As women have a hypergamous instinct, a polygamous society would be a disaster. It would essentially mean that the most attractive men would each have multiple partners, while a huge stack of men would be left lonesome.

This is what happened in societies of the past. Kings would have entire harems of women, while the majority of men were slaves or lower class.

Lonesome and discarded men turn violent and cause revolutions. They end up turning the society upside down in order to rearrange the hierarchy.

As much as I enjoy the idea of having multiple girlfriends, to me, a polygamous society would be a dangerous experiment.