The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox proposed by Amartya Sen which shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:
The unrestrictedness condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally, The Pareto condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and Liberalism, or L (from which the theorem derives its gist): all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, anyone can exert their freedom of choice at least in some decision with tangible results. Sen's result shows that this is impossible. The three, rather minimalistic, assumptions cannot all hold together. The paradox—more properly called a proof of contradiction, and a paradox only in the sense of informal logic—is contentious because it appears to contradict the classical liberal idea that markets are both Pareto-efficient and respect individual freedoms. Sen's proof, set in the context of social choice theory, is similar in many respects to Arrow's impossibility theorem and the Gibbard–Satterthwaite theorem. As a mathematical construct, it also has much wider applicability: it is essentially about cyclical majorities between partially ordered sets, of which at least three must participate in order to give rise to the phenomenon. 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.
Pareto efficiency
Definition 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. Put another way, an outcome is not Pareto-efficient if there is a way to improve at least one person's situation without harming anyone else. For example, suppose a mother has ten dollars which she intends to give to her two children Carlos and Shannon. Suppose the children each want only money, and they do not get jealous of one another. The following distributions are Pareto-efficient:
However, a distribution where the mother gives each of them $2 and wastes the remaining $6 is not Pareto-efficient, because she could have given the wasted money to either child and made that child better off without harming the other. In this example, it was presumed that a child was made better or worse off by gaining or losing money, respectively, and that neither child gained or lost by evaluating her share in comparison to the other. To be more precise, we must evaluate all possible preferences that the child might have and consider a situation as Pareto-efficient if there is no other social state that at least one person favors (or prefers) and no one disfavors.
Use in economics Pareto efficiency is often used in economics as a minimal sense of economic efficiency. If a mechanism does not result in Pareto-efficient outcomes, it is regarded as inefficient, since there was another outcome that could have made some people better off without harming anyone else. 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. As a result, these results often feature prominently in conservative libertarian justifications of unregulated markets.
Two examples
Sen's original example Sen's original example used a simple society with only two people and only one social issue to consider. The two members of society are named "Lewd" and "Prude". In this society there is a copy of a Lady Chatterley's Lover and it must be given either to Lewd to read, to Prude to read, or disposed of - unread. Suppose that Lewd enjoys this sort of reading and would prefer to read it rather than have it disposed of. However, they would get even more enjoyment out of Prude being forced to read it. Prude thinks that the book is indecent and that it should be disposed of, unread. However, if someone must read it, Prude would prefer to read it rather than Lewd since Prude thinks it would be even worse for someone to read and enjoy the book rather than read it in disgust. Given these preferences of the two individuals in the society, a social planner must decide what to do. Should the planner force Lewd to read the book, force Prude to read the book or let it go unread? More particularly, the social planner must rank all three possible outcomes in terms of their social desirability. The social planner decides that they should be committed to individual rights, each individual should get to choose whether they, themself will read the book. Lewd should get to decide whether the outcome "Lewd reads" will be ranked higher than "No one reads", and similarly Prude should get to decide whether the outcome "Prude reads" will be ranked higher than "No one reads". Following this strategy, the social planner declares that the outcome "Lewd reads" will be ranked higher than "No one reads" (because of Lewd's preferences) and that "No one reads" will be ranked higher than "Prude reads" (because of Prude's preferences). Consistency then requires that "Lewd reads" be ranked higher than "Prude reads", and so the social planner gives the book to Lewd to read. Notice that this outcome is regarded as worse than "Prude reads" by both Prude and Lewd, and the chosen outcome is therefore Pareto inferior to another available outcome—the one where Prude is forced to read the book.
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