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You have it backwards.

This formal game-theoretic notion of fairness acknowledges that power disparity exists and that having less power than your counterparty allows them to inflict greater disutility on you without you being able to inflict disutility on them in turn to discourage this.

On the other hand, fairness "in the usual sense", pretends power disparity doesn't exist and that, say, an armed robber is not allowed to take your stuff when you have nothing to defend yourself with. Which in reality only works as long there is a powerful third party (the state) that will inflict disutility on the robber for it.

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In reality people never have equal power over anything (what would that look like, physically?) so something like nash bargaining is an attempt to get closer to a notion of fair given this inequality
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I don't think the difficulty of equal power is a good excuse to pretend power doesn't exist.

One way we solve it in the real world is that the negotiators also have power - including, possibly, the power to force the party most OK with the status quo to come to the negotiating table, and reject exploitative proposals.

That isn't foolproof either, of course. But it beats rhetoric trying to convince the weaker party to submit.

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I didn’t say it doesn’t exist, rather that it’s already taken into account. I’m also not sure what you are proposing- if mediation is required, and someone has more power than someone else, why would they voluntarily engage with a mediator who will reduce that power? Or if they are forced to use this mediator (eg by the state) then this means they never had the power in the first place
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