FAIRNESS AS PROPORTIONALITY

Though we consider alternative accounts below, we begin with the idea that fairness demands proportionality. Many writers draw this connection,Footnote ⁴ but its best-known development is in the work of John Broome,Footnote ⁵ which has been much discussed elsewhere.Footnote ⁶

According to Broome, fairness consists in the proportionate satisfaction of claims. Note that Broome distinguishes fairness from what we should do all things considered. Since we focus on the former, claims about ‘what should be done’ should be read with the implicit qualifier ‘as a matter of fairness’. Here, C ₁ has a claim to d ₁ and C ₂ has a claim to d ₂. Proportionality implies that, if these claims are of equal strength, C ₁ will receive m.d ₁/(d ₁ + d ₂) and C ₂ will receive m.d ₂/(d ₁ + d ₂); for instance they may each receive half of what they are owed. If C ₂'s claim is twice the strength of C ₁'s, then Broome holds that C ₂ should receive twice the satisfaction (proportionately), e.g. if C ₁ has a claim to £20 of strength 5 and C ₂ has a claim to £40 of strength 10, then a fair distribution of £20 would be to give £4 to C ₂ (20% satisfaction) and £16 (40% satisfaction) to C ₂. We focus on cases where claims are of equal strength for simplicity since showing that no theory is aggregative when dealing with claims of equal strength shows that no theory is aggregative in general.

That Broome's theory is non-aggregative can be demonstrated by example. Suppose d ₁ = 80, d ₂ = 40 and m = 60. Here D can only repay half his debts, so (according to Broome) the fair thing for him to do, if the claims are equal, is to pay 40 to C ₁ and 20 to C ₂, giving each creditor half of what she was owed. Now introduce a second debtor, D*. Suppose d ₁* = 40, d ₂* = 80 and m* = 90. Broome's theory would have D* pay 30 to C ₁ and 60 to C ₂, giving each three-quarters of what they are owed. The aggregate result of these two transactions is that C ₁ was owed 120 (d ₁ + d ₁*) and received 70, while C ₂ who was also owed 120 (d ₂ + d ₂*) received 80. Thus, C ₁ receives greater satisfaction than C ₂, though ex hypothesi both were owed the same amount and had equal claims.

This inequality between C ₁ and C ₂ seems unfair. Consider the case where D = D*. Here this single individual owes a total of 120 (d ₁ + d ₁*) to C ₁ and 120 (d ₂ + d ₂*) to C ₂ and has 150 (m + m*) with which to repay these debts. In this case, the fair thing to do is to repay 75 to each of C ₁ and C ₂, reflecting their equal claims to equal amounts. The surprise is that we get a different solution where D ≠ D*. In other words, Broome's theory is non-aggregative. It focuses on the distribution of particular goods on particular occasions, what we might call a ‘narrow’ view, but – as our example shows – the outcome of several fair transactions may be unfair, and vice versa, when we take a wider, overall view.

A response might be that the nature of the claims determines how they should be met. If C ₁ is owed money by D and D* separately, then they should repay separately, whereas if the money is owed by D and D* together then they should repay collectively. But this response appears unsatisfactory.

First, it is intuitively unfair if C ₁ receives greater satisfaction than C ₂ though they have equally strong claims to the same amount (albeit against different debtors), so an adequate theory of fairness should be able to account for this. Any theory that tells us this situation is fair is, at best, an incomplete theory of fairness as it fails adequately to account for fairness in such cases.

Second, Broome defined claims as duties owed to particular individuals,Footnote ⁷ but he did not specify that they must be owed by particular individuals. Thus, if C ₁ has a claim to £80 and a claim to £40, then C ₁ does have claim to £120, even if no single agent owes this amount to her. We do not mean that C ₁ has a single claim to this amount, but only that she has claim to this amount. In our debtor/creditor example, it is natural to separate C ₁'s claims into claims against D and D*. However, this strategy would not necessarily work for all claims, such as claims based on need,Footnote ⁸ which are not necessarily claims against any particular individual. (One could suggest that claims of need are directed against specific individuals, for instance whoever is best-placed to meet that need. But this appears to run into problems if that individual does not respond to the claim.) In these cases, a fair response to an individual's claims depends on whether they are treated piecemeal or collectively.

Third, nothing in our schematic example specifies whether D = D* or D ≠ D*. It is possible that D and D* are parts of some larger organization, such as the state, or even the same person, perhaps with two separate debts incurred in different currencies or at different times. This person could act fairly in settling one debt at t ₁ and fairly in settling another debt at t ₂ but, once again, we would probably want to know whether his actions were fair in aggregate.

Our example suggests that two acts that are both fair in isolation may nonetheless result in unfairness when aggregated. As we shall show, this feature is not unique to Broome's theory of fairness. We will not repeat the points in this paragraph and the previous three, but they should be understood as applying to all our subsequent discussion.

FAIRNESS AS SHORTFALL MINIMIZATION

An alternative theory of fairness states that D should divide m between C ₁ and C ₂ so as to minimize the largest shortfall suffered by either creditor.Footnote ⁹ This has been termed ‘shortfall minimization’, though it might more accurately be called ‘minimax shortfall’. If D owes 900 to C ₁ and 600 to C ₂ but has only 500 then, on this theory, he should pay 400 to C ₁ and 100 to C ₂, leaving both creditors with a shortfall of 500.

Abstracting from this example, assume without loss of generality that d ₁ ≥ d ₂. Two types of cases must be distinguished. If m ≤ d ₁ − d ₂ then the shortfall minimization theory would have D give all his money to C₁ and none to C ₂. Alternatively, if m ≥ d ₁ − d ₂ then D should give C ₁ amount (m + d ₁ − d ₂)/2 and C ₂ amount (m + d ₂ − d ₁)/2; since m ≥ d ₁ − d ₂ and d ₁ ≥ d ₂ both amounts are non-negative. The formulas return the same answer in the overlapping case where m = d ₁ − d ₂.

Again, this theory can be shown to be non-aggregative by example. Suppose that a second debtor, D*, owes 600 to C ₁ and 900 to C ₂ but has only 200. This theory directs him to give all 200 to C ₂, since this is the most he can do to reduce C ₂'s shortfall, which remains larger than C ₁'s. The aggregate result is that C ₁ receives 400 from D and 0 from D*, leaving her with a total shortfall of 1,100, while C ₂ receives 100 from D and 200 from D* leaving her with a total shortfall of 1,200. However, had D and D* acted as a collective agent distributing m + m* then both C ₁ and C ₂ would have received 350, leaving each of them with a shortfall of 1,150. Once again, if each agent acts in isolation, applying the theory of fairness only to his own distributive problem and without regard to what others will do, the outcome differs from when they act in concert, even though they act as the theory directs.

THE GENERAL CASE

Similar aggregation problems arise for any non-trivial theory (as defined below), as we now prove. Suppose that a theory of fairness would have D repay C ₁ amount F ₁(m, d ₁, d ₂) and C ₂ amount F ₂(m, d ₁, d ₂), where F ₁ and F ₂ are functions of the stated variables. By stipulation, m < d ₁ + d ₂. Also,

\begin{equation*} \begin{array}{*{20}{c}} {{F_1}\left( {m,{d_1},{d_2}} \right){\rm{ }} + {F_2}\left( {m,{d_1},{d_2}} \right) \le m}&\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,m,{d_1}\,{\rm{and}}\,{d_2}} \end{array} \end{equation*}

and in particular

\begin{equation*} \begin{array}{*{20}{c}} {{F_1}\left( {m,{d_1},{d_2}} \right) \le m}&\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,m,\,{d_{1{\rm{ }}}}\,{\rm{and}}\,{d_2}} \end{array} \end{equation*}

\begin{equation*} \begin{array}{*{20}{c}} {{F_2}\left( {m,{d_1},{d_2}} \right) \le m}&\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,m,\,{d_{1\,}}\,{\rm{and}}\,{d_2}} \end{array} \end{equation*}

The first inequality, from which the second and third follow (since the amounts in question are non-negative), expresses the fact that D has no resources beyond the amount m. Note that these last two inequalities imply the equations

(Eq1) \begin{equation} \begin{array}{*{20}{c}} {{F_1}\left( {0,{d_1},{d_2}} \right) = {\rm{ }}0}&\,\,\,\,\,\,\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,{d_1}\,{\rm{and}}\,{d_2}} \end{array} \end{equation}

(Eq2) \begin{equation} \begin{array}{*{20}{c}} {{F_2}\left( {0,{d_1},{d_2}} \right) = {\rm{ }}0}&\,\,\,\,\,\,\,\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,{d_1}\,{\rm{and}}\,{d_2}} \end{array} \end{equation}

since the values of F ₁ and F ₂ are non-negative.

Two further facts about the functions F ₁ and F ₂ are needed.

(Eq3) \begin{equation} \begin{array}{*{20}{c}} {{F_1}\left( {m,{\rm{ }}0,{d_2}} \right){\rm{ }} = {\rm{ }}0}&\,\,\,\,\,\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,{d_2}\,{\rm{and}}\,m} \end{array} \end{equation}

(Eq4) \begin{equation} \begin{array}{*{20}{c}} {{F_2}\left( {m,{d_1},{\rm{ }}0} \right){\rm{ }} = {\rm{ }}0}&\,\,\,\,\,\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,{d_{1{\rm{ }}}}\,{\rm{and}}\,m} \end{array} \end{equation}

(Eq3) says that if D owes C ₁ nothing – C ₁ has no claim on D – then the fair thing for D to do vis-à-vis C ₁ is not to pay her anything; similarly for (Eq4) if D owes C ₂ nothing.

Finally, we define precisely two items of terminology introduced earlier. A function is said to be trivial if it is identically zero, i.e. it outputs 0 for any inputs:

\begin{equation*} \begin{array}{*{20}{c}} {F\left( {m,{d_1},{d_2}} \right) = 0}&\,\,\,\,\,\,\,\,\,\,\,{{\rm{for}}\,{\rm{all}}\,m,\,{d_1}\,{\rm{and}}\,{d_2}.} \end{array} \end{equation*}

To introduce the second definition, recall that, intuitively, repayment functions aggregate over individual cases if the amount C ₁ should receive in Case 1 is the same as she receives in Case 2, and likewise for C ₂. More formally, we say that the function F ₁ is aggregative just when, for all m, m*, d _1,d _2,d ₁*, d ₂*,

\begin{equation*} F_1 \left( {m + m^*,d_1 + d_1 ^* ,d_2 + d_2 ^* } \right){\rm } = F_1 \left( {m,d_1 ,d_2 } \right){\rm } + F_1 \left( {m^*,d_1 ^* ,d_2 ^* } \right) \end{equation*}

and similarly for F ₂. With this terminology and conditions in place, the result will now follow.

The general proposition is that if (Eq1) and (Eq3) are satisfied, then F ₁ is aggregative only if it is trivial. Similarly for F ₂ given (Eq2) and (Eq4). The proof proceeds simply by noting that if F ₁ is aggregative then

\begin{equation*} F_1 \left( {m,d_1 ,d_2 } \right){\rm } = F_1 \left( {m,{\rm }0,d_2 } \right){\rm } + F_1 \left( {0,d_1 ,{\rm }0} \right) \end{equation*}

By (Eq3), F ₁(m, 0, d ₂) = 0. By (Eq1), F ₁(0, d ₁, 0) = 0. Hence for all m, d _1,d ₂

\begin{equation*} F_1 \left( {m,d_1 ,d_2 } \right){\rm } = {\rm }0{\rm } + {\rm }0{\rm } = {\rm }0. \end{equation*}

A parallel argument can be run for F ₂ using (Eq2) and (Eq4).

It is worth briefly pointing out two further generalizations of this already general result. The first generalization is to the case of M debtors and N creditors, which follows immediately from the two-Debtor, two-Creditor case. If a fairness function (which outputs the amount the i ^th Debtor should repay the j ^th Creditor) is aggregative in the M-Debtor, N-Creditor case then it must also be aggregative when restricted to any two of these Debtors and any two of these Creditors (setting all other debts equal to 0). Our result tells us it must be trivial for this pair, and since this is true of any pair then by aggregation the general function must also be trivial.

The second generalization is to the case in which claims differ not only in size but also in strength. As noted, we have so far assumed equal strength of claims. Broome holds that stronger claims should receive more satisfaction, but weaker claims should still receive some satisfaction.Footnote ¹⁰ Another approach would be to say that claims should be satisfied in order of strength, so a claim of strength X receives no satisfaction until all stronger claims are fully met.Footnote ¹¹ This view is obviously non-aggregative (the result of D and D* acting independently will differ from that of their acting in concert if e.g. all of the claims on D* are stronger than those on D). Our proof does not incorporate strength of claims, so it directly concerns only cases where claims are equal in strength. Nonetheless, our result applies in this more general setting too, since a counterexample to a restricted version of a theory (in this instance, in which all claims are of equal strength) also serves as a counterexample to the general theory. (This holds on the natural assumption that, on a single occasion, D's owing £M to C with strength X and £N to the same creditor C with equal strength X is equivalent to D's owing £M + N to C with strength X.) Aggregation thus fails for any (non-trivial) theory of fairness that takes strength of claims into account. This applies also to a fairness function including any other parameters, assuming these can be set to the same value.

ILLUSTRATIONS

Although we have concentrated on Debtor–Creditor cases for simplicity, our discussion has much greater application. Most obviously, as mentioned earlier, the good in question can be any divisible good whatsoever (time, amount of a divisible commodity, amount of care, quality, etc.) or the monetary equivalent. Further, the agents in question need not be individuals; they may also be groups, corporations, nation states, etc. Moreover, as we now briefly illustrate, aggregation can occur in various ways, for instance we may aggregate over agents, over time and over goods.

First, aggregation might occur across different agents. This was the case in our earlier example, where two debtors each acting fairly produce an aggregate outcome that seems unfair. To give another example, if Mother and Father each treat their children fairly, but do so independently, the result may differ from what would happen if they pooled their resources and divided the total fairly.

Second, aggregation might also occur over time. For instance, one agent may respond fairly to two claims on Monday and then respond fairly to two claims on Tuesday, but the overall result may be different from what would have occurred had she made one single distribution of goods. This case straightforwardly parallels that of different agents since, in effect, we can regard the agent on Monday and the agent on Tuesday as if they were different agents.

Third, aggregation may occur across different goods to be distributed. For instance, suppose a debtor has debts in both dollars ($) and pounds (£). Dividing the dollars and the pounds that she has fairly between her creditors may result in a different outcome than would arise were she to change all her funds to a single currency, and then divide the whole fairly between her creditors.

COMMENTS

Our result is entirely general, and rests only on the mildest assumptions on the ‘fairness functions’. We may paraphrase it by saying that non-trivial functions are not aggregative. Its import, however, must not be exaggerated. The claim is not that, for any non-trivial theory of fairness, aggregation cannot be achieved in any circumstances whatsoever. It is rather that, for any non-trivial theory of fairness, aggregation fails in at least some cases. Indeed, for Broome's theory, aggregation fails in almost all cases. (Specifically, when the denominators in the fractions below are non-zero, aggregation holds for Broome's proportionality theory in the two-Debtor, two-Creditor case just when either m/m* = (d ₁+ d ₂)/(d ₁* + d ₂*) or d ₁/d ₂ = d ₁*/d ₂*. If m, d ₁, d ₂, m*, d ₁*, d ₂* take values in the positive real numbers, then the equations define two surfaces each of six-dimensional volume 0 in a space with infinite six-dimensional volume.) In sum, aggregated individual fair allocations and collective fair allocations can be identical; what our result shows is that the range of cases for which this holds must necessarily fall short of all the cases, for any non-trivial theory of fairness satisfying the innocuous conditions set out above.

Second, our result should not be confused with the more familiar point that, sometimes, an act that would be deemed unfair if taken in isolation is needed in order to produce a fair outcome overall. Our result supports this point, but is distinct from it, as well as the argument that usually supports it. The usual argument is that an unfair action taken in isolation is often required to redress some previous unfair actions; for instance, affirmative action programmes may give goods to those with weaker claims, in order to redress prior unfairness. Our point is that a series of individually fair actions may give rise to an outcome that would be judged unfair had it resulted from a single action. Thus, an action that is unfair (taken in isolation) may be required to preserve an overall pattern of fairness, even when all previous actions were fair (taken in isolation). This argument for the necessity of redress is much less familiar and interestingly different from the usual one.