User:Max Qaly/sandbox

The Paretian-liberal impossibility theorem is a well-known result in social choice theory proved by the Indian economist and philosopher Amartya Sen. The theorem was proved in 1970 paper published in the Journal of Political Economy. The theorem states that it is impossible to simultaneously satisfy both a certain principle of liberalism and the Pareto Criterion.

Result
The two main axioms that Sen shows to be inconsistent are the following:

Weak Pareto: For any alternatives (x, y), if everyone prefers x to y, then x is better than y.

Liberalism: For each individual, there it at least one pair of alternatives, say (x, y), such that
 * 1) if this individual prefers x to y, then society prefers x to y, and
 * 2) if this individual prefers y to x, then society prefers y to x.

Sen also assumes an additional axiom called Unrestricted Domain, according to which any profile of individual preferences are possible.

The result can be stated as follows:

Paretian-Liberal Impossibility Theorem: There are no transitive societal preferences that satisfy Weak Pareto, Liberalism and Unrestricted Domain.

Sen's example
Sen uses illustrates the theorem using a famous example involving the (at the time) controversial book Lady Chatterrly's Lover. In the example, there are two individuals and three alternatives:

Suppose that individual 1 is a prude that prefers z (no one reads it), but prefers x to z (Sen writes that "Prudes, I am being told, tend to prefer to be censors than being censored). On the contrary, individual 2 prefers that someone reads it rather than no one and takes delight in the thought of individual 1 reading it. Thus, individual 2 prefers x to y and y to z. From a liberal perspective, society should prefer y to z and z to x (and therefore y to x). However, from a Paretian perspective, society should prefer x to y. But this means that society both prefers y to x and x to y, which is impossible.
 * Alternative x: Individual 1 reads the book.
 * Alternative y: Individual 2 reads the book.
 * Alternative z: No one reads the book.

Discussion
The Paretian-Liberal Impossibility Theorem has been widely discussed in the economics literature, including in textbooks.