The Problem

A ‘good society’ is one which is governed by a number of prized social virtues. Amongst these virtues, surely, must be counted a deference to the values of personal liberty and inter-personal equity. (These, after all, are two of the three values embodied in the French Revolution's stirring exhortation to ‘Liberty, Equality, and Fraternity’!) In urging, or professing, an acceptance of these values, there is the implicit judgment that the acceptance entails no possible problem of internal coherence or logical consistency. In this small essay, it will be shown that the apparently unproblematic judgment just mentioned could prove to be suspect. In particular, the reader is invited to consider that there are plausible, but mutually incompatible, ways in which the principles of ‘liberty’ and ‘equity’ can be formulated. The essay draws on the conventions and methods of a body of knowledge called ‘social choice theory’, which lies at the intersection of philosophy, political science, and economics.

Background: Social Choice Theory and ‘Impossibility Theorems’

One of the commonest problems faced by any society is the following one: given the preference rankings of a group of individuals over a set of alternatives, how should these rankings be aggregated into a single ranking which may be taken to be a reasonable ranking of the alternatives for the society as a whole? How, that is, may a society arrive at a social ranking of alternatives from information on the individual rankings of these alternatives? One method, which we are all very familiar with, is the elementary majority decision rule, or, as it is sometimes called, the Method of Majority Decision (MMD). MMD simply requires that, in the pair-wise choice from any pair of alternatives x and y, if a majority of individuals prefer x to y, then x should be socially preferred to y. Simple though it is, MMD is vulnerable to a problem of internal consistency, one which is popularly known as the Paradox of Voting. Here is the paradox.

Imagine a society comprising three individuals, whom we shall call 1, 2 and 3 respectively. Suppose there are three alternatives – think of these as candidates in an election – named x, y and z respectively. Suppose further that individual preferences are described by the following: 1 prefers x to y and y to z; 2 prefers z to x and x to y; and 3 prefers y to z and z to x. Notice now that a majority of individuals (1 and 2) prefer x to y, and a majority (1 and 3) prefer y to z. MMD will dictate that x be socially preferred to y, and y socially preferred to z. The logical property of transitivity will now demand that x be socially preferred to z. (Transitivity is a necessity of logic imposed on triples: for example, if we are talking of the relation ‘taller than’, and if it is given that Ashoo is taller than Babloo and Babloo is taller than Chutki, then transitivity will entail that Ashoo is taller than Chutki.) However, a majority of individuals (2 and 3) prefer z to x, so MMD will require z to be socially preferred to x. But, clearly, we can't have both x socially preferred to z and z socially preferred to x at the same time – and it is this contradiction, precipitated by the Method of Majority Decision, which is encompassed in the Paradox of Voting.

‘Social choice theory’ is concerned with an examination of the logical problems of preference aggregation – the Paradox of Voting is an example – in a variety of settings. A number of distinguished names have been associated with the evolution of this branch of study – including Jean-Charles de Borda and the Marquis de Condorcet in the 18^th century, Charles Lutwidge Dodgson (somewhat better known as Lewis Carroll) in the 19^th century, and Duncan Black, Kenneth Arrow, Michael Dummett, and Amartya Sen, among others, in the 20^th and 21^st centuries. Arrow, in particular, proved a quite remarkable result, incongruously named ‘The General Possibility Theorem’ (which appeared in a book, published in 1963, called Social Choice and Individual Values Footnote ¹). The result pointed to the logical impossibility of there being any rule of preference aggregation which could satisfy a small number of apparently very reasonable, undemanding, and innocuous principles of social choice.

Subsequently, Sen (in a brief paper titled ‘The Impossibility of a Paretian Liberal’ which appeared in a 1970 issue of the Journal of Political Economy Footnote ²) established that there is a specific and well-defined sense in which a widely employed criterion of ‘efficiency’ employed by economists, the so-called Pareto Principle, is incompatible with a very weak criterion of liberty which he called Minimal Liberalism. Social choice theory, for reasons of a great proliferation of demonstrations of this nature, has acquired a reputation for being the natural home of ‘impossibility results’.

Sen was also the author of a simple and appealing principle of equality called the Weak Equity Axiom, which appears in a book of his called On Economic Inequality, published in 1973Footnote ³. The axiom essentially demands that, in a distribution of income between two individuals, a larger share ought to go to the uniformly more disadvantaged person (such as in the division of an income of given size between an able-bodied and a physically-challenged person, the assumption being that at every level of income the latter individual is worse-off than the former). Peter Hammond is an economist who has offered an axiomatic justification of a certain celebrated criterion of justice due to the philosopher John Rawls. In the process, he provided a generalization of Sen's Weak Equity Axiom to social choice situations which go beyond pure income-distribution problems (Hammond's essay, ‘Equity, Arrow's Conditions, and Rawls’ Difference Principle', appeared in a 1976 issue of the journal Econometrica Footnote ⁴). In what follows, we shall employ Sen's principle of Minimal Liberalism and a version of Hammond's Weak Equity principle in order to analyze what might be called the Problem of the Possibility of Liberal Egalitarianism.

On Two Specific Types of Pairs of Alternatives

In formulating our principles of liberty and equity, we shall take the assistance of Alan Gibbard (‘A Pareto Consistent Libertarian Claim’, published in 1974 in the Journal of Economic Theory Footnote ⁵). But first, alternatives, or social states, will, as is customary, be taken to be complete descriptions of society including each person's state of being and doing in it. Typically, social states will be designated x, y, z, etc. Individuals will be designated 1, 2, 3, …,i,…,n.

A pair of states x and y will be termed i-variants if they differ only with respect to a feature which may be deemed to be personal to individual i. Thus, if x and y differ from each other only in the respect that in x person 1 is whistling a tune in his bathroom and in y he is not doing so, then x and y are a pair of 1-variants; if w and z differ only in the respect that in w person 3 is drinking grape juice and in z she is drinking orange juice, then w and z are a pair of 3-variants; and so on.

Similarly, a pair of states x and y will be termed j, k-variants if they differ only with respect to features which may be deemed to be personal to individuals j and k. Thus, if s is a state in which person 2 is singing in his bathroom and person 4 is drinking grape juice, while t is a state which is identical to s in all respects save that in t person 2 is silent in his bathroom and person 4 is drinking orange juice, then s and t are a pair of 2,4-variants.

Minimal Liberty

Now it seems reasonable to demand, in a libertarian world, that if x and y are a pair of i-variants, then i's preference over the states x and y ought to count as the social preference over that pair: after all, the two states differ only in a matter of personal relevance to i, so i's personal preference ranking of the states ought to be socially decisive. It may be construed as being minimally liberal to allow at least two individuals such social decisiveness over a pair of alternatives each. Minimal Liberty, therefore, can be taken to demand the following: there should be at least two individuals j and k and two pairs of alternatives {x,y} and {w,z}, with x and y being a pair of j-variants and w and z being a pair of k-variants, such that if j prefers x to y (respectively, y to x), then society should prefer x to y (respectively, y to x), and if k prefers w to z (respectively, z to w), then society should prefer w to z (respectively, z to w).

Weak Equity

Suppose x and y to be a pair of j,k-variants. Since x and y differ only with respect to features which are personal to j and k, it can be argued that it is only the preferences of j and k over the alternatives x and y which ought to matter for the social ranking of x and y. How might an equity-conscious ethic determine the manner in which j's and k's preference rankings of x and y should influence the social ranking of x and y? Consider the following plausible generalization of the spirit underlying Sen's Weak Equity Axiom. Suppose j is uniformly more advantaged than k, in the specific sense that everybody in society would rather be person j than person k in the state x, and would similarly rather be person j than person k in the state y. That is to say, suppose that no matter in which state, it is unanimously held to be better to be j than to be k. Then, it seems to be compatible with the requirements of equity to demand that it is the uniformly more disadvantaged individual k's preference ranking over the pair of states {x,y} which should determine the social ranking of the pair. This (with some modifications) captures Hammond's formulation of a generalization of Sen's axiom. Weak Equity, therefore, may be taken to demand the following: for any pair of j,k-variants x and y, if j is considered by everybody in society to be better off than k in each of the states x and y, and if j prefers x to y while k prefers y to x, then y should be socially preferred to x (in ‘equity-sensitive’ partisanship with the relatively disadvantaged individual k).

Liberty, Equity, Impossibility

It can be shown that in a situation which does not proscribe any logically possible pattern of individual preferences, and in which the social ranking of alternatives is required to be transitive, the principles of minimal liberty and weak equity, as we have defined them, are mutually incompatible. A precise statement of the relevant results, with proofs, are available in a recent paper by this authorFootnote ⁶. For the present, the following example, which takes some liberties with a Wodehousean situation, should suffice to highlight the essential nature of the underlying difficulty.

Police Constable Oates would give anything to ensure that the dog Bartholomew does not accompany his mistress Stephanie Byng on her walks, for there is between man and dog a deep-seated enmity. If Bartholomew makes unfriendly noises at Oates it is, in Miss Byng's view, because Oates is aggravating the dog by doing his beat on a bicycle when the right thing to do would be for him to walk which, according to Miss Byng (although Oates does not regard this as being the ‘point at tissue’), would also help in knocking off some of the policeman's fat.

We can now define the following social states. x is a state in which Miss Byng is accompanied on her walks by her dog, and Oates does his beat on a bicycle; y is identical to x in all respects save that in y Oates does his beat on foot; and z is identical to y in all respects save that in z Miss Byng leaves her dog behind at home.

Clearly, the states x and y, being a pair of ‘Oates-variants’, are in Oates' ‘protected sphere’, while similarly, the states y and z, being a pair of ‘Miss-Byng variants’, are in Miss Byng's ‘protected sphere’. Since other things equal the policeman prefers travelling by bicycle to traveling on foot, and Miss Byng prefers travelling with Bartholomew to traveling without him, the principle of Minimal Liberty assures us that x is socially preferred to y and y is socially preferred to z whence, by transitivity, x should be socially preferred to z.

Assume further (i) that Miss Byng prefers state x to state z (her overriding concern is that the dog Bartholomew should accompany her on her walks); (ii) that Oates prefers state z to state x (never mind if he has to do his beat on foot, so long as he does not have to have his day blighted by an encounter with that dog); and (iii) that in any neutral ethical observer's eye Oates' welfare level (possibly by virtue of his being a policeman which, according to Wodehouse, makes for a generally jaundiced view of life) is lower than Miss Byng's in both states x and z. Given this, and noting that the pair of states {x,z} is a pair of ‘Byng,Oates-variants’, the principle of Weak Equity will dictate that z is socially preferred to x. But this contradicts ‘x socially preferred to z’, derived earlier. In the improbable contingency that Sir Watkyn Basset, Justice of the Peace for Totleigh-on-the-Wold, cared to serve the interests of both equity and individual liberty, he should find himself in a tight spot.

Concluding Observations

I conclude by desisting from addressing queries – which I leave to the reader (if s/he is still there) to ponder - such as (i) what is the source of the contradiction established in this paper? (ii) what are some possible ‘escape routes’ from the liberty-equity dilemma? and (iii) what significance might the impossibility result have for our notions of liberty and equality?