Editing Liberal paradox (section)

==The theorem==
Suppose there is a society ''N'' consisting of two or more individuals and a set ''X'' of two or more social outcomes. (For example, in the Alice and Bob case, ''N'' consisted of Alice and Bob, and ''X'' consisted of the four color options ⟨Blue, Yellow⟩, ⟨Blue, Green⟩, ⟨Red, Yellow⟩, and ⟨Red, Green⟩.)

Suppose each individual in the society has a [[connected relation|total]] and [[transitive relation|transitive]] preference relation on the set of social outcomes ''X''. For notation, the preference relation of an individual ''i''∊''N'' is denoted by ≼<sub>''i''</sub>. Each preference relation belongs to the set ''Rel(X)'' of all total and transitive relations on ''X''.

A social choice function is a map which can take any configuration of preference relations of ''N'' as input and produce a subset of ("chosen") social outcomes as output. Formally, a social choice function is a map

<math>F : {Rel}(X)^N \rightarrow \mathcal{P}(X)</math>
<!--all possible combinations of preference relations for members of ''N''!-->

from the set of functions between ''N''→''Rel(X)'', to the power set of ''X''. (Intuitively, the social choice function represents a societal principle for choosing one or more social outcomes based on individuals' preferences. By representing the social choice process as a ''function'' on ''Rel(X)''<sup>''N''</sup>, 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.)

The liberal paradox states that every social choice function satisfies ''at most one'' of the following properties, never both:
# '''Pareto optimality''' (collective efficiency): whenever all individuals of a society strictly prefer an outcome ''x'' over an outcome ''y'', the choice function doesn't pick ''y''. 
#* Formally, a social choice function ''F'' is Pareto optimal if whenever ''p''∊''Rel(X)''<sup>''N''</sup> is a configuration of preference relations and there are two outcomes ''x'' and ''y'' such that  ''x''⪲<sub>''i''</sub>''y'' for every individual ''i''∊''N'', then ''y''∉ ''F(p)''.
#* Intuitively, Pareto optimality captures an aspect of collective efficiency: the social choice is made so that everyone is collectively as well off as possible, to the extent that every available tradeoff would make someone worse off.
# '''Minimal liberalism''' (individual freedom): More than one individual in the society is '''decisive''' on a pair of social outcomes. (An individual is decisive on a pair of social outcomes ''x'' and ''y'' if, whenever they prefer ''x'' over ''y'', the social choice function prefers ''x'' over ''y'' regardless of what other members of the society prefer. And similarly whenever they prefer ''y'' over ''x'', the social choice function prefers ''y'' over ''x''.)  
#* Formally, a social choice function ''F'' respects minimal liberalism if there is more than one individual ''i''∊''N'' for which there exists a pair of outcomes ''x''<sub>''i''</sub>, ''y''<sub>''i''</sub> on which they are decisive—that is, for every configuration of preference relations  ''p''∊''Rel(X)''<sup>''N''</sup>, ''y''<sub>''i''</sub>∊ ''F(p)'' only when ''x''<sub>''i''</sub>≼<sub>''i''</sub>''y''<sub>''i''</sub> (and similarly, ''x''<sub>''i''</sub>∊ ''F(p)'' only when ''y''<sub>''i''</sub>≼<sub>''i''</sub>''x''<sub>''i''</sub>).
#* As an example of decisiveness: in the Lewd/Prude case, Lewd was decisive on the pair of outcomes ⟨"Lewd reads", "No one reads"⟩ and Prude was decisive on the pair of outcomes ⟨"Prude reads", "No one reads"⟩.
#* Intuitively, minimal liberalism captures an aspect of individual freedom: for some issues, if you prefer x over y (or vice versa), then society respects your preference for x over y even if everyone else is against you. Sen's example is your personal preference for sleeping on your back or your side: on at least one innocuous personal area like this, a liberal society ought to prioritize your individual preference even if everyone else in society would prefer you to sleep another way. The formal requirement is that at least two people are decisive in this way, to rule out the possibility of a single person who dictates society's preferences.

In other words, the liberal paradox states that for every social choice function ''F'', there is a configuration of preference relations ''p''∊''Rel(X)''<sup>''N''</sup> for which ''F'' violates either Pareto optimality or Minimal liberalism (or both). In the examples of Sen and Gibbard noted above, the social choice function satisfies minimal liberalism at the expense of Pareto optimality.