We are, overall, a law-abiding country. We believe in obeying the laws and we generally seem to believe that people who do not obey our laws should be punished. Punishment is, by the way, the primary way we encourage people to obey laws. Punishment is our ‘cure’ for the ill.
Here is where a law against polygyny runs in to a fundamental problem. We’ll assume for the moment that polygyny is illegal because it causes some kind of harm. Under most state laws the husband in a polygynous relationship is the criminal. The wives and children are usually not. Let’s assume a family of one husband, 3 wives, and 12 children. The husband works and maybe one of the wives works while the other 2 wives take care of the children and home. They are not on welfare, other than polygyny are law-abiding, and their children are healthy, educated, and well-mannered. This scenario is, BTW, fairly typical of polygynous families.
Well, this guy is breaking the law so we have to punish him for it. What are we to do? Throw him in jail? That will sure be of benefit to the family and to society. Order him to leave 2 of his wives? That works well. Make him pay a fine? Remove the wives and kids from their home? Even community service creates the problem of taking a father away from his children while at the same time communicating to them that according to society their father is a criminal. We’ve either harmed their relationship with their parents or their relationship with society. And to what end?
Just how much harm is caused by polygyny itself that we apparently believe is so atrocious that we are willing to cause these other harms?