### On contractualism, reasonable compromise, and the source of priority for the worst-off

Different policies introduced by a social planner, whether the government of a country or the head of an institution, lead to situations in which different peoples' lives go better or worse. That is, in the jargon of this area, they lead to different distributions of welfare across the individuals they affect. If we allow the unfettered accumulation of private wealth, that will lead to one distribution of welfare across the people in the country where the policy is adopted; if we cap such wealth, tax it progressively, or prohibit it altogether, those policies will lead to different distributions. The question I want to think about in this post is a central question of social choice theory: How should we choose between such policies? Again using the jargon of the area, I want to sketch a particular sort of social contract argument for a version of the prioritarian's answer to this question, and to show that this answer avoids an objection to the standard version of prioritarianism raised by Alex Voorhoeve and Mike Otsuka. But for those unfamiliar with this jargon, all will hopefully become clear.

Disclaimer: I've only newly arrived in ethics and social choice theory, so while I've tried to find versions of this argument and failed, it's quite possible, indeed quite likely, that it already exists. Part of my hope in writing this post is that someone points me towards it!

A cat with whom there is no reasonable compromise |

# 1. Two approaches to social planning: axiological and contractual

There are two sorts of situation in which the social planner might find themselves: in the first, they are certain of the consequences of the policies they might adopt; on the second, they are not. Throughout the post, I'll assume there are just two people in the population that the social planner's choices will affect, Ada and Bab, and I'll write $(u, v)$ for the welfare distribution in which Ada has welfare level $u$ and Bab has $v$. The following table represents a situation in which the social planner knows the world is in state $s$, and they have two options, $o_1$ and $o_2$, where the first gives Ada $1$ unit of welfare and Bab $9$, while the second option gives Ada and Bab $4$ units each. $$\begin{array}{r|c} & s \\ \hline o_1 & (1, 9) \\ o_2 & (4,4)\end{array}$$And this table represents a situation in which the social planner is uncertain whether the world is in state $s_1$ or $s_2$, and the welfare distributions are as indicated: $$\begin{array}{r|cc} & s_1 & s_2 \\ \hline o_1 & (2, 10) & (7,7) \\ o_2 & (4,4) & (1,20)\end{array}$$

There are (at least) two ways to approach the social planner's choice: axiological and contractual.

An axiologist provides a recipe that takes a distribution of welfare at a state of the world, such as $(1, 9)$, and aggregates it in some way to give a measure of the social welfare of that distribution, which we might think of as the group's welfare given that option at that state of the world. If there is no uncertainty, then the social planner ranks the options by their social welfare; if there is uncertainty, then the social planner uses standard decision theory to choose, using the social welfare levels as the utilities---so, for instance, they might choose by maximizing expected social welfare.

Average and total utilitarians are axiologists; so are average and total prioritarians.

The average utilitarian takes the social welfare to be the average welfare, so that the social welfare of the distribution $(1, 9)$ is $\frac{1+9}{2} = 5$;

The total utilitarian takes it to be the total, i.e., $1+9=10$.

The average prioritarian takes each level of welfare in the distribution, transforms it by applying a concave function, and then takes the average of these transformed welfare levels. The idea is that, like utilitarianism, increasing an individual's welfare while keeping everything else fixed should increase social welfare, but, unlike utilitarianism, increasing a worse-off person's welfare by a given amount should increase social welfare more than increasing a better-off person's welfare by the same amount; put another way, increasing an individual's welfare should have diminishing marginal moral value, just as we sometimes say that increasing an individual's monetary wealth has diminishing marginal welfare value. So, for instance, if they use the logarithmic function $\log(x)$ to transform the levels of welfare, the social welfare of $(1, 9)$ is $\frac{\log(1) + \log(9)}{2} \approx 1.099$. Notice that, if I increase Bab's 9 units of welfare to 10 and keep Ada's fixed at 1, the average prioritarian value goes from $\frac{\log(1) + \log(9)}{2} \approx 1.099$ to $\frac{\log(1) + \log(10)}{2} \approx 1.151$, whereas if I increase Ada's 1 units to 2 and leave Ada's fixed at 9, the total value goes from $\frac{\log(1) + \log(9)}{2} \approx 1.099$ to $\frac{\log(2) + \log(9)}{2} \approx 1.445$.

The total prioritarian takes the social welfare to be the total of these transformed welfare levels. So, if our concave function is the logarithmic function, the social welfare of $(1,9)$ is $\log(1) + \log(9) \approx 1.099$.

The second approach to the social planner's choice appeals to social contracts. For instance, Harsanyi, Rawls, and Buchak all think the social planner should choose as if they are a member of the society for whom they are choosing, and should do so with complete ignorance of whom, within that society, they are. These theorists differ only in what decision rule is appropriate behind such a veil of ignorance. Others think you should choose a policy that can be justified to each member of the society, where what that entails can be spelled out in a number of ways, such as minimizing the worst legitimate complaints members of the affected population might make against your decision, or minimizing the total legitimate complaints they might make, and where there are different ways to measure the legitimate complaints an individual might make.

# 2. The Voorhoeve-Otsuka Objection: social planning for individuals

One of the purposes of this blogpost is to bring the axiologists and contractualists together by showing that a certain version of contracturalism leads to a certain axiological approach that resembles prioritarianism, but avoids an objection that has troubled that position. Let me spell out that objection, which was raised originally by Alex Voorhoeve and Mike Otsuka. In it, they ask us to imagine that the social planner is choosing for a population that contains just a single person, Cal. For the sake of concreteness, let's say they face the following choice: $$\begin{array}{r|cc} & 50\% & 50\% \\ & s_1 & s_2 \\ \hline o_1 & (2) & (3) \\ o_2 & (1) & (5)\end{array}$$Then prioritarianism tells the social planner to maximize expected social welfare: for $o_1$, this is $\frac{1}{2}\times \log(2) + \frac{1}{2} \times \log(3) \approx 0.896$; for $o_2$, it is $\frac{1}{2}\times \log(1) + \frac{1}{2} \times \log(5) \approx 0.804$. But standard decision theory says that Cal themselves should maximize their expected welfare: for $o_1$, this is $\frac{1}{2} \times 2 + \frac{1}{2} \times 3 = 2.5$; for $o_2$, it is $\frac{1}{2} \times 1 + \frac{1}{2} \times 5 = 3$. So the social planner will choose $o_1$, while Cal will choose $o_2$. That is, according to the prioritarian, morality requires the social planner to choose against Cal's wishes. But, Voorhoeve and Otsuka contend, that can't be right.

# 3. Justifying compromises to each

*squared Euclidean distance*(

*SED*):$$\mathrm{SED}(a, b) = |a-b|^2.$$ So the distance from $a$ to $b$ is just the square of the difference between them.

*generalized Kullback-Leibler divergence*(

*GKL*): $$\mathrm{GKL}(a, b) = a\log \left ( \frac{a}{b} \right ) - a + b.$$

*geometric compromise*.

# 4. Properties of Geometric Compromise

*Welfarism*: like utilitarianism, prioritarianism, and egalitarianism, according to geometric compromise, when there is no uncertainty about the outcomes of different policies, the social planner's ranking of those policies depends only on the welfare distributions to which they give rise.

*Anonymity*: like utilitarianism, prioritarianism, and egalitarianism, according to geometric compromise, if one welfare distribution is obtained from another by changing only the identity of the individuals who receive the different levels of welfare, then both distributions have the same social welfare. That is because $\sqrt{u \times v} = \sqrt{v \times u}$.

*Pigou-Dalton*: like prioritarianism and egalitarianism, but unlike utilitarianism, according to geometric comprise, if one welfare distribution is obtained from another by taking a particular amount of welfare from a better-off individual and giving it to a worse-off individual in a way that leaves the latter worse-off, then the latter distribution has higher social welfare. That is because, if $\varepsilon > 0$ and $u + \varepsilon < v - \varepsilon$, then $\sqrt{u \times v} < \sqrt{(u+\varepsilon) \times (v - \varepsilon)}$.

*Person Separability*: like prioritarianism and utilitarianism, but unlike egalitarianism, according to geometric compromise, the order of the social welfare of two distributions depends only on the welfare of the individuals who have different welfare in those two distributions. That is because $\sqrt{u \times v} \leq \sqrt{u \times v'}$ iff $\sqrt{v} \leq \sqrt{v'}$.

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