Gnomonic projection

The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox discovered by Amartya Sen which purports to show that no social system can simultaneously (1) be committed to a minimal sense of freedom, (2) always result in a type of economic efficiency known as Pareto efficiency, and (3) be capable of functioning in any society whatsoever. This paradox is contentious because it appears to contradict the classical liberal claim that markets are both efficient and respect individual freedoms. If, as Sen claims, classical liberalism means Pareto efficiency by the word efficiency, there is a paradox, then this liberal claim cannot be true.

The paradox is similar in many respects to Arrow's impossibility theorem and uses similar mathematical techniques.

Pareto efficiency

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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 only want money, and they do not get jealous of one another. The following distributions are Pareto efficient:

Carlos	Shannon
$5	$ 5
$10	$0
$2	$8

But a distribution that gives each of them $2 and where the mother 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 money, 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 no other social state that at least one person prefers and no one disprefers.

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 which 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 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 Chatterly'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 himself rather than have it disposed of. However, he 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 that he, himself 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 she should be committed to individual rights, each individual should get to choose whether he, himself 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.

Gibbard's example

Another example was provided by philosopher Allan Gibbard. Suppose there are two individuals Alice and Bob who live next door to one another. Alice loves the color blue and hates red. Bob loves the color green and hates yellow. If each were free to choose the color of their house independently of the other, they would choose their favorite colors. But Alice hates Bob with a passion, and she would gladly endure a red house if it meant that Bob would have to endure his house being yellow. Bob similarly hates Alice, and would gladly endure a yellow house if that meant that Alice would live in a red house.

If each individual is free to chose their own house color, independently of the other, Alice would choose a blue house and Bob would choose a green one. But, this outcome is not Pareto efficient, because both Alice and Bob would prefer the outcome where Alice's house is red and Bob's is yellow. As a result, giving each individual the freedom to choose their own house color has led to an inefficient outcome—one that is inferior to another outcome where neither is free to choose their own color.

Mathematically, we can represent Alice's preferences with this symbol: ${\succ }_A$ and Bob's preferences with this one: ${\succ }_B$. 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) ${\succ }_A$ (Red, Yellow) ${\succ }_A$ (Blue, Green) ${\succ }_A$ (Red, Green)

And Bob's are:

(Red, Green) ${\succ }_B$ (Red, Yellow) ${\succ }_B$ (Blue, Green) ${\succ }_B$ (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 made of $N$ individuals where $N\ge 2$ (Lewd and Prude, or Alice and Bob in the above examples). Suppose there is a set of social outcomes $X$ with at least two outcomes. In the example with Lewd and Prude $X=\{$Lewd reads, Prude reads, No one reads$\}$, and in the example with Alice and Bob $X=\{$(Blue, Yellow), (Blue, Green), (Red, Yellow), (Red, Green)$\}$.

Each person, denoted by $i$, in $N$ is endowed with a preference relation over the set of social outcomes $X$. Each individual's preference is represented by a relation ${\succsim }_i$ over $X$ which is complete and transitive.

A benign social planner must choose one or many "best" social options using the information about the individuals' preferences. The planner is represented as a function from the set of all possible combinations of preferences for the members of society to a subset of $X$. Because we are representing this as a function, it is presumed that the social planner is prepared for all possible preferences of community members (this is sometimes called the Universal Domain assumption).

Mathematically, we represent by $P$ the set of all complete and transitive preference orderings over $X$. The social planner is represented as a function $F:P^N\to \mathcal{𝒫}(X)$ (where $\mathcal{𝒫}(X)$ is the power set of $X$).

There are two properties for this social choice function:

1. A social choice function respects the Paretian principle (also called Pareto optimality) if whenever every individual strictly prefers outcome $x$ to outcome $y$, then the social planner cannot choose $y$. Mathematically,
   • For all societies $p\in P^N$, if for all $i$, $x{\succ }_{i}y$, then $y\notin F(S)$
2. A social choice function respects minimal liberalism if there are at least two individuals, each of whom has at least one pair of alternatives over which he is decisive. That is, where regardless of other individuals' preferences, the social planner will not select the decisive individual's least preferred outcome from that pair. For example, there is a pair $x,y$ such that if he prefers $x$ to $y$, then the society should also prefer $x$ to $y$. Mathematically,
   • There are at least two individuals $i,j$ such that for each there exists a pair of outcomes $x_i,y_i\in X$ (or $x_j,y_j\in X$) such that for all societies $p\in P^N$, $y_i\in F(p)$ only if $y_i{\succsim }_i{x}_i$ (or $y_j\in F(p)$ only if $y_j{\succsim }_j{x}_j$).

In the example above, Lewd was decisive for the pair ("Lewd reads", "No one reads") and Prude was decisive for the pair ("Prude reads", "No one reads").

Sen's impossibility theorem establishes that it is impossible for the social planner to satisfy both conditions. In other words, for every social choice function there is at least one set of preferences that forces the planner to violate either condition (1) or condition (2).

Ways out of the paradox

Because the paradox relies on very few conditions, there are a limited number of ways to escape the paradox. Essentially one must either reject the universal domain assumption, the Pareto principle, or the minimal liberalism principle. Sen himself suggested two ways out, one a rejection of universal domain another a rejection of the Pareto principle.

Universal domain

Julian Blau proves that Sen's paradox can only arise when individuals have "nosy" preferences—that is when their preference depends not only on their own action but also on others' actions. In the example of Alice and Bob above, Alice has a preference over how Bob paints his house, and Bob has a preference over Alice's house color as well.

Most arguments which demonstrate market efficiency assume that individuals only care about their own consumption and not others' consumption and therefore do not consider the situations that give rise to Sen's paradox. 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. Externalities arise when the choices of one party affect another. Classic examples of externalities include pollution or over fishing. Because of their nosy preferences, Alice's choice imposes a negative externality on Bob and vice versa.

To prevent the paradox, Sen suggests that "The ultimate guarantee for individual liberty may rest not on rules for social choice but on developing individual values that respect each other's personal choices." Doing so would amount to limiting certain types of nosy preferences, or alternatively restricting the application of the Pareto principle only to those situations where individuals fail to have nosy preferences.

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.

Pareto

Alternatively, one could remain committed to the universality of the rules for social choice and to individual rights and instead reject to universal application of the Pareto principle. Sen also hints that this should be how one escapes the paradox: 31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

Minimal liberalism

Most commentators on Sen's paradox have argued that Sen's minimal liberalism condition does not adequately capture the notion of individual rights. Essentially what is excluded from Sen's characterization of individual rights is the ability to voluntarily form contracts that lay down one's claim to a right.

For example, in the example of Lewd and Prude, although each has a right to refuse to read the book, Prude would voluntarily sign a contract with Lewd promising to read the book on condition that Lewd refrain from doing so. 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.

In this vein, Gibbard provides a weaker version of the minimal liberalism claim which he argues is consistent with the possibility of contracts and which is also consistent with the Pareto principle given any possible preferences of the individuals.

References

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