In order to produce analytically tractable results, we simplify by assuming throughout that there are only two types of males, rich and poor, with rich males being a fraction u of the population. All rich are identical, as are all poor. The rich males are indexed by r and the poor by p. We now demonstrate two theoretical results with the potential to resolve the polygyny paradox. First, diminishing returns to additional wives arising from causes other than necessity to share a husband’s rival material wealth will reduce the number of wives acquired by each rich male. Second, because of this fact, a highly unequal wealth distribution with few extraordinarily rich men may produce little polygyny, while a less unequal wealth distribution with a larger fraction of rich men may produce a greater extent of polygyny. Two rich men, for example, can be expected to have more wives in total than one very rich man whose wealth equals their combined wealth. For this same reason, the Gini coefficient—see table 2 for a definition—is not a sufficient statistic for the analysis of the relationship between polygyny and wealth inequality. We take up each of these results, in turn, before assessing if our empirical estimates are consistent with this explanation.If we assume that male demand is limiting, then equation determines the number of wives each rich man will have. It is clear from inspection of equation that a greater extent of diminishing fitness returns to additional wives produces a lower male demand for additional wives. This is demonstrated mathematically in the electronic supplementary material. Determining the effect of greater diminishing returns to additional wives when female supply is limiting is more challenging. As noted above, if female supply is limiting,1 litre square plant pots the value of n* implied by the polygyny threshold inequality in equation has no closed form solution.

To address this challenge, we proceed as follows.Then a reduction in d, holding all other terms constant, would reduce the right-hand side of the equation—the fitness of each of the n wives of a rich man—while having no effect on the left-hand side—the fitness of a singleton wife. Thus, holding all else equal, an offsetting decrease in n would be required to restore the equality. This is demonstrated in the electronic supplementary material by differentiating equation with respect to d. This means that a man who was just barely rich enough so that an unpaired woman would choose to marry him as wife number under the initial d, would, under the lower d, be unable to secure the unpaired woman’s partnership. As such, an increase in the extent of diminishing returns to additional wives will reduce both male demand for, and female supply to, polygynous marriage. Our results imply that if d , 1, a larger quantity of moderately rich men can be expected to have more wives in total than a smaller quantity of even richer men, holding constant the total wealth held by the rich across these cases. This first finding will interact with our second finding, discussed below, concerning the effects of the population density of the rich class of men on the frequency of polygyny and the level of wealth inequality. In the electronic supplementary material, we present an alternative approach to account for diminishing marginal returns to increasing number of wives and find that our insights do not depend on the specific way in which diminishing fitness returns to increasing number of wives are modelled.To address prediction P1, we present empirical estimates of m and d . These values are estimated using a multi-level regression model fit to our individual level data; methodological details are provided in the electronic supplementary material. In all but four of the populations in our sample, the estimated d coefficient is reliably less than 1. This result provides cross-cultural empirical support for the first of the two conditions needed to generate a transition to a greater degree of monogamy with increasing wealth inequality.

Note two further results also shown in figure 5. First, our estimates for m are quite low, particularly across the agricultural economies. Second, our estimates of d 2 m are positive in almost all populations, including those that are concurrently polygynous and those that are serially monogamous. The consistently small values of m across all of our samples, even the monogamous ones, was unexpected. However, these low values reflect changes in male fitness per wife. Because of biological limits to the rate of reproduction in human females, significant increases to wealth are constrained to have less than proportional effects on fitness per wife. The effects observed here are more likely to reflect the ability of males with more than a threshold level of resources per wife to minimize offspring mortality, rather than to significantly enhance their own fertility. Though not discussed in detail here, our data suggest that male wealth impacts male fitness primarily by increasing the rate of wife acquisition rather than by increasing reproductive success per wife . Our second point addresses the possible concern that our estimates of d may be low, in part, because we use times married as our measure of polygyny. While it is true that men can accumulate a greater maximal number of marriage years through concurrent polygyny than serial monogamy, figure 5a demonstrates that the use of times married is an appropriate measure of polygyny for our purposes. Across almost all populations, the elasticity of fitness with respect to times married, d 2 m, is positive and reliably non-zero. Because these estimates measure the population-specific effects of cumulative number of wives on reproductive success, they demonstrate that an increased number of marriages leads to increased reproductive success in both types of marriage systems—concurrently polygynous and serially monogamous.We have established that there exists a strong cross-cultural pattern of decreasing—but reliably non-zero—fitness returns to increasing number of wives for reasons beyond rival wealth sharing.

We now turn our attention to testing if the transition to agriculture is associated with a decreasing fraction of wealthy males. In our theoretical model, we assume a discrete two-class wealth distribution, but empirical wealth data typically have continuous distributions. To deal with this issue, we consider two proxy measures for per cent rich in our empirical data: the minimum percentage of men that account for a fraction f of the total wealth and the frequency of men with more than c wealth, where c is the empirical midpoint in each population between the average wealth of males with one wife and the average wealth of males with two wives. More details about these metrics are included in the electronic supplementary material. Table 3 provides population-level posterior estimates of the completed wealth and completed polygyny measures, with the mean estimates by subsistence type shown in the bottom panel. To address prediction P2, we calculate empirical estimates of the fraction of rich men by production system . We find that agricultural populations have a significantly reduced frequency of wealthy individuals relative to horticultural populations. All four panels show reliable differences in mean per cent rich between the horticultural and agricultural subsistence modes. This lesser fraction of wealthy individuals suggests a decreased number of men both able and willing to take second wives. This in turn leads to reduced levels of per cent female polygyny in contexts where large wealth differentials are not able to underwrite large differentials in wives due to the existence of diminishing fitness returns to such additional wives. A limitation of this last result is that it is based on data from only four agricultural populations, three of them concentrated in a restricted region and time period . Moreover, a more informative dataset would come not from agricultural populations in the time period between the 1700s and 2000s, but rather from the agricultural populations in which monogamy actually began to emerge denovo. In our main analysis, we use estimates derived from the individual-level records available in the populations shown in table 1; in the electronic supplementary material,macetas por mayor we present comparable analyses that include 14 additional wealth distributions from historical agricultural populations. The results of this supplementary analysis are consistent with our arguments here—and in fact show stronger and more reliable effects in the direction predicted by P2. These supplementary data, however, are based on sometimes contested reconstructions of the historical wealth distributions pieced together by archaeologists and economic historians;they must be appreciated within the constraints associated with such forms of data.Using individual-level data from 29 populations, we show evidence of a general cross-cultural pattern of decreasing marginal fitness returns to increasing number of marriages. Further, using these same 29 datasets , we demonstrate the existence of an increasingly skewed distribution of material wealth in class-based agricultural societies . Both of these empirical findings are consistent with our model-based explanation for the decline of polygyny in societies engaged in agricultural production.We use cross-cultural data and a new mutual mate choice model to propose a resolution to the polygyny paradox. Following Oh et al., we extend the standard polygyny threshold model to a mutual mate choice model that accounts for both female supply to, and male demand for, polygynous matchings, in the light of the importance of, and inequality in, rival and non-rival forms of wealth.

The empirical results presented in figures 5 and 6 demonstrate two phenomena that are jointly sufficient to generate a transition to more frequent monogamy among populations with a co-occurring transition to a more unequal, highly stratified, class-based social structure. In such populations, fewer men can cross the wealth threshold required to obtain a second wife, and those who do may be fabulously wealthy, but—because of diminishing marginal fitness returns to increasing number of marriages—do not acquire wives in full proportion to their capacity to support them with rival wealth. Together, these effects reduce the population-level fraction of wives in polygynous marriages. Our model demonstrates that a low population-level frequency of polygyny will be an equilibrium outcome among fitness maximizing males and females in a society characterized by a large class of wealth-poor peasants and a small class of exceptionally wealthy elite. Our mutual mate choice model thus provides an empirically plausible resolution to the polygyny paradox and the transition to monogamy which co-occurred with the rise of highly unequal agricultural populations. We, however, cannot yet explain the causes of the unexpectedly substantial decreasing marginal fitness returns to increasing number of marriages. A purely statistical explanation of our results could be that we have missed some important rival form of wealth, which if accounted for would result in a larger estimate for m and hence a reduced estimate of the degree of diminishing returns to additional wives for reasons other than the sharing of rival wealth. Another possibility, already mentioned, is that in some of our datasets the very wealthy could be deliberately limiting their reproductive success , which would also drive m downwards. In addition to these possible statistical effects, there are a number of other plausible causes of the diminishing returns to additional wives observed in our populations. One possibility is that a male’s time and attention are rival inputs to his own fitness. This situation is likely to arise when paternal investment is essential to offspring survival and well-being. A male’s time can also be rival in other fitness relevant ways. For example, it may be difficult for a single wealthy man to effectively mate guard a large number of wives. With a wealth ratio of mr ¼ 2 and a per cent rich of u ¼ 0.5, a single rich man will have to monopolize his two wives in the face of challenges from a single unmarried man on average; however, with a wealth ratio of mr ¼ 10 and a per cent rich of u ¼ 0.1, a single rich man will have to defend his 10 wives from nine unmarried men on average. As the wealth ratio grows even more skewed, this situation could become increasingly difficult to manage . A related possibility is that a growing number of unmarried men could socially censure wealthy polygynous males, imposing costs on them that reduce male demand for and/or female supply to polygynous marriage. A third possibility is that sexually transmitted infection burden could diminish returns to polygyny, if polygyny enhances infection rates.