Liberal paradox

The liberal 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 libertarian claim that markets are both efficient and respect individual freedoms. If, as Sen claims, there is a paradox, then this libertarian claim cannot be true.

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

Pareto efficiency

Main article: 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 persons 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 fact that markets produce Pareto efficient outcomes is regarded as an important and central justification for capitalism and is the central result of an area of study known as general equilibrium theory. 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 to 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 outcome than "Prude reads" by both Prude and Lewd, and the chosen outcome is therefore Pareto inferior to another available outcome.

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 chose the color of their house independently of the other, they would chose 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 chose a blue house and Bob would chose 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, given each individual the freedom to chose their own house color has led to an inefficient outcome -- one that is inferior to another outcome where neither is free to chose 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 $N\geq 2$ (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 $N$ in endowed withe a preference relation over the set of social outcomes $X$. Each individuals 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 chose even in a tie with others $y$. 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 individual's 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 $x,y\in X$ such that for all societies $p\in P^{N}$, $y\in F(p)$ only if $y\succsim _{i}x$.

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).

Another example

Suppose Alice and Bob have to decide whether to go to the cinema to see a 'chick flick', and that each has the liberty to decide whether to go themselves. If the personal preferences are based on Alice first wanting to be with Bob, then thinking it is a good film, and on Bob first wanting Alice to see it but then not wanting to go himself, then the personal preference orders might be:

• Alice wants: both to go > neither to go > Alice to go > Bob to go
• Bob wants: Alice to go > both to go > neither to go > Bob to go

There are two Pareto efficient solutions: either Alice goes alone or they both go. Clearly Bob will not go on his own: he would not set off alone, but if he did then Alice would follow, and Alice's personal liberty means the joint preference must have both to go > Bob to go. However, since Alice also has personal liberty if Bob does not go, the joint preference must have neither to go > Alice to go. But Bob has personal liberty too, so the joint preference must have Alice to go > both to go and neither to go > Bob to go. Combining these gives

• Joint preference: neither to go > Alice to go > both to go > Bob to go

and in particular neither to go > both to go. So the result of these individual preferences and personal liberty is that neither go to see the film.

But this is Pareto inefficient given that Alice and Bob each think both to go > neither to go.

		Bob
		Goes		Doesn't
Alice	Goes	4,3	→	2,4
		↑		↓
	Doesn't	1,1	→	3,2

The diagram shows the strategy graphically. The numbers represent ranks in Alice and Bob's personal preferences, relevant for Pareto efficiency (thus, either 4,3 or 2,4 is better than 1,1 and 4,3 is better than 3,2 – making 4,3 and 2,4 the two solutions). The arrows represent transitions suggested by the individual preferences over which each has liberty, clearly leading to the solution for neither to go.

Liberalism and externalities

The example shows that liberalism and Pareto-efficiency cannot always be attained at the same time. Hence, if liberalism exists in just a rather constrained way, then Pareto-inefficiency could arise. Note that this is not always the case. For instance if one individual makes use of her liberal right to decide between two alternatives, chooses one of them and society would also prefer this alternative, no problem arises.

Nevertheless, the general case will be that there are some externalities. For instance, one individual is free to go to work by car or by bicycle. If the individual takes the car and drives to work, whereas society wants him to go to work by bicycle there will be an externality. However, no one can force the other to prefer cycling. So, one implication of Sen's paradox is that these externalities will exist wherever liberalism exists.

Ways out of the paradox

There are several ways to resolve the paradox.

• First, the way Sen preferred, the individuals may decide simply to "respect" each other's choice by constraining their own choice. Assume that individual A orders three alternatives (x, y, z) according to x P y P z and individual B orders the same alternative according to z P x P y: according to the above reasoning, it will be impossible to achieve a Pareto-efficient outcome. But, if A refuses to decide over z and B refuses to decide over x, then for A follows x P y (x is chosen), and for B z P y (z is chosen). Hence A chooses x and respects that B chooses z; B chooses z and respects that A chooses x. So, the Pareto-efficient solution can be reached, if A and B constrain themselves and accept the freedom of the other player.
• A second way out of the paradox draws from game theory by assuming that individuals A and B pursue self-interested actions, when they decide over alternatives or pairs of alternatives. Hence, the collective outcome will be Pareto-inferior as the prisoner's dilemma predicts. The way out (except Tit for tat) will be to sign a contract, so trading away one's right to act selfishly and get the other's right to act selfishly in return.
• A third possibility starts with assuming that again A and B have different preferences towards four states of the world, w, x, y, and z. A's preferences are given by w P x P y P z; B's preferences are given by y P z P w P x. Now, liberalism implies that each individual is a dictator in a least one social area. Hence, A and B should be allowed to decide at least over one pair of alternatives. For A, the "best" pair will be (w,z), because w is most preferred and z is least preferred. Hence A can decide that w is chosen and at the same time make sure that z is not chosen. For B, the same applies and implies, that B would most preferably decide between y and x. Furthermore assume that A is not free to decide (w,z), but has to choose between y and x. Then A will choose x. Conversely, B is just allowed to choose between w and z and eventually will rest with z. The collective outcome will be (x,z), which is Pareto-inferior. Hence again A and B can make each other better off by employing a contract and trading away their right to decide over (x,y) and (w,z). The contract makes sure that A decides between w and z and chooses w. B decides between (x,y) and chooses y. The collective outcome will be (w,y), the Pareto-optimal result.
• A fourth possibility is to dispute the paradox's very existence, as the concept of demonstrated preference, as explained by Austrian economist Murray Rothbard, would mean the preferences that other people do certain things are incapable of being shown in action.

And we are not interested in his opinions about the exchanges made by others, since his preferences are not demonstrated through action and are therefore irrelevant. How do we know that this hypothetical envious one loses in utility because of the exchanges of others? Consulting his verbal opinions does not suffice, for his proclaimed envy might be a joke or a literary game or a deliberate lie.

— Murray Rothbard