Responses on G+ to some of what I posted on polygamy raise an interesting consequentialist argument against it—that wealthy men would "buy up" too many wives, leaving a surplus of unmarried single men likely to cause social problems such as increased levels of crime.
The argument takes it for granted that polygamy mostly means polygyny, multiple wives rather than multiple husbands. Historically that has been the pattern. Monogamy is the most common marital arrangement, polygyny next, polyandry rare. But it raises the question of why that pattern existed and whether it would persist in a modern society where polygamy was legal and common enough to have a significant effect on the marriage market.
One answer is that men, for reasons probably hardwired by evolution, want to know which children are theirs in order that they can avoid spending their scarce resources on other men's children. Prior to modern times, maternity was an observed fact, paternity a conjecture. The obvious way of strengthening the conjecture was to arrange matters so that a woman had sex with only one man, a condition satisfied by monogamy and polygyny but not by polyandry. Modern paternity testing, which I like to refer to as the stealth reproductive technology, changed that. It no longer requires a wise child to know his father, merely a properly equipped lab.
A second possible answer is that under pre-modern conditions, with high rates of both infant mortality and death in childbirth, one woman could not be counted on to produce as many children as several husbands would want. That again has changed. In a world where infant mortality is close to zero, a fertile woman who enjoys producing and rearing children, supported by the income of multiple husbands, should be able to produce enough offspring for all of them. And it is worth noting that a second function of marriage is sex, and women are less limited in that regard than men.
All of which suggests that, in a modern context, polyandry might turn out to be as common as, or more common than, polygyny, in which case the objection vanishes or even reverses, becomes an argument in favor of polygamy rather than an argument against it.
How could one find data to test the theory? One possibility would be to study modern polygamy not in contexts such as the FLDS, where it represents the survival of old marital patterns, but in the context of polyamory, where it appears as the growth of new ones. I do not know if anyone has attempted a census of polyamorous households—there are obvious difficulties, since many have reasons to keep a low profile—but the results would be interesting.