 It matters whether the valuations are known to the other party. Such knowledge can be used to advantage by the one who cuts. First consider the case that your valuation is unknown to me, and vice versa. Then if I cut, the best I can do is pick sets \(A, B \subseteq X\) with \(A \cap B = \emptyset\), \(A \cup B = X\), and \(V_m(A) = V_m(B)\). If you choose, you will use \(V_y\) to pick the maximum of \(\{ V_y(A), V_y(B) \}\). It follows immediately that cutting guarantees a fair share, but no more than that, while choosing holds a promise for a better deal. So if you ever get the choice between cutting and choosing in a situation where both parties only know their own valuation, then it is to your advantage to leave the cutting to the other person.