Editing Liberal paradox (section)

===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.