Question title says it all really. Polygamy is illegal in most US states, here in Britain and also no doubt many countries elsewhere, but why? What are the political motivators for this, especially in a country which has a separation of church and state?
The main current purpose those laws serve is to placate a vast majority of the citizenry who object to polygamy (to the point of, about 100+ years ago, trying to physically destroy Mormons).
Ironically, this is as fully bipartisan issue as they come:
- progressive feminists see polygyny (which is what most polyamorous relationships seem to be in history) as the ultimate expression of evil patriarchy (it's beyond the scope of the answer whether they are right or not)
- right wing social conservatives see it as contradictory to mainline Christianity and New Testament rules (conveniently ignoring the whole King David, and especially king Solomon, precedents from Old Testament :) and generally a sexually deviant thing (it's beyond the scope of the answer whether they are right or not).
- just to make this political bed-meeting weirder (and poly? :), the less-successful (or less full of themselves) people in MRA/etc... scene also object to polygyny, as it enables women to indulge in hypergamy even easier (it's beyond the scope of the answer whether they are right or not). The only time they see eye to eye with hard-core third wave feminists, I suppose. But stranger things happened.
Let's get down to the numbers. Even in these, extremely sexually liberal times, with polygamy being "morally acceptable" # as high as it's ever been in USA, only 17% population supports it, according to Gallup 2017 poll. That's up from only 7% in 2003.
If that seems "high" - compare it to 63% supporting gay/lesbian sexual relations.
However, an argument may be made that those laws are in fact socially a good thing, in both theoretical and practical way:
In theory, in a society with permitted polygamy, there's a high chance that a large portion of less-successful (straight) males would end up with no female partnership. This is a recipe for social disorder if there ever was one - it's well known from sociological and psychological research that males in long term relationships are far less likely to engage in risky, asocial, or criminal behavior; and are more likely to be more socially productive. Or, like, not engage in mass violence and rioting.
In practice, there is tons of research that polygamy has bad outcomes, although in all honesty, I haven't vetted ANY of the following links and don't know how solid (or biased) the research is. YMMV, PATYR.
The Perils of Polygamy has some in depth discussion (note: it seems to be a socially conservative think tank, so they clearly have an axe to grind).
The pros and cons of polygamy from Psychology Today. They often have progressive bent in my limited experience (as in, publish articles how conservatism is a mental disorder, in full seriousness) so probably also have a different axe to grind.
Slate lists a study highlighting problems. Again, progressive axe grinding is likely.
The Atlantic had an article critical of polygamy from societal standpoint, largely drawing from non-poly-related study of male-surplus Asian societies in 2004 book "Bare Branches: Security Implications of Asia's Surplus Male Population" by Valerie M. Hudson and Andrea M. den Boer.
Having said that, the above reasoning and arguments are NOT the purpose of anti-poly laws, although they may be used by defenders of such laws. The purpose stated above is the only one - to placate the majority of population that for one reason or another supports such laws.
 and if that 63% people who find homosexual relations morally acceptable seems low - that's only 6% less than 69% who find sex outside marriage morally acceptable. As in, only 6% more people don't accept homosexuality than "living in sin"
 Then again, one has to wonder at how much social acceptability bias affected that poll, since a staggeringly low 9% considers extramarital affairs "morally acceptable", compared to between 20 and 70% estimates of people actually cheating.
Currently, anti-polygamy laws serve to "protect" traditional marriage (at least, traditional in the eyes of the Christian West) the same way that laws against same-sex marriage were supposed to. Note that these laws apply only to the legal institution of marriage, not the religious institution.
Unlike prohibitions against same-sex marriage, I doubt that prohibitions against polygamous marriage will be overturned anytime soon for the following reasons:
Same-sex marriages don't introduce the legal headaches for custody, power of attorney, property rights, inheritance, etc., that polygamous marriages would. How would you apportion assets in a poly divorce when a partner leaves? When multiple partners leave? How about parental rights?
Right now, polygamy (at least in the US) is most closely associated with some groups that are, let's face it, pretty far out on the fringe socially and politically, and it's not the polygamy that makes them so fringey.
There are arguments that poly marriages (especially polygyny, multiple wives for a single husband) truly are bad arrangements in a number of objectively measurable ways, although I don't know how solid those arguments are. But they're not the silly slippery slope arguments employed against allowing same-sex marriage ("why, next thing you know, people will be marrying their pets").
To ration females to males in equal share for everyone. Prohibition of polygamy, if combined with no sex outside marriage, will ensure the poorest ugliest dumbest males will still be able to get a wife.
Why "society" bother doing that? Well, the poorest, ugliest, dumbest males can vote. Politicians want their vote.
If polygamy are allowed, guys like Donald Trump/Bradd Pitt can get 5-10k wives, and many guys will get none. That's because men, if allowed, will aim for quantity. Look at Sultans and emperors. To prevent that, the laws "ration" females in equal share for everyone.
Yes. Women are still legal to choose poor men. However, if those women can choose to share richer smarter men instead, will they? It's like employers can still legally hire Americans. But if those employers can hire cheaper immigrants instead, will they?
Men will instinctively sabotage their competitors. So marriage laws in democratic countries allow women to pick only singles.
Matt Ridley in his book Red Queen says something along, "Far from laws protecting women ... anti polygamy statues really protect men"
Here is the quote and the whole argument
In democratic countries, the majority rules.
The majority of people are not alpha males.
So they choose to ban polygamy.
In general, we get free competition, if a small number of people compete for the benefit of majority.
For examples, most businesses compete with each other. The majority of people are benefited from their competition and hence force businesses to compete.
However, competing is not a good idea for profit.
If the majority is the competitor, we usually have trade restrictions.
The trade restrictions are in favor of majority.
For example, we got many trade restrictions in labor market. That's because laborers are majority compared to employers. Laborers are significant "voting bloc" so to speak.
Majority of men are poor or middle classes. So marriage laws in many democratic countries tend to kick out the rich from mating market.
For example, within marriage, alimony tend to be proportional to wealth of a man. This discourage richer men from getting married. Using money to get girls tend to be heavily discouraged. Explicit use of using money to get girls, called prostitution, is prohibited.
Like all marriage, the states heavily influence who can marry who.
In states that allow polygamy, who would stop the law makers from saying that girls can only marry guys that vote for me or embrace my religions? That means the religious leaders, or the alpha males, pretty much become pimp of all girls. Only by obeying the whim of those alpha males, usually called religion, a man can get laid.
No wonder, middle east often have war.