Editing Pareto efficiency (section)

=== Ex-ante Pareto efficiency ===
When the decision process is random, such as in [[fair random assignment]] or [[random social choice]] or [[fractional approval voting]], there is a difference between '''ex-post''' and '''ex-ante Pareto efficiency''':
* Ex-post Pareto efficiency means that any outcome of the random process is Pareto-efficient.
* Ex-ante Pareto efficiency means that the ''lottery'' determined by the process is Pareto-efficient with respect to the ''expected'' utilities. That is: no other lottery gives a higher expected utility to one agent and at least as high expected utility to all agents.
If some lottery ''L'' is ex-ante PE, then it is also ex-post PE. ''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{{snd}} 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 {{nobr|(2/2 + 3/2) {{=}} 2.5}} for Alice and {{nobr|(2/2 + 9/2) {{=}} 5.5}} 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 {{nobr|(2 + 3/3) {{=}} 3}} for Alice and {{nobr|(9 × 2/3) {{=}} 6}} for George. Again, both allocations are ex-post PE.

While both lotteries are ex-post PE, the lottery&nbsp;1 is not ex-ante PE, since it is Pareto-dominated by lottery&nbsp;2.

Another example involves [[dichotomous preferences]].<ref name=":02">{{Cite journal |last1=Bogomolnaia |first1=Anna |last2=Moulin |first2=Hervé |last3=Stong |first3=Richard |date=2005-06-01 |title=Collective choice under dichotomous preferences |url=http://www.ruf.rice.edu/~econ/papers/2003papers/09bogomolnaia.pdf |journal=Journal of Economic Theory |language=en |volume=122 |issue=2 |pages=165–184 |doi=10.1016/j.jet.2004.05.005 |issn=0022-0531}}</ref> There are 5 possible outcomes {{nobr|(''a'', ''b'', ''c'', ''d'', ''e'')}} and 6 voters. The voters' approval sets are {{nobr|(''ac'', ''ad'', ''ae'', ''bc'', ''bd'', ''be'')}}. 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.