  We study implementation of a social choice correspondence in the case of two
players who have von Neumann - Morgenstern utilities over a finite set of
social alternatives, and the mechanism is allowed to output lotteries. Our main
positive result shows that a close variant of the popular approval voting
mechanism succeeds in selecting only Pareto efficient alternatives as pure Nash
equilibria outcomes. Moreover, we provide an exact characterization of pure
Nash equilibria profiles and outcomes of the mechanism. The characterization
demonstrates a close connection between the approval voting mechanism and the
notion of average fixed point, which is a point that is equal to the average of
all points that it does not Pareto dominate.
