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. singlepeakedness. Assume that the set of alter­ However, note that the Pareto-extension rule natives is some subset of9\1' the real line. For gives everyone a 'veto' - if any individual example, the alternatives could be different political parties, arranged from left to right prefers x toy, then x is socially at least as good in terms of their political ideology. Alterna­ as y. The existence of an individual with ~ tively, the elements of A could represent tax veto does not represent an asymmetric distri­ bution of power whereas a dictatorship clearly rates corresponding to various levels of go v­ ern men t spending. means that power is distributed very un­ evenly. However, under rules such as the A preference ordering R is singlepeaked if Pareto extension rule the social preferences there exists a* E A such that for all a, b E A, are typically indecisive, since it requires a*PbPa whenever (a* < b < a) or (a < b < a*). unanimous consent for x to be declared In other words, preferences are singlepeaked strictly better than y. Unfortunately, it was if there is a most-preferred outcome a*, and subsequently shown that the existence of a preferences decline the further away the al­ vetoer is an unavoidable consequence of any ternative is from a* Now, if individual pref­ attempt to resolve the Arrow problem by erences are singlepeaked and if the number weakening the transitivity of social prefer­ of individuals is odd, then the method of ence to quasi-transitivity. majority rule will yield transitive social pref­ erences. Another route out of the impossibility result is to restrict the domain of the Arrow social An alternative interpretation of the single welfare function. It can be argued that in peakedness is that in every triple of alterna­ many practical applications, all logically pos­ tives {x,y, z}, all individuals agree that some sible orderings cannot possibly be meaning­ outcome is not the worst. For instance, if x,y, ful individual preferences. To take an eco­ z is the order in which the alternatives are nomic example, suppose A is the set of all arranged, then for i for whom xRiy, it must possible allocations of a fixed endowment of be the case that yPi z. So, this condition is a certain number of goods. It makes sense to equivalent to demanding that for all i, xRiy assume that individuals only care about their and zRiy do not both hold simultaneously. own component of the allocation, and that In other words,y is not the worst outcome in they prefer more of any good to less. These {x,y, z} for any individual i.

------~~~------RE-S-O-N-A-N-C-E--I-J-u-n-e--20-0-0 100 RESEARCH I NEWS

Sen showed that the sufficient condition for individual utility functions, instead of as a transitivity of social preferences under ma­ function of n-tuples of individual orderings. jority rule (for an odd number of individuals) The use of utility functions permits the in­ could be weakened considerably to one of troduction of various measurability and com­ value restriction. This essentially requires that parability assumptions. in any triple of alternatives {x,y,z}, there is Two different assumptions about the nature at least one alternative, say y) such that every­ of individual utilities are made in econom­ one agrees thaty is 'not the worst' or 'not the ics. If individual utility is ordinal, then an best' or 'not the middle'. Further, Sen showed individual can only compare the levels of that value restriction is sufficient to ensure utility (or satisfaction) associated with dif­ that the social preference relation is quasi­ ferent alternatives. In other words, the transitive irrespective of whether the num­ individual's utility function can be viewed as ber of individuals is odd or even. In joint a real-valued representation of his preference work with Prasanta Pattanaik, Sen went on ordering. So, if u represents individual i's to characterize the domains of preferences i utility function, then so does any positive under which (i) the majority rule will yield monotonic transformation of u • Cardinal util­ transitive social preferences, and (ii) the ma­ i ity is information ally richer. If utility is car­ jority rule will yield social preferences which dinal, then an individual can compare the are acyclic. (Acyclicity means that there is difference in utility between alternatives x {xI' ... , x } X X ... no set k such that 1Pz 2 Pzxk and and y with those between z and w. So, if x kPtX 1') So if preferences are acyclic over A, individual utility is cardinal, then any posi­ then the choice set will be non-empty. tive affine transformation of ui represents the Unfortunately, subsequent work has shown same underlying tastes and preferences. that for subsets of higher dimensional Eu­ So, the utility information which is to be clidean spaces these sufficient conditions for used to generate a given social ordering R is transitivity or even acyclicity are extremely not a single n-tuple of utility functions, but a stringent. This implies that the Arrow im­ set of n-tuples of individual utilities which possibility theorem cannot be evaded by seek­ ing for plausible restrictions on individual are informationally identical. Of course, the preferences. analysis also has to specify the extent to which utilities are comparable across individuals. Perhaps, this motivated Sen and others to The measurability (that is, whether utilities pursue another route of escape from the are ordinal or cardinal) and comparability straightjacket imposed by the Arrow theo­ assumptions can be incorporated by impos­ rem. This is to enrich the informational base ing a class of invariance requirements which of social choice by making the social prefer­ demand the same social ordering for each of ence relation R a function in n-tuples of the n-tuples of utility functions that reflect

------~------RESONANCE I June 2000 101 RESEARCH I NEWS the same underlying reality. Definition: For any (u\' ... , u*n) E L, it is required that I consists of all n-tuples (up A social welfare functional specifies exactly ... ,u) such thatui = tflJu") for all i, for some one social ordering ov~r A for any given n­ n-tuple of transformations (lJIp ... , lJIn) satis­ tuple (u (. ), '" , u (. )) of individual utility 1 n fying the following alternative restrictions. functions each defined over A. Let L be the set of all possible functions u: A ~ 9\1' So, L ONC: Each ~ is a positive, monotonic trans­ is the set of possible utility functions. Let L formation.

= L n be the set of all possible n-tuples of OLe: ~ = lJI for all i, where lJI is a positive, utility functions. Then a social welfare func- monotonic transformation. tional is a mapping F: L ~ 9\.

CUC: Each ~ is a positive, affine transforma­ The invariance requirement takes the gen­ tion, where ~(.) a + be. ) with b > 0, the eral form of specifying that for any two n­ = i same for all i. tuples in the same comparability set L, re­ flecting the assumptions of measurability and In the case of ONC, the comparability set is comparability, the social ordering R must be 'large' because the absence of interpersonal the same. I will give below three types of comparability means that different transfor­ comparability sets. mations can be applied to each individual utility function. Moreover, since individual First, note that the Arrow framework as­ utilities are assumed to be ordinal, all mono­ sumes that individual utilities are ordinal tonic transformations are permissible. An and that there is no possibility of interper­ alternative framework would be to assume sonal comparability. Let us call this ordinal that individual utilities are cardinal but that non comparability (ONC). Second, suppose there is no interpersonal comparability of utilities are ordinal, but the levels of indi­ utilities. Sen actually showed that this would vidual utilities are comparable. Ordinal level not make any difference to the Arrow theo­ comparability (OLC) implies that statements of the following kind are meaningful - indi­ rem. vidual i in state x is better off than individual Interpersonal comparability is incorporated j in the state y. Finally, suppose individual into the framework by imposing the restric­ utility is cardinal unit comparable (CUC); tion that the comparability set only includes this implies that it is possible to compare the transformed n-tuples of utility functions if change in individual irs utility with that of the individual transformations are restricted individual j's in moving from x to y. Then, in specific ways. So, when levels of utilities the comparability sets L corresponding to are comparable and utilities are assumed to ONC, OLC and CUC are specified in the be ordinal (the case of 0 LC), the same mono­ following definition. tonic transformation has to be applied. This

1-Q2------~------R-E-SO--N-A-N-C-E-I-J-u-n-e--20-0-0 RESEARCH I NEWS ensures level comparability since ui(x) > uj(y) choice as being in a person's 'protected implies If/(u 1 (x)) > If/(uj(y)). sphere', and then allowing each person to determine the social preference over pairs of Notice that the in variance requirement ONC alternatives in the person's protected sphere. is the strongest. This has often been 'blamed' Define a person i as strongly decisive over (x,y) for precipitating the impossibility theorem. if xPy holds whenever xPiy holds. It is worth pointing out that in the presence of OLC, the Rawlsian rule of judging x to be Minimal liberty condition: At least two persons socially at least as good as y if and only if the are strongly decisive over one pair of social worst-off individual in x is at least as well-off states each. Sen went on to prove that there is as the worst-off individual in y, satisfies ap­ no choice rule satisfying Arrow's indepen­ propriate modifications of the other Arrow dence of irrelevant alternatives, the Pareto axioms. Similarly, CUC permits consider­ condition, minimal liberty and yielding a ation of classic utilitarianism in which xRy if social preference relation which is quasi-tran­ and only if sitive .

U .I r i (x)?: I;' i ZE:l" 10;\ Although this result was proved as early as 1970, there are attempts even today to re­ Various illuminating characterizations of solve the conflict between minimal. liberty these and other rules are possible in this and the Pareto principle. richer informational framework. Amartya Sen has been the leading advocate of the use of Finally, no account of Sen's contribution to interpersonal comparability in social welfare the theory of social choice is complete with­ judgements, and has also contributed sub­ out mentioning his classic book Collective stantially to the characterization results. Choice and Social Welfare,. written in 1970. Even today, it serves as the best introduction A theorem of Sen which has generated a huge to the subject. amount of literature is Impossibility of the Paretian Liberal. Sen's purpose was to show Bhaskar Dutta, Indian Statistical Institute, that even a very mild requirement of indi­ New Delhi 110016, India. vidual rights or liberty can conflict with the

Pareto principle. The particular form of lib­ Reproduced with permission from Current Science, Vol. 76, erty is based on identifying certain types of No.6, 25 March 1999.

Technology is a gift of God. After the gift of life it is perhaps the greatest of God's gifts. , I' ~ - Freeman Dyson

-RE-S-O-N-A-N-C-E--I-J-u-ne---20-0-0------~------103--