Proof: suppose that one of the ex-post outcomes x of L is Pareto-dominated by some other outcome y. Then, by moving some probability mass from x to y, one attains another lottery L that ex-ante Pareto-dominates L. The opposite is not true: ex-ante PE is stronger that ex-post PE. For example, suppose there are two objects a car and a house. Alice values the car at 2 and the house at 3; George values the car at 2 and the house at 9. Consider the following two lotteries: With probability 1/2, give car to Alice and house to George; otherwise, give car to George and house to Alice. The expected utility is for Alice and for George. Both allocations are ex-post PE, since the one who got the car cannot be made better-off without harming the one who got the house. With probability 1, give car to Alice, then with probability 1/3 give the house to Alice, otherwise give it to George. The expected utility is for Alice and for George. Again, both allocations are ex-post PE. While both lotteries are ex-post PE, the lottery 1 is not ex-ante PE, since it is Pareto-dominated by lottery 2. Another example involves dichotomous preferences. There are 5 possible outcomes and 6 voters. The voters' approval sets are . All five outcomes are PE, so every lottery is ex-post PE. But the lottery selecting c, d, e with probability 1/3 each is not ex-ante PE, since it gives an expected utility of 1/3 to each voter, while the lottery selecting a, b with probability 1/2 each gives an expected utility of 1/2 to each voter. Bayesian Pareto efficiency Bayesian efficiency is an adaptation of Pareto efficiency to settings in which players have incomplete information regarding the types of other players. Ordinal Pareto efficiency Ordinal Pareto efficiency is an adaptation of Pareto efficiency to settings in which players report only rankings on individual items, and we do not know for sure how they rank entire bundles. Pareto efficiency and equity Although an outcome may be considered a Pareto improvement, this does not imply that the outcome is satisfying or equitable. It is possible that inequality persists even after a Pareto improvement. Despite the fact that it is frequently used in conjunction with the idea of Pareto optimality, the term "efficiency" refers to the process of increasing societal productivity. It is possible for a society