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Amartya Sen and the Math- Ematics of Collective Choice

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Amartya Sen and the Math- Ematics of Collective Choice RESEARCH I NEWS Amartya Sen and the Math­ traced back to the mathematical analysis of elections and committee decisions which ematics of Collective Choice started roughly two centuries ago. Various authors, including Charles Lutwidge Bhaskar Duua Dodgson (alias Lewis Carrol, the author of Alice in Wonderland) had discussed proper­ Sen's work on the causes offamines, gender dis­ ties of different voting systems with the aim crimination and social and economic issues deal­ of proving the superiority of one voting sys­ ing with upliftment of the poor is largely known. tem over another. However, he is better known among professional economists for his contributions in social choice Their interest in this line of research was theory. motivated by the following problem. Given the various individual preferences of mem­ AMARTYA Sen has been a very well-known bers of a society (or committee or collective), economist even amongst laypersons long be­ how do we arrive at a collective or social fore the decision of the Royal Swedish Acad­ choice between various options? Various vot­ emy to award him the 1998 Nobel Prize for ing methods are obviously ways of aggregat­ Economics. His fame has been largely due to ing individual preferences into a social deci­ his influential work on issues which are of sion. However, it was realized long ago that interest to all concerned citizens. Some ex­ the method of majority rule, which is per­ amples are: the causes of famines, gender haps the most common voting method, could discrimination, and more generally social and lead to inconsistencies of the following kind. economic issues dealing with the upliftment of the poor. Not for nothing has he been Example: Consider a committee N = (1,2,3) labelled the 'conscience-keeper' amongst which has to choose one option out of the economists. feasible setA = {a, b, c}. Suppose that the pre­ ferences of individuals in N are as follows: The social relevance of his less technical work cannot be overestimated. However, profes­ I : abc sional economists hold him in high esteem 2: b c a mainly because of his theoretical contribu­ 3: cab tions in social choice theory and the related areas of measurement of inequality and pov­ (Here, preferences are indicated by the order erty. The purpose of this essay is to provide a from left to right; so, 1 prefers a to b to c, and brief introduction to social choice theory for so on.) the non-economist, focusing on Sen's contri­ If majority rule is used to derive the commi­ butions. ttee's (binary) preference relation over pairs The origins of social choice theory can be of alternatives, then a is preferred to b since I R-E-S-O-N-A-N-C-E--I-J-u-ne--2-0-0-0--------------~-------------------------------- RESEARCH I NEWS and 3 prefer a to b, and only 2 prefers b to a. Similarly, the committee must prefer b to c and c to a. Hence, there is no obvious 'best' Let 9\ be the set of all possible orderings choice for the committee since the commi­ over A. ttee's preferences cycle over {a, b, c}. Definition: An Arrow social welfare function The birth of modern social choice theory is is a mappingf:9i n --t R. due to Kenneth Arrow, the Nobel Laureate The interpretation is that an Arrow social for Economics in 1971. In what has been welfare function is a procedure which maps rather inappropriately named the 'general every logically possible n-tuple of individual possibility theorem', Arrow showed that ev­ preferences into a social preference ordering. ery 'reasonable' method of aggregating indi­ It will be convenient to use the following vidual preferences to a social preference or­ notation in the subsequent discussion. Given dering suffered from the same problem of , any n-tuple of individual preferences (Rp R 2 inconsistency associated with the majority ... ,Rn) and an Arrow social welfare function rule. Here is a brief description of what has f, let us use R E 9\ to denote the social been called the Arrowian framework. preference orderingf(Rl' R 2, ... ,Rn). Simi­ larly, R' will denote the social preference Let N be a finite set of individuals, with IN I orderingJtR'p R' 2' ... , R'n)' Just as Pi is used = n ~ 2, andA be a set of feasible alternatives to denote the asymmetric factor of Ri' let us or options with IA I ~ 3. I will assume that A use P and P' to denote the asymmetric is finite, purely for heuristic convenience. However, much of the subsequent discus­ factors Rand R'. sion depends crucially on the assumption Arrow imposed the following relatively mild that N is finite. Each individual in N has a conditions onf preference ordering Ri over A. A binary rela­ tion B on a set X is called an ordering if (i) for Condition 1 (Independence of irrelevant alterna­ tives): For any two n-tuples (Rp R , .•. , Rn) all x,y E X, xBy andyBx (the connectedness 2 and (R'pR'2' ... ,R'n) in the domain off, and condition); and (ii) for any x, y, Z E X, xBy and yBz implies xBz (transitivity). Let us any pair of options x and y such that xRjY if and only if xR'j y, we must have xRy if and interpret R j as the 'weak' preference relation 'at least as good as'. With each Ri is associated only if xR'y. a strict preference relation Pi) the so-called Condition I says that the social preference asymmetric factor of R i, which stands for between any pair x andy should depend only 'strictly better than'; namely xPiy for x,Y E A on the individual preferences between x ifxR.y but notyR.x. The symmetric factor of ! ! andy. 1., R.,t denoted t stands for the relation 'is in- different to'. So, for any x,y E A,xIiy holds if Condition D (Non-dictatorship): There is no ________AAA~AA,~-- ____ 98 ~v V VV V v RESONANCE I June 2000 RESEARCH I NEWS i E N such that for all CRp R z' ... ,Rn) E 9\n, Much of modern social choice theory can be for all pairs x andy inA,xPy holds whenever viewed as attempts to see whether consistent xPjy. decision-making is possible if the original conditions are relaxed. Sen has been one of Condition D rules out the existence of an the leading researchers in this area. individual whose preference dictates social preference between every pair of alternatives. For instance, it can be argued that consistent decision-making is possible even when the Condition P (Pareto): For any CRp R z' ... , Rn) social preference relation is not an ordering. E 9\n and x,y E A if xPjy for all i then xPy. What is required for consistent social choice Arrow Impossibility Theorem: There is no Ar­ is the existence of an alternative which is the row social welfare function satisfying condi­ 'best' in the feasible set in terms of the social tions I, D and P. preference relation. So, let R be any social preference relation, and A' ~ A. Define the Remark: Notice that the restriction that A choice set CCA',R) = {x E A' IxRy for ally E contains at least three elements is crucial. For A'}. That is, the choice set consists of the if A contains only two elements, then major­ pairwise-best elements inA' according to the ity rule satisfies all of Arrow's conditions. binary relation R. Indeed, all sensible voting rules will yield the same social preference relation in this case. Now, consider the notion of ' quasi-transitiv­ The reason why the impossibility theorem ity', which means that the strict preference requires A to contain at least three alterna­ relation is transitive. Formally, tives is that transitivity becomes a vacuous Definition: The binary relation R is quasi­ concept if there are only two elements. transitive over A iffor x,y,z EA,xPy andyPz This theorem posed aformal challenge to the implies xPz. possibility of arriving at consistent social de­ Now, the set of socially optimal outcomes cisions in a democracy. A 'democratic' gov­ can be taken to be the choice set correspond­ ernment should be responsive to the com­ ing to the social preference relation R, and mon will of its citizens. Hence, it will want to this set will be non-empty if R is connected evaluate various policy options from the and quasi-transitive. standpoint of the 'representative' citizen, where the representative citizen's preference It can be seen that given an n-tuple CRt' ... , ordering combines or aggregates the views of Rn) of individual preference orderings of the all the citizens in the society. Unfortunately, n members, we can define a quasi-transitive one interpretation of the Arrow theorem is social preference relation by the so-called that the representative citizen's preferences Pareto extension rule: given x, YEA, xPy if cannot be represented as an ordering. and only if xPjy for all i = 1, ... , n. -RE-S-O-N-A-N-C-E--I-J-u-ne---20-0-0--------------~-------------------------------99- RESEARCH I NEWS The Pareto-extension rule declares the alter­ plausible assumptions can then be used to native x to be socially preferred to y only if restrict the domain of the aggregation proce­ everyone in the society prefers x to y. This dure. rule, and more generally relaxation of the requirement of transitivity of the social pref­ One preference restriction which has been erence to quasi-transitivity, provides a route used in a number of applications is called to escape the Arrow impossibility result.
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