Utilitarianism with and without expected utility

by Aaron Gertler4 min read24th Jul 20204 comments

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Research SummaryMoral Philosophy
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The paper this post is based on: McCarthy, D., Mikkola, K., Thomas, T., Utilitarianism with and without expected utility. Journal of Mathematical Economics 87 (2020): 77-113.   (Online Article | PDF)


A common criticism of utilitarianism is that it makes strong assumptions about the nature of welfare and about welfare comparisons. It assumes that welfare should be understood as something like pleasure or desire satisfaction, and that we can measure welfare precisely. But to many people, that assumption is implausible. They think that welfare is made up of different goods that are difficult to compare, and certainly different to measure.

Moreover, many people think that social welfare comparisons should reflect the social consensus about individual welfare comparisons. But the social consensus about individual welfare comparisons arguably falls a long way short of full agreement, and rules out pleasure or desire-based accounts. This is one of the objections that defenders of contractualism make against utilitarianism. For example, Rawls (1982) gives an especially strong statement of a version of this criticism. His disagreement with utilitarianism on this point is one of the central features of his contractualist theory.

Our paper explains how utilitarianism can respond. This post will also connect this topic with uncertainty.

1. Harsanyi’s utilitarianism

Criticisms of utilitarianism, including Rawls’ famous separateness of persons criticism, are too often directed at old fashioned versions of the view. But the worry about its assumptions about welfare applies to the sophisticated version presented in the 1955 utilitarian theorem of John Harsanyi. I understand this to be the version of Harsanyi’s result that assumes interpersonal comparisons.

Harsanyi’s utilitarian theorem assumes a constant population, and is stated for circumstances involving risk. In other words, it assumes that uncertainty is modelled by standard point-valued probabilities that are objective or universally agreed. It rests on some simple and natural premises, and shows that these assumptions imply that individual welfare comparisons uniquely determine social welfare comparisons.

In one way of presenting Harsanyi’s result, welfare comparisons, both intrapersonal and interpersonal, are encoded in a single preorder that we call the individual preorder. (A preorder is a binary relation that is reflexive and transitive.) In those terms, the premises of Harsanyi’s utilitarian theorem are as follows: (a) the individual and social preorders satisfy the expected utility axioms; (b) the strong Pareto principle; and (c) an impartiality axiom.

Strong Pareto says that (i) if two social lotteries are equally good for every member of the population, they are equally good; and (ii) if one social lottery is at least as good for every member of the population, and better for some members of the population, then it is better than the other. Impartiality essentially expresses indifference to permutations of individuals. But here I want to focus on expected utility.

But first, let’s quickly address what utilitarianism has to assume about the nature of welfare. Harsanyi himself proposed to identify welfare with the satisfaction of ideally rational preferences. He has a long and complicated argument to back that up which is not very convincing, and has been amply criticized by Broome (1993) and others.

However for the purposes of the utilitarian theorem, this view is inessential: any account of the content of welfare — such as more objective accounts, like Rawls’ own account, that take into account such things as resources or capabilities — is compatible with the theorem. What I want to focus on is what Harsanyi assumes about the structure of welfare comparisons, or more precisely, the structural assumptions he makes about the individual preorder. They are that it satisfies the expected utility axioms.

2. Expected utility and welfare comparisons

Expected utility theory is the best known and developed theory for rational decision making under conditions of risk. But this is where the old-style criticisms of the assumptions classical utilitarians make about welfare comparisons applies equally to Harsanyi. Expected utility theory contains three basic axioms: completeness, continuity, and independence. But the assumption that the individual preorder satisfies these axioms means the following.

1. Completeness implies that all goods, and even lotteries over goods, are comparable.
2. Continuity implies that no goods are infinitely more valuable than others.
3. Independence implies a precise way of making welfare comparisons under risk.

The first of these means that the standard objection to classical utilitarianism applies directly to Harsanyi’s version. The second and third provide further ways in which Harsanyi’s utilitarianism does not allow for much flexibility about welfare comparisons.

3. Information and uncertainty

The following is another feature of Harsanyi’s framework.

4. Framing the problem in terms of risk means that the theorem only applies when probabilities are objective or agreed.

This last point is not directly about welfare. But it is connected.

Amartya Sen taught us to think about different ethical theories in terms of how much information they need in order to function; see e.g. Sen 1985. Both classical utilitarianism and Harsanyi’s utilitarianism require a lot of precise information about welfare comparisons. But in only applying to risk, Harsanyi’s utilitarianism also requires a lot of information about uncertainty. But in real-world situations, we often face uncertainty without having any idea about what the probabilities are.

In Sen’s terms, we can summarize 1 to 4 as follows. Harsanyi’s utilitarianism demands a very high, and perhaps unattainable, amount of information. It is not a theory which we can use in most of the situations where we need ethics.

4. A utilitarian solution

The main result of our paper manages to avoid all of these difficulties. We present three simple and plausible axioms. These are much weaker than Harsanyi’s axioms, but they still preserve the utilitarian flavor of his approach. Moreover, our axioms still imply that individual welfare comparisons uniquely determine social welfare comparisons.

In addition, we do not assume any of the expected utility axioms. In fact, they are all allowed to fail. This means that our approach is much more flexible than Harsanyi’s in the welfare comparisons it handle.

In particular,

1′. Our result can accept that some (or all!) goods are incomparable.
2′. It allows some goods to be infinitely more valuable than others.
3′. It has room for a wide range of welfare comparisons involving risk.

For simplicity, most of our paper follows Harsanyi in working in the framework of risk. But we also explain how to extend our main result. In particular,

4′. Our approach allows for a very wide range of ways of representing uncertainty.

For example, it can allow for so-called imprecise probabilities in all kinds of variations; see Bradley (2019) for an excellent introduction to that topic. In Sen’s terms again, our version of utilitarianism needs very little information.

In my view, this means that standard criticism of utilitarianism we started with is off-target. But I would hesitate to put this by saying that utilitarianism is right and contractualism is wrong. We could also say that the result allows utilitarianism to incorporate some contractualist ideas.

Our paper is very long, and contains much else besides, including an extension of the result sketched above to variable populations, and proofs that our assumptions lead to very general versions of the standard additive form of classical and Harsanyi-style utilitarianism. But explaining how utilitarianism can cope with all kinds of limitations on welfare comparisons and many forms of uncertainty is at least how I see its central philosophical point.

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