incompleteness and reasoned choice

AMARTYA SEN

INCOMPLETENESS AND REASONED CHOICE

The subtitle of Isaac Levi’s book, Hard Choices, explains the nature of theproblem that he addresses in that classic work: ‘Decision Making underUnresolved Conflict’.1 We have learned greatly from Levi’s analyses ofwhy, despite our best efforts, the valuational conflicts that we face maynot always be fully resolved when the point of decision making comes,and how we may nevertheless use systematic reasoning to decide whatone should sensibly do despite the presence of unsettled conflicts. Indeed,through a variety of contributions stretching over several decades, IsaacLevi has powerfully illuminated the challenges of decision making in thepresence of imperfect information, conflicting evidence, divergent values,discordant commitments, and other sources of internal dissension.2

I seize the wonderful occasion of celebrating Isaac Levi’s work and ac-complishments by presenting a series of observations on rational decisionmaking with incompletely resolved internal dissensions. In that context, Icomment also on the nature and use of incomplete valuational orderingsand the ways and means of extending their reach. I pursue these issues inthe form of addressing a series of questions. Sometimes I draw on Levi’swork, and at other times, I comment on differences that we may still have.As will be obvious, even when we disagree, my understanding of theseissues is strongly influenced by Levi’s thinking.

1. IS NONCOMMENSURABILITY THE PRIMARY REASON FOR

UNRESOLVED CONFLICT?

Noncommensurability of different types of values is often seen as a reason– indeed the main reason – for incompleteness of an overall (“all thingsconsidered”) ranking. Indeed, the belief that commensurability is neces-sary for arriving at a completely ordered ranking of different optionsseems to have considerable following in the literature. Since I have alwayshad some problem in grasping the reasoning behind that belief, I shouldperhaps try to articulate where my difficulty lies.

Synthese 140: 43–59, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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Commensurability of two distinct objects stands for their being measur-able in terms of each other. Noncommensurability is present when “severaldimensions of value are irreducible to one another”.3 In the context ofevaluating a choice, commensurability requires that in assessing its results,we can see the values of all the relevant results in exactly one dimension– measuring the significance of distinct outcomes in a common scale – sothat in deciding what would be best, we need not go beyond “counting” theoverall value in one homogeneous metric. Since the results are all reducedto one dimension, we need do no more than checking which option willgive us how much of the “one good thing” to which every value is reduced.

We are, certainly, not likely to have much difficulty in choosingbetween two alternative options both of which offers just the same goodthing, but one offers more than the other. This is an agreeably trivial case,but the belief that whenever the choice problem is not so trivial, we musthave “great difficulty” in deciding what we should sensibly do seems pe-culiarly feeble (it is tempting to ask, how “spoilt” can one get!). Indeed, ifcounting one set of real numbers is all we could do, then there would notbe many choices that we could sensibly and intelligently make. Whetherwe are deciding between buying different commodity baskets, or makingchoices about what to do on a holiday, or deciding whom to vote for in anelection, we are inescapably involved in evaluating alternatives with non-commensurable results. Noncommensurability can hardly be a remarkablediscovery in the world in which we live. And it need not, by itself, make itvery hard to choose sensibly.

For example, a fine mango may give us nutrition as well as some palatalor olfactory pleasure, whereas buying the record of a good song may offera very different reward (not immediately reducible into the dimensions ofthe other), and given a budget constraint we could quite possibly face thechoice of having one or the other. This involves choosing between non-commensurable results. And yet we may have no great difficulty in optingfor the mango when immensely hungry or starved, and going for the song,when well endowed with tasty food but short of melodious entertainment.The choice need not be hard to make in many situations, despite the non-commensurability involved. The distinct dimensions of values may not bereducible into one another, and yet there may be no problem whatsoever indeciding what one should sensibly do when our priorities or weights overthese values are clear enough.

Making choices with noncommensurable rewards is like speaking inprose. It is, in general, not particularly hard to speak in prose (even if M.Jourdain in Moliere’s Le Bourgeois Gentilhomme may marvel at our abilityto perform so exacting a feat). But this does not negate the recognition

INCOMPLETENESS AND REASONED CHOICE 45

that speaking can sometimes be very difficult (for example, when one isoverwhelmed by emotions), but that is not because expressing oneself inprose is in itself arduous. The presence of noncommensurable rewards onlyindicates that choice decisions will not be trivial (reducible just to countingwhat is “more” and what is “less”), but it does not at all indicate that it isimpossible – or even that it must be particularly difficult.

What we have to ask is: why are some choices “hard”? This must bethe case when the diverse values involved are both in conflict and difficultto weigh relative to each other, that is, when (as Isaac Levi discusses withsuch care) it is hard to “resolve” their conflicting pulls. The exercise ofoverall evaluation may not only involve more than counting, the processof non-trivial aggregation may be complex and challenging in some casesin a way it may not be in others. To say that “noncommensurability is theprimary cause of incompleteness” amounts to missing the specific causesof incompleteness in favour of an ever-present precondition that has littlediscriminating relevance. To use an analogy, it would be like saying that wefeel hungry primarily because we have a stomach. Certainly, it is hard toexplain hunger without presuming something about the stomach, but it isnot particularly useful to concentrate on the existence of the stomach in thehuman body in trying to explain hunger in the world. While the stomach isgenerically involved, we are likely to get more help in explaining hunger ifwe try to examine instead how difficult it may be for a person – particularlya poor person – to fill his or her belly. Similarly, even though the presenceof noncommensurable results is generically involved in incompleteness,we have to investigate whether it is difficult – or easy – to weigh thedifferent types of values involved and to resolve their conflicts.

2. IS THERE ULTIMATELY ONLY ONE SOURCE OF IMPORTANCE?

Noncommensurability does not in itself take us very far in understandingthe nature of the decisional problem to be addressed. It is, therefore, inmany ways a pity that so much attention has been heaped in the literatureon the existence of noncommensurability as such, which is an omnipresent– and ordinary – feature of the world, compared with the distinct reasonsthat make value conflicts so serious in some cases and not at all in oth-ers. But perhaps there is a different, more dialectical, explanation of whyseveral perspicacious thinkers have chosen to emphasize the importance ofnoncommensurability. The attention paid to noncommensurability is partlya critique of the belief – often implicit – that there must ultimately beonly one source of significance. By focusing on noncommensurability, thecritics of such reductionism assert something of importance regarding the

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absence of a single standard of value.4 Indeed, the hold of the traditionof reducing everything to one homogeneous virtue may explain why themundane recognition that there are different values that are not reducible toeach other has been seen as worth asserting. Even a humdrum cognizancemay have a positive – and indeed a constructive – role in the dialectics ofconceptual diagnosis.

Adam Smith complained more than two hundred years ago about thetendency of some philosophers (he had separated out Epicurus, but clearlyhad others in mind as well, including – circumstantial evidence indicates– his friend, David Hume) to look for a single homogenous virtue in termsof which all values could be explained:

By running up all the different virtues to this one species of propriety, Epicurus indulgeda propensity, which is natural to all men, but which philosophers in particular are apt tocultivate with a peculiar fondness, as the great means of displaying their ingenuity, thepropensity to account for all appearances from as few principles as possible. And he, nodoubt, indulged this propensity still further, when he referred all the primary objects ofnatural desire and aversion to the pleasures and pains of the body.5

There are indeed schools of thought which insist, explicitly or by implic-ation, that all the appearances of value must be reduced ultimately to asingle source of importance. This claim about the nature of value is of-ten supplemented by the further thought, in which commensurability doescome in, that in order to be able to choose rationally, all values must bereduced to one. Evidently, the protagonists of this view believe that humanbeings can count but cannot evaluate.

The “counting freaks” (if I may call them that, without intending anydisrespect) include some – but not all – utilitarians. There is indeed aversion of utilitarian reasoning that takes the form of arguing that thereis no noncommensurability at all in what ultimately matters, to wit utility.In the example considered earlier, if we contingently have no difficultyin choosing between the mango and the song, it is because (the argumentruns) both can be judged by their respective ability to generate utility. Inthe last analysis (the argument insists), we count, not judge.

I shall not scrutinize here this particular argumental loop, but only notethat utility may be far from a homogenous magnitude (a recognition thathad not escaped that great utilitarian, John Stuart Mill), and also that therecan be other reasons for choice (other than the pursuit of utility), no mat-ter how utility is substantively defined. If, however, utility is not definedindependently, but only as the real-valued representation of the binary re-lation underlying choice behaviour (as is common in modern economics),then utility is not only not one good thing, it is not a thing at all, but amere phantom of representation. Furthermore, even a consistent phantom


may not exist when choice behaviour is non-binary. I have discussed theseissues elsewhere, and will desist from pursuing them further here.6

3. HOW USEFUL IS LEVI’S GENERAL CONCEPT OF

“V-ADMISSIBILITY” IN UNDERSTANDING THE DEMANDS OF

RATIONAL DECISION MAKING WITH UNRESOLVED CONFLICTS?

V-admissibility, which is a central concept in Levi’s investigation, is in-deed very useful. Even though I shall presently argue (in Section 5) thatparticular questions can be raised about some applications of the idea ofV-admissibility that Levi endorses, let me first discuss why the concept is,I think, important and helpful.

In a set of options (let me call it the “menu”), the V-admissible optionsare those that “have not been prohibited by the agent’s value commitmentsfrom being chosen by the agent”.7 They are, thus, “admissible relative tothe agent’s valuations of the feasible options as better or worse, all thingsconsidered”. The concept of V-admissibility introduces, in two distinctways, a useful “gradualism” in the process of systematic choice. First, ifan option is not V-admissible, then it would not be sensible to choose it,but if it is V-admissible, there may still be further questions to be askedas to whether it is optimal or not. In being definitive in exclusion but onlypermissive in terms of inclusion, it makes the choice process capture animportant asymmetry. Weeding out the clearly “unchoosable” options canbe a good way to begin, but even when the “rotten” ones have been weededout, the remainder may call for further – and closer – attention.

Second, V-admissibility itself is a parametric concept: the criteria ofadmissibility are not already ingrained in the very idea of V-admissibilityand they have to be additionally defined, and can be varied. As the criteriaare more and more specified, the constraints imposed by V-admissibilitycan be gradually tightened. Sequential tightening can be helpful in comingto grips with hard choices.

Let me illustrate by recharactering Levi’s investigation in the form offour distinct steps associated with the idea of V-admissibility (drawing ondifferent parts of Levi’s work).

(1) Single-dimensional valuation. It is, as discussed already, trivial todecide what to do when the different results are all fully commensurable.We can, in this very special case, simply “count” our way to checkingwhich option – or options – are the best (that is, have most of the onegood thing to which everything else is reduced). With single-dimensionalvaluation, V-admissibility is both trivial and decisionally definitive: in onestep we can reject all the rejectable alternatives and go straight to what we

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should choose.8 If this does not work – and mostly it will not – then wemust get into non-trivial exercises.

(2) Congruent multi-dimensionality. When distinct and disparate valuesall move together, in the same direction with each other, the ranking ofoptions can simply follow the shared ordering of all the distinct value com-mitments. As Levi notes, “when the agent’s value commitments generateno conflict but constrain the agent to evaluate his feasible options in an un-equivocal manner, the agent is obligated by those commitments to restricthis choice to one of the feasible options which is optimal according to themandated ranking”.9 The V-admissible alternatives are those that could notbe eliminated on the ground of their being inferior to some feasible optionin terms of each of the value commitments involved. This is also a simpleenough case, but it does not presuppose any commensurability at all.

(3) The Weighted Average Principle: So far nothing more than ordinalranking of each value commitment has been invoked, but we must now gobeyond that. Let v1, . . . , vn be the n numerical valuational functions thatgive the value of each option according to each of the n value commitmentsrespectively. It is assumed that each of these vi is cardinally measurable –unique up to a positive affine transformation.10 Levi confines attention nowto the weighted sum of all the vi as reflecting the aggregate value v of therespective options, for a set of non-negative weights (w1, . . . , wn).11 So wehave: v = � wi vi, and depending on the weights we choose for the respect-ive value commitments, we get a corresponding set of aggregate values ofeach option. If we had a uniquely specified set of such weights (w1, . . . ,wn), then of course the valuational exercise would be over. But so long asthe conflict is unresolved, this could not be presumed. So, at this stage,Levi concentrates attention on the entire class of non-negative weights.However, some options may end up having a lower aggregate value thananother under every permissible set of non-negative weights, and in thatcase, the uniformly lower valued alternative can be eliminated as beingnon-admissible. This way the afforced criterion of V-admissibility elim-inates some “non-admissible” options in the menu.12 This is a criticallyimportant and innovative step in Levi’s reasoning, and I shall examine thepros and cons of going this way in Section 5, when I specifically scrutinizethe weighted average principle.

(4) Categorical preference and optimality. It is possible to go beyondvaluational conflicts when aggregate valuations are in conformity witheach other, at least in part. For example, if x is “strictly preferred in value toy according to all permissible valuation rankings”, then x is “categoricallypreferred” to y.13 The notion of categorical preference is useful in checkingadmissibility, but it can also in some cases – with luck – help to identity an


optimal choice. When it turns out that there exists an option x in the menusuch that for every permissible valuational ranking, x is at least as good asevery other alternative in the menu, then x is declared to be “V-optimal”.If, furthermore, the rankings are such that any option y in the menu thatis regarded as indifferent to an optimal option x is also optimal (but y isdefinitely not optimal when x is ranked above y), then we get to the moreregular idea of full “optimality” rather than only V-optimality.14

The possible existence of optimal or V-optimal options can make de-cisions easier to take despite unresolved conflicts, but the broader ideaof V-admissibility gets us, in general, part of the way, even when we donot arrive at a categorically justified optimal decision. In the gradualistapproach investigated by Isaac Levi, the role of each of the finely definedconcepts is explored and explained.15

4. IS THERE AN ALTERNATIVE SYSTEMATIC APPROACH TO RATIONAL

DECISION MAKING UNDER UNRESOLVED CONFLICTS?

There is, and indeed, to a great extent, Levi’s strategy can be seen as areasoned departure from an older approach that focuses on “maximization”based on a possibly incomplete ordering derived from the “intersection” ofdifferent value commitments. That approach can be seen in terms of its twoconstitutive components, viz. maximization and intersection.16 I shall callit (not terribly imaginatively) “intersection maximization”.17

The idea of maximization as choosing an option that is no worse thanany other feasible option can work with incomplete orderings. To qualifyas “maximal” an option need not be shown to be at least as good as all theother feasible options – only that it is not strict worse than any. The ideaof maximality in this broad sense has been well formalized by Bourbaki,Debreu and others.18 If decision making is based on maximization (ratherthan “optimization” – choosing an option that is at least as good as everyother option), it is not a requirement that all value conflicts be resolvedbefore a reasoned choice can occur. The incomplete ordering to be used formaximization can be derived on the basis of the intersection of the differentvaluational rankings. Even when unresolved conflicts exist, an incompleteordering can be identified that ranks two options in the overall ranking in acertain way if and only if those options are ranked in the same way by allthe different values involved. So, if x is ranked above y according to eachof these values, then x is ranked overall above y. Similarly, if x and y tiein terms of each of these values, then x and y are taken to be indifferentoverall. The “intersection” partial ordering generated in this way can thenbe used, along with the approach of maximization, for systematic choice

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despite the presence of unresolved conflicts (which can make the overallranking incomplete).

The combined use of maximization and intersection is basically iso-morphic with Levi’s identification of a “mandated ranking” related toV-admissibility before we get to his “weighted average principle”.19 Sincethe older approach does not invoke the “weighted average principle”, wecan perhaps ask what alternative principle does it use? The answer is noth-ing: it stops right there with maximization based on the intersection partialordering. In this sense, intersection maximization is an ineloquent theory (Ishall presently comment on the strategic use of this lack of eloquence). The“maximal” set is the set of all options that are undominated by the inter-section partial ordering. It can be shown that maximization, in this broadersense (broader than optimization, and also as it happens, broader thanV-admissibility with Levi’s “weighted average principle”) yields many im-portant – and rather far-reaching – properties in the discipline of choice,and it can be defended both on analytical and substantive grounds.20 Thegeneral approach of maximization can also incorporate various additionalfeatures that were not part of the old structure of maximality identified byBourbaki and Debreu: such as incorporating actions of agents as a part ofthe “comprehensive outcome” of choice, which allows us to re- examinethe dividing line between consequentialist and deontological reasoning.Various other properties can also be optionally imposed by utilizing thecapacious format of maximization, and making use of the room for furtherarticulation left open by the ineloquent form of that approach.21

We have to examine whether Levi’s addition of “weighted average prin-ciple” and its implications for “categorical preference” can also be sensiblyadded on to the basic structure of intersection maximization. I take up thatissue next.

5. IS THE CRITERION OF “WEIGHTED AVERAGE PRINCIPLE” FOR

V-ADMISSIBILITY PERSUASIVE AND ACCEPTABLE?

There are certainly good arguments in favour of using some additionalstructure, like Levi’s “weighted average principle”, to extend the reach ofintersection maximization. There is, first, a manifest need here, and second,Levi’s particular constructive proposal has some conspicuous merit.

I begin with the issue of need and the motivation for Levi’s departure.The maximal set based on intersection can be quite large. Consider Levi’sexample of a hapless office manager, called Jones: I shall call her Ms.Jones. She “wants to hire a secretary who is a good typist and a goodstenographer as well” (and thus has “two value commitments”), and con-


siders three candidates: Jane, Dolly and Lilly. Jones finds that in terms oftyping skill, Jane is better than Dolly and Dolly superior to Lilly, but instenographic competence, Lilly is better than Dolly, who is better thanJane. The intersection of these two rankings is empty, and intersectionmaximization based on these two orderings would suggest that all threeare maximal.

Levi, not surprisingly, sees this as hopelessly inarticulate. He wants toask such questions as whether Dolly, the middle skilled in each, is the“second best typist” and “second best stenographer”, or merely the “secondworst typist” and “second worst stenographer”. Dolly is, of course, allthese, in ordinal terms, but Levi wants to be able to get and use moreinformation about where Dolly figures vis-a- vis Jane and Lilly. This takesLevi to cardinal valuational functions vi and then to the weighted aver-age principle. With this motivation for wanting a more informed – andthrough that a more articulate – choice, I am in total sympathy.22 The issueto be examined is whether the particular procedure developed by Levi isadequate.

The “weighted average principle” is a way of cutting down theintersection-based maximal set by eliminating some options that wouldhave a lower value than an alternative option in terms of all non-negativeweights that can be placed on the cardinal valuational functions associatedwith the respective value commitments. The Levi approach goes beyondthe older maximization approach both (i) by introducing cardinal meas-urement associated with each value commitment, and (ii) by using thestrategy of weeding out any option as V-inadmissible if it gets dominatedby all possible weighted aggregate values by some other feasible option (interms of the cardinal representation chosen). This is where the departurecomes, and we have to ask whether it is convincing.

The second argument, which takes us beyond motivation, is thatLevi’s reasoning certainly has considerable plausibility (which – as I shallpresently discuss – is not the same thing as its being, all things considered,convincing). The plausibility can be illustrated by giving valuational num-bers to the two skills of the three candidates each. Consider the followingvaluations:

Candidates Typing skill Stenographic skill

Jane 10 1

Dolly 2 2

Lilly 1 10

Given the cardinal characteristics of this measurement, any positive affinetransformation of these numbers will serve as well.

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We can add up the two numbers of each with any set of non- negativeweights. But it is clear that no matter what these chosen weights are, Dollywill get trumped by either Jane or Lilly, since Dolly is nearly as bad a typistas Lilly and nearly as bad a stenographer as Jane. To see this consider thepossibility that Dolly gets a higher total score than Jane. In that case theweights must be such that the one- point advantage in stenography thatDolly has over Jane (with her mark of 2 against Jane’s 1) outweighs the8-point advantage that Jane has over Dolly in typing (with their respectivefigures of 10 and 2). So the weight on stenographic skill must be at least8 times as great as on typing skill. But, then, clearly Lilly will beat Dollyhollow with her 8-point advantage over Dolly in stenography, despite herone-point deficit vis-a-vis Dolly in typing. If Dolly beats Jane, then Lillybeats Dolly, and it can also be readily checked that if Dolly beats Lilly,then Jane will beat Dolly. Dolly is, thus, not V-admissible, given Levi’s“weighed average principle”. The reasoning not only takes us beyond thepurely ordinal comparisons of intersection maximization, it also has someevident appeal.

Though I have not been retained by luckless Dolly, let me now argue,first, against this particular conclusion, and then, against the principle ofweighted average in general. The central issue is this: can we do an overallevaluation of the candidates without asking how the importance of thetwo skills may vary as we consider different levels of achievement of thecandidates? Suppose that a skill level of 1 in typing, which Lilly has, doesnot make it possible for letters to be typed well enough to be despatched,but level 2, which Dolly has, makes that possible, though it is nowhere nearthe superb level of typing skill that Jane’s mark of 10 indicates. Supposealso that a skill level of 1 in stenography, which Jane has, leads to totalchaos in the office (“say that a third time please, Ms. Jones”), whereasDolly’s level 2, modest as it is, prevents that, even though the stenographicwork would be massively better with a skill level of 10, which Lilly enjoys.Faced with these considerations, Ms. Jones may drop Jane to prevent chaosin office, and she may also let Lilly go so that some letters can actuallyget typed. Dolly satisfies the qualifying level in each skill, and may wellbe chosen on these grounds (despite her V-inadmissibility in terms of theweighted average principle). I rest my case for Dolly – or more accuratelyfor not giving Dolly the boot without checking what the importance of thedifferent levels of skill are for the choice at hand (and not just how muchskill there is).

I turn now to the more general issue of the acceptability of the weightedaverage principle. Underlying my scepticism is a question about the inter-


pretation of the valuational numbers related to each value commitment.What do the numbers given by vi stand for? Do they:

(1) measure the levels of accomplishment in field i, or(2) measure the importance of these accomplishments for the purpose for

which the choice decision is being taken?

Levi’s language suggests that he takes the former interpretation, so that thenumbers represent the levels of skill of Dolly and others. If so, we still haveto ask how valuable is that skill for the decision at hand, and this cannot beanswered independently of the levels of that skill and the presence of otherfactors that allow or facilitate or hinder the use of that skill. For example,Jane’s excellent typing skill (mark 10 – no less) can be quite valuelessin terms of Ms. Jones’s decisional problem if the woeful nature of herstenographic skill makes her an impossible holder of the office position inquestion. We need not only a valuation of the respective skills (and moregenerally, of the extent of accomplishment in each value commitment), butalso an evaluation of the contingent importance of the exact level of skill(or accomplishment) given everything else that is involved in the decisionalpicture.

If, on the other hand, the second interpretation is taken, then we haveto ask how can the importance of a skill (or more generally, of a valueaccomplishment) for the decision at hand be represented independently ofother factors in the choice? How can we, for example, determine that thechoice- context importance of Jane’s typing skill, in an additive framework,is invariably 10 for this office job, no matter whether she can do anythingelse at all that may also be a part of the job in question? Something orother is clearly missing in this framework, and the operation of weightedaddition, which leads the way to the weighted average principle, can bedeeply problematic.

Perhaps an analogy from social choice theory can help to clarify theissue. Consider a welfarist framework, in which the social value of a stateof affairs is seen as a function of the welfare levels of all the people inthat state.23 We can think of each person’s well-being as a kind of valuecommitment, in terms of an analogy with Levi’s framework. To do thisexercise, we need:

(1) to evaluate each person’s well-being (measured on its own);(2) to make interpersonal comparisons of the values of different person’s

well-being (putting them in a comparable scale);(3) to decide on the weights to be put on increments or diminutions in each

person’s well-being respectively vis-a-vis those of others (reflecting

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what importance we want to attach respectively to distributive andaggregative considerations).24

These distinct exercises cannot be combined in one assignment of vi

functions (unique up to a positive affine transformation) which are thensimply added together with non-negative weights. No matter how wechoose the values of vi functions, something or other would not have beenaddressed. I would argue that a similar lacuna exists in the frameworkof V-admissibility with vi functions, limited to the class of fixed affinetransformations and fixed non-negative weights. Judging the decisional im-portance of the accomplishments of different value commitments requiresmore than a weighted averaging of the different accomplishments.

6. CAN WE USE V-ADMISSIBILITY WITHOUT THE WEIGHTED

AVERAGE PRINCIPLE?

Even though the particular proposal of using the weighted average prin-ciple with V-admissibility seems problematic that does not negate themotivation that led Isaac Levi to seek an extension of the reasoning under-lying intersection maximization. The maximal set given by the intersectionof all the value commitments can be unhelpfully large, and it is sensibleenough to try to see how that set can be reduced in size to give more biteto the decisional process. Dolly may be hard to eliminate from the list ofthree through the weighted average principle in particular, but that need notindicate that all three must be seen as fine appointees merely because theintersection partial ordering is empty. Levi’s motivation in seeking morestringent criteria in the form of tighter requirements of V-admissibility isin general just right.

Consider a more general specification of the problem of going beyondintersection maximization through V-admissibility, by combining Levi’smotivation with formal investigations pursued in social choice theory.25

We follow Levi in considering valuation functions vi related to each valuecommitment, but do not necessarily constrain the uniqueness propertiesof each vi to the class of positive affine transformations. Depending onthe type of values involves, the class may be wider (and the extent ofmeasurability correspondingly less, e.g., ordinal), or narrower (and cor-respondingly have more measurability, e.g., ratio scale), and it is possiblealso to consider various intermediate possibilities of a hybrid kind.26 Wecan proceed from there to consider aggregation functions v = fk(v1, . . . ,vn), and impose invariance conditions that reflect the particular extentsof measurability and any comparability restrictions that we may want to


impose.27 We can consider a number of such aggregation functions fk, withk = 1, . . . , m, in a class F. If for some feasible y, for all fk in F, v(x) is lessthan v(y), then x is not V-admissible.

This general structure can accommodate Levi’s weighted average prin-ciple if the valuation functions vi are cardinal and if the aggregationfunctions fk in F are restricted exactly to the class of weighted aver-ages. Similarly, by choosing ordinal vi functions and an ordinal classF of aggregation functions fk, we can get back to simple intersectionmaximization. But there are a great many other possibilities as well, andV-admissibility can be used to consolidate the minimal articulation of in-tersection maximization, and then to go beyond that to the extent that ispermitted by our actual ability to marshall information and to scrutinizeour assessments. Unresolved value conflicts leaves open many possibilitiesof methodical use of decisional reasoning, and V-admissibility can serveas a general format that accommodates as much articulation as we canreasonably justify.

7. IS INCOMPLETENESS ALWAYS TENTATIVE, AWAITING

RESOLUTION?

In the preceding discussion the focus has been on reducing incomplete-ness as much as possible. But what about any remaining incompleteness?Should we see it as an embarrassment, or as a defect that calls for waysand means of total elimination. I would argue that the answer must dependon the nature of the remaining incompleteness.

In many cases the incompleteness is best described as “tentative”. Itawaits resolution (with more information, or deeper analysis, or closerscrutiny, or whatever), whether or not the resolution actually occurs. Thiskind of incompleteness has to be contrasted with the idea of “assertiveincompleteness”.28 This category separates out cases of incompleteness inwhich the lack of completeness is positively asserted, yielding statementssuch as x and y cannot be ranked. It is radically different from incom-pleteness that is tentatively accepted, while awaiting – or working for –completion. The partial ranking may simply not be “completable”, andmay not even be “ideally completed”. Rather, incompleteness may be theright answer in these cases.

It is useful to distinguish between three different types of assertiveincompleteness related to the source of the “incompletability” involved.First, when the inquiry concerns a specific field of ethical or decisionaljudgment (such as “justice”), the assertive incompleteness may relate tothe domain of that type of judgment (for example, the reach of a theory

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of justice). The recognition that incompleteness of judgments can be aconstituent part of the definite conclusions advanced by a complete theoryof justice can be quite momentous for practical reason.

Indeed, even a complete theory of justice can yield – and assert –incomplete rankings of justice. Affirmed incompleteness (e.g., “x and ycannot be ranked in terms of justice”) may be as much a definite conclu-sion as the decision that x and y can be ranked in terms of justice (andin particular that, say, x is more just than y, or that x is just, whereas yis not). However, it must also be noted that even when it is assertivelyconcluded that x and y cannot be ranked in terms of justice, this does notentail that they cannot be ranked in terms of some other type of ethicalor political concern. There is a distinction here between this kind of field-specific conclusion (important as it may be) and more ambitious claims ofassertive incompleteness for “all things considered” decisions.

Second, even for an “all things considered” evaluation, a partial ranking(or a partial partition) may be a crucial assertion as the end product at a par-ticular stage of a multi-stage exercise. A theory may assert incompletenessfor a well-defined purpose, leaving room for a possible extension throughan appropriate subsequent stage. If some decisional issues are not decid-able by, say, general (or foundational) ethical reasoning, the incompleteranking emerging from general ethical reasoning may be an appropriateassertion for that general stage. This may perhaps be supplemented bya subsequent stage using other procedures (for example, some kind of ademocratic decision mechanism) that extend the general ethical reasoningby choosing between alternative courses of action that are all consistentwith general ethical reasoning.29 The affirmation of incompleteness at onestage of the process may be critically important for the necessity andviability of the next stage.

Third, aside from field-specific and stage-specific assertive incomplete-ness, there is the possibility of an unqualified assertion of incompleteness.It is possible that incompleteness may be a durable and definitive partof the end product of “all things considered” and “all stage included”evaluation. It may be as far as we can proceed with reasoned discrim-ination, given the information that we can conceivably have. If so, theincompleteness will not await “completion” at a later stage or over a widerfield of reference, and will yield such statements as: “x and y definitelycannot be ranked for decisional purposes”.30 There is a need to see “as-sertive incompleteness” as a conceptual category of its own. For example, afine-grained discrimination between two value commitments with exactlyspecified weights may simply be beyond the reach of reasoned ethicalscrutiny, or may demand information that may not only be contingently


lacking but also impossible to obtain even in principle. For example, itcan be argued that interpersonal comparisons of well-being, though by nomeans impossible, may not take us all the way to invariance conditionsthat yield one-to-one correspondence of everyone’s well-being numbersvis-a-vis each other.31

I end by noting that the recognition of the possibility of assertive incom-pleteness does not reduce in any way the value of scrutiny and investigationaimed at reducing the extent of tentative incompleteness through continuedscrutiny of unresolved conflicts. Nor, for that matter, does it rule out thepossible usefulness of challenging whether a putative claim of assertiveincompleteness is indeed justified. Assertive incompleteness – no less thantentative incompleteness – is well within the domain of “inquiry and jus-tification” which Isaac Levi has done so much to clarify for us. We knowfrom Isaac’s work how magnificently capacious that domain is.

NOTES

1 Isaac Levi, Hard Choice: Decision Making under Unresolved Conflict, CambridgeUniversity Press, Cambridge (1986).2 Isaac Levi, Gambling with Truth, Knopf, New York (1967); ‘Information and Inference’,Synthese 17, 369–379 (1967); ‘On Indeterminate Probabilities’, Journal of Philosophy 71,391– 418 (1974); The Enterprise of Knowledge, MIT Press, Cambridge, MA (1980); HardChoices (1986); ‘Conflicts and Inquiry’, Ethics 102, 814–834 (1992); The Covenant ofReason, Cambridge University Press, Cambridge (1997). Levi has been concerned withdecisions in epistemology as well as practical reason. The focus of this essay is, how-ever, exclusively on the latter, even though some of the basic issues have relevance toepistemology as well.3 The Covenant of Reason (1997, p. 236).4 As Isaac Levi explains, “when the perspective on inquiry and justification is combinedwith a rejection of a single fixed standard of value, the key elements of the pragmatism ofPeirce and Dewey which I admire are identified” (The Covenant of Reason, p. 218).5 Smith, The Theory of Moral Sentiments, revised edn., VII.ii.2.14 (1790). Republished,Clarendon Press, Oxford (1976, p. 299).6 See Choice, Welfare and Measurement, Blackwell, Oxford (1982); Harvard UniversityPress, Cambridge, MA (1997); On Ethics and Economics Blackwell, Oxford (1987); andRationality and Freedom, Harvard University Press, Cambridge, MA (2002).7 Levi, Hard Choices (1986, p. 10).8 I am abstracting here from the problem of infinite sets in which even a complete orderingneed not necessarily yield a best or an optimal alternative. That raises a different rangeof issues, with which I shall not be concerned in this essay. In the present investigation,we may as well assume that the set of options is finite, so that a complete ordering willalways identify a best alternative. Indeed, even an acyclic but complete ranking will dothat; on this and related results, see my Collective Choice and Social Welfare, Holden-Day, San Francisco (1970). Republished, North-Holland, Amsterdam (1979, Chap. 1∗),

58 AMARTYA SEN

and ‘Choice Functions and Revealed Preference’, Review of Economic Studies 38 (1971);Hans Herzberger, ‘Ordinal Preference and Rational Choice’, Econometrica 41 (1973); andLevi, Hard Choices (1986).9 Levi, Hard Choices (1986, p. 9).10 An affine transformation is of the form: a + b.vi, and the positivity of such a trans-formation refers to b being positive. On the characteristics and uses of aggregations basedon cardinal values (with uniqueness up to positive affine transformations), see my Col-lective Choice and Social Welfare (1970); Kotaro Suzumura, Rational Choice, CollectiveDecisions and Social Welfare, Cambridge University Press, Cambridge (1983); Levi, HardChoices (1986).11 The weights w1, . . . , wn are not only non-negative each, but also they must sum to 1.12 Levi, Hard Choices (1986, Section 5.4, pp. 77–79). See also the motivating discussionon pp. 69–77.13 This may require us to go beyond the weighted average principle. An extreme casewould be one in which there is a unique set of appropriate weights (w1, . . . , wn). Butcategorical preferences over a subset of alternatives can arise in many different ways, andmay even be precipitated, in some cases, by the weighted average principle.14 Levi, Hard Choices (1986, pp. 83–55). Levi discusses these structures in the context ofexamining “values revealed by choices”, but they throw light also on the movement fromvalues to choices, and not merely from choices to revealed values.15 There are many other structural features which Levi has investigated, which I shall notexamine in this essay.16 Combined use of maximization and intersection can be found in many exercises inpublic economics. See for example A. B. Atkinson, ‘The Measurement of Inequality’,Journal of Economic Theory 2 (1970); Amartya Sen, On Economic Inequality, ClarendonPress, Oxford (1973); extended edition, with a joint Annexe with James Foster (1997).17 In On Economic Inequality (1973) it was called “the intersection approach”, takingmaximization for granted.18 N. Bourbaki, Elements de Mathématique, Herman, Paris, and Theory of Sets, Addison-Wesley, Reading, MA (1968); Gerard Debreu, The Theory of Value, John Wiley, New York(1959).19 Levi, Hard Choices (1986, p. 9).20 On this see my ‘Maximization and the Act of Choice’, Econometrica 65 (1997).21 Some of these possibilities have been explored in my ‘Maximization and the Actof Choice’ (1997) and ‘Consequential Evaluation and Practical Reason’, Journal ofPhilosophy 97 (2000).22 I have a natural sympathy here for various reasons, including the fact that I had a similarmotivation in trying to move social choice theory towards accommodating more informa-tion on cardinality and interpersonal comparability of well-being and also including morecognizance of freedoms and liberties, in Collective Choice and Social Welfare (1970).23 This is, of course, a very limited model of social choice, but adding further consid-erations, such as freedoms, liberties or rights, will not make the problem at hand anyeasier.24 For the distinctions involved see my Collective Choice and Social Welfare (1970) andChoice, Welfare and Measurement (1982).25 For assessments of alternative contributions to the ways and means of using more in-formation in social evaluation, see my ‘Social Choice Theory’, in Kenneth Arrow and


Michael Intriligator (eds.), The Handbook of Mathematical Economics, North-Holland,Amsterdam (1986); and Kenneth Arrow, Amartya Sen and Kotaro Suzumura (eds.), SocialChoice Re-examined, Elsevier, Amsterdam (1997).26 See my Collective Choice and Social Welfare (1970, Chaps. 7∗ and 8∗).27 The form of fk may or may not be additive.28 I have discussed the notion of assertive incompleteness in ‘Maximization and theAct of Choice’, Econometrica 65 (1997); and in ‘Justice and Assertive Incompleteness’,Rosenthal Lectures (Lecture 2), Northwestern University Law School (1998).29 The fact that despite my arguing for the use of the “capability perspective” in comparingindividual advantages and my attempt to highlight the relevance of some basic capabilitiesin particular (for example in ‘Well-being, Agency and Freedom: The Dewey Lectures1984’, Journal of Philosophy 82 (1985)), I have failed to specify a fixed list of distinct“capabilities” (with specified weights or other ways of prioritization) has been the sourceof some chastisement I have received. However, if such a fixed list with fixed priorities andfixed weights were indeed arrived at by general ethical reasoning, it is not clear to me howthis would be consistent with the democratic process of setting priorities and precedence.30 Assertive incompleteness must not be confused with an assertion of indifference. If it isclaimed that x and y cannot be ranked, then that is what it says, not that they can be rankedas equals. Indeed, in some ways incompleteness is the “opposite” of indifference. Considertwo possible claims: (1) x is at least as good as y, and (2) y is at least as good as x. If xand y are indifferent, then both (1) and (2) are true, whereas if their ranking is assertivelyincomplete, then (1) and (2) are both denied. On this distinction, see my Collective Choiceand Social Welfare (1970, Chap. 1∗), and ‘Maximization and the Act of Choice’ (1997).31 I have tried to discuss the class of limited – though possibly quite extensive –informational discrimination in ‘Interpersonal Aggregation and Partial Comparability’,Econometrica 38 (1970), and in Collective Choice and Social Welfare (1970), and alsoin ‘On Weights and Measures: Informational Constraints in Social Welfare Analysis’,Econometrica 45 (1977).

Trinity CollegeCambridge, CB2 1 TQU.K.andHarvard UniversityCambridge, MA 02138U.S.A.

incompleteness and reasoned choice

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