Given σ, a family ofchoice problems, subsets ofR n representing the payoff vectors (measured in von Neumann-Morgenstern utility scales) attainable by a group ofn players, asolution f on σ associates to everyS in σ a unique elementf (S) ofS. Amonotonicity axiom specifies how the solution outcome should change when the choice problem is subjected to certain geometric transformations while anindependence axiom requires, in similar circumstances, the invariance of the solution outcome. A number of such axioms are here formulated and the logical relationships among them are established.Strong monotonicity is shown to be the strongest axiom and strongly monotonic solutions are characterized.