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shminux comments on Poly marriage? - Less Wrong
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It has died some time ago in more socially progressive countries, but none recognize a group marriage, so, I agree, clearly social mores are not the only or even the main issue.
My guess would be that, just like in Physics, in Law the general n-body problem (ahem) is much less tractable than the two body problem. I suspect that there are many relevant legal issues that have not been even touched and will have to be addressed, before a reasonable group marriage framework can be constructed. Child custody/support/rights, potential for spousal abuse are some of those.
On general grounds, one would also expect the stability of such a union be much less than that of a 2-person one, thus reducing the need for legal protection. Most polys freely admit that they have a "primary" and one or more secondaries, who are not in an equal position. Again, the analogy with orbital motion is quite interesting, but I will not push it further.
I don't have statistics on poly relationships, and I doubt that anyone has good statistics.
Still, so far as raising children is concerned, I think you're conflating two situations. A poly relationship might be considered unstable if one or more people leaves, but if a core group remains raising children, it might be more stable for children than a couple that breaks up.