My question is concerning the uniqueness of the arbitration pairs on a Bargaining set. On every problem I have done, the solution tries to find an arbitration pair on every region of the bargaining set, not stopping if one was found, but at the end only one exists.
For me, it makes no sense because Nash's Theorem gives us uniqueness, or at least I understand it that way, so we could stop looking for arbitration pairs if we already found one, because that's the only one existing on the bargaining set. I'm referring to the theorem below:
Theorem 2.4 (Nash): There exists a unique arbitration procedure $\phi$ satisfies the Bargaining axioms.
Bargaining axioms are a set of axioms that my Bargaining set is satisfying every time.
Am I missing something?
Thanks for your time.