If women had their way, would there be a lot more societies with polyandry?

Based on the number of children who are not the biological child of their mother’s husband, it seems like there is a lot of fooling around going on out there. One estimate I saw said that 22% of men and 14% of women had extra-marital affairs as of a decade ago. This article suggests the rate might be higher now.

I’ve copied a quote from the first article that suggests a reason why women might be unfaithful:

“Being unfaithful enables females to have their cake and eat it,” says the zoologist and author Matt Ridley. “In many species of bird, the female needs a male to help bring up the kids, but she’s unlikely to get the best male in the world to hang around for that. She’s much more likely to end up with a boring one with a straggly beard. But why should she settle for his genes?” With infidelity, she needn’t. And there is much evidence to show that she doesn’t.

We know that there is a fair amount of polygamy in the world. There are countries where it is legal for a man to have several wives. Based on this, I think it is fair to say that if men could get away with the double standard, there are some who would probably do it.

However, women, too, benefit from having multiple male partners. So my question is, if women were running the world, would there be a few nations that would legalize polyandry? Or, if not polyandry, at least remove the stigma from feminine infidelity?