A set solution concept for a family of bargaining games F is a function ? associating every bargaining game (S, d) in F with a subset of S (which may contain more than a single point). Let f : R4 → R be a function. Define a set solution concept ? as follows:
Ψ(S, d) = argmaxx∈S f(d,x).
(a) Give an example of a function f for which ?(S, d) is not always a single point.
(b) Prove that ? satisfies independence of irrelevant alternatives. In other words, if T ⊇ S and x ∈ ?(T,d) ∩ S, then x ∈ ?(S, d).