Let (S,c) be a bargaining game, and let (T,d) be the image of (S,c) under the positive affine transformation x #→ ax + b on R2; that is, T = aS + b and d = ac + b.
Prove that the set argmaxy∈T (y1 - d1)(y2 - d2) is the image of the set argmaxx∈S(x1 - c1)(x2 - c2) under this transformation.