Greater wealth inequality, less polygyny: rethinking the polygyny threshold model

Monogamy appears to have become the predominant human mating system with the emergence of highly unequal agricultural populations that replaced relatively egalitarian horticultural populations, challenging the conventional idea—based on the polygyny threshold model—that polygyny should be positively associated with wealth inequality. To address this polygyny paradox, we generalize the standard polygyny threshold model to a mutual mate choice model predicting the fraction of women married polygynously. We then demonstrate two conditions that are jointly sufficient to make monogamy the predominant marriage form, even in highly unequal societies. We assess if these conditions are satisfied using individual-level data from 29 human populations. Our analysis shows that with the shift to stratified agricultural economies: (i) the population frequency of relatively poor individuals increased, increasing wealth inequality, but decreasing the frequency of individuals with sufficient wealth to secure polygynous marriage, and (ii) diminishing marginal fitness returns to additional wives prevent extremely wealthy men from obtaining as many wives as their relative wealth would otherwise predict. These conditions jointly lead to a high population-level frequency of monogamy.


Notes on the data taken from the Standard Cross-Cultural Sample
In two cases, we needed to revise the Standard Polygamy Code (variable 861): for Toda, and for Basseri.
The Toda were coded as 0 ("polyandry," even though Cultural Basis for Polygamy (variable 860) is coded as 2 ("monogamy preferred but exceptional cases of polygyny"); we thus changed Standard Polygamy (variable 861) to 2 ("Monogamy preferred, but exceptional cases of polygyny") on basis of White [1], who refers to exceptional cases of polygyny.
The Basseri have a missing value for variable 861. From White [1] and Barth [2, page 107], we coded (conservatively) Standard Polygamy 861 as 3 ("Limited polygyny <20% of married males") on the basis of: "wealthy herd owners, with additional labor needs, frequently have plural wives who extend a man's fecundity in a way that saps his wealth." It could plausibly be recorded as 4 (>20%) [2].

Wealth proxies
A possible concern related to the cross-cultural compatibility of our estimates is that our rival wealth proxies vary between populations and productions systems-see Table 1 from the main text. As such, it remains possible that vari-ation in the wealth measures used is responsible for variation in our estimates. For example, had we chosen a different rival wealth proxy for a given population-e.g., the value of household items instead of land owned-we may have obtained a different estimate of µ, and hence δ, in that population. In cross-cultural projects as wide-ranging as this one, however, there is rarely a single variable that can be compared directly across populations-instead, we have relied on ethnographic accounts to identify which sources of wealth are most relevant to production and reproduction in each society, and attempted to build a cross-culturally comparable data set by using the most locally relevant measures of wealth in each population.
Our proxy of rival wealth in foraging societies (weight) is especially problematic insofar as it undoubtedly also captures important elements of non-rival wealth, such as health or foraging skill; however, our choice here is constrained by the very fact that foragers hold small and relatively unvarying amounts of material wealth. We therefore have little choice but to treat weight as a proxy for access to material resources. Little in our overall argument, however, is affected by this methodological choice in foraging populations.
Among agriculturalists, the Polish sample is notable for its relatively low Gini coefficient on rival wealth (specifically, land). Similiar estimates have been shown in other studies [3]. This study site is situated in an area with poor soils and long, hard winters, and the area was never particularly well-suited for large-scale agriculture for these reasons [4]. More than 80% of farming households in the sample own land that is classified in the lowest official grades [4]. Most importantly, the population has a long history of small-holder farming and a partible inheritance system [4]. In concert, these social norm reduced plot sizes over time since the late 1700s, and have likely contributed to the smaller-than-typical Gini coefficient on rival wealth.

Percent female polygyny at equilibrium
If a man marries n women, the non-rival wealth available to each wife is g and the rival wealth available to each wife is m−cn n . Here, m refers to the total rival wealth of a male, and c refers to the cost of mating investment. In the Oh et al. [5] model, each wife produces offspring as a function of the wealth she has been provided by the male, adjusted for the importance-γ and µ-of each type of wealth to fitness. The fitness, w, of a male is then given by the effective number of wives acquired by the male multiplied by their average fitness: If we consider only two classes of men, the rich and poor, with the rich males being indexed by r and the poor by p, then females can optimize their reproductive success by pairing with a (possibly married) rich male so long as the following condition is satisfied: tions to derive (see [5]) an analytic expression for the equilibrium number of wives of the rich men, n * , Now, by differentiating the right-hand side of Eq. 5 with respect to δ, we have: The value g γ r n (δ−µ−1) (m r − cn) µ log(n) is positive so long as n > 1-which will always be true if the prospective bride is to marry as a co-wife-and m r > nc-which must also be true if the original equality were satisfied.
Since this derivative is positive for all plausible model parameters, a decrease in δ will decrease the prospective female's fitness with the wealthy man below what it was under the higher δ value.
It is then apparent from Eq 6 that a lower δ would, holding other terms constant, decrease the supply of females to polygynous marriage.
A man who was just barely rich enough so that an unpaired woman would choose to marry him as wife number (n+1) under the initial δ, would, under the lower δ, be unable to secure the unpaired woman's partnership. An increase in the extent of diminishing returns to additional wives (lower δ) therefore reduces both male demand for, and female supply to, polygynous marriage.
Under the Oh et al. model [5], if at least some women marry monogamously, and if Eq.
3 holds, then there is a closed form solution for P at equilibrium, given by: Conversion factor n * , wives per rich man (7) This follows from the fact that in a population with N m males and N f females, and a given n-polygyny level, n * , there will be θN m males who marry a total of θN m n * females. Percent female polygyny is then:  (9) where λ gives the importance of each female's biological contribution to offspring production, l, and α controls the extent to which additional wives diminish a given female's ability to reproduce. When α = 0, this model reduces to the Oh et al [5] model, but when α > 0, additional wives diminish the fitness of other wives for reasons other than dilution of rival wealth resources.
If we assume that male demand is limiting, then n * can be written as: but there is no simple solution for the female supply condition. If we look at the of derivative of n * in Eq. 10 with respect to α: we find that it is always negative. Increasing the extent to which females' biological contributions to reproduction are diminished by addi-tional wives, drives down male demand for additional wives. This model leads to the same qualitative findings as in the main analysis. Empirically, it is possible that polygyny might enhance female reproductive rates (e.g., α < 0) [9], but evidence for such an effect in humans is very limited [10]. In the main text, we state that the partial derivative of the Gini coefficient with respect to θ is negative whenever: To calculate this, we write: (13) and then elvauate when the right-hand side of Eq. 13 is less than 0.
From the main text, if male demand is limiting, then percent female polygyny is given by: The Gini coefficient on rival wealth is given by: The ratio of percent female polygyny to wealth inequality, P/G, is thus: so long as the parameters are such that 0 < P < 1. The derivative of Eq. 16 with respect to θ is: This value is positive as long as δ > µ, m r > 1, and s, c > 0, which are fundamental assumptions of the model. As such, the ratio of percent female polygyny to the Gini coefficient on rival wealth is monotonic and decreasing as θ, the fraction of rich males, decreases towards 0.
As wealth becomes concentrated by a small, rich elite, there will be lower levels of polygyny for a given level of wealth inequality.

Estimating percent rich
In our theoretical model, we assume a discrete two-class wealth distribution, but empirical wealth data typically have continuous distributions. To measure percent rich, we consider the rival wealth distribution, M , of a single population with N males, sorted in decreasing order, and define the condition: Then, for a given φ ∈ (0, 1), we can calculate the minimum number of men in the population, In the final, empirically motivated estimateψ in the main text-we calculate the average wealth of men with one and two wives, and define ψ to be the percentage of men who have more wealth than the average of these two numbers. Our analysis is robust to inclusion or exclusion of the two populations where income rather than wealth estimates were provided.

Historical agricultural populations
[ Table 1 about here.] We replicate Fig. 6 from the main text in Fig.   2. We find that our results in the this supplementary analysis are qualitatively similar to our findings in the main analysis, but our confidence intervals are now much narrower in support of our predictions.

Estimating wealth elasticities
To estimate the importance of rival wealth and wives to reproductive success in each population, p, we use a Cobb-Douglas function, which implies that the log of predicted reproductive success in male i, Λ [i] , is given by: where each male's exposure time to risk of reproductive success (i.e., number of years lived in the age range between 13 and 60 years) is given by E [i] , number of marriages is given by In cases where the vectors containingM [i,1] or 2] contain zeros, we add a small constant. In the case of foragers, where weight was used as a proxy for wealth, we subtract a constant slightly less than the minimum weight value in order to yield a reasonable location for the zero of the wealth vector.
The shadow price parameters are given lognormal priors: The parameters controlling reproduction outside of marriages are given uniform priors on the unit interval:η The inverse scale parameters are given weak but proper, positive-constrained priors: where the symbol T [0, ] indicates truncated support > 0.
We use a multi-level model to estimate the elasticity parameters: The elements of Θ are given weak priors: The covariance matrix, Ω, is defined as: where the elements of σ have weak but proper, positive-constrained priors: 10 Age-adjustment In order to use all cohorts of the adult male population, relevant measures-wives and wealthare age-adjusted to represent their predicted values at the age of 60 years. To age-adjust wealth, we model: where: Note thatĒ = 60 gives the maximum possi-  [ Figure 3 about here.]

Replication and open science
Interested readers can replicate our analysis using the model code and data included at: https://github.com/ctross/ publications/polygynypuzzle. Analysis is conducted using R [15] and rstan [16].  List of Tables   1 A sample of historical agricultural and/or material wealth limited populations. The numerical columns reflect the value of the frequency of rich males, θ, when the value of φ, the cumulative wealth owned by the rich class, is set at various values. . 19 Table 1: A sample of historical agricultural and/or material wealth limited populations. The numerical columns reflect the value of the frequency of rich males, θ, when the value of φ, the cumulative wealth owned by the rich class, is set at various values.