Editing Liberal paradox (section)

==Two examples==

===Sen's original example===
Sen's original example<ref name="SenJPE">{{cite journal|first=Amartya|last=Sen|title=The Impossibility of a Paretian Liberal|journal=Journal of Political Economy|year=1970|volume=78 |issue=1|pages=152–157|jstor=1829633|doi=10.1086/259614|s2cid=154193982|url=https://dash.harvard.edu/bitstream/1/3612779/4/Sen_ImpossibilityParetian.pdf}}</ref> used a simple society with only two people and only one social issue to consider. The two members of society are named "Lewd" and "Prude". In this society there is a copy of a ''[[Lady Chatterley's Lover]]'' and it must be given either to Lewd to read, to Prude to read, or disposed of - unread.  Suppose that Lewd enjoys this sort of reading and would prefer to read it rather than have it disposed of.  However, they would get even more enjoyment out of Prude being forced to read it.

Prude thinks that the book is indecent and that it should be disposed of, unread. However, if someone must read it, Prude would prefer to read it rather than Lewd since Prude thinks it would be even worse for someone to read and enjoy the book rather than read it in disgust.

Given these preferences of the two individuals in the society, a social planner must decide what to do. Should the planner force Lewd to read the book, force Prude to read the book or let it go unread?  More particularly, the social planner must rank all three possible outcomes in terms of their social desirability. The social planner decides that they should be committed to individual rights, each individual should get to choose whether they, themself will read the book. Lewd should get to decide whether the outcome "Lewd reads" will be ranked higher than "No one reads", and similarly Prude should get to decide whether the outcome "Prude reads" will be ranked higher than "No one reads".

Following this strategy, the social planner declares that the outcome "Lewd reads" will be ranked higher than "No one reads" (because of Lewd's preferences) and that "No one reads" will be ranked higher than "Prude reads" (because of Prude's preferences). Consistency then requires that "Lewd reads" be ranked higher than "Prude reads", and so the social planner gives the book to Lewd to read.

Notice that this outcome is regarded as worse than "Prude reads" by both Prude ''and'' Lewd, and the chosen outcome is therefore [[Pareto efficiency|Pareto inferior]] to another available outcome—the one where Prude is forced to read the book.

===Gibbard's example===
Another example was provided by philosopher [[Allan Gibbard]].<ref name="Gibbard">{{cite journal|last=Gibbard|first=Allan|title=A Pareto Consistent Libertarian Claim|journal=Journal of Economic Theory|year=1974|volume=7|issue=4|pages=388–410 |doi=10.1016/0022-0531(74)90111-2}}</ref> Suppose there are two individuals Alice and Bob who live next door to each other.  Alice loves the color blue and hates red. Bob loves the color green and hates yellow. If each were free to choose the color of their house independently of the other, they would choose their favorite colors. But Alice hates Bob with a passion, and she would gladly endure a red house if it meant that Bob would have to endure his house being yellow. Bob similarly hates Alice, and would gladly endure a yellow house if that meant that Alice would live in a red house.

If each individual is free to choose their own house color, independently of the other, Alice would choose a blue house and Bob would choose a green one. But, this outcome is not Pareto efficient, because both Alice and Bob would prefer the outcome where Alice's house is red and Bob's is yellow. As a result, giving each individual the freedom to choose their own house color has led to an inefficient outcome—one that is inferior to another outcome where neither is free to choose their own color.

Mathematically, we can represent Alice's preferences with this symbol: <math>\succ_A</math> and Bob's preferences with this one: <math>\succ_B</math>. We can represent each outcome as a pair: (''Color of Alice's house'', ''Color of Bob's house'').  As stated Alice's preferences are:

:(Blue, Yellow) <math>\succ_A</math> (Red, Yellow) <math>\succ_A</math> (Blue, Green) <math>\succ_A</math> (Red, Green)

And Bob's are:

:(Red, Green) <math>\succ_B</math> (Red, Yellow) <math>\succ_B</math> (Blue, Green) <math>\succ_B</math> (Blue, Yellow)

If we allow free and independent choices of both parties we end up with the outcome (Blue, Green) which is dispreferred by both parties to the outcome (Red, Yellow) and is therefore not Pareto efficient.