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.

An example of the paradox

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

The formal statement of the theorem is as follows.

Suppose there is a set of social outcomes $X$ with at least two alternatives and that there is a group of at least two people each with individual preferences $\succsim _{i}$ over $X$.

A benign social planner has to choose a single outcome from the set using the information about the individuals' preferences. The planner uses a social choice function, which selects a choice $x\in X$ for every possible set of preferences.

There are two desirable properties for this social choice function:

1. A social choice function respects the Paretian principle (also called Pareto optimality) if it never selects an outcome when there is an alternative that everyone strictly prefers. So if there are two choices, $(x,y)$ such that $y\succ x$ for all individuals, then the social choice function does not select $x$.
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. For example, there is a pair $x,y$ such that if he prefers $x$ to $y$, then the society should also prefer $x$ to $y$.

The Paretian principle assumption is that there exists an individual $i$ and a pair of alternatives $a,b$ such that if $i$ strictly prefers $a$ to $b$, then the social choice function cannot chose $b$ and vice-versa.

Similarly there must be another individual called $j$ whose preferences can veto a choice over a (possibly different) pair of alternatives $c,d$. If $c\succ d$ then the social choice function cannot select $d$.

The Minimal Liberty assumption is that there exist at least two individuals who may each independently decide at least one thing (for instance, to decide whether to sleep on his belly or back.) This is a very weak form of liberalism - in reality, almost everyone can decide whether to sleep on his belly or back without society's input. However, even under this weak-form assumption, the social choice cannot reach a Pareto efficient outcome.

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 (there exists at least one situation in the social preferences sets) that forces the planner to violate either condition (1) or condition (2).

Sen's example

The following simple example involving two agents and three alternatives was put forward by Sen.

There is a copy of a certain book, say Lady Chatterly's Lover, which is viewed differently by individuals 1 and 2. The three alternatives are: that individual 1 reads it ($x$), that individual 2 reads it ($y$), that no one reads it ($z$). Person 1, who is a prude, prefers most that no one reads it, but given the choice between either of the two reading it, he would prefer that he read it himself rather than exposing the gullible Mr. 2 to the influences of Lawrence. (Prudes, I am told, tend to prefer to be censors than being censored.) In decreasing order of preference, his ranking is $z,x,y$. Person 2, however, prefers that either of them should read it rather than neither. Furthermore he takes delight in the thought that prudish Mr. 1 may have to read Lawrence, and his first preference is that person 1 should read it, next best that he himself should read it, and worst that neither should. His ranking is, therefore, $x,y,z$.

Suppose that we give each individual the right to decide whether they want or don't want to read the book. Then it's impossible to find a social choice function without violating "Minimal liberalism" or the "Paretian principle". "Minimal liberalism" requires that Mr. 1 not be forced to read the book, so $x$ cannot be chosen. It also requires that Mr. 2 not be forbidden from reading the book, so $z$ cannot be chosen. But alternative $y$ cannot be chosen either because of the Paretian principle. Both Mr. 1 and Mr. 2 agree that that they prefer Mr. 1 to read the book ($x$) than Mr. 2 ($y$).

Since we have ruled out any possible solutions, we must conclude that it's impossible to find a social choice function.

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