Why Sen’s Liberal Paradox is Unsound
André van Dokkum, 1 December 2014
In his book Collective Choice and Individual Welfare, Amartya Sen (1970: ch. 6*) derives a mathematical inconsistency, labelled the “liberal paradox” that is interpreted as showing “liberalism” in conflict with two other concepts combined, namely the Pareto condition (P) and the condition of universal domain (U). Within a context of a number of alternatives to choose from by a certain collective of individuals, condition P entails that when all individuals (of the collective) prefer some alternative x over another alternative y, the collective (social) preference is such that x is preferred over y. Condition U entails that all logically possible combinations of individual preference orderings over the alternatives are allowed as input for establishing collective preference relations (41). “Liberalism” is defined as condition L as follows (87):
“For each person i there is at least one pair of distinct alternatives (x, y) such that he is decisive in the social choice between them in either order, i.e., xPiy → xPy, and yPix → yPx.” (The Ps here stand for “preference” and are not to be confused with the condition P.)
Sen proves (87-88) that the conditions U, P, and L are inconsistent in the sense that not always a social top preference (cf. 52) can be identified because of the requirement of “acyclicity” (no cycles in the collective preference relations concerning the alternatives). This requirement can be violated in certain combinations of individual preference orderings. To see this it is best to quote the crucial part of Sen’s proof of the inconsistency for four distinct alternatives and a condition L* which is a special case of condition L concerning “at least two” individuals (j and k) who can each decide over a pair of alternatives:
“Assume […] that xPky, zPjw and i: (wPix & yPiz). […] By condition L*, xPy & zPw. By condition P, wPx & yPz. But this […] violates acyclicity.”
Thus L* cannot be satisfied when P and U also have to be satisfied. The same result follows immediately for condition L. “Liberalism” in the form of condition L as given in the proof, cannot co-exist with P and U together.
“Liberal values” are given as “to permit [someone] to do what he wants, everything else remaining the same” (87). This, however, is not captured by the definition of condition L. It is not visible from individual k’s preference between x and y that this individual makes only a personal choice to do what he wants, everything else remaining the same. That there is a choice to make between x and y says nothing about the internal structure of the alternatives x and y. This is made clear by Sen himself in the book’s chapter 6 (without asterisk) where a personal choice is given as against a background of circumstances, called omega, that would have no influence on the personal choice to make and vice versa. Consequently, this omega may vary (79) and the personal choice to make may vary, each without impacting on the other. For determining the collective outcome this means that individuals may exercise their right to decide a pair of personal options with background omega more than once as soon as omega knows more than one manifestation. The proof given above, however, would suggest that such a right is to be exercised only once, over a pair of two internally undifferentiated alternatives to be dealt with collectively. For convenience, let us study an example given by Sen in the chapter 6 without asterisk.
A person A is confronted with a choice between sleeping on his back or on his belly, say x and y respectively, and a person B is confronted with a choice between painting his kitchen walls pink or crimson, say z and w respectively (79). Sen explicitly couples these choices; B’s options function as the omega for A. I add that A’s options function as the omega of B. Mutually, the omegas should not affect the collective outcome if A and B are allowed to enforce their respective choices of sleeping mode and kitchen walls, according to liberalism in the form of condition L (or L*). Now what are the social (collective) states, the alternatives, to be ordered here? They are not x, y, z and w. There is no collective choice to be made between A’s sleeping on his belly and B‘s painting’s the walls pink, for instance. Rather we have to remember that the respective omegas have to be reckoned with as part of the social states to be ordered. Thus the social states to be ordered are combinations of x, y, z and w as follows:
p: x & z
q: x & w
r: y & z
s: y & w
Suppose that A prefers x over y and B prefers z over w. The choices to be made are, however, between p, q, r and s as given in the table, and this should not be confused, as Sen appears to do, with choices between x, y, z and w. By “liberal values”, we then get as collective preferences: (1) p > r; (2) p > s; (3) q > s; (4) p > q; (5) r > s. The collective preference between q and r cannot be determined, because A’s preference for x would force B to abandon the preference for z, and vice versa, in conflict with liberal values as defined. In this sense condition U is violated. Nevertheless it is easy to see from results (1)-(5) that p constitutes the choice set (optimum collective decision) of the set of four alternatives and A and B have their way both as they like.
We can conclude that condition L (or L*) conflicts with condition U directly, not just through intermediation via condition P as Sen’s theorem has it. There is nothing paradoxical about this. Condition U is about what should count as an admissible (and if admitted and submitted, also required) preference ordering to be processed; if any admissible preference ordering in a certain social environment contains effects in which one person’s ordering should effect another person’s preference ordering, the conclusion to draw is simply that that social environment is not, or not wholly, liberal in the sense of the definition of “liberal values”.
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