neither is free to choose their own color. Mathematically, we can represent Alice's preferences with this symbol: and Bob's preferences with this one: . 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) (Red, Yellow) (Blue, Green) (Red, Green) And Bob's are: (Red, Green) (Red, Yellow) (Blue, Green) (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. 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 total and transitive preference relation on the set of social outcomes X. For notation, the preference relation of an individual i∊N is denoted by ≼i. 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 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)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.) 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)N is a configuration of preference relations and there are two outcomes x and y such that x⪲iy for every individual i∊N, then y∉ F(p). Intuitively, Pareto optimality captures an aspect of collective efficiency: the