Figure 1. We interpreted significance in GLMM model outputs where 95% credible intervals did not span zero (red bars; otherwise black bars denote non-significant results), and points denote posterior mean. Our observed results (A1-2) suggest that female birds who divorce social partners regularly engage with more extra-pair males than those who maintain social monogamy. Likewise, the number of broods initiated also significantly predicted an increase in extra-pair males, without affecting the proportion of extra-pair offspring, presumably by virtue of opportunity (A2; red bars). Simulated breeding events (B1-2), where extra-pair males and extra-pair offspring were permuted between females in our system were not significant. Random effects are given in the shaded box (Dam and Cohort) for each model. Intercept and residuals for each model are not shown in the figure: A1, intercept -1.65 (-2.12 – -1.1), residuals 0.61 (-2.11 – 0.92); A2, intercept -1.65 (-2.11 – -1.8), residuals 0.53 (3.47 – 0.72); B1, intercept -0.5 (-8.78 – -0.14), residuals 0.81 (6.54 – 0.95); B2, intercept 1.26 (8.53 – 1.66), residuals 1.06 (9.37 – 1.2).

We empirically tested an assumption of the intrasexual antagonistic pleiotropy hypothesis - that extra-pair behaviour is linked to another trait beneficial to female fecundity, in this case, solicitation of social partners (Halliday and Arnold 1987; Wang et al. 2020), which we measured as the rate of divorce within years. Our results suggested that female divorce is linked with females engaging more extra-pair males. However, we found no effect of divorce on the proportion of extra-pair offspring. We also found a link between the number of broods initiated and female engagement of extra-pair males, which we consider a by-product of the opportunity for more extra-pair paternity over a longer breeding period.

Opportunity has been demonstrated to increase extra-pair behaviour in several systems (see Maldonado-Chaparro et al. 2018), including empirical studies suggesting that extra-pair copulations may be obligate, where they are facilitated by opportunity (Fossøy, Johnsen, and Lifjeld 2006). Liker, Freckleton, and Székely (2014) suggested that the sex ratio of the wider social environment also influences the divorce rate, driven by female-biased sex ratios and infidelity, by male-biased sex ratios (but also Culina, Hinde, and Sheldon 2015). However, opportunity cannot explain what drives extra-pair decision-making, instead, that must come from selective advantage to those who engage in extra-pair copulations. Our results support the suggestion by Wang et al. (2020) that female social solicitation behaviours are linked with extra-pair mate solicitation, and therefore, support intrasexual antagonistic pleiotropy as a mechanism for female extra-pair behaviour. However, we suggest that other, probably post-copulatory, processes control the proportion of extra-pair offspring born to females within a breeding year. Equivalent attention has also been given towards intersexual pleiotropic effects, where female extra-pair behaviour may be linked with a trait which benefits male reproductive success, these studies have found little support in captivity (Wang et al. 2020), or in wild populations (Zietsch et al. 2015; Reid and Wolak 2018). Where quantitative genetic studies have sought to demonstrate a heritable basis for male extra-pair behaviour, required for intersexual pleiotropy to drive female extra-pair behaviour, estimates are low (Reid et al. 2011; Reid and Wolak 2018; Grinkov et al. 2020). However, Dobson et al.. (2023) found that the inclusion of social partner indirect genetic effects (those derived from the behaviour of another) improved model fit for the heritability of both male and female extra-pair behaviour, implying a role for the wider social environment in the plasticity of extra-pair behaviour.

We used divorce rate within a female breeding year, between multiple broods, as a measure of solicitation, but not between breeding years - more akin to a classic definition of divorce behaviour. Divorce is more broadly defined as where both partners are alive, and at least one of them has paired with a new social partner (Black and Hulme 1996; but also see Culina, Radersma, and Sheldon 2015). However, the motivations for female sparrows to divorce social partners between broods may be like those described elsewhere for divorce, which has been demonstrated to represent an adaptive strategy between sexes across monogamous birds (Choudhury 1995; Culina, Radersma, and Sheldon 2015; Mercier, Yoccoz, and Descamps 2021; Wilson, Nguyen, and Burley 2022). Previous works have also considered divorce and extra-pair behaviours as linked traits in the context of indirect benefits (Cezilly and Nager 1995; Choudhury 1995), that is, both provide females with a mechanism to improve the quality of their social partner, by divorcing or cuckolding, lesser quality males. However, empirical evidence for indirect benefits derived from extra-pair behaviours is scarce (Akçay and Roughgarden 2007; Arct, Drobniak, and Cichoń 2015), and multi-paternity broods were not linked to the divorce rate Ramsay et al. (2000).

Although we consider our data to be near-complete, our phenotypic study is still subject to some bias (Hadfield 2008). For example, we sampled chicks for paternity at day two, (see Dunning et al. 2023), but this may still exclude an invisible fraction of those pairs, females (Kidd et al. 2015) or eggs (Yuta et al. 2018; Assersohn et al. 2021) who fail early. As a result, our study measures extra-pair paternity, and not extra-pair copulations, which could be more frequent than is reflected in paternity analysis (Fossøy, Johnsen, and Lifjeld 2006; Girndt 2018). Females that exhibit extra-pair behaviours may not always produce extra-pair offspring, but their within-pair offspring will inherit her genes, which may determine the extra-pair behaviour. Future work may support intrasexual pleiotropy as a mechanism for female extra-pair behaviour by either including measures of copulation attempts (rather than paternity) or, through further quantitative genetic methods.

We demonstrate that females who swap mates more frequently within breeding years engage more extra-pair males, but do not have greater proportions of extra-pair offspring. Our study contributes an empirical example to the growing body of research that supports non-adaptive phenomena as mechanisms for why females engage in extra-pair behaviours.

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