Liberal paradox
Since the idea is about pure mathematics and logic, similar arguments abound much further afield.
They, for example, lead to the necessity of the fifth normal form in relational database design.
The history of the argument also goes deeper, Condorcet's paradox perhaps being the first example of the finite sort.
A particular distribution of goods or outcome of any social process is regarded as Pareto-efficient if there is no way to improve one or more people's situations without harming another.
For example, suppose a mother has ten dollars which she intends to give to her two children Carlos and Shannon.
The view that markets produce Pareto-efficient outcomes is regarded as an important and central justification for capitalism.
This result was established (with certain assumptions) in an area of study known as general equilibrium theory and is known as the first fundamental theorem of welfare economics.
Sen's original example[4] used a simple society with only two people and only one social issue to consider.
As stated Alice's preferences are: And Bob's are: 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.
Suppose each individual in the society has a total and transitive preference relation on the set of social outcomes X.
By representing the social choice process as a function on Rel(X)N, we are tacitly assuming that the social choice function is defined for any possible configuration of preference relations; this is sometimes called the Universal Domain assumption.)
In the examples of Sen and Gibbard noted above, the social choice function satisfies minimal liberalism at the expense of Pareto optimality.
In fact, this shows a strong relationship between Sen's paradox and the well known result that markets fail to produce Pareto outcomes in the presence of externalities.
Because of their nosy preferences, Alice's choice imposes a negative externality on Bob and vice versa.
Note that if we consider the case of cardinal preferences—for instance, if Alice and Bob both had to state, within certain bounds, how much happiness they would get for each color of each house separately, and the situation which produced the most happiness were chosen—a minimally-liberal solution does not require that they have no nosiness at all, but just that the sum of all "nosy" preferences about one house's color are below some threshold, while the "non-nosy" preferences are all above that threshold.
Since there are generally some questions for which this will be true—Sen's classic example is an individual's choice of whether to sleep on their back or their side—the goal of combining minimal liberalism with Pareto efficiency, while impossible to guarantee in all theoretical cases, may not in practice be impossible to obtain.
If someone takes the Pareto principle seriously, as economists seem to do, then he has to face problems of consistency in cherishing liberal values, even very mild ones.
While the Pareto criterion has been thought to be an expression of individual liberty, it appears that in choices involving more than two alternatives it can have consequences that are, in fact, deeply illiberal.
In such a circumstance there was no violation of Prude's or Lewd's rights because each entered the contract willingly.
Similarly, Alice and Bob might sign a contract to each paint their houses their dispreferred color on condition that the other does the same.
Marc Masat hints that this should be another way out of the paradox: If there's, at least, one player without dominant strategy, the game will be played sequentially where players with dominant strategy and need to change it (if they are in the Pareto optimal they don't have to) will be the firsts to choose, allowing to reach the Pareto Efficiency without dictatorship nor restricted domain and also avoiding contract's costs such as time, money or other people.
