Lotteries, Equal Consideration and Unequal Outcomes

Here's something I've been writing recently, drawing largely on discussion with Jerry Cohen after his last two lectures:

Amartya Sen’s seminal essay ‘Equality of What?’ distinguishes between utilitarian equal consideration of equal utilities and views that aim at equality of some outcome, whether equal total utility, resources or his proposal capabilities. If we believe, as almost all now do, in the equality of all people, then that grounds something like equal concern. This seems to be the form of equality common to utilitarianism, Dworkin’s ‘equal concern and respect’ and Rawls’ original position – none of which end up endorsing any equality of outcome, except coincidentally. Indeed, I fail to see how equality of outcome necessarily follows. There may be some goods where equality of outcome is a necessary consequence of equal consideration, for example votes – ‘one person, one vote’ is so appealing because if you have two votes then my vote is worth less. There may be a similar danger with great inequalities of money – if you have a great fortune, it may increase prices such that I can buy less with what I have. In general, however, this is not the case when it comes to distributing things – even if you have three cars, it doesn’t make my car any less useful to me. As such, I see no objection to inequality where it reflects equal concern for each person.

We are concerned not simply with formal equality. Arguably the Nazis showed equal concern for each of the Jews they gassed – the problem was, it wasn’t enough concern – and John Broome notes that, where we can save only one of two people, equality is satisfied by saving neither. What we must do is show positive – indeed, I would argue, maximal – concern for each. So, suppose we have eight units of good to distribute between two persons. It would be wrong, I claim, though equal, to give them two units each and destroy the other four. We care about the distribution of this stuff because it is a good, each person would prefer more of it, and so that is a reason for us to prefer greater totals.

If we care about the amount of good each person gets, this can lead us away from equality. Suppose we can bring it about that person A has up to four units of good, and that person B has up to eight units – but what each has is independent of what the other gets (we cannot redistribute between them). In this case, we should show each maximal concern by fulfilling their potential – that is, giving A four and B eight, or, as I shall write, (4,8). This departs from equality, but (4,4) would be little better than (3,3) – it fails to show B maximal concern, because it gives her less than she could have, though it does no benefit to A. But, since we have shown maximum concern for A, by giving him as much as possible too, there is no reason for this Pareto suboptimal levelling down.

Now suppose that we are dealing with ‘lumpy’ – or not continuously divisible – goods, such that the only possible distributions are (4,8) and (8,4). In this case, equal concern for each party is reflected by tossing a coin. If we can do this, we may still regret that goods weren’t perfectly divisible, but there is no reason to prefer (4,4) to randomly allocating the extra four units to either of the parties. To choose the lower total, when either of them could have had extra good, would not be to show either of them maximum concern. (Remember that I am talking about goods where A’s larger holding doesn’t diminish the value of B’s).

It may be objected that, if we aim to maximize, we will collapse back into some form of consequentialism. That is, suppose our maximum feasible distributions are (5,4) or (4,7) – with the highest possible equality (4,4) – in this case, it follows from my argument that we should reject the Pareto inferior equality. Does it follow that we have to prefer (4,7)? No, for this would be to accept impersonal maximization that ignores the ‘separateness of persons’. Rawls is right to point out that B’s gain is no benefit or compensation to A. If our only options were (4,4) or (4,7) then choosing the latter would be to benefit B without in any way harming A. If we choose (4,7) over the available alternative of (5,4), however, we deny A some potential benefit, simply to bestow a greater benefit on B.

Here, as in a choice between (5,4) and (4,5), I believe we should toss a coin. This gives each of A and B an equal chance of their maximal satisfaction. Fairness does not require us to maximize aggregate good – simply because B could have even more is not always reason for A to accept less than he could have. This approach has the advantage of restricting the need for interpersonal comparison – there is no need for one to be able to measure whether B’s potential is actually slightly higher than A’s for, provided they are on a par, it is fair to randomize between them. Thus, we can have most of the information we need to choose between social states, simply by looking at ordinal individual preferences: we only need to know that state 1 is better for A and state 2 better for B, without having to say how much so.

I do not think, however, that we should reject all interpersonal comparisons and consign ourselves only to making Pareto improvements. Suppose the case were one better represented as a choice between (5,4) and (4,100) – i.e. a case where we could make A slightly better off or B vastly better off. In this case, I do not think we should toss a coin – but I think A should recognize that B has a greater claim. Interpersonal comparisons may be vague or fuzzy, but sometimes a pairwise comparison will be clear, and in these cases one person’s greater claim can trump another’s.

Lotteries Against Equality of Outcome

I have said there is no reason to prefer (4,4) to a lottery over the possibilities (5,4) and (4,5). Those who insist on some value to people sharing the same fate will resist even this, but insofar as we are maximally concerned for each person I see no justification for settling for less good, especially when we can give each an equal chance to receive the extra (presumably, both parties would agree to the lottery).

One in favour of equality may, however, ask why I only employ the lottery to distribute the odd extra benefit. Why randomize between (5,4) and (4,5) rather than the possibilities (9,0) and (0,9) or tossing a coin for each unit of good individually? If I prefer to randomly allocate only the ninth, odd unit, doesn’t the initial presumption in favour of distributing the first eight units (4,4) display some concern for strict equality of outcome? My answer is no.

Firstly, I do not think we should always favour randomizing (5,4) as opposed to (9,0). Sometimes it may be that a certain threshold is crossed in between five and nine, such that both parties would rather have a 50% chance of nine – even if the alternative was zero – rather than the guaranteed minimum of four. Suppose, for example, that the two parties are explorers lost in a mountainous wilderness, and the nine units represent their nine days’ worth of food. If they know rescue is a week away, then distributing their food (5,4) or (4,5) will result in both of them starving. Maybe their bond of solidarity is such that there is something to be said for this, but surely it would not be irrational or unjust for them to agree to toss a coin and let the winner survive. Suppose, in fact, they hold important military secrets, and it is vital one of them lives to pass these on to their rescuers. Now, it seems that at least one of them should get at least seven days’ food. Perhaps they could still randomize (7,2) rather than (9,0), but two days’ worth of food is of little benefit to one who will starve anyway, whereas it might help the designated survivor (e.g. if aid ends up taking longer than expected). In deciding how to distribute our good, we need to look at what benefit it does and what the parties concerned want.

But imagine a less extreme case. Suppose our Pareto optimal possibilities are (3,5), (4,4) and (5,3) – is there any reason to favour the equality rather than tossing a coin between the two inequalities? Well, either of these options seems to show equal and maximal concern for each party, so in that respect I would argue both are fair. Our reason for preferring (4,4), if we do, is I think the suspicion that it is a better outcome – if, for example, the good exhibits diminishing marginal utility (in the case of resources) or diminishing marginal moral importance (in the case of utility). Of course, this cannot always justify a preference for equality – indeed, if we have reason to suspect increasing marginal importance, then we will have reason to prefer inequality, but this is precisely what I argued in my previous paragraph.

Can a more general reason be given why we are only inclined to randomize the odd unit of good, and not each unit individually? Well, in some circumstances it may simply be that it is quicker and easier to anticipate the expected outcome than to keep tossing coins, but again this defence seems too contingent. However, I want to suggest, the reason that we prefer equality is because we only toss a coin to decide between equal claimants. Suppose someone suggested that we begin by tossing a coin simply for the first unit of the good, and it goes to A. Now, when it comes to allocating the second unit of good, we are no longer faced with equal claimants – rather, A already has one unit and B has none. As I have already suggested, how we respond to this difference depends on the good in question. If we have reason to favour an unequal distribution – as in the starvation case described above – then A’s winning the first unit may be reason to give him the rest. That is, the randomization of the first unit effectively becomes the allocation of it all, as we take possession of this as reason to give him the rest too.

Alternatively, in the kind of cases where we prefer equality, the fact that A got the first unit can be reason to simply give B the second. This does not show any presumption towards equality of outcome, but simply results from showing equal concern for equal claims. If the two parties have equal claims, consistency requires them to be equally satisfied. When neither party has anything, they have equal claims to the first unit – which is why we toss a coin – but once A has that, B may now have a stronger claim to the second unit, since her claim is still unmet whereas A’s has been partly satisfied. Thus the distribution of our nine units may proceed as follows: the first is randomly given to A, the second is directly given to B, the third is randomly given to A, the fourth directly to B, the fifth randomly to B, the sixth directly to A, the seventh randomly to A, the eighth directly to B (thus achieving (4,4) so far), and finally the odd unit gets randomly distributed to A – and, if there was a tenth unit, we would give it to B next. While we are distributing units that come in pairs, there is no reason to actually randomize, since we know whoever gets the first, the other person will get the second. Thus, in practice, we only need to randomize the odd unit.