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Can Liberals Value Efficiency? : Sen’s Paradox

By Ishan Kashyap Hazarika


Rights are paramount to a liberal society, where individual liberty is heralded as supreme. Having been tormented for centuries in the Dark Ages, the European capitalist declared his freedom on his ascendance to power in the seventeenth and the eighteenth century as the industry took hold of the feudal society. Great philosophers in the Age of Enlightenment such as Locke held life, liberty, and property as natural rights. The moral defense of liberal society was complemented by equally important arguments by Classical Economists such as Smith and Mill who too, believed in the freedom of the individual. They argued that an economic system based on freedom and property rights would only ensure efficiency. Their arguments contributed to the downfall of the feudal economy— based on serfdom and oppression by the feudal lords and the rise of the free market. From the concurrent development of the social and the economic defenses of liberalism, the belief in individual liberty and the pursuit of efficiency appear complementary. It is, in fact, the case that numerous liberals do hold both the values dear as complementary. Even if not complementary, most, I presume, would not support one and oppose the other. In this article, I seek to explore the connections between the two.

The belief in Individual Liberty, for this article, simply implies that every individual has a ‘private sphere’ of choices. This may include my choice of whether to sleep on my back or my belly for instance. I would not want society to dictate which way I sleep in my bedroom. While this is a rather innocent choice to make, depending on who you ask, one’s personal sphere can include anything from the right to read any book (regardless of what others feel about the book) to eating a certain kind of food, which may be controversial. While what is and is not in one’s personal sphere is a contentious issue, this is irrelevant for this article. In this article, we simply say that a personal sphere exists, which consists of at least one choice (my choice of sleeping on my back or belly or whether I wear blue or green sandals at my home for instance), nothing more. This, I safely believe, is not controversial.

The belief in efficiency can on the other hand be formalized as Pareto Optimality— the impossibility to make one person better-off without making another worse-off. This also appears completely uncontroversial— if we are in a situation where at least one person can be made better-off without making anyone else worse-off, the current situation is not optimal.

Before we explore how these conditions interact, however, we must gain a deeper insight into how societies make choices. Let us say that a society consists of N individuals 1, 2, …., N. The society can be in different states. Let S be the set of all the possible states of society. In some states I sleep on my belly and wear blue sandals at home, in some states, I sleep on my belly again but wear green sandals. In some other states, I sleep on my back! There can be innumerably many such states in the set S. Next, we can define a ‘Social Choice Function’ that takes all the N preference orderings of the N people in the society and converts it into one ‘Social Preference’ ordering. Whichever feasible state is the most preferred by society is selected as a whole by society.

So how do our conditions of ‘Liberalism’ and ‘Pareto Efficiency’ fit in this framework? Let us start with Liberalism—

The ‘L’ Condition: For every individual i, there exist at least two states x and y such that if i prefers x over y, then the society prefers x over y and if i prefers y over x then the society prefers y over x.

This means that there is at least one choice, over which i is decisive— that choice can even be the color of the walls of his house instead of the color of his sandals, but there is at least one such choice. That choice is in i’s ‘private sphere’. To be more clear, the words “the society prefers x over y” does not mean that everyone or even the majority “thinks that x is better than y”. It does not even mean that one person other than i “thinks x is better than y”. It simply means that state x will be chosen over state y by the society if the person i prefers x over y regardless of what anyone else prefers, because this is in i’s personal sphere. It is hereby clear, I presume, why the question of what constitutes the private sphere is irrelevant for this article— the condition simply demands that the personal sphere is non-empty, does not dictate what should or should not be in the sphere.

The other condition or the condition of Pareto Optimality can similarly be formally stated. But to make our case even stronger, we instead state the Weak Pareto Condition— the impossibility that everyone can be made better-off. If we are in a situation where there is a scope of making everyone better-off, then the situation is not optimal.

The ‘P’ Condition: For all states x and y, if every individual prefers x over y, then the society prefers x over y.

For example, if every person in a room prefers watching football over watching ice-hockey, then the room, if given a chance to choose between watching football and ice-hockey, would choose to watch football. If the TV has ice-hockey on it, everyone can be made better off by switching to a channel telecasting a football match. Thus, watching football satisfies the ‘P’ Condition while watching ice-hockey violates it.

Finally, we introduce a technical condition, called the ‘Condition of Unrestricted Domain’—

  1. The ‘U’ Condition: All possible N-tuples of preference orderings are included in the domain of the Social Choice Function.

This condition simply states that the society will be able to decide in every situation (possibly including doing nothing). This appears fairly obvious because it is practically impossible for the society to “not be in any state at all” at any given time

.his condition simply states that the society will be able to decide in every situation (possibly including doing nothing). This appears fairly obvious because it is practically impossible for the society to “not be in any state at all” at any given time.

Now that we have set-up the basic framework to understand the social choice, we can proceed to understand how liberalism (the ‘L’ condition) interacts with efficiency (the ‘P’ condition). Let us consider a society that satisfies both liberalism and efficiency. It then has a Social Choice Function that satisfies both the ‘L’ and the ‘P’ conditions, along with the ‘U’ condition.

In that society, let us consider two individuals 1— with states x and y in his personal sphere and 2—with w and z in her personal sphere. Note, that it is impossible to have {x, y} = {w, z} because the ‘L’ condition will be violated whenever 1 and 2 have opposite preferences.

Thus, let us first assume that x=z and that the rest of the states are distinct. Suppose 1 prefers x to y and y to w and 2 prefers y to w and w to x. x and y are in 1’s personal sphere, so society prefers x to y. Similarly, w and z(=x) are in 2’s personal sphere, so society prefers w to x. Therefore, by the ‘L’ condition, w is the most preferred. But this violates Pareto efficiency, as both 1 and 2 can be made better-off by choosing y instead of w.

Finally, the only possibility left is that all four states— x, y, w, and z are distinct. Here, suppose 1 prefers w to x, x to y and y to z, and 2 prefers y to z, z to w, and w to x. By the ‘P’ condition, society must prefer w to x and y to z. But by the ‘L’ condition, it must prefer x to y and z to w. These again contradict each other. We thus find that in these situations, the society would not be able to choose any state at all, because the conditions contradict each other, which is impossible given the ‘U’ condition.

We can, therefore, conclude that there exists no Social Choice Function with the universal domain, which satisfies both the ‘L’ condition and the ‘P’ condition— implying the impossibility of a Paretian Liberal—who believes in both liberalism and Pareto efficiency. This, itself is the startling conclusion of the paper “the Impossibility of a Paretian Liberal” by Dr. Amartya Sen in 1970, called the ‘Liberal Paradox’ or ‘Sen’s Paradox’ today. This result is considered to be one of the most influential contributions of Sen to Economics. In fact, it has been commented that in terms of its profoundness and impact on Social Choice Theory, this result is second only to Arrow’s Impossibility Theorem.

Now, we can be content that early philosophers, in the absence of mathematical rigor, we're unable to recognize the possible contradiction between individual liberty and efficiency, and that Sen’s result corrects their position. But how do we reconcile this result with the First Fundamental Welfare Theorem, which states that individual decisions in a perfectly competitive market lead to Pareto efficient outcomes? Isn’t the market mechanism also just a Social Choice Mechanism that would satisfy the theorems of Social Choice Theory? The answer lies in seeing under which scenarios the ‘L’ and the ‘P’ conditions contradict. In the First Welfare Theorem, it is assumed that all agents derive utility only from their consumption (that is, there are no externalities)— and thus, other people’s fates do not concern any agent. But in Sen’s Paradox, the contradictions occur when there are ‘nosy preferences’— observe that 1 has strict preferences over states in 2’s personal domain too and vice versa. Thus, one person’s utility depends also on other people’s fate. This is analogous to externalities in the market— under which even the market mechanism fails to attain Pareto efficiency— thus, free markets attain Pareto efficiency in a restricted domain, not an unrestricted domain.

Sen’s paradox has thus, received widespread attention from scholars across disciplines in the social sciences, philosophy, and mathematics. While numerous solutions have been suggested to overcome the problem, including by Sen himself, and numerous rebuttals to the implications have been made, this remains a paradox, and one with deep implications for society and philosophy.


Ishan is a student of Economics from Hansraj College, University of Delhi. He is interested in behavioral economics, decision theory, and the emergent properties of systems.

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