipulated. The former explanation is a costly signalling model (Zahavi & Zahavi, 1999), whereas the latter does not require costly signals to maintain signal reliability (Lachmann, Szamado, & Bergstrom, 2001). According to the above framework, testing these two explanations in specific signalling systems would involve demonstrating that the mechanism that limits Δs is unrelated to the cost of production.