Andy Blunden June 2004

Justice, Amartya Sen and mathematics

Amartya Sen. Rationality and Freedom, Harvard University Press, 2002.

This article is supseded by: Amartya Sen on Well-being and Critical Voice.

One way of approaching the problem of justice is to be able to answer the question as to whether this or that state of the world can be judged better than another, such as its present state, by consulting the preferences of everyone concerned.

The whole problem then becomes a problem of how to rationally compare the goodness of two given states of the world. As is well-known, Utilitarianism answered this question by saying that one must compare the sum of happiness (or utility), and whichever state of the world has the greatest sum of utility is judged the better; the task of justice, the good life, is then to increase the sum of utility in the world. This utilitarian maxim expresses in the language of ethics, the laws of political economy, and the perversity of its implications I have dealt with in my earlier review of Amartya Sen. For just one example, if a millionaire hires a thief for $100 to rob a poor person of their life savings, then this is just, because the world is better off by that $100, the transfer of the poor person’s life-savings having been “goodness neutral.”

This formulation turns out to have problems for the kind of people who take this stuff seriously. Firstly, it is unproven that the poor person’s savings are worth the same to the millionaire as they are for the poor person, and in fact the logical positivists claimed that interpersonal comparisons of happiness are in principle unintelligible (a proposition which Sen rightly rejects, and his work elsewhere on “capability” goes to resolving that problem). The solution they derived for this one solves also the second problem, that there is no way of giving actual expression to the sum of human happiness, and the problem is in fact resolved every day by the citizens engaging in exchange, and so long a you have a free market, people will make, and only make, those exchanges which increase the “happiness” (or “utility”) of each party.

Thus, the formula of utilitarianism is given a new, more “logical” expression in the concept of Pareto Optimum. The Pareto Optimum is a state of the world in which there remain no exchanges between any two citizens which would increase the happiness of both (each in their own terms) left unmade (or more generally, any single change which does not reduce the “happiness” of anyone while increasing the “happiness” of at least one person).

This concept of Pareto turns out not only to be more “logical,” it is also a better expression of how the market works, and further, it gives a strong justification for the neo-liberal concept of the “ideal market” in which no mutually beneficial exchange is left unmade. The strength of this “modelling” of the capitalist market is reflected in the fact that it mirrors the well-known outcomes of the market. A famine is for example, a Pareto optimum — no-one has any money, having spent everything they have buying food at inflated prices trying to stay alive, there’s no food and everyone starves.

This is like a mountaineer who sets off from London to find the highest mountain on Earth, and ends up at Highgate Hill, with nowhere to go but down. In a world with more than one peak, unless he sets off from Delhi, he will never find Mount Everest — he may see higher places on the horizon, but unable to take a step downwards, can never get there.

It is intuitively obvious even to the most rabid utilitarian that something has gone wrong here. No-one believes that a universal famine is the best of all possible worlds, so a solution to the problem needs to include a caveat that any state of the world which no-one thinks is the best, ought not to be regarded as just. This is called the “No Irrelevant state” condition. Universal famine may be a Pareto optimum, but it is ruled out because no-one thinks it is the best outcome.

So for example, if a group of people are considering how to divide up a pie, and, all being self-centred, selfish economists, none of them think that a 3-way equal division of the pie is the best outcome, because each wants all the pie. An equal three-way division is therefore ruled out on principle. Each thinks that a 50-50 division of the pie with one of the others is at least second best; it is also better for the other person they are considering sharing with, so in fact any 50-50 division of the pie has majority support from these characters. This is, of course, just the kind of thing that goes on in the world of corporate take-overs. This ability of utilitarian “ethics” to mirror market behaviour is its real claim to validity, even though its claim to be a moral philosophy is highly dubious!

Thus the “No Irrelevant state” caveat rules out an equal division, but we have another problem: which of the three characters is to be wiped out by a deal between the other two depends on who speaks first, and even if A and B make a deal to cut out C, C can pipe up and suggest a 40-60 division with B and cut A out of the deal, who counter-offers C a 45-55 deal, and so on without end. Thus, majority voting is in principle incapable of determining which division of the pie is best, not because the majority ruthlessly cutting out a minority is unjust — this is no problem for the utilitarians since this is just normal business — it is the intransitivity of the majority voting, which means that no stable outcome can be determined without giving preference by choosing whose vote to count first. So, to solve this problem of justice we need an additional condition to rule out intransitivity, a condition, by the way, which eliminates all forms of majority voting where there are more than two options on offer. This is a genuine problem which is well-known to anyone who has been seriously involved in running elections: the result depends on how the votes are counted just as much as it depends on how the people vote.

The utilitarians, lacking any criteria of justice other than the untrammelled satisfaction of people’s desires or preferences, or at least the choices they would make given a chance (which is the reality of a laissez faire capitalist society), open up the need to incorporate an unlimited domain of possible preferences. So, according to these guys, if we are to determine whether a state is better than another, we have to take account of people who, for example, care more about someone else having less than they care about themselves having more, or people who care only that they have the same or conversely something different, from what their neighbour has. These people can be very difficult to satisfy. If each of these characters fighting over the pie are going to be happy only so long as the person on their left (around a circular table) have less than they do, then this leads to a vicious circle, in which there can be no agreement. Likewise, if just two neighbours are in a position where one just wants to paint their house the same colour as the other, who in turn only wants their house to be different from the first’s, then these people are never going to be happy — for each suggested outcome, either can suggest an improvement, and so on forever. This condition is called “circularity” and leads to inconsistent results, and our utilitarians have find a method of searching for a means of comparing the virtues of states of the world which avoids circularity and inconsistency.

Slowly, step by step, we have been led into a mathematical puzzle. We are relying on the preferences of all our citizens (however defined, it makes no difference whatsoever to the mathematics); we place no limits on their preferences, so even if we have someone who actually prefers an equal three-way split of the pie, this does not help the mathematicians, because “What if they didn’t?” The problem for these guys is that to start with they have no basis whatsoever for saying what a person should prefer — that would be begging the question, the question being what is the preferred state of the world. Once that question is answered, people are free to prefer it, but in the meantime, the good citizens can only be guided by what the others prefer, or would choose if they had the chance.

So the situation we have in mind is a set of agents, each with a menu of possible states of the world, which they have arranged in order of preference, and have submitted to a hypothetical “returning officer” to determine, from these preferences, a group preference.

Now, because of the fact that our laissez faire friends have so far allowed people to express whatever preference they like (such as desiring that their neighbour would drop dead), it seemed not unreasonable to allow a notion of “minimum liberalism,” that is, that people have a private sphere, their own life for example, which, whatever anyone else said, they could express a preference over, and so long as it was possible — “me alive” as against “me dead” (for example) would always be allowed, other things being whatever. The illustrious mathematician, Kenneth Arrow, first made his name as a young PhD student, by including the minimum conceivable degree of liberalism, namely that there should be just two people in the world who are granted the right to this “personal space” in the study of social-decision theory first begun by Condorcet. Arrow further incorporated the liberal ideal into his reasoning by the criterion of “No dictatorship.” What this “no dictatorship” criterion means is that, given that there is disagreement about at least something in the world (and since all conceivable preferences are to be allowed this is certainly the case), no person should have their preference met in every single question; i.e., the group preference that Arrow was looking for, had to reflect some kind of compromise, and simply picking one of the ballot papers out of the pile at random was not an acceptable way of finding an outcome.

The first 500 pages of Sen’s book is devoted to the 1951 proof by Arrow that there is no way of reliably finding a group preference given: Unlimited domain of possible preferences, no Dictatorship, no Irrelevant options (options which no-one thought was best when they put in their ballot papers), Minimum liberalism (i.e., at least two people with each at least one preference in their private space) and at least three citizens. Obviously, if there are more than 3 voters and more than 2 of them with a private space, etc., the outcome is even more impossible, so to speak. Sen is very concerned about this situation, and so it appears are thousands of other mathematicians and economists beavering away in universities around the world, who have been sweating over Arrow’s result for the past 53 years.

In the earlier Development as Freedom, Sen said:

The informational base for this class of rules, of which the majority decision procedure is a prominent example, is this extremely limited, and it is clearly quite inadequate for making informed judgments about welfare economic problems. This is not primarily because it leads to inconsistency (as generalised in the Arrow theorem), but because we cannot really make social judgments with so little information.

Acceptable social rules would tend to take notice of a variety of other relevant facts in judging the division of the cake: who is poorer than whom, who gains how much in terms of welfare or of the basic ingredients of living, how is the cake being “earned” or “looted” and so on. The insistence that no other information is needed (and that other information, if available, could not influence the decisions to be taken) makes these rules not very interesting for economic decision making. Given this recognition, the fact that there is also s problem of inconsistency — in dividing a cake through votes — may well be seen not so much as a problem, but as a welcome relief from the unswerving consistency of brutal and informationally obtuse procedures. [p. 252, Development as Freedom, Sen 1999]

If Rationality and Freedom is intended to prove that “social choice theory” (in the form known under this name) cannot contribute to understanding the problems of social choice and justice, as this quote from the earlier book seems to imply, then Sen is going a long way around to prove it.

Sen tries a few moves to find a way out of the situation which is interpreted as meaning that under the given conditions, there is no rational way of reliably determining whether one situation is more just than another, throws in a few criticisms of his own and considers some of the suggested resolutions (not all of which have any interest for me, so I will confine myself only to those which I judge to have some interest).

What if, Sen hypothesises, some of our characters have some notion of justice other than their own narrowly conceived self-interest, what if they would actually prefer it if the pie was divided equally three ways? Well, in this arena, this is a quite irrelevant consideration. Our egalitarian friend is welcome to have such a preference, just like our envious characters who only want to have more than the person to their left, or the squabbling conformist/individualist neighbours. But since in principle no-one’s preferences are ruled out — no-one knows in advance that an egalitarian division of the pie is more just, that is to be determined; and since all possible preferences are to be admitted, the egalitarian adds nothing to the situation at all.

What if, Jürgen Habermas might ask, the citizens decide that the outcome ought to be worked out between them, through discourse. In the closing paragraph of India: Development and Participation, Sen himself says: “Democracy is government by discussion,” and “not the same thing as majority rule.” Perhaps, even though an equal three-way division of the pie was not put forward in the first place, after discussing with one another, and reading about the possible outcomes of this difficult problem, our characters might all decide to put an egalitarian division on the top of their lists?

Sen’s answer to this one is startling: could people arrive at an agreement through interaction? That’s a question which can only be answered empirically, he says. He sees no ethical dimension to this suggestion at all, just an empirical question as to whether interaction between citizens could lead to a restricted domain of preferences which would fail to generate the contradictions and indeterminacy anticipated mathematically by Arrow. Sen sees nothing unethical in preferring that the neighbour to your left should have less than you have, or in deciding to split a pie two ways to eliminate a third person; ethical positions, whether arrived at by discussion, by cultural experience or by contract are just preferences like any other, and none of them are any better than any other.

More than this. One critic suggested that people could establish a contract between them, such as “I'll put in for a 1/3 share if you two do as well.” Sen regards this suggestion, going beyond discussion to a negotiated settlement for mutual benefit, as actually unethical and dangerous! His argument is the same as that which Hegel used to dismiss the possibility of common property: so long as there exist independent wills, people can always break their promises if that is their preference, and to enforce the contracts and ensure justice, an oppressive bureaucracy is needed. It’s anti-competitive. A just outcome must be freely made on the basis of one’s own will, otherwise, it can always be broken.

And isn’t it unjust, one might ask, that someone may desire for their own enjoyment, another person’s suffering, and can it be legitimate to weigh that preference along with other, unselfish, preferences? Again, with no criterion for deciding what is just, such propositions have to be ruled out. If you want a world in which most people are not self-centred and selfish, then include that in your preferences, and likewise, if you want a world in which people consider other people’s suggestions, are open to negotiation and keep their promises, include that in your preferences.

Sen gives some thought to the idea that process may be a better criterion than outcome, i.e., deontology, that an action may be good or bad in its own right, independently of the outcome of that action. But if a process produces an outcome which is patently unjust, then isn’t it thereby an unjust process, or conversely, if someone chooses a certain “strategy” to attain their ends, and brings about a result which they don’t like, is that just their business, or is it unjust despite it being of their own doing? Maybe they just weren’t skilled players, or would have preferred to play with different rules?

Which brings Sen to a consideration of the game-theoretic conception of justice. That is, instead of trying to compare infinitely many possible states of the world, and people’s subjective preferences for one or another (an impossible problem anyway), isn’t it more logical to define what are legitimate strategies people can utilise to cope with the conflicting aims of other people, and then rest assured that whatever outcome results from the combined action of everyone’s chosen strategy is the best of all possible worlds, because it is the outcome which people have freely chosen, albeit indirectly, by their own action.

Although Sen does not exhaustively investigate all possible game theory solutions of the problem of justice, he does not believe this approach can resolve the problem. In the first place, everyone agrees that if the outcome is patently unjust, perhaps because the players did not know how to play well, or perhaps because their preferences were so at odds with each other, that mutually assured destruction was the only outcome possible without arbitration.

There are a number of game theory scenarios which demonstrate this, such as the Prisoners’ Dilemma, in which, people have to choose between betraying an other, or rely on making a pact with an other they have no reason to trust. Against both game theory solutions and the logical positivist rejection of classical utilitarianism, and against deontological ethics, Sen is insistent that if the process produces an outcome which is not the preferred one, then the process cannot be deemed to be just, and therefore, the problem of determining whether a given state of the world is more or less just than another state of the world remains, and game theoretic, Pareto or deontological formulations of the decision-making process can only be formulations of a means of finding the group-preference; that still leaves the problem of knowing whether it is just when you get there.

So, for example, if someone has committed a murder and the citizens have to decide whether to put him in jail, or perhaps jail one of the witnesses, how do we judge the preferred outcome? Well, most likely the decision will be made by a great majority to jail the culprit, and if there is to be one person with an entitlement to a “person space,” it certainly won’t be this villain, as in selecting the first person to enjoy “minimum liberalism” the criminal will be denied the normal rights of a citizen. While jailing a convicted murderer presents no special problem, it does raise the issue, doesn’t it, of where the citizens got their preferences from, if not from some concept of justice. If the witness in question was innocent of complicity, but failed to intervene and prevent the murder, on what basis can we hold that they are not just as worthy of being jailed? The outcome is not better or worse; whichever party is jailed, we leave one person in jail and another dead. Does it matter whose action brought about the problem? Is there any criterion for distinguishing one and the same state of the world according to who contributed what to bringing it about? If there is, surely this implies that the preferences people bring to the “returning officer in the sky” already contain a valid conception of justice, and a conception which would justify overriding the preference of the murderer, for example?

Sen adds a genuinely interesting observation to the problem of considering the relative merits of a given state of the world: the healthier a population (such as in Communist-governed Kerala, as opposed to neighbouring Tamil Nadu), the higher is the number of people who report being in poor health. Consciousness of the difference between health and ill-health and the feasibility of doing something about it are an integral part of a good health system. Likewise, teachers know that the classes who achieve the highest level of education are the most likely to be critical of the education they received, the capacity to be critical being the foremost aim of education.

In other words, people cannot be deemed to be the best judges of their own welfare. So what status do their preferences have then?

Continuing in this theme, Sen neatly dismisses cultural relativism on the basis that any culture has its own critics and dissenters, people offering internal criticisms of their culture. There being no basis for determining what is good in such a society on the basis of established opinion, and being assured that dissenters at least will be aware of the ways of life and values of other societies, there can be no basis for withholding “outside” criticism of a culture on the basis of deference for cultural relativism.

So, when the returning officer in the sky settles down to work with everyone’s expressed preferences, how is He to take into account the indisputable fact that people cannot justify their own expressed preferences? If dissenters have been denied air-time by the billionaire media magnates, it is (as we know) highly likely that the citizens will agree that after all, they do live in Candide’s “best of all possible worlds.”

What criterion is to be used to decide whether a given set of preferences, selected by the mathematically-gifted returning officer in the sky, is countable as a group preference, the “winning” proposal? We know that the answer cannot be “would receive a majority vote in a ballot against all other states,” and we know that “No-one can see any single-step change which would improve the selected state” does not suffice, and we know that we can’t simply pick one citizen’s preferred state of the world without taking account of everyone else’s in some way.

Why not some arbitrary algorithm, like “take ballot paper number 75 and invert the 15th selection"? In this case, the answer is that by definition all the citizens and their preference lists are unordered; in the hypothetical society of atomised, sectarian preference-holders, there is no order, there is no citizen No. 1 or 75, and no option 1 or 15. The sought-after procedure has to be able to produce a consistent result whatever order proposals and preferences are taken in. And no proposals are ruled out as inadmissible until after the process has been completed.

As I understand it, Arrow’s Impossibility Theorem disposes of any criterion. Even a decision to find a set of preferences which was least favoured would come to the same impossible result.

Nevertheless, every democratic institution in world history has had some procedure for deciding elections and referenda; all the “players” know the rules, select their strategy, and make up their tickets and preference lists accordingly, according to a kind of game-theoretic conception of the election campaign and voting counting, and hopefully accept the outcome as the group preference, so long as the returning officer has followed the agreed procedure. If the rules are changed, the players change their tactics, and the result will possibly be different. Awareness of the relative arbitrariness of the result, given the equal arbitrariness of the system of voting (an outcome of historically antecedent processes which cannot have been any better placed) would cause us to treat rhetoric such as “The people have spoken” with a grain of salt.

But is justice just a matter of personal preference, or of due process? Is the conception of justice, taking account of the views of everyone affected, necessarily a self-contradictory notion?

And is the exclusive focus on “voters” with fixed views, protected from all ethical scrutiny in themselves, really a sound basis for consideration of the problem of determining a criterion for what is a just state of the world?

In moral philosophy there is a long history of writers who have rejected such a conception (a longer history in fact than that of Utilitarianism). Kant gets frequent mention in Sen’s book, but Kant’s main conclusion from his moral philosophy — that one cannot ethically adopt a maxim unless you can will that everyone else adopts it — is never considered relevant. The 2000-year-old Christian ethic — “Do unto others as you would have them do unto you” is not mentioned. Jürgen Habermas is touched upon in passing and dismissed without any consideration of his recommendation that Kant’s metaphysical dictum needs to operationalised (in Agnes Heller’s version): “What I do unto you, and what you do unto me, shall be decided by you and me,” let alone my own dictum: “what we do must be decided by us.”

In Arrow’s ethics, in principle, nothing a person may prefer can be unethical; even the most outrageous preferences sit alongside the most altruistic. Any discussion between the parties, any effort to find a compromise or a “win-win situation” is for Sen unethical, for the reasons mentioned above. If the “returning officer in the sky” did manage, by good luck, to determine a group preference, there is no reason to suppose that a voter who acted against this outcome was acting unethically. Their action would be contrary to the group preference, but there is no ethical reason to abide by the group preference as opposed to their own preferences.

On what basis would the “group preference” be binding on anyone? Sen has already stated that he regards any commitment to do other than whatever you want, any binding “contract,” social or otherwise, to be immoral and unstable. One is led back to the original question which Arrow was answering, which was to do with international negotiations, and whether the representative of a country or group of people of any kind, had any rational basis for saying what the group they represented preferred in negotiations with others. There is a sense in which one could say that the group preference is binding on someone who is a group representative (including government leaders, diplomats, civil servants, etc.) but it is not binding on anyone else. It is, as they say, “for information only.” It is difficult to see why any citizen should be guided by such information, in that it includes preferences which have been subject to no “vetting” whatsoever. Why should a woman respect a conception of justice which took equal account of her husband’s desire for her to be dead, or a neighbour’s desire only that she should be worse off than him? Even if it were possible to decide upon such a “group preference.”

And where did these arbitrary preferences come from? Were the voters born with these preferences? According to Sen, this is just an “empirical question,” not relevant to the in-principle problem facing the “returning officer in the sky.” And it is only impossible because the domain of preferences is in principle unlimited. As soon as we are allowed to restrict what is counted as an admissible preference, as soon as the citizens are allowed to consult with one another, and seek a just outcome, the impossibility theorem breaks down. And isn’t that exactly what actually happens?

Let us look at an example which is as near as possible to the conditions envisaged in Arrow’s theorem. A change in the law has forced three trade unions (the Airline workers, the Bus workers and the Car workers) which have equal memberships, to amalgamate and draft a single structure to represent their members in the tribunal. Negotiators sit down with firm mandates to defend the interests of their own members, who have for many years been at odds with one another. The law allows only a single seat for the amalgamated union on all representative bodies and tribunals and by law, such representatives cannot be mandated, so selection of a member of one of the unions is a point of considerable sensitivity. Each of the representatives, a, b and c, has a preference list with a, b and c respectively preferred to occupy the hot seat, and b, c and a respectively as their second preference. Their preference lists are thrown into the ballot box and the returning officer is asked to determine the outcome.

The result depends entirely on how the returning officer decides to resolve the first round tie, and it can be fairly agreed that nothing he might devise could in any sense do justice to the preferences expressed. A judgment of Solomon is required to force someone to give ground, force the parties to negotiate or compromise in some way, maybe widening the scope of the discussions so as to allow room for trade-off, make a pact to rotate the position or whatever.

What does the initial impossibility tell us? That majority voting, even in a fair contest, is not a valid way of determining what is just? But we knew that already, without conundrums like this one. Majority voting is as fallacious as a criterion of justice as is the classical utilitarian “sum of happiness” or the Pareto Optimum.

I believe it tells us that justice is something which has to be actively sought, and is not and cannot be the automatic, “logical” outcome of the unrestricted, uninformed preferences of all the people concerned. Justice is a concept relevant only to beings who actually seek it. Flora and fauna cannot know justice, even if we can think of animals as having and being able to express preferences.

Personally, I think far too much time has been given to an old curiosity of mathematics. No mathematical model can be used to study a phenomenon in nature or society unless and to the extent that the mathematical operations to which the mathematical entities are subjected mirror real processes in nature and society by means of the which the real entities and structures modelled are combined or transformed.

The kind of process which goes on inside the counting room after voting in an election or referendum, which is modelled in the “social-choice theory” discussed by Sen, actually happens, and can actually happen, only in a very narrow domain of social life. The will and preferences brought into any such election, which have their gestation in social life generally, are the real processes where ethical conflicts are posed and resolved. If they are not resolved in discussion and social conflict, they will not be resolved by the ballot box. Unless people actively seek justice, then there is actually no meaning to justice. Justice as determined by a “returning officer in the sky” is no better than justice determined by an imaginary God, who at least is not troubled by impossibility theorems.

This is not to say that mathematics is inherently useless. Of course not. But if mathematics is to be applied to the affairs of society, including ethics, then the results can only be as good as the methods chosen. Quoting mathematical results against social and natural reality (“according to aerodynamics the bumble-bee cannot fly”) is a useful exercise only to the extent that it should cause mathematicians and those who try to apply mathematics to be more cautious. Arrow’s impossibility theorem is no different in that.

Personally, I doubt that social-decision theory has very much to offer moral philosophy (if Sen, Arrow and their associates are any guide), but if it is to contribute something, then it must use a tenable model of decision-making, which must, as a minimum, include the capacity of any person to imagine a better world, alongside the certainty that whatever people prefer or aspire to at any given moment, will fall short of what the collective efforts of everyone concerned are able to produce cooperatively. Instead of treating differences as fixed and mutually indifferent, the important thing is how a contradiction is overcome.

Habermas’s ideal discourse theory comes closer to the required approach, but his concept lacks the collaborative activity or project which provides the only reason for talking in the first place and the only possibility for participants in a discourse to come to agreement.

The whole crux of the problem lies in the interaction between people and their collective search for justice, the construction of the ideas and forms of organisation necessary for justice and against those who defend the status quo.