A decision scheme makes the probabilities of alternatives depend on individual strong orderings of them. It is strategy-proof if it logically precludes anyone's advantageously misrepresenting his preferences. It is unilateral if only one individual can affect the outcome, and duple if it restricts the final outcome to a fixed pair of alternatives. Any strategy-proof decision scheme, it is shown, is a probability mixture of schemes each of which is unilateral or duple. If it guarantees Pareto optimal outcomes, it is a probability mixture of dictatorial schemes. If it guarantees ex ante Pareto optimal lotteries, it is dictatorial.