Abstracting a bit from marriage as a social, cultural institution, one might make the case that the state's role in marriage is in contract enforcement. A marriage is a bit like a firm: the basic rules of organization are circumscribed by common law, but the specific arrangements are left to the founders of the organization. The state recognizes the contract and honors its terms, including implied boilerplate like rights of inheritance, powers of attorney, et al. The recent tussle over same-sex marriage has cast a spotlight on the common-law circumscriptions of the institution. I do think it's fair to ask what are reasonable limits on the type of marrigae contracts that can be written.
Luckily, the common law already provides some pretty good guidance. Adults of sound mind absent coercive pressures are able to write contracts. If you think of a family as a species of firm, you really have to seriously ask why organizational contracts are limited to a joint partnership of only one man and only one woman. What reason is there to prohibit plural marriage?
Lest I appear to be baking two cakes in the same oven, I must stress that this is indeed an entirely different question than same sex marriage. Plural marriage considerations question the validity of the prescribed form, but same sex marriage considerations are an appeal to what North, Wallis and Weingast refer to as impersonal exchange: that the class identity of the contracting parties is irrelevant. The only things that should matter are considerations of competence and duress. Plural marriages challenge the universe of contract forms that can be written. This is a categorically different question. But still one worth considering.
Down to the brass tacks of the EE, then. Once again, I don't know if I have a good answer for you. It strikes me that (at least in the language we use here) that what we observe is a violation of conventional capacity to trade. But this is exactly the point we're trying to address, so that objection just plain begs the question. Again here, I think it ultimately boils down to a question of aesthetics. Plural marriage is weird, so people (the median voter) opposes it. Polygamists are low status, so they're less likely (at least, for now) to garner much popular support.
So I close with these simple questions: is plural marriage non-euvoluntary? If so, what condition is violated? If not, why the continued ban? If you do end up grilling your friends, family or students about this, hold their feet to the fire. I expect you'll find their arguments end up teetering on a precipice of visceral revulsion and not much else. If I'm wrong about that, do please let me know.