There were a few threads touching on the topic of polygyny, but since there are so many issues within that topic to cover, I thought I'd create a broader thread.
ECONOMIC ARGUMENTS
One of the main arguments against polygyny is that it will destroy civilization or, at best, make it run less efficiently. A small number of men will hoard all the available women, removing the incentive of the rest of the men to be productive.
Beta bux scenario
Let's assume that women are primarily interested in a man's beta bux, and will go with the guy who brings the most money to the table. And suppose society tolerates polygyny. Adam, a guy who makes $50K/year to support himself and his wife, realizes he can support a second wife if he increases his income to $75K/year. He works harder/smarter until he reaches that income goal, and gets his second wife.
Meanwhile, there's another guy, Bob, who is making $25K/year to support himself, and would have been interested in raising his income to $50K/year if he could've gotten married. But because Adam already took the woman Bob wanted off the market, Bob doesn't bother to work harder/smarter. Instead, he focuses on leveling his character in World of Warcraft.
How is society poorer because of this? How does this destroy civilization? The market measures productivity in dollars. The $25K/year increase in Adam's productivity to be able to support his second wife offsets the $25K/year in productivity that Bob didn't feel motivated to pursue. In fact, society might become richer if, rather than giving up, Bob competes with Adam to be the most productive and bids up the price of wives.
Of course, one could argue that as men, we shouldn't support this, since it's going to make us have to pay more to get a wife. It depends on which you value more: the security of being able to have one wife even if your income is below average, or the opportunity to have more than one wife if your income is higher than average. Personally, I'd rather live in a society where by taking risks, I can hit the jackpot, even if there's a chance I could instead lose everything.
Alpha fux scenario
On the other hand, suppose that women choose men based on their alpha fux. Therefore, they all gravitate toward the alphas (regardless of how much those men earn), leaving the betas without any incentive to be productive.
Suppose Adam is making $50K/year to support himself and his wife, and because he's so alpha, he's able to attract another woman to be second wife, based solely on his confidence, masculinity, etc. Adam doesn't feel the need to increase his income to $75K/year, because he can get two women to marry him even when he's only making $50K/year. Meanwhile, Bob doesn't bother to increase his income above $25K/year, because Adam took the woman that he would've married. Theoretically, society just lost $25K/year in productivity because of polygyny.
This is still offset, though, if one of Adam's wives enters the workforce and earns $25K/year to help support the family. With three parents in the household, two of them can work while the third stays home to cook, clean, and take care of the kids. There is no need for every woman in a polygynous household to be a housewife.
SOCIAL STABILITY ARGUMENT
The other argument against polygyny is that some of the men who can't get laid are going to become violent because they have little to lose. I don't see this as necessarily a bad thing. It depends on whom they're directing their violence against. If they're risking their lives to overthrow bad leaders whose policies are keeping them poor (and therefore celibate), it could be a good thing.
Monogamy arguably makes society too stable, by creating too much equality in how wives are distributed. Even if the government is corrupt in such a way that the politically powerful are enriched at the expense of the oppressed, men may be reluctant to rise up and rebel, because they are content as long as they are able to have a wife and family. The government uses monogamy like bread and circuses, to keep the masses appeased by offering them enough to keep them happy.
A little more revolution might help society function more efficiently in the long run, or even save civilization from bad leaders whose actions would otherwise go unchecked until it's too late. As Thomas Jefferson wrote, "What country can preserve its liberties if their rulers are not warned from time to time that their people preserve the spirit of resistance? Let them take arms. The remedy is to set them right as to facts, pardon and pacify them. What signify a few lives lost in a century or two? The tree of liberty must be refreshed from time to time with the blood of patriots and tyrants. It is its natural manure."