Solomonic Judgment vs. Sophists, Economists and Calculators [1] [2]

Given the choice, would you accept to live in a society where happiness and prosperity is guaranteed for all on the condition that one single person be kept permanently unhappy? Is the well-being of thousands of people “worth” the sacrifice and suffering of a single innocent child? Such is the dilemma to which the inhabitants of the utopian city of Omelas are confronted in Ursula Le Guin’s philosophical short-story “The Ones Who Walk Away from Omelas”. In her parable, most people are ultimately able to come to terms with the atrocity. The few citizens who cannot end up walking away from the city — nobody knows where they go and they are never heard from again.

Consider another parable, the biblical story of the Judgment of Solomon. King Solomon must rule between two women claiming to be the mother of a child. Though the initial verdict is that the boy be split in two, each woman receiving half of a dead child, the final judgment is to reward the woman who gave up her claim to the child upon hearing of this initial verdict. Only a true (good) mother would want to save the child’s life at all costs, even if it meant that another woman would get to raise him. A bad mother, on the other hand, would let her feelings of jealousy prevail over the well being of the child.

Welfare economics evaluates the desirability of different social states resulting from different allocations of resources amongst different members of society. Do the stories of Omelas or Solomon’s judgment depict a socially desirable outcome? The now traditional approach in the economics literature, exemplified by Arrow’s impossibility theorem (Arrow, 1950), is framed in terms of the determination of the appropriate aggregation mechanism to produce a social choice on the basis of individual rankings (preferences) over the shared alternatives. This conceptual framework, as outlined in MWG (Chapter 21), first defines as a formal matter the kind of rule that assigns a social preference to possible profiles of individual preferences (technically a social welfare functional[3] or social welfare aggregator [Def. 21.B.1]), and the kinds of restrictions to be placed on such a choice function.

The concept of a welfare function to be defined over social states of affair is rather abstract. Should social welfare be characterized in terms of a ranking of social states of affairs (e.g. a set of alternative complete descriptions)?[4] Does it make sense to postulate an objective welfare function or functional which society ought to maximize? What is this “society” that we speak of? We may ask, as George Orwell does in his essay, “The Lion and the Unicorn”, whether there are really such things as nations. “Are we not forty-six million individuals, all different? And the diversity of it, the chaos!” If societies are not separate organic entities making decisions (see for example Buchanan and Tullock 1962), in what sense is an objective welfare function being “pursued”? To quote Amartya Sen from the opening line to Collective Choice and Social Welfare (Sen 1970a):

“There is something in common between singing romantic songs about an abstract motherland and doing optimization exercises with an arbitrary objective function for a society.”

That said, we must accept the inevitability of social choice and collective action. Social choices (i.e. actions on the part of societal institutions which have implications for all) are being made de facto as a matter of practical necessity, even if as a consequence of disorganized processes rather than coherent and deliberative ones (let alone due to the existence of ‘organic’ or group minds[5]).

Arrow’s impossibility theorem holds that if the number of alternatives being considered is three or greater, a social welfare functional cannot exist which simultaneously satisfies a number of reasonable properties, namely: universal domain[6], Paretian property[7], non-dictatorship[8], independence of irrelevant alternatives[9] (also known as the pairwise independence condition), and collective rationality.[10] This result is not intuitive, and could not really have been anticipated without Arrow’s axiomatic formalization of the problem of social choice (it might therefore be thought of as an example of a proposition that is both true and non-trivial[11]). It is a puzzling, troubling result that shows the incoherence or undesirability of many practical preference aggregation mechanisms, including majority voting — indeed, it expands Condorcet’s Paradox to a much wider range of types of preference aggregation mechanisms.[12]

But is this deep trouble? While Arrow’s theorem shows the impossibility of aggregating preferences without violating seemingly mild axioms of reasonableness, does it establish that reasonable social choice is altogether impossible?

Ways Out?

MWG draw our attention to procedural concerns, as the order in which pairwise decision rules such as majority voting are applied will be relevant to the final social aggregation outcome: “What Arrow’s theorem does tell us, in essence, is that the institutional detail and procedures of the political process cannot be neglected.” (MWG, p799). This is hardly a resolution to Arrow’s impossibility but rather suggests a method of coping: preferring the outcomes generated by specific procedures seemingly overcomes indeterminacy in a specific situation without addressing the fundamental unsatisfactoriness of the choice procedure. More fundamentally, MWG raise the possibility that the impasse of Arrovian social choice may be addressed by loosening some of the imposed axioms. MWG suggest two such ‘possibility results’ in Section 21.D. The first involves the potential relaxation of the strong version of the transitivity requirement imposed on social preferences, acknowledging that social aggregation rules can lead to potential intransitivity.[13] The second involves dropping the universality condition to focus on specific types of individual preferences. The best-known result along this line assumes “single peaked” preferences.[14]

Why stop at social rationality and universal domain? We may wish to consider the desirability of other axioms in Arrow’s framework. For instance, the independence of irrelevant alternatives is not uncontroversial.[15] The Borda count, whereby the rank of an alternative depends on the rank of every other alternative, is an example of a sensible ranking procedure that violates pairwise independence. Though the method can be criticized on the basis, for instance, that it may result in choosing alternatives that have broad mild support rather than those strongly preferred by the majority, it is not evidently unsound.

One final possibility is to drop the Pareto criterion. At first glance it seems like an innocuous assumption, as it is difficult to imagine how it would not be a desirable property of a social decision making procedure. That said, Sen (1970b) provides a somewhat provocative take on the Pareto criterion, showing that it is inconsistent with some degree of minimal liberalism. This result (the ‘impossibility of a Paretian liberal’) establishes that it is not possible to apply the Pareto criterion while allowing individuals to be decisive with respect to at least one pair of personal alternatives (more on this below).

These so-called ‘possibility results’ retain the Arrovian framework, which refuses to make explicit value judgments regarding interpersonal utility comparisons (or indeed interpersonal comparisons of involving non-utility information). Recognizing this points to another interpretation of Arrow’s impossibility theorem, which is that it may instead highlight a fundamental difficulty in making aggregate social welfare judgments solely on the basis of individual subjective preference orderings – a crucial point, unaddressed by MWG, to which we shall return below.

Judgment Contra Aggregation

Is it justified to conceive of social choice as being formed solely on the basis of the aggregation of individual preferences?[16] This limited informational landscape excludes taking into account utility levels and gains associated with social states, thereby precluding interpersonal utility comparisons, as well as any other type of information that is not reflected in utility being drawn upon when arriving at social choices. It confines social choice procedures to, broadly speaking, voting processes.

One example from Sen (1995) illustrates the limitation of the Arrovian approach in this respect. We are to split a cake between two people A and B, with three potential cake allocations: [99,1] (where A receives 99%), [50,50] (equal division), and [1,99] (where B receives 99%). The parties’ preference rankings, assuming that each person prefers to receive as much of the cake as possible, are as follows (where > indicates preference):

A: [99,1] > [50,50] > [1,99]

B: [1,99] > [50,50] > [99,1]

Although it is not obvious, the Arrovian framework only allows us to consider either [1,99] or [99,1] as the most socially desirable outcomes. This result is derived from the “Field-Expansion Lemma”, outlined in Sen (1995) as part of a strategy of proof for Arrow’s impossibility theorem, which provides that if a person is decisive over a particular pair (in the sense that his preference over an alternative prevails regardless of other persons’ preferences over the same alternative), that person will be decisive over other pairs as well, and in effect become a dictator.[17] It follows that the only social outcomes to be considered ultimately are the outcomes preferred by either party, precluding some “middle-ground” solution where neither party gets what they originally wanted but some sort of a compromise is reached. This is not to say that [50,50] is the best social outcome, but rather that it is unclear why it is altogether excluded from consideration, given its evident merits.

Another question is whether it is always desirable to reflect individual preferences (or utilities) in arriving at social judgments. If individual preferences are relevant, are they uncategorically and exclusively so? What if all the citizens of Omelas had been comfortable with the idea of living a good life at the cost of the torture of one single person. Would the overwhelming support make it a socially desirable outcome? We can imagine real life scenarios raising similar issues. Consider for example the will of a deceased father who wishes to leave all family wealth to his only son. The other children, all daughters, wish to respect the father’s decision. All parties at hand are in agreement that the son should inherit everything. Yet it could be argued that a thoughtful social evaluator[18] might favor other allocations whereby all children receive something, in spite of the fact that this goes against the will of the family (indeed, in many jurisdictions disinheriting a child is altogether illegal).

Let us begin to explore some of these issues by considering the story of Solomon’s judgment, which involves the following social outcomes:

  • A: Real mother gets custody
  • B: “Bad” mother gets custody
  • C: The child is split in half (and therefore killed)

The parties’ preference rankings are:

  • Real mother’s preference: A > B > C
  • Bad mother’s preference: B > C > A

Both parties agree that B is preferred to C, so the Pareto criterion requires that the social preference reflect this as well. However, considering only pairwise comparisons (as required by the independence of irrelevant alternatives), what should be the social preference between A and B, and between A and C? The preferences of the real mother and bad mother are in conflict in relation to both of these pairs. If the social preference reflects the real mother’s preference in either case, then she is dictatorial (by the application of the field expansion lemma, as earlier). The same point can be made if the social preference reflects the bad mother’s preference in either case (similar arguments can be made under different configurations of preferences, highlighting the fundamental nature of the difficulty). This difficulty cannot necessarily be avoided by allowing for the possibility of indifference. Social indifference is hardly a helpful concept here, since Solomon’s task as a judge is to choose, and to do so for good reasons if they are available. Should Solomon really choose by tossing a coin? There is a further and more devastating objection to this approach within the social choice framework, however. The social preference must reflect indifference between (A,C) and (A,B) if neither woman is to be decisive with respect to either pair. However, in that case by transitivity the social preference relation must also register indifference between B and C, which conflicts with the Pareto criterion.

A reasonable basis for social choices does exist here (as reflected in Solomon’s judgment) but it is ruled out by the Arrovian framework! By choosing A, Solomon acts as if he were choosing in accordance with a ranking of social states that violates some Arrovian axioms: the social preference is either dictatorial (social preference relation A > B > C), or does not respect the Pareto criterion (A > C > B). Yet would anybody disagree with the view that Solomon’s judgment is for the best? As such, the set of supposedly desirable properties identified in the Arrovian framework offer no guidance at all in understanding the merits of Solomon’s decision:

  • The existence of a real mother “dictator” is less harmful or objectionable than we might have suspected a priori (for the very good reason that good sense in general, and justice in particular, does not always involve ‘splitting the difference’).
  • The Pareto criterion plays little role in the decision, as it merely establishes that a bad mother is universally concurred to be preferable to a dead child (B > C), but it does not help identify the best mother.
  • Solomon’s judgment relies on the fact that the bad mother would rather see the child dead than raised by the other woman and as such violates the independence of irrelevant alternatives (individuals’ preferences between B and C play a role in determining the social preference between A and B, by revealing something about the nature of the person who holds the preferences in question). This is an entirely appropriate violation of the axiom, but the Arrovian framework wholly forbids such violations.

The deeper issue here is that the problem of social choice cannot be reduced to one of mere computation given a profile of individual preferences and a set of desirable axioms. In order to reach a wise decision the judge has to engage in an assessment of what is right to do which draws on the information provided by individual preferences (the bad mother’s preference of C over A is not only horrifying but also most relevant to the determination of what to do) and goes beyond applying any mere mechanism of aggregation, because it depends on the substantive meaning of these preferences. We could conceive of situations involving a similar pattern of preferences (A > B > C and B > C > A), but where A, B and C stand for different social outcomes, in which Solomon would (and should) rule differently. (Consider for instance a dispute between a nature lover and a sports enthusiast about whether or not to leave a nearby nature reserve alone (A), to build a swimming pool in it (B) or to build a tennis court in it (C). In this case, Solomon’s judgment might very reasonably go in either direction, and depend on other relevant considerations such as whether the nature reserve is the last such plot of land or one of very many, on whether there are other sports facilities already in place, or perhaps on whether there are species which would go extinct if it is developed). Solomon interprets the preference rankings in light of his view of what constitutes a good mother and what the rankings tell us in that regard. As this example illustrates, there is a stark difference between adopting a computational approach to social decision-making, and applying considered judgment. Can an approach of the former kind ever adequately encompass the demands of the latter? This is not a question even asked by MWG.

How should we understand what Solomon is doing? It might be thought that in the case of Solomon’s judgment the pair (A,B) should be decided upon by the real mother (assuming we know who she is), and that her views should prevail when the well-being of her child is at stake. Solomon might be asserted in this respect simply to be implementing an approach which recognizes a sphere of rights. However, we can conceive of a situation in which the biological mother is the “bad mother” (the parties’ preferences are switched), so that the judge should rule in favor of the non-biological mother. It seems that inescapably, Solomon must be viewed as engaged in an exercise of judgment, and not as mechanically implementing any set of rules. Solomon in part faces an epistemic problem (who is the real mother?) which he can only resolve through a contextually informed assessment, in this case concerning the actual content of the social states under consideration (e.g. alternative C standing for the child being killed, as opposed to for example, the child being raised by a third person) and what preferences tell us about those who hold them.

Solomon’s judgment illustrates a difficulty in social choice which, though very different in terms of the substantive issues at stake, is related to what Sen (1970b) refers to as the ‘impossibility of a Paretian liberal’, namely that it may not be possible simultaneously to respect the Pareto criterion, consider preferences over a universal domain, and allow individuals to be decisive with respect to at least one pair of ‘social’ alternatives. Sen focused on situations in which, as a guarantee for individual liberty individuals ought to exercise complete discretion regarding certain social outcomes that might be viewed as best relegated to a sphere of purely personal decision-making (e.g. whether to paint one’s walls pink rather than white, or whether to sleep on one’s stomach or not). The difficulty can arise in such situations because of the ‘interfering preferences’ of others over outcomes within this private sphere. In Sen’s example, which concerns who will read Lady Chatterley’s Lover, the two members of society (named “Lewd” and “Prude”) display interfering preferences in that they each have a view regarding whether the other person should read the book, which moreover conflicts with the other person’s preference.[19] A commitment to individual liberty would require that each person be permitted to decide for themself whether to read the book or not, but this will conflict with the Pareto criterion (precisely due to the existence of interfering preferences).[20]

The normative rationale for such interfering preference to be discounted has to do with the case for a sphere of personal liberty, in light of which such preferences must be seen as ‘inappropriately nosy’ or at any rate inappropriate to take into account in social choice. However, there is no way of recognizing this quality without additional information beyond individual preference orderings (Blau 1975) and the appropriate application of relevant evaluative criteria. (In Solomon’s judgment, thepreferences of the bad mother must also be discounted, albeit for different reasons having less to do with ‘nosiness’ than with the best interest of the child). For such preferences to be discounted in social decision making would require a consideration of their substantive character and contextual meaning and not merely their ordinal content, thus involving a departure from the Arrovian framework. Specifically, in the case of the Paretian liberal we must be satisfied that the criteria relevant to determining that a sphere of rights is appropriate to define are satisfied, for instance that there are no externalities of a kind which would place an appropriate limit on a person’s actions. Determining whether preferences are to be regarded as unduly meddlesome or otherwise inappropriate to defer to when determining social outcomes cannot be determined in the absence of information regarding the specifics involved as well as the application of sound judgment. Should it be within a protected sphere of rights to purchase a painting and then decide to destroy it?[21] Is the potential sense of loss experienced by art lovers an externality deemed relevant or irrelevant to the definition of the sphere of rights? In either case, how does this differ from Prude being upset and offended by Lewd’s private reading of Lady Chatterley’s Lover? Does it matter if the painting is a recognized masterpiece as opposed to an ordinary work? Similar issues arise in a very wide range of contexts.

In the case of the Paretian liberal, as in the case of Solomon’s judgment, it is necessary to depart from the Arrovian framework, by taking note of the substantive descriptions of the alternatives being ranked in order to make sense of the situation and determine how best to proceed. The ordinal Arrovian framework requires that we ‘see no evil’ and ‘hear no evil’ but alas this does not imply that we speak no evil nor that we do no evil. Examples of this kind merely illustrate the larger point that whatever the specific evaluative issues involved, judgment is inescapable and rules of aggregation are no substitute for it. This is, alas, our human predicament.

Wikimedia Commons. Copyright © 2003 David Monniaux

Untying Our Hands

How can the informational framework of social choice theory be broadened? What information should be considered relevant to formulating claims about society as a whole? One possibility is to take cardinal aspects of individual utility (capturing for instance the strength of preference) into account. For this, we may conceptually turn to the concept of a Bergson-Samuelson social welfare function (described in Chapter 22), which aggregates individual utilities experienced in each social situation. This raises the issue of how to engage in meaningful comparisons of utility amongst members of society, and in particular whether the utility of different persons can be expressed in a common metric. The scientific legitimacy of such an endeavor has of course been criticized on the basis that utility is highly psychological and subjective, and thus in essence unverifiable. This critique was highlighted in the 1930s by Lionel Robbins (Robbins 1935), operating within the general frame of logical positivism and its insistence that the sole stuff of science consists in empirically verifiable propositions (interpersonal utility comparisons being, as he famously remarked, a case of “thy blood or mine”[22]). This highly influential critique led to the attempts to construct a welfare economics that takes preference rankings as the sole relevant information, relying on the use of the Pareto criterion instead of utility summation (now deemed meaningless). The influence of the Robbins critique can be seen in Arrow’s approach to social choice which asks, in effect, can we still form social welfare judgments with one arm tied behind the back?

There are practical difficulties associated with establishing a common denominator of states of mind, of course, but this does not mean that it is an impossible task, or that it is always unwarranted to make the attempt. Peter Singer, for example, approaching ethical dilemmas from a utilitarian perspective, has suggested that one has the moral duty to accept small sacrifices (e.g. donate to humanitarian relief causes) in order to reduce suffering experienced by deprived starving individuals (Singer 1972). This does suggest that it is meaningful to compare, to some extent, levels of well-being across persons (wealthy and poor), and accept some tradeoff, rather than dismiss the exercise entirely on grounds that utility is completely subjective. Is comparison of such a kind utterly ‘meaningless’ regardless of how rich the rich person and how severely deprived the deprived person in the comparison? If it is not, then what are the guideposts for determining what comparisons are meaningful as guides to judgment and decision?

Another possibility is to look beyond the utility framework. It might be possible to compare social states in terms of some other criteria for evaluating them, including for instance non-utility information about the quality of human experience. A person’s use of a particular resource, or his or her expressed level of contentment, may not always be an adequate measurement of the person’s well being. For instance in situations of ‘adaptive preference’, a person may identify herself as possessing a higher level of well-being than warranted given her economic or social deprivation as recognized from a more ‘objective’ standpoint. More generally, errors in judgment, for instance concerning what conduces to well-being as the individual herself might judge from a more distanced standpoint, may call for the use of other information. In such situations, looking at what conduces to people’s actual well-being and not simply their subjective preference-based orderings over individual and social states of affairs, may be a more fruitful basis for public policy making. This requires a framework for thinking about well-being as potentially distinct from a person’s ability to derive satisfaction from a particular good, and also raises fundamental questions concerning how to define and assess quality of life, to address which both philosophical reasoning and social scientific information of diverse kinds are needed.

One approach, as in Rawls’ theory of justice as fairness, is to focus on the ownership of basic goods that most people would consider to be essential to their well-being, with ‘primary goods’ defined as things which a rational person wants whatever else he wants (Rawls, 1971: 92). The external assessment regarding welfare here involves a reference to the requirements of a rational individual, whatever the goals pursued. Amartya Sen’s “freedom-based” capability approach takes the analysis a bit further by focusing on a person’s capability to convert particular resources into achievements, and not just on the ownership or control of said resources (see for instance Sen 1979, 1992). Sen’s work on poverty and deprivations suggests that it is possible (and indeed necessary) to look beyond income levels, or indeed any materially defined poverty line, and understand poverty as capability deprivation. More generally, the conceptualization of well-being requires an adequate account of human flourishing, in which freedoms to achieve diverse functionings define a person’s capabilities. The approach has been highly influential in development policy, contributing to the creation of the Human Development concept and more recently in guiding economic and social policy assessment generally. For example, the Commission on the Measurement of Economic Performance and Social Progress (the Stiglitz-Sen-Fitoussi Commission) put forward the idea of a ‘dashboard’ consisting of a variety of indicators to evaluate social welfare and sustainability, as opposed to a single measure of well-being (Stiglitz, Sen and Fitoussi 2009). These practical efforts have emphasized the narrowness of an income-based approach to well-being evaluation as opposed to the conceptual deficiencies of utility-based evaluation. However, a capability-based approach to social assessment seems to provide a more natural foundation for capturing concerns for diverse freedoms to achieve.

This informational basis for assessments of social choice and welfare – focusing on the kinds of effective freedoms (encompassing positive and negative liberties) needed for people to achieve a fruitful life is very different from that adopted by the Arrovian framework, where such effective freedoms can only be accounted for to the extent individuals view them as relevant, forming preferences accordingly. Even if utility is given a cardinal meaning, rights and capabilities can only be taken into account on the basis of the utility generated rather than their own intrinsic merits. Still more significantly, as we discuss next, broadening the informational basis of well-being assessment creates a potentially more robust basis for interpersonal comparisons, necessary in order to make meaningful comparisons of social states and to avoid Arrow’s paradox. The impossible becomes possible: all that is needed is to untie one’s hands!

Thy Blood and Mine: Distributive Justice and Interpersonal Comparison

Should distributive justice play any role in analyses of social welfare and choice? Social judgment (including in regard to social welfare and social justice) would seem to require that we promote fair consideration of all individuals’ interests, at least by way of the symmetric treatment of individuals in the procedures of social choice. However, we must ultimately ask, in the words of Amartya Sen (Sen 1979), equality of what?

A libertarian might resist engaging in any assessment of the desirability of a particular social allocation, stressing instead whether it was brought about by free exchange, or more generally whether proper procedures were followed in arriving at an outcome, regardless of the inequalities that may be generated (e.g. Nozick 1974). In contrast, if the ‘pattern’ of outcomes realized is taken to be most relevant in assessing a state of affairs, what distributive criteria should be used to rank different outcomes? If interpersonal utility comparisons are taken to be meaningful, different types of social welfare functions aggregating individual utilities differently will result in different social allocations. Going back to Sen’s cake allocation example, a utilitarian social planner will divide the cake so as to maximize total utility (at the point of equality of everyone’s marginal utility).[23] An egalitarian planner would seek to achieve a social outcome whereby each person derives equal utility from eating. A Rawlsian-inspired planner would likely assign a larger portion of the cake to the poorest person, so as to maximize the utility of the worst-off individual.[24] Allocation problems can also be addressed in the framework of axiomatic bargaining (see our prior entry Thirteen Ways To Split a Cake), where utilities resulting from the cake allocation would be assessed on the basis of utility differences from a threat point, as opposed to absolute values.

As we noted above, there are other possible conceptual frameworks that do not involve utility comparisons. The capability approach allows for interpersonal comparisons that take forms other than utility comparisons, with differences in capabilities amounting to differences in individuals’ abilities to live freely and well. Though the capability approach does not suggest any specific formula for policy decision (for example it does not suggest that a social planner should necessarily seek to equate everyone’s capabilities at all costs), it does provide an informational framework for assessing the goal of promoting individuals’ opportunities to achieve outcomes, deemed socially valuable.[25] A greater share of cake going to a starving person might be desirable to the extent it alleviates hunger, not merely because it generates utility or well-being, but because it is intrinsically and instrumentally linked to greater opportunities for individuals to live free and flourishing lives. Where Robbins excoriated interpersonal comparisons on the ground that they involved questions of thy blood or mine, the capability approach begins with the presumption that flourishing lives are valuable: thy blood and mine.

Adopting capabilities or another non-utility space of evaluation of individual lives helps with the problem of interpersonal comparison for two reasons. Firstly, information about individual flourishing in the space of capabilities may lend itself to interpersonal comparisons since it is more tangible and less metaphysical than utility (pace Robbins). To ask, for instance, about anthropometric information on individuals which tells us something about their future prospects for good health and a long life is decidedly to employ measurable and interpersonally comparable information, even if there are methodological challenges which arise in this as in all cases of comparison. Secondly, concern for adequacy, or equality, in the space of capabilities may have determinate implications that are distinct from those which arise in other spaces (e.g. resources). Going back to our prior example of the family inheritance, the capability approach would place less emphasis on the children’s own preferences (the only son should get everything), and instead consider the distribution of opportunities amongst children.[26] A capability-based approach to social judgment (taking into account interpersonal comparisons in valued capabilities and the ability to transform resources into them) can both facilitate interpersonal comparison and thereby sidestep the Arrovian paradox, which is based on a restriction to ordinal subjective preference information. Whether such an approach, or any other, is warranted or not must however ultimately be judged on the basis of underlying evaluative commitments. The scientific approach desired by Robbins must be allied with sound evaluative judgments if we are to overcome the impasse.

The Impossibility of a Computational Democracy

It has been suggested that Arrow’s impossibility theorem establishes the ‘impossibility of a democratic computer’ (Mirowski, 2002: 302). What, in light of our discussion, should we make of this result? Does it establish an inadequacy of computation or of democracy?

Arrow’s result was a foundational development in formal social choice theory. The axiomatic reasoning approach in social choice theory makes it possible to identify specific conditions that may or may not be mutually satisfied, and therefore what conditions we must seek to weaken or to dispense with, and in this respect places us in a “well-lit space as to what we are doing” [Amartya Sen in the Second Annual Kenneth Arrow Lecture]. Signally, it alerts us to the limitations intrinsic to viewing social welfare solely in terms of outcomes determined by a voting process — majority voting or other such procedure need not deliver consistency let alone (as we have seen above) satisfactory decisions. One simple, final example will illustrate this latter point: A proposal of taking the income of the poor minority in a country, cutting it in half and giving the other half to the rest of the population would strike us as unsound social judgment, and therefore a poor basis for policy, though it may well receive the approval of majority voting.[27] Indeed, if the poor minority is reduced to one person from whom we take almost everything, we get the story of Omelas.[28]

MWG’s approach to Arrow’s negative result focuses on how to accommodate the difficulties highlighted by Arrow’s theorem through various possible relaxations of the framework, but it focuses not at all on the role of broadening the informational basis of social judgment by leaving the realm of subjective preference. The Arrovian framework views the alternatives in a fashion which is content-neutral, but that restriction makes it impossible to look beyond individual orderings of social states of affair to form social judgments and choices which are warranted by considered reasoning, whether or not they rise to the level of Solomonic wisdom. Those who take the Arrovian framework as the last word in social judgment and decision cannot step outside the framework to assess such decisions from an independent perspective, and in this respect try to wash their hands of the problem altogether. Arrow’s theorem tells us that, much as this would be nice if it were possible, it is not. As such we might suggest that Arrow establishes the impossibility of a computational democracy, pointing instead to the need for non-algorithmic bases for judgment — an inadequacy of computation rather than of democracy. Viewing Arrow’s impossibility theorem as an invitation for informational enrichment opens up new possibilities for constructive social choice theory, freeing our hands to build and create.


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[1] We are very grateful to Wulf Gaertner for commenting on this text – without of course holding him responsible for its weaknesses.

[2] Edmund Burke famously wrote, in his Reflections on the Revolution in France (1790): “But the age of chivalry is gone. That of sophisters, economists, and calculators has succeeded; and the glory of Europe is extinguished forever.”

[3] A social welfare functional is a rule that assigns to any profile of individual ‘rational’ preference relations (a profile of individual rankings defined over a set of alternatives), a social preference relation (a unique social ranking of such alternatives).

[4] It is possible to approach formal social choice in a manner which allows procedural concerns (e.g. enforcement of contracts, protection of rights, etc.) to be taken into account, alongside social outcomes as done in the formal literature on liberties and rights (in which a foundational contribution is Gaertner, Pattanaik, and Suzumura 1992).

[5] Some of the recent philosophical literature on group minds considers the hypothesis that collectivities may have a claim in social ontology as entities in their own right, with psychological autonomy. See for example Petit (2003) for the claim that thinking of a collectivity as an intentional subject displaying consistency and rationality in decision-making can help solve a ‘discursive dilemma’ in situations where a judgment on a final issue is determined by a number of separate considerations. If a collective view is reached on the basis of first aggregating individual views on each premise (premise-centered voting) and deriving the logical consequences, the outcome may not necessarily coincide with what would have been reached had individuals directly voted on the final outcome (conclusion-centered voting). It is claimed that thinking of the collectivity as an intentional subject displaying consistency in decision-making helps to avert this dilemma.

[6] There must be no limitations on the sets of preferences of individuals to which the functional can be applied, i.e. the social welfare functional must yield a social ordering for every possible combination of individual preferences.

[7] The social welfare functional must respect unanimity of strict preferences on the part of individuals. If every individual prefers x to y, the social welfare functional must yield a social ordering where x is also preferred to y.

[8] There is no one individual in the society whose preference relation determines the social preference.

[9] The social preference between two alternatives must be dependent only on individual preferences over the same alternatives. Individual preferences with respect to other alternatives are irrelevant.

[10] The social welfare functional is by definition taken to be ‘rational’ (complete and transitive in the definition of MWG).

[11] Samuelson, P.A. (1969), “The Way of an Economist”, in International Economic Relations: Proceedings of the Third Congress of the International Economic Association, London: Macmillan, pp. 1–11.

[12] A related result had already been known since the late 18th Century as the Condorcet Paradox in the specific case where the social aggregation mechanism consists of majority voting. In a work published shortly before the French Revolution, Condorcet (a central figure in the Enlightenment who also played a leading role in the French Revolution) showed that pairwise majority voting can lead to cyclic patterns and intransitive social preferences in situations where three alternatives or more are under consideration (Condorcet 1785, 1788). Incidentally, Condorcet’s work on constitutional issues ultimately led to his demise. He was branded a traitor during the Terror for criticizing Constitution’s Commission, and subsequently arrested. He died in prison in an apparent suicide.

[13] A possible approach is to replace the transitivity requirement with the quasi-transitivity of the social preference relation, on which see MWG p800.

[14] If the domain is restricted to single peaked preferences, majority voting will result in social preferences being identical to those of the median voter, which will depend on the distribution of preferences in society. As such, the social aggregation rule will be transitive because under the specified assumptions the median voter’s preferences are transitive (the median voting theorem).

[15] The pairwise independence condition seems to impose some kind of menu independence requirement with respect to the formulation of social choice, but it is not immediately obvious why the social preference between two alternatives should be unaffected by individual preferences over other alternatives. MWG (p794) gives us three possible lines of justification for this condition: a normative appeal (not immediately obvious, for instance because the presence or absence of other alternatives could potentially matter if choice is thought of as being driven not just by intrinsic properties of objects, but also by a range of external considerations – on this, see Sen 1997); a concern for practicality, in that the assumption helps to “separate problems” (in which case could Arrow’s impossibility potentially be overcome if one was ready to accept practical difficulties?); and an incentive issue connected with the need to provide inducements for truthful revelation of individual preferences (this point is not immediately obvious – is it so critical that the social choice function provide for such an inducement mechanism?).

[16] This point is not necessarily obvious. For instance, it may be argued that the concept of volonté générale (general will) developed by Jean-Jacques Rousseau (Rousseau 1762) is meant to capture a notion of common good and public interest that cannot be reduced to whatever decision can be reached with majority rule, or indeed to the ‘sum’ of individual wills arrived at through any ‘static’ aggregative scheme. Rather it appears to involve the idea of the formation of the general will through a dynamic procedure involving the encounter of individual wills in a social whole (through which process the individual will itself can change). The general will is moreover not described even thereafter as standing in any specific aggregative relationship to individual wills.

[17] For a proof of the Field-Expansion Lemma, please refer to Sen (1995). In the particular example of the cake division, if the social preference ranks [50,50] over [99,1], we are allowing B’s preference to prevail and B may be deemed decisive over that specific pair. However, by the Field-Expansion Lemma, B must then be decisive generally and B’s preference of [1,99] over [50,50] must get priority in the determination of social preference as well. By an analogous argument involving A, if the social preference ordering ranks [50,50] over [1,99], then [99,1] must be ultimately deemed socially preferred to [50,50] as well. In either case, [50,50] cannot ultimately be deemed socially preferred. In this particular example, one of the parties is necessarily decisive and indeed dictatorial.

[18] We can think of such an evaluator in terms of Adam Smith’s impartial spectator in The Theory of Moral Sentiments, or Harsanyi’s impartial observer (Harsanyi 1953, 1955). Given that observations and judgments are always formed on the basis of a specified viewpoint, we might expect that the social arbitrator aspire to ‘positional objectivity’, which requires some degree of interpersonal independence once the particular parameters of an observational position are fixed or indeed to ‘trans-positional objectivity’ which requires that all positional perspectives be adequately taken into account (Sen 1993).

[19] Prude would prefer not to read the book (which he views as indecent), but Lewd would rather Prude read the book than not (he gets enjoyment out of Prude having to read it), and vice-versa; Lewd would rather read the book than not, but Prude thinks this would not be in Lewd’s best interest.

[20] Considering three social outcomes “Lewd reads” (L), “Prude reads” (P) and “No one reads” (N), in Sen’s example the personal preferences are (where > stands for strict preference): P > L > N for Lewd, and N > P > L for Prude. By the condition of universal domain, this pattern of preference must yield an acyclical social preference, but it cannot comply with both the Pareto condition and that of minimal liberty, which would require that the social preference reflect Lewd’s personal preference to read the book (L > N), and that Prude should not be forced to read it (N > P). The contradiction arises not just because interfering preferences conflict with the other person’s private choice, but also because of the intensities of the reciprocal interests (Blau 1975). In Sen’s example both parties feel more strongly about opposing the other person’s private decision than fulfilling their own. If, on the other hand, interfering preferences still contradict personal choice (Lewd still wants Prude to read Lady Chatterley’s Lover, and Prude still wants to censor Lewd), but one of the parties is ultimately more interested in advancing his or her own interest than in interfering with the other (for example, ultimately Lewd places his own interest in reading the book above Prude’s), Lewd would be permitted to read the book without violating the Pareto condition (as Lewd and Prude would no longer be in agreement that P > L).

[21] An apposite example involves best-selling author Patricia Cornwell, who was the subject of controversy for allegedly destroying a painting by Walter Sickert in an attempt to establish the identity (DNA) of Jack the Ripper.

[22] “If we disagree about ends it is a case of thy blood or mine”. Robbins (1935: 150).

[23] Note that it is also possible to adjust the profiles individual utilities by attaching decreasing weight to successive units of individual utility with the generalized utilitarian social welfare function such that, for instance, a person with a uniformly high capability of enjoyment (e.g. a “utility monster” who never experiences diminishing marginal utility) would not attract as high a share of resources as his or her utility function would suggest (see MWG p827). One could conceive of other possible adjustments – e.g. from a complementary point of view, a person who derives lower levels of enjoyment from an available resource (e.g. due to some disability) could receive a compensatory allocation. However, such adjustments would seem to involve identifying the non-utility reasons for differences in the ability of different persons to translate resources into utilities.

[24] The maximin (Rawlsian) social welfare function reformulates Rawls’ maximin principle in terms of utility. However, we note that Rawls’ original analysis in Rawls (1971) was framed in terms of ownership of primary goods rather than utilities, and also defined as an institutional principle aiming at bringing about ongoing outcomes as opposed to a resource allocation rule meant to be applied in single instances.

[25] There is a growing literature which implicitly invokes interpersonal comparisons on the basis of capability sets. The development of the Human Development concept and index is the best known applied effort. See also e.g. among others, Gaertner and Xu (2006) or Reddy and Pogge (2010).

[26] If one child was disabled and required a much higher income to maintain a quality of life on par with that of her siblings, an application of the capability approach aiming toward equalization or at least sufficiency of capabilities might suggest that the disabled child should be allocated a higher share of the family wealth so as to have equal or at least adequate freedom to achieve well-being.

[27] This is an example directly taken from Sen’s Second Annual Kenneth Arrow Lecture.

[28] Majority voting is not the only candidate for a social aggregation rule, of course. This example merely illustrates how a meaningful exercise in social judgment cannot be reduced to a mere aggregation of individual preference rankings (whether this is done via majority voting or another aggregation method) but must rather integrate other concerns relevant to social choice, such as rights. Indeed any consideration relevant to individual and social well-being not properly captured in the parties’ ordinal preferences falls outside of the standard informational framework of social choice theory and must be separately noted and integrated into social assessment and decision making, if it is to play any role at all.

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