Tests of reproductive skew theory: a review and prospectus

Reproductive skew refers to the unequal partitioning of reproduction by same-sex members of a social group. The concession, restraint and tug-of-war skew models attempt to predict which social and ecological factors influence skew, whilst assuming different power dynamics and making contrasting predictions. The factors underpinning within-species variation in skew remain unclear, despite decades of research. Individuals vary in both the benefit of leaving the group to breed (outside option) and in the contribution they make to group productivity if they stay. Variation is likely among several aspects of individual quality, so that individuals and species may show different associations between the subordinate's outside option and group productivity. Yet, no existing skew models allow for variation in this association, here named the quality-productivity coefficient (QPC). If high subordinate quality is associated with being a good helper, for instance if both help and quality improve with age, the QPC would be positive. If, instead, there is a trade-off or other contrast between breeding ability and helping effect, then the QPC will be negative. Here, we incorporate the QPC and derive new predictions for the concession, restraint and tug-of-war models, that sometimes changed their predictions. We review the empirical tests of the association of skew with relatedness, dominant’s competitive advantage, group productivity and subordinate’s outside option. From 16 studies available to calculate an effect size, relatedness correlated positively with skew; yet overall most of 45 studies reported no association. The association of skew with the outside option was negative or null, fitting the restraint and tug-of-war models. The mostly non-significant effect of group productivity supported the three models. Our review highlights future fruitful avenues for the study of cooperative breeding, especially more work to assess the association between subordinate’s outside option and subordinate’s effect on group productivity.


Introduction
Cooperative breeding is when the parents are not the only caregivers of infants (Cockburn 1998).Ecological and social factors shape the evolution of cooperative breeding (Boomsma 2009;Dillard and Westneat 2016;Field and Cant 2009;Lin et al. 2019), as they influence the inclusive fitness associated with cooperation and solitary breeding (Johnstone 2000).In cooperative groups, dominant individuals tend to monopolise reproduction to some extent, leading to reproductive skew: the unequal partitioning of reproduction by same-sex members of a social group (Verhencamp 1983).
Variation in skew and helping among cooperative breeders has long been a subject of research into how animals decide whether-and to what extent-to help and to breed (Koenig and Dickinson 2016).Models of reproductive skew seek to understand how individuals should divide up reproduction (Johnstone 2000;Verhencamp 1983).Three main types of skew models have been developed: the concession, restraint and tug-of-war models (Table 1).Table 1 shows the assumptions and the predictions of these three models.In the two transactional models − restraint and concession models − individuals can leave the group and so subordinates could breed on their own (i.e. they have an "outside option").In the concession model, the dominant concedes a certain amount of reproduction to the subordinate and the subordinate decides whether to stay and help, whereas in the restraint model the subordinate restrains themselves to a certain amount of reproduction and the dominant decides whether to evict them from the group.In the tug-of-war model, individuals engage in a competition over the share of reproduction, which is determined by their investment in the competition and their relative competitive ability.
The concession model predicts that skew should correlate positively with the relatedness between the subordinate helper and the dominant's offspring, as related subordinates get more indirect fitness benefits and therefore require a lower reproductive share to stay than unrelated subordinates (Reeve and Ratnieks 1993) (Table 1).By contrast, the restraint model predicts that skew should correlate negatively with relatedness, as related dominants can afford to share a higher proportion with the subordinate since they will get higher indirect fitness benefits than with an unrelated subordinate (Keller and Reeve 1994;Reeve and Ratnieks 1993).The subordinate efficiency version of the tug-of-war model predicts no association between skew and relatedness (Reeve and Keller 2001;Reeve Emlen and Keller 1998a, b) (Table 1).
To test the predictions of skew models using empirical data, researchers have focussed on three factors that influence the benefits and costs of cooperative behaviours (Hellmann and Hamilton 2018;Ragsdale 1999;Reeve and Emlen 2000): (1) genetic relatedness between dominants and subordinates; (2) subordinate's outside option, which may depend on intrinsic quality or environmental conditions such as nest site availability and (3) the dominant's competitive advantage which may depend on relative individual qualities or a status-related factor such as coalitional support from others.We may discern patterns of skew by investigating a wide range of associations in empirical tests (e.g.Langer et al. 2004;Lu et al. 2011), that include the critical factors included in models of reproductive skew (Johnstone 2000;Verhencamp 1983), in both within-and between-species tests.
Reviews of many tests have suggested that relatedness does predict skew across species, in birds (Riehl 2017), Polistes wasps (Liebert and Starks 2006;Nonacs 2006), and other taxa (Nonacs and Hager 2011).However, these reviews did not use effect sizes but relied on counting the number of positive, negative and null effects.The empirical tests of skew in all species were last reviewed 10 years ago (Nonacs and Hager 2011)

Table 1
Assumptions and predictions of the three main models of skew: concession, restraint and tug-of-war.Predictions of the effect of variables on skew are adapted from Liebert and Starks (2006), Reeve and Ratnieks (2001) and Nonacs (2006).Sources are listed inside brackets and are the original models of skew: (1) Reeve and Ratnieks, (1993)

Relatedness r
Positive (1) Negative ( 2 numerous more recent empirical tests, within-species variation in skew is not well understood.A new review focussing on effect sizes would quantify, and hence potentially clarify, the link between skew and relatedness. Our review is non-exhaustive partly because the data are so patchy, so we do not aim to unambiguously assign each taxa to supporting one model of reproductive skew.We also acknowledge that it may not be possible in many cases given that the concession, restraint and tug-of-war models may actually represent different facets of a continuum of reproductive strategies (Johnstone 2000).
That said, our aims herein are: (1) to provide an update and a summary of the three types of model of reproductive skew and their predictions, including predicting when there will be non-significant effects; (2) determine whether particular taxa support particular models which would indicate that we can understand the selective pressures; and (3) to identify gaps in the literature of empirical tests to help target future research.

Original concession, restraint and tug-of-war models
The three types of models of skew are based on different assumptions about the mechanism of control and make different predictions (Table 1).One assumption in all three models is that the parameters are assumed to be fixed and independent from one another.This means that the models are not self-consistent (sensu Kokko et al. 2006), as one factor does not dynamically influence the payoffs.However, in nature the factors are likely to interact.Differences in breeding ability, survival outside the group or other factors such as age might cause variation in subordinate outside options.These differences might also create variation in the contribution that the subordinate makes to group productivity via its helping.In this case, the subordinate's outside options may correlate with its contribution to group productivity.
Here, we introduced a link between the subordinate's outside option-assumed to perfectly correlated with intrinsic quality-and the benefit they provide to the group productivity, named the quality-productivity coefficient (QPC).If high subordinate quality is associated with being a good helper, subordinates' quality would correlate positively with group productivity.A positive quality-productivity coefficient may occur in apostle birds (Struthidea cinerea), as older helpers contribute more than younger ones, and outside options are better for older individuals because the juveniles have poorer body conditions (10% less mass) (Woxvold 2004;Woxvold et al. 2006).Similarly, subordinate Seychelles warblers (Acrocephalus sechellensis) that do not help have lower body condition than helpers, and likely lower outside options (van de Crommenacker et al. 2011).Another example is the El Oro parakeet (Pyrrhura orcesi) because subordinate quality, measured as heterozygosity, correlated positively with clutch size and offspring body mass (Klauke et al. 2013).Heterozygosity is positively associated with individual condition and reproductive success (Seddon et al. 2004;Wetzel et al. 2012).With positive QPC, whether subordinates develop into helper or breeders will depend on the conditions (Tibbetts et al. 2018).
If instead subordinates either have high breeding ability or high helping effect then the QPC will be negative.This is the case in the paper wasps Polistes dominula where experimentally increasing subordinates' outside option decrease their cooperative foraging effort (Grinsted and Field 2017), which suggests a negative correlation between subordinate quality and helping effect.

Incorporating the quality-productivity coefficient
Here, we describe how we refined the standard three models.Note that we standardised the parameter symbols, so they do not all match those of previous presentations.The dominant and the subordinate share a symmetrical relatedness r, the subordinate quality is x, the proportion of reproduction that is by the subordinate is y, and k is group productivity (Johnstone 2000), (Table 2).In the concession model, the minimum proportion of reproductive share that a subordinate will accept to stay in the group and not breed independently is defined as y min (Johnstone 2000).We extended the simple tug-of-war model (Reeve et al. 1998a).We used the subordinate efficiency in converting resources into reproduction as a proxy for subordinate outside options (quality) (Johnstone 2000), whilst keeping the focus on the tug-of-war model by setting the minimal and maximal subordinate shares as 0 and 1, respectively.Group stability varied with subordinate outside option and the quality-productivity coefficient (Figure A2).

Concession
The dominant's quality is assumed to be unity, which is the maximal subordinate quality.Hence, the solitary breeding fitness of a subordinate is its quality (i.e.outside option) x plus its relatedness to the dominant r.The minimal reproductive share conceded to the subordinate by the dominant that will induce them to stay is the value of y (y min ) at which its inclusive fitness if it stays in the group (l.h.s.) and if it leaves to breed independently (r.h.s.) are equal: By re-arranging Eq. (1), we find

Restraint
In the restraint model the subordinate takes a maximum proportion p max of the group reproduction, above which the dominant would do better by breeding alone and so evicts the subordinate from the group (Reeve and Ratnieks 1993;Reeve et al. 1998b;Verhencamp 1983).The maximum subordinate's reproductive share is the value of y at which the dominant's inclusive fitness if it accepts the subordinate (l.h.s.) and if it evicts the subordinate and breeds alone (r.h.s.) are equal.
Rearranging Eq. ( 3) gives the maximum subordinate reproductive share

Tug-of-war
In the tug-of-war model the dominant and subordinate compete over the share of reproduction by investing in the reproductive conflict with d and s effort, respectively.The dominant has a relative competitive ability, which here we assume to be equal to their relative quality ( x ∕ b ).The decision is assumed to be simultaneous.The equilibrium levels of d and s (d* and s*) are those at which the dominant and subordinate can do no better by changing it.

Incorporating the quality-productivity coefficient
We derived the predictions of the concession, restraint and tug-of-war models after including an association between subordinate quality and their helping effect (i.e.QPC, a) by assuming a linear relationship, and that the dominant's quality is added to the effect of the subordinate's quality where b is the dominants' quality, m is the baseline benefit cooperating, x is subordinate's quality and a is the QPC (Table 2).
We replaced k as given by Eq. ( 5) in Eq. ( 2) to get the optimal reproductive share in the concession model We replaced k as given by Eq. ( 5) in Eq. (3) to get the optimal reproductive share in the restraint model In the tug-of-war model, k is the multiplier of the payoff that subordinate and dominants invest s and d, respectively, into the struggle for reproduction, which reduces group productivity.With subordinate ability being x/b, the evolutionary stable share of the subordinate is.
Which simplifies to.

Predictions of the modified models
Numerous empirical tests have found non-significant results for the association of most interest: the effect of relatedness r on reproductive skew y (Nonacs and Hager 2011), which may be due to insufficient statistical power if the effects are weak.That is, any evolved response to be measured may be slight or the selective pressure may be too weak to cause the evolution of a response.It may be that there is no selection at all, or that there is selection to not respond.However, no study to our knowledge has quantified the predictions of the models of skew, so models have never predicted when there will be only a weak effect of included parameters.Here, we quantified the relative effect size and identified when the three models might predict a weak and so statistically non-significant effect (Online Appendix, Tables A6, A7).To do this, we chose two values for each parameter and calculated the difference in the skew y relative to its magnitude (i.e.standardised effect size).We then found when the standardised effect size is less than a value assumed to be empirically detectable.We chose 10% for the standardised effect size, but the insights are not sensitive to the particular value used.
Incorporating the QPC alters the predictions of the concession, restraint and tug-of-war models (Fig. 1).All three models predict a positive or no association between relatedness and skew in some circumstances (Fig. 1a, f, k).The concession model predicts that skew will correlate positively (purple areas) with relatedness when subordinate quality is high and the QPC is negative (Fig. 1a), and when subordinate quality is low and QPC high.The concession model predicts skew correlates negatively with relatedness when the values of subordinate quality and QPC are equivalent (white area, Fig. 1a).The restraint model makes the opposite predictions to the concession model about the effect of relatedness on skew (Fig. 1f).
The concession model predicts a positive correlation between skew and subordinate quality, unless both relatedness and QPC are strongly positive, in which case x and skew correlate negatively (Fig. 1b).Thus, there are situations where increasing outside options increases the dominant's share.The predictions of the restraint model about x also change if the QPC is positive, from greater skew to less skew if x increases (Fig. 1g).
The three models predict dominant quality b should not influence skew (Fig. 1c,h,m).Indeed, as a relatedness over 0.75 is not biologically realistic, the negative link between skew and b predicted by the concession model is deemed irrelevant.
The link between group benefit m and skew was different for each model but not sensitive to a nor x.The concession model predicts skew decreases (Fig. 1d,e), whereas the restraint model predicts skew increases with group benefit m (Fig. 1i, j).The tug-of-war model predicts no correlation between group benefit and skew.
Adding the QPC did not change the predictions of the tug-of-war model about the skew, that skew should increase with the dominant's competitive advantage b and decrease (or do not correlate) with relatedness r (Reeve and Keller 2001;Reeve et al. 1998a).This was despite their effect on the competitive effort (d* and s*; Eq. 2.0 and 2.1).

Associations with relatedness
For studies for which we could find effect sizes we analysed the overall effects of parameters on skew.For many studies we could only get a direction of the effect (positive, negative, or non-significant), so for these we analysed counts.

Direction of effects
The sampling effort did not differ between insects and vertebrates (Binomial test, p = 0.471, N = 45).Most studies reported no correlation between skew and relatedness (Binomial test, p = 0.011, N = 45), which supports the tug-of-war model (Fig. 1k).There were more studies on females than on males (Binomial test, p = 0.027, N = 41).

Size of effects t
We now consider studies of the association between skew and relatedness for which computing the effect size was possible (N = 16) (Figure A1, see Methods in Online Appendix).The average Hedges' g for all species (mean = 0.533 ± SD 0.798) is significantly positive (one-sample t test t 16 = 2.85, p = 0.011, N = 16).The positive effect sizes averaged to 0.721 (± SD 0.622) and the two negative ones averaged to − 0.780 (± SD 0.764).Removing the large effect sizes (> 1 or < − 1) does not change the results, as the average Hedges' g for all species (mean = 0.382 ± SD 0.346) is still significantly greater than zero (one-sample two-tailed t test t 11 = 3.15, p = 0.010, N = 11).This suggests reproductive skew increases with relatedness (Fig. 3), which is predicted by more of parameter space in the tug-of-war and concession models than the restraint model Fig. 1a,f,k).The studies which reported no significant correlation between relatedness and skew still show a large positive effect (0.41 ± SD 0.441, N = 10).A very large positive effect (1.41 ± SD = 0.570, N = 4) and a large negative effect (− 0.600 ± SD = 1.018,N = 2) were found in studies that reported a positive and a negative effect, respectively.
The effect did not differ with sex (ANOVA F 14,15 = 1.11, p = 0.311).However, female vertebrates and insects did not differ (ANOVA F 11,10 = 2.27 × 10 −5 , p = 0.996), and no insect male data was available.Male insects might differ from female insects, so there may be a hidden sex difference (Fig. 3a).

Associations between skew and other factors
For factors other than relatedness there were insufficient data on effect sizes (Table 3), so we analysed direction data only.

Outside options and skew
The association between skew and outside options (subordinate quality) was examined in fewer insect than vertebrate studies although this was not significant (Binomial test, p = 0.125, N = 7) (Table 3), and in as many female as male studies (Binomial test, p = 0.508, N = 9).Similar numbers of studies reported a nil, positive and negative, associations between outside options and skew (χ 2 = 1.273, df = 2, p = 0.529, N = 9) (Fig. 4a).The direction of the association between subordinate quality and skew was not biased by taxa (Fisher's exact test, p = 0.679, N = 9) nor sex (Fisher's exact test, p = 0.740, N = 9).Only one study examined haplodiploids.

Group productivity and skew
There were equal numbers of studies of the association between group productivity k and skew was equal in insects and vertebrates (Binomial test: p = 0.316, N = 19) (Fig. 4c), but there were more studies of female than males (Binomial test: p = 0.0413, N = 20) (Table 3).
In summary, the dominants' competitive advantage b and group productivity k were not associated with skew in insects (Binomial test, p = 0.07, N = 17) (Fig. 4c), suggesting that other factors influence skew Fig. 1c, h, m).In vertebrates, by contrast, the association between b and k on skew was not clear which may imply strong between-species differences in the selective pressures and current mechanisms to decide skew.

Test of co-occurrence of positive/negative associations
We tested the co-occurrence of different directions for the associations to detect potential patterns (N = 24, Table A3, A6).No clear pattern emerged, as e.g.within birds no species showed similar directions for relatedness and skew nor for outside options and skew.The association between relatedness and skew was independent of the association between group productivity and skew (Fisher's test, p = 0.564, N = 11).Similarly, the results of the association between relatedness and skew were independent to that of the association with outside options (χ 2 = 5, df = 4, p = 0.287, N = 11).

Empirical support for models of skew
We focused on the associations of skew y with: relatedness r, outside option x, dominant's advantage b and group productivity k, for which our new models makes clear predictions.Overall, for the baseline parameter values (dotted lines in Fig. 2), 67.5% of empirical studies matched the predictions of the tug-of-war model, while 26.0%fit the predictions of the concession model and 18.2% of the restraint model (Table A2).However, we have shown that the QPC may affect these predictions (Fig. 1), making it essential to measure the QPC empirically.

Summary of support for the models
A previous review found that 21 over 27 studies (78%) reported no correlation between skew and relatedness, and concluded that skew theory did not apply to within-species variation in skew (Nonacs and Hager 2011).In our review, 31 over 45 studies (69%) showed no correlation between relatedness and skew.However, as we have shown, the models will sometimes predict non-significant effects, such as the concession and restraint models when subordinates are high quality and the QPC is medium (Fig. 1a, f).By incorporating the QPC, we are in a position to better understand the variation in empirical findings.

Relatedness
The analysis of effect size indicated a positive association between relatedness and skew which supports the restraint model, or the tug-of-war model for low subordinate quality (or the concession model for high subordinate quality and low QPC) (Fig. 1a,f,k).By contrast, the comparison of count data found that most studies showed no correlation, which supports, for high subordinate quality only, the tug-of-war model (or both concession and restraint models).The conflicting results are likely due to the methodological differences, as effect sizes capture not only the direction but the magnitude of the association (Haddaway et al. 2020).The conflicting results may also arise from the mere difference in sample size (N = 16 versus 45), since few studies reported sufficient details to calculate the effect sizes.The meta-analysis is more reliable than 'count voting' (Haddaway et al. 2020), yet the small sample size may prevent from assessing if the association is nil or positive.However, neither method supported a negative correlation, so the concession model is overall not supported (except for specific cases where bad helpers have high outside options) (Fig. 1a).The new predictions suggest that skew should correlate positively with relatedness in most of the parameter space of the models (Fig. 1a,f,k).
Relatedness and skew were not significantly correlated in any mammalian studies, but were positively correlated in several bird studies.Philopatric mammalian social groups are often highly related and subordinates may engage less in extra-pair mating than birds, potentially because the social groups are further apart and/or less tolerant (Beekman et al. 2003;Chen et al. 2021;Kingma et al. 2011;Sharp and Clutton-Brock 2011).Consequently, individuals may have fewer opportunities to breed as subordinates (whilst avoiding inbreeding).Furthermore, mammalian helpers are seldom unrelated individuals (Isler and van Schaik 2012) as they typically are the offspring or siblings of the dominants, so variation in relatedness may simply be insufficient to get significant effects.Since in several birds species good helpers also have good quality, and that some species showed a positive effect of relatedness and skew, the restraint or the tug of model may apply to avian systems (Klauke et al. 2013;van de Crommenacker et al. 2011;Woxvold et al. 2006).

Competitive advantage
The association between dominant's competitive advantage and skew was positive in vertebrates, which does not match the new predictions for any of the models.By contrast, the lack of correlation between dominant's competitive advantage and skew in insects fits with the three models.As dominant's competitive advantage encompassed very diverse measures spanning from size difference to canine length, the studies may have not grasped fully the trait that characterises dominant's competitive advantage.Indeed, measuring a range of traits may be needed to get data that accurately reflects how animals perceive the competitive advantage and make decisions.

Subordinate quality
The direction of the association between subordinate quality and skew was not significantly biased, overall, per sex nor per taxa.This may suggest that the data of subordinate quality are not based on equivalent measures of outside options, but may also reflect that each model makes a range of predictions and so species with different QPC and relatedness may make different predictions.Furthermore, the association between subordinate quality and skew may be more complex than a linear function.
Our results highlight the need to conduct meta-analyses (with standardised effect sizes) for as many empirical tests as possible.A definitive test would be to look at interactions between parameters.For example, measuring skew for high and low quality related and unrelated individuals would help distinguish between the models as they make differing predictions about this interaction.

Group productivity
While most of the examined studies reported no correlation between group fecundity and skew, taxa differed.This between-species variation may be linked to the two main evolutionary pathways to cooperative breeding, the defence of resources in a stable environment or the buffering of adverse variable environments (Lin et al. 2019).

Variation in the quality-productivity coefficient
Our review reveals a lack of empirical tests of the association between skew and the quality-productivity coefficient (QPC), despite the theoretical implications of including the dependence of quality on group productivity.The QPC is the parameter that transforms subordinate quality into its helping benefits.A high QPC means that the strong subordinates are very helpful.Variation in the value of the QPC may explain the inconclusive results regarding the association between skew and relatedness, as the predictions of the concession and the restraint models changed with QPC.Our findings suggest QPC is useful to disentangle subordinate's quality from its effect on group productivity, and that empirical studies would benefit from assessing it (beyond helper's effect).Further studies on QPC are needed to determine where in the prediction plot (Fig. 1) the species is, and determine its association with skew.
As most of the species for which several tests have been conducted fit the predictions of two or three models (Table 4), it proves impossible to assign them to a model without further data on subordinate quality, relatedness and QPC.However, most of the wasp species seem to support the tug-of-war model, which is in line with a review that indicated that the concession and restraint models do not explain skew in paper wasps Polistes (Nonacs 2006).One species of wasp fit with the predictions of the concession model in pairwise (but not between nest) comparisons of skew.
Incorporating the QPC influenced the predictions of the tug-of-war model, as skew did not correlate positively with dominant's competitive advantage (Fig. 1m).In line with the predictions of the original tug-of-war model, relatedness did not correlate with skew-although only for strong subordinates (Fig. 1k).The tug-of-war model implicitly links subordinate quality and their contribution to group productivity, as the individual investment in the reproductive conflict decreases group productivity (due to an assumed trade-off).Unlike in our model, in the tug-of-war model group productivity does not influence skew, as individuals lack outside options.Indeed, in the tug-of-war model the quality did not refer to subordinate outside options, but to the subordinate's ability to fight or to invest in group productivity, which suggests that other models with QPC may be more comparable to our model.
In the general costly-young model, individuals can cooperate to raise the young by breeding cooperatively but compete over the share of reproduction (Shen et al. 2011).This model of cooperation explored the link between quality and group productivity.The dominant has lower costs to produce a young than the subordinate.The general costly-young model predicts that group productivity increases and skew decreases with subordinate investment into care (Shen et al. 2011).This prediction, contrary to the predictions of the tug-of-war model, suggests that subordinate quality correlates positively with group productivity.
Incorporating the QPC changed the predictions of the concession and restraint skew models concerning subordinate quality.In the presence of a positive correlation between skew and subordinate quality, as was found in meerkats (Suricata suricatta) (Cram et al. 2019), we would have concluded that the restraint model applies without incorporating the QPC, but because of the QPC we conclude that this finding is not sufficient to determine which model applies (Table 4).Indeed, we cannot rule out the concession model as it predicts such positive correlation in their parameter spaces, and the restraint model predicts a negative correlation and no correlation in some of the parameter space.Consequently, variation in the QPC might explain the positive and negative correlations between female meerkats skew and subordinate quality (Clutton-Brock et al. 2001;Cram et al. 2019).
The QPC altered the predictions of the link between skew and relatedness, which changed the interpretations of empirical findings.For instance, in white-winged choughs (Corcorax melanorhamphos), where skew correlates positively with relatedness (Heinsohn et al. 2000), we would have concluded that the concession model fits this species (and rejected the restraint and tug-of-war models) without incorporating the QPC.Yet, because of the QPC, the restraint and tug-of-war models are applicable in a larger portion of the parameter space than the concession model.We would need the value of the QPC, relatedness and subordinate outside options to determine whether the date supports the concession, tug-of-war and restraint models.In a nutshell, our new theoretical findings change the conclusions that can be drawn from the empirical literature, which highlights the importance of considering different aspects of individual variation as distinct and potentially correlated.

Which empirical tests do we need?
We did not find sufficient, good-quality evidence to judge whether one model of skew is the most applicable, and to which taxa.We showed here that making group productivity dependent on helper's outside options altered and complicated the predictions of the original model.Indeed, where one original model predicted a single direction, most of the new predictions vary with subordinate quality or relatedness.However, the predictions of the link between skew and benefit to group productivity of one subordinate (i.e.helping effect) give clear-cut distinctions between the concession, restraint and tug-of-war models.Importantly, the direction of the association between skew and group benefit should vary with the processes/strategies (i.e.model), but not depend on subordinate quality nor on how much help a subordinate provides as a function of their quality (QPC).These unambiguous predictions make it possible to test which model applies by measuring genetic data to determine skew and the increase in group productivity provided by one helper to determine group benefit.Future data collection of this group benefit, in a wide range of taxa, may be useful to understand diversity in cooperative breeding.We suggest that unrecognised variation in factors such as the quality-productivity coefficient makes evidence support several models (i.e. one empirical association predicted by one model and another by another model).Hence, adding the quality-productivity coefficient might give more realistic predictions that may better match empirical data, and may especially explain sex differences in skew with sex differences in QPC or other factors.For instance, the bird Porphyrio porphyrio fits the original predictions of all models: male skew decreased with relatedness, supporting the restraint model, whereas female skew increased with relatedness supporting the concession model.The higher skew with higher constraints (lower x and lower b) fits with the concession and restraint models.P. porphyrio may behave following our new restraint model, if subordinate quality is associated with very high benefits (high QPC), and if females have higher subordinate outside option than males.
This review sheds some light on the need to collect data for both males and females for each taxa.In the only taxa for which we were able to calculate the size of the association between relatedness and skew for both sexes, birds, females had a smaller effect than males.This may suggest that males are operating within the tugof-war window of the synthetic model, whereas females sometimes operate within the restraint model window-conversely these results may stem from between-species variation.Sexual selection and the sex-specific costs of reproduction may drive females and males to make decisions based on different parameters when deciding whether to disperse and how much to help within a group (Boomsma 2009;Creel and Creel 2015;van Boheemen et al. 2019).Coalitions of brothers for instance may cause one male to monopolise reproduction and get related helpers (Gottelli et al. 2007;Krakauer 2005), who can then become dominant more easily than if they were on their own (Schülke et al. 2010).To our knowledge no models of skew have been tested on male insects, although sex-specific selection forces may be strikingly different in male and female insects due to haplodiploidy.We found only one study on female vertebrates (for 11 on males) which tested the association between relatedness and skew.Further empirical tests of the association between relatedness and skew of female vertebrates and male insects (termites) will help disentangle the potential taxa and/or sex differences.Empirical tests of females and males would strongly inform our understanding of social evolution across taxa.
Our review revealed the gaps in tests of the models of skew in some insects and in fish.Although some well-established study sites and species have provided good insights and built our understanding on particular species (e.g.Polistes), further tests of all cooperative breeders would ensure the veracity of the models on a range of taxa.To push the boundaries of this field we need empirical tests in termites where only one empirical tests exist, in other insects and in cooperatively breeding shrimps.
The field of social evolution would benefit from precise tests of the theory of skew in species where long-term field work exists.Indeed, the genotypic data for calculation of the effect sizes on long-term field projects is not readily available yet, either because it was not measured, or it is unpublished (Table A5).Over the 45 studies reporting a (negative, positive or nil) association between skew and relatedness, only 16 provided sufficient data to calculate the effect (e.g.only the average skew was reported, not the skews for related vs unrelated).To test the predictions accurately with meta-analysis, studies should collect and report the reproductive skew for each category (average ± SD or standard error, e.g.related vs unrelated) along the sample size of each category, and detail the statistical tests.We need more information about the mating systems (who reproduces with whom) and the parental care systems (who takes care of the offspring) (Kappeler 2019) to further advance our understanding of the factors that influence the diversity in skew and cooperative breeding (Makuya et al. 2021).
The optimal skew has been shown to vary with environmental, social and individual factors, which combined with our updated predictions of the three models of skew suggests that instead of matching empirical results to a simple single prediction per model, we should aim to determine the parameter space in which species lie.Empirically, environmental quality such as food availability and the number of helpers significantly correlates with breeding success and skew in female meerkats (Field and Cant 2009) and banded mongoose (Nichols et al. 2012).Harsh environments by contrast imply lower solitary breeding success (variation in dominant's competitive advantage b) and higher benefits by the helpers (variation in QPC and/or group benefit m).Rather than looking at a static image of reproductive skew in one (or several) population(s), and study reproductive skew in isolation, it may be fruitful to consider all the factors from the original models of skew, allowing for correlations among the parameters such as individual quality and group productivity.

Fig. 1
Fig. 1 Direction of predicted association between skew and parameters of the skew models: a-e concession, f-j restraint, k-o tug-of-war.Areas indicate a positive (dark purple), negative (white) or non-significant (yellow) difference in y* for two values of: a, f, k relatedness r; b, g, l subordinate quality x; c, h, m dominant's competitive advantage b; d, i, n group benefit m when outside option and group productivity are not associated a = 0, m = k; e, j, o group benefit m when outside option and group productivity are positively associated a = 0.5.Each area is the comparison between two values for a given parameter that is indicated in the top column: r = 0, 0.5, x = 0.35, 0.65, m = 0, 0.5.The difference was considered non-significant if its absolute value was smaller than 10% of the skew at the lower value of each parameter: e.g. if |y * (r=0.5)−y* (r=0)| y * (r=0) < 0.1 .Horizontal dotted lines indicate a = 0, so show the predictions of the original models, and m = 0.6 − a as otherwise groups are always preferred (for any y) if a is large and never preferred if a is small

Fig. 2
Fig. 2 Number of species showing a positive, negative or nil link between skew and relatedness, as a function of a sex, b group and c taxa.N = 45

Fig. 3
Fig. 3 Effect (mean Hedges' g ± 1 SE) of the link between skew and relatedness as a function of a sex (females: light red, males:dark red), b group and c taxa.Insects: dark green, vertebrates: yellow.N = 16

Fig. 4
Fig. 4 Number of species showing a positive, negative or nil link between skew (y) and a subordinate quality x, b dominant's competitive advantage (quality) b (N = 17) and c group productivity k (N = 19) in insects and vertebrates (N = 7)

Table 2
Parameters used in the models of reproductive skew and the range of values; baseline values

Table 3
Sample sizes of the effects empirically tested in insects and invertebrates.Studies with data for males and females were shown as two data points Unclear results are not displayed (n > 1 studies per group, N = 111 total with all links and unclear results.)

Table 4
Empirical tests of our new predictions (species with ≥ 2 different factors tested) Models are the skew models whose predictions fit with the empirical tests once the QPC is incorporated.Factors are r,