The semantics of stability: evolutionarily stable strategy in biology and economics literature

Originally coined by Maynard Smith and Price, the term evolutionarily stable strategy has been adopted by a broad repository of disciplines across the spectrum of natural and social sciences (Piel, 2019(Piel, , 2020;;Leimar & McNamara, 2023). An ESS, as a subset of Nash equilibrium (NE; Apaloo et al., 2015), states that if a population adopts a strategy in a given environment, it cannot be invaded by an alternative strategy that is initially very rare (Nakamaru, 2023, Chapter 1). As described by Bishop & Cannings (1976), in a population where most individuals use a strategy p against a mutant strategy q, ∀ □□, □□ ≠ □□, for p to be an ESS, for every possible q, a) the average pay-off, in terms of inclusive fitness, the ultimate utility (Levin & Grafen, 2019), of using strategy p against itself is greater than the pay-off of q against p, □□(□□, □□) > □□(□□, □□), or b) If the pay-off of p against p is equal to the pay-off of q against p, i.e., □□(□□, □□) = □□(□□, □□), then p must have a higher pay-off against q than q does against itself, □□(□□, □□) > □□(□□, □□). Notably, Taylor (1989) further refined ESS by providing a mathematically rigorous definition in the context of continuous one-parameter models under weak selection by introducing two conditions required for ESS, namely a) m-stability, stability which favors convergence by causing a population near an evolutionary equilibrium to gravitate toward the equilibrium (this concept of convergence stability is analogous a situation if p were rare and invading the more common q in Bishop & Cannings, 1976), and b) δ-stability, local stability that causes a population at the equilibrium to remain there and resist deviations. An equilibrium □□ * can be said to be m-stable if a variant □□ + δ has a positive fitness increment □□(□□, δ) when □□ + δ is on the same side of m as □□ * , and a negative one when on the opposite side; this is represented as ∂□□/ ∂δ(□□ * , 0) = 0, and (∂ 2 □□)/ ∂m ∂δ(□□ * , 0) < 0. An equilibrium □□ * is δ-stable if for small δ ≠ 0, the fitness increment □□(□□ * , δ) is negative; this is represented as ∂□□/ ∂δ(□□ * , 0) = 0, and (∂ 2 □□)/ ∂δ 2 (□□ * , 0) < 0. The principles of ESS, initially defined at the individual level, could apply to populations, given additional assumptions (Van Cleve, 2023). Using correct terminology would be more historically accurate, better representing the original authors (Maynard Smith & Price, 1973). However, "evolutionarily" often mutates into "evolutionary" when this acronym is unshortened or mistyped. Interestingly, the first documented substitution occurred in Parker (1974), a paper with over 2,000 citations. It is impossible to determine if subsequent errors was influenced by this mistake or arose independently. In an ironic twist of evolutionary memetics (Fomin, 2019), "evolutionary stable strategy" had become established and entrenched in the population like an invasive allele. Adding to the confusion, "evolutionary stable" is sometimes used without the context of "strategy." Hence, it might be unclear to some members of a general audience whether "evolutionary stable" refers to a well-defined ESS concept or a notion that is both stable and evolutionary. A similar analogy in economics is the term "socially responsible investing," sometimes miswritten as "social." Using "social responsible" without the word 'investing' can lead to confusion (e.g., Alda, 2019) by blurring where the responsibility rests: with the investor or society. Using search engine data, I examined: a) whether there were trends in the usage of the incorrect ESS form over time, b) how the trend of ESS misuse compared to similarly comparable misnomers, and c) the prevalence of incorrect ESS usage in biology versus economics literature. The purpose was not to criticize authors using the incorrect form, especially those with unrelated backgrounds or facing linguistic challenges (Labrador, 2022). Rather, the aim was to improve scientific communication quality and discourse.This study examined the annual usage of correct and incorrect ESS terms from 1973 through 2022, based on Google Scholar search results. The analysis investigated the number of papers using "evolutionary stable strategy" as a proportion of those using either "evolutionary stable strategy" or "evolutionarily stable strategy." The same procedure was applied for the term ES ("evolutionary stable" versus "evolutionarily stable") and two similarly comparable phrases: ECP ("evolutionary conserved protein" versus "evolutionarily conserved protein") and ESU ("evolutionary significant unit" versus "evolutionarily significant unit"). Because Google Scholar does not offer disciplinebased filtering, JSTOR, an online repository, was utilized to quantify the frequency of misnomers in biological sciences compared to economic and social sciences.Due to the time-series nature of the data and the outcomes of the Shapiro-Wilk and Breusch-Pagan tests (see Table 1), normality and homoscedasticity could not be assumed. Hence, a nonparametric method was chosen to detect long-term trends in the data. The Mann-Kendall test (Kendall & Gibbons, 1990); Yue et al., 2002;Bronaugh et al., 2023) was applied to determine whether there was a significant upward or downward trend over time (Bürger, 2022). The test calculated Kendall's tau rank correlation coefficient, which measures the strength and direction of the relationship between variables in the time series, with values closer to -1 or 1 indicating a stronger trend (Rahman & Dawood, 2017). Figure 1 also provides a visual representation of the identified trend. The results of the Mann-Kendall test indicated significant positive monotonic relationships between ESS Incorrect Percentage (□□ = 0.57, p < .001) and ES Incorrect Percentage (□□ = 0.54, p < .001) with time, implying strong and substantial increasing trends respectively. No significant monotonic relationship was found for ECP incorrect percentage (□□ = 0.032, p = .78), suggesting the absence of a discernible trend. ESU incorrect percentage showed a significant positive monotonic relationship with time (□□ = 0.47, p < .001), indicating a moderate increasing trend. To compare the ESS with the ES incorrect percentage, a Wilcoxon Signed-rank test was conducted on a sample of 50 years, finding no significant difference between the ESS and ES incorrect percentages (W = 468.50, p = 0.22). The social sciences had a greater proportion of incorrect terminology than the biological sciences. The Mann-Whitney U test showed a significant difference in the distribution of the ESS incorrect percentage (U = 2,500,966, p < .001), as well as the ES incorrect percentage (U = 13,836,430, p < .001). Statistical analyses were performed using Posit Cloud (2023; formerly RStudio Cloud) and plotted using Python. All raw data, Python, and R codes for analyses are accessible in the data repository linked in the Data Availability Statement section.Although I cannot offer a good hypothesis to explain the higher incidence of incorrect spelling in economics literature compared to biological literature, I could propose one for the greater prevalence of the misnomer in ESS relative to similarly comparable terminologies. The incorrect spelling of ESS might be less deleterious because it is less confusing and, as a result, faces less scrutiny in the publication process. Consider Sperlich & Uriarte (2019), where the authors referenced "the evolutionary stable mixed strategy Nash equilibrium of the game to build an economic model of linguistic behaviour." Is this a new, specialized subset of NE? Upon closer inspection, it becomes clear that this is just an elaborate way of saying ESS. Similarly, in the study by Migot & Cojocaru (2021), they explored "stability for the replicator dynamics towards an evolutionary stable state." This phrase raises questions regarding what state is simultaneously evolutionary and stable. Could it refer to games involving more than three strategies where there is a continuous orbit around a cyclic attractor fixed point (Adami et al., 2016;Kuhn et al., 2023) without ever achieving a stable state? However, an examination of Google Scholar's preview window quickly clarifies this; it is merely a misspelling. In contrast, consider the example of the article titled "proteome-wide discovery of evolutionary conserved sequences in disordered regions" (Nguyen Ba et al., 2012). This phrasing causes confusion about what is simultaneously evolving and what is being conserved. Is it possible that the non-functional parts of the protein evolve while the functional parts remain conserved? Could the changing keto-enol tautomerization equilibrium ratios of DNA bases (Gheorghiu et al., 2020) change over time? Alternatively, one might wonder whether the transition probabilities in a hidden Markov model may have evolved over time (Nystrup et al., 2017), whereas the amino acid sequences remain static. Did the proteins evolve somatically (Wang & Tsien, 2006) while their germline counterparts remained conserved? Without opening and reading the article, it is difficult to determine whether the term "evolutionary" is a typographical error. Regardless of terminological variation, the concept is not generally clouded by whether "evolutionarily" or "evolutionary" is used, although the correct spelling does improve searchability, an important advantage. However, this variation can potentially lead to confusion for casual readers who are not necessarily interested in elaborate bio-realistic models. This is because the incorrect form, which juxtaposed "evolutionary" and "stable," seems to push the reader to imagine that the ESS of a typical game is evolutionary (constantly changing). This is not impermissible in nature, as the expected pay-offs of a game could indeed constantly change if there is new knowledge or information about costs and benefits (Leimar & McNamara, 2019), perceived or speculated relatedness (Faria et al., 2018;Madgwick et al., 2019), frequency (Rubin, 2016), or other external clues (Mühlenbernd et al., 2022), not to mention the fact that an equilibrium of phenotypic strategies exists does not preclude evolutionary changes in genetic frequency. An evolutionary equilibrium that is m-stable but not locally δ-stable (Taylor 1989;Christiansen 1991) will also tend to become polymorphic with more evolutionary dynamics of variation. Still, as most models concede to simplifications (Grodwohl & Parker, 2023), these lifelike nuisances are not typically included. They could be a source of distraction for general audiences and learners, preventing them from focusing on the mathematics. From a grammatical perspective, "evolutionarily stable strategy" is the clear winner as it unambiguously conveys the intended meaning. It features an adjective phrase (Berg, 2019) modifying the noun "strategy." Here, "evolutionarily" is an adverb modifying the adjective "stable," and the phrase "evolutionarily stable" modifies the noun "strategy." In this manner, it conveys the robustness of the strategy, which is a key attribute of an ESS. In contrast, "evolutionary stable strategy" confuses due to two interpretations. The words "evolutionary" and "stable" could be read as coordinating or cumulative adjectives. 1) As coordinating adjectives: These could be separated by a comma or "and," such as "evolutionary, stable strategy" or "evolutionary and stable strategy." Here both adjectives independently describe "strategy," misrepresenting the intended meaning of a strategy stable in an evolutionary context. 2) As cumulative adjectives: If considered as cumulative adjectives, "evolutionary" would modify "stable strategy." This portrayal can be misleading, as it would imply that the strategy is stable first and foremost, and then evolutionary, which could suggest a strategy that changes stably. This confusing notion is not the original intended meaning of ESS.The findings demonstrate a growing trend of improper ESS terminology usage over time, with significantly higher misuse in economics literature compared to biology. The rate of incorrect usage for ESS is higher than that for similarly comparable terminologies, possibly attributed to the lower editorial penalties associated with such a misspelling. However, the correct form should still be encouraged, as it reduces confusion, maintains consistency, and improves searchability. The incorrect version of the terminology resembles an allele that is slightly deleterious (or arguably inconsequential at best) with no conservation significance in a species' genome. Authors are encouraged to use "evolutionarily stable" instead of "evolutionary stable" in their writing.

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