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GuySrinivasan comments on If you don't know the name of the game, just tell me what I mean to you - Less Wrong
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I feel like mu isn't the really important part... it's more like mu = A x B where A encodes the translation from "one util" in U1 to "one util" in U2 and B encodes the relative amounts the agents matter in the deal. It seems like A is the bit that remains fairly constant across deals and short periods of time, while B can be differently bargained each time relative to the specific deal in question.
If B differes over time, you'll have outcomes that are not Pareto optimal in total. Idealised utility maximising agents should establish μ once and for all at the beggining; each change in μ is paid for in decreased utility.
I'm not claiming that human agents do, or should, behave this way.