I wonder when all the dudes who bitched about a woman who expressed frustration over some men who don't do enough housework will chime in here and denounce the objectification and othering of women that's happening in this thread.
For that matter where are our MRAs boo-hooing that marriage is a raw deal for men and that's why they're abandoning it in droves, rending the polygamy point moot?
And where is this polygamy-cum-puratinism idea coming from that once there's polygamy unmarried men will have no hope of getting laid?
And for pity's sake where do women's actions in this structure come into it?
Look, let's look into two anecdotal examples:
1. Old-school sultan and his harem polygamy. (My understanding of this is probably based on old racist movies and someone should feel free to correct me.) Enormous power difference means small number of men get all the girls. On the one hand, not so bad because more men were dying from wars and fighting. Those remaining probably more mad about wildly uneven distirbution of power and lack of girls just icing in the cake. From women's perspective, uneven distribution of power seems to mean they don't have much choice though for some chances of security of offspring may be considered better than chances in peasantry.
2. China and other areas with sex-selective abortion. There have been a lot of concerns voiced about social destabilization of men not being able to find mates, comparing it to wild west and tendency (sorry, guys) of heavily-male populations to be inclined towards violence and crime. First, destabilization is conjecture, we don't have evidence yet. Second, the issue is not just about men not having potential mates but about women, and their supposed feminine settling influence over a culture, being missing entirely.
So in the case of #1, say what the you will about the 1%, economic conditions are not nearly as bad as that. Women who want to be with a poor-but-honest guy still have a good chance of raising their children in safety and security.
In fact look at the number of unwed mothers in lower-income places. Plenty of them would be happy to have a partner if they could find one who would stick by them and wasn't incarcerated (tabling political discussion of high rates of incarceration for now). It seems unfair to say they can't have the option to join a polygamas marriage when no one wants them now.
For that matter, I think a lot of women may actually become wary of very successful and attractive guys knowing they may have to compromise and share him. (I'm imagining that she has to give legal consent to his marriage for it to happen but that once she's older she has less power in the relationship to say no.)
Then think of divorce laws. A man may not be left with half his assets, but only a third. It becomes riskier for him.
So I'm imagining that in a world where women are equal, in terms of education and the economy, that very few are going to go for this. I imagine there will be a type of marriage contract which promises monogamy in the future and that most women will accept only this as a "real" marriage. There will certainly be some women who would rather have a higher-caliber man no matter what than settle for a lower-caliber guy she has all to herself. (Sorry to rather insensitively speak in terms of a ranking system. I know it's a bit like ranking women's looks on a scale 1-10.) But frankly we sort of have this sort of imbalance among available mates already due to young women being willing to marry older men who are successful rather than marrying their contemporaries*. In fact, in the most evolutionary-based sense we already have it an imbalance due to the biology that men are sexually viable over a longer period of their lifespan than women, so you always have more men looking than there are desirable women available.
To the extent that some women do go for it and there's an imbalance, I feel like people basing their world view on Heinlein might agree that plenty of men don't want to be tied down to marriage at all, and that might be the equal to the number of women not available. There are still going to be women you can date and women you can fuck (some for a price), but I suspect that any men who want to get married will still be able to do so.
*In fact, now that I think of it, if your husband is the type of guy to pick up a second wife because you got old and wrinkly, he's also probably the type of guy to just divorce your ass when that happens, so in a sense polygamy might actually provide more security.