7Separability

7.1The construction of utility

When a possible outcome looks attractive, then this is usually because it has attractive aspects. It may also have unattractive aspects, but the attractive aspects (the “pros”) outweigh the unattractive aspects (the “cons”). In this chapter, we will explore how this weighing of different aspects might work.

Take a concrete example. You are looking for a flat to rent. There are two options. \(A\) is a small and central flat that costs £800/month. \(B\) is a larger flat in the suburbs for £600/month. You might draw up a lists of pros and cons for each option, and give them a weight, like so:

You might then determine the total utility of each option as the sum of these numbers, so that \(\U (A)\) is +2-1-3 = -2, while \(\U (B)\) is -2+3-1 = 0.

Is this a reasonable approach? It looks OK in this example. But we have to be careful. Suppose you had drawn up the following table.

Now \(\U (A)\) comes out as \(0\) and \(\U (B)\) as \(-2\). Do you see what’s wrong with this table?

The problem is that the first three criteria in the list aren’t independent. Once you’ve taken “good location” into account, you shouldn’t additionally take into account “short commute” and “can get up later”. Location, size, and costs are independent criteria. Location and commute time are not.

But what, exactly, does independence mean here? There is no logical connection between “good location” and “short commute”. And there may well be a strong statistical connection between (say) location and costs.

7.2Additivity

Let’s stick with the flat example. We assume that you care about certain aspects of a flat: size, location, and costs. We’ll call these aspects attributes. Let’s assume that size, location, and costs are all the attributes that ultimately matter to you. Your preferences between possible flats is then determined by your preferences between combinations of these attributes. If two flats perfectly agree in each of the three attributes then you are always indifferent between them. If you prefer one flat to another, that’s always because you prefer the combined attributes of the first to those of the second.

Instead of talking about the desirability of a particular flat, we can therefore talk about the desirability of its attributes. We’ll write combinations of attributes as lists enclosed in angular brackets. ‘\(\t {40 \m ^2, \text {central}, \text {£500}}\)’, for example, would represent any flat with a size of 40 \(\m ^2\), central location, and monthly costs of £500. We are interested in the utility you assign to any such list.

Strictly speaking, of course, utility functions don’t assign numbers to lists, or even to flats. When I say that you prefer one kind of flat over another, what I really mean is that you prefer living in the first kind of flat over living in the other. In full generality, we should speak about attributes of worlds, not of flats. To keep things simple, we currently assume that the only thing you ultimately care about is what kind of flat you are living in (or going to live in). A list like \(\t {40 \m ^2, \text {central}, \text {£500}}\) therefore settles everything you ultimately care about. It represents one of your “concerns”, in the terminology of section 5.4.

In the example from section 5.4, we assumed that you care about two things: being free from pain and being admired. We pretended that these are all-or-nothing matters. The resulting four concerns could be represented by the following lists: \[ \t {\emph {Pain}, \emph {Admired}}, \t {\neg \emph {Pain}, \emph {Admired}}, \t {\emph {Pain}, \neg \emph {Admired}}, \t {\neg \emph {Pain}, \neg \emph {Admired}}. \] Here, there are two attribute, each of which can take two value. The first attribute specifies whether you are in pain, and the answer is either yes or no. The second attribute similarly specifies whether you are admired. If we allowed for different degrees of pain, then the first attribute would have more than two possible values. We could, for example, distinguish \(\t {\emph {Little Pain}, \emph {Admired}}\) from \(\t {\emph {Strong Pain}, \emph {Admired}}\).

In the flat example, we have three attributes, each of which can take many different values: size, location, and costs. Your intrinsic utility function assigns a desirability score to all possible combinations of these values.

If you’re like most people, we can say more about how these scores are determined. For example, you probably prefer cheaper flats to more expensive flats, and larger flats to smaller flats. The “weighing up pros and cons” idea suggests that the overall score for a flat is determined by adding up individual scores for the flat’s properties. Let’s spell out how this might work.

We want to compute the utility of any given attribute list as the sum of numbers assigned to the elements in the list. We’ll call these numbers subvalues. A size of 40 m\(^{2}\) might have subvalue \(V_{S}(40\, \m ^2) = 1\). Central location might have subvalue \(V_{L}(\text {central}) = 2\). Monthly costs of £500 might have subvalue \(V_{C}(\text {£500}) = -1\). Note that we have three different subvalue functions: one for size, one for location, one for costs. The overall value (utility) of \(\t {40\, \m ^2, \text {central}, \text {£500}}\) would then be the sum of these subvalues: \[ \U (\t {40\, \m ^2, \text {central}, \text {£500}}) = V_{S}(40 \m ^2) + V_{L}(\text {central}) + V_{C}(\text {£500}) = 2. \] If \(\U \) is determined by adding up subvalues in this manner, then it is called additive relative to the attributes in question.

Additivity may seem to imply that you assign the same weight to all the attributes: that size, location, and price are equally important to you. To allow for different weights, we could introduce scaling factors \(w_S, w_L, w_C\), so that \[ \U (\t {40\, \m ^2, \text {central}, \text {£500}}) = w_s \cdot V_{S}(40\, \m ^2) + w_L \cdot V_{L}(\text {central}) + w_C \cdot V_{C}(\text {£500}). \] For convenience, we will omit the weights by folding them into the subvalues. We will let \(V_S(200\, \m ^2)\) measure not just how awesome it would be to have a 200 \(\m ^2\) flat, but also how important this feature is compared to cost and location.

Subvalue functions are typically defined over propositions that don’t have uniform utility. Recall that, strictly speaking, ‘\(200\, \m ^{2}\)’ expresses the proposition that you are going to live in a 200 \(\m ^{2}\) flat. Some of the worlds where you live in such a flat are great. Others are bad. That’s because you also care about location and costs, and the \(200\, \m ^{2}\) worlds differ in these respects. An (improbable) world in which you rent a 200 \(\m ^{2}\) central flat for £100/month is better than a (more probable) world in which you rent a 200 \(\m ^{2}\) flat in the suburbs for £1000/month. As a result, the utility of 200 \(\m ^2\) may be low, even though the subvalue is high.

Informally, the utility of 200 \(\m ^{2}\) measures the desirability of the relevant proposition. Would you be glad to learn that you are going to rent a 200 \(\m ^{2}\) flat? Perhaps not, because the large size indicates high costs and bad location. The subvalue of 200 \(\m ^{2}\) is not sensitive to your beliefs. It measures the intrinsic desirability of that size, no matter what it implies or suggests about other attributes. It measures how much a size of 200 \(\m ^{2}\) contributes to the overall desirability of a flat, holding fixed the other attributes.

7.3Separability

Under what conditions is intrinsic utility determined by adding subvalues? How are different subvalue functions related to one another? We can get some insight into these questions by following an idea from the previous chapter and study how intrinsic utility might be derived from preferences.

The main motivation for starting with preferences is, as always, the problem of measurement. We need to explain what it means that your subvalue for a given attribute is 5 rather than 29. Since the numbers are supposed to reflect, among other things, the importance (or weight) of the relevant attribute in comparison to other attributes, it makes sense to determine the subvalues from their effect on the overall ranking of attribute lists.

So assume we have preference relations \(\succ \), \(\succsim \), \(\sim \) between lists of attributes. (We aren’t interested in lotteries or gambles this time, only in complete concerns.) To continue the illustration in terms of flats, if you prefer a central 40 \(\m ^2\) flat for £500 to a central 60 \(\m ^2\) for £800, then we have \[ \t {40 \m ^2, \text {central}, \text {£500}} \succ \t {60 \m ^2, \text {central}, \text {£800}}. \]

If, like most people, you prefer to pay less rather than more, then your subvalue function \(V_C\) is a decreasing function of monthly costs: the higher the costs \(c\), the lower \(V_C(c)\). This doesn’t mean that you prefer any cheaper flat to any more expensive flat. You probably don’t prefer a 5 \(\m ^2\) flat for £499 to a 60 \(\m ^2\) flat for £500. The other attributes also matter. But the following should hold: whenever two flats agree in size and location, and one is cheaper than the other, then you prefer the cheaper one.

Let’s generalize this idea.

Consider an attribute list \(\t {A_{1}, A_{2}, \ldots A_{n}}\), and let \(A_{1}'\) be an alternative to \(A_{1}\). If, for example, the first position in an attribute list represents monthly costs, then \(A_{1}\) might be £400 and \(A_{1}'\) £500. We can now compare \(\t {A_{1}, A_{2}, \ldots A_{n}}\) to \(\t {A_{1}', A_{2}, \ldots A_{n}}\) – a hypothetical flat that’s like the first in terms of size and location, but costs £100 more. If \[ \t {A_1,A_2,\ldots ,A_n} \succ \t {A_1',A_2,\ldots ,A_n}, \] we say that you prefer \(A_1\) to \(A_1'\) conditional on \(A_{2},\ldots ,A_{n}\).

Suppose you prefer \(A_{1}\) to \(A_{1}'\) conditional on any way of filling in the remainder \(A_{2},\ldots ,A_{n}\) of the attribute list. In that case, we can say that your preference of \(A_{1}\) over \(A_{1}'\) is independent of the other attributes.

In the flat example, your preference of £400 over £500 is plausibly independent of the other attributes: whenever two possible flats agree in size and location, but one costs £400 and the other £500, you plausibly prefer the one for £400. (We are still assuming that size, location, and costs are all you care about.)

We can similarly consider alternatives \(A_{2}\) and \(A_{2}'\) that may figure in the second position of an attribute list, and alternatives \(A_{3}\) and \(A_{3}'\) in the third positions, and so on. If we find that your preferences between \(A_{i}\) and \(A_{i}'\) are always independent of the other attributes, we say that your preferences between attribute lists are weakly separable.

Weak separability means that your preference between two attribute lists that differ only in one position does not depend on the attributes in the other positions.

Consider the following preferences between four possible flats. \begin {gather*} \t {50 \m ^2, \text {central}, \text {£500}} \succ \t {40 \m ^2, \text {beach}, \text {£500}}\\[0.5em] \t {40 \m ^2, \text {beach}, \text {£400}} \succ \t {50 \m ^2, \text {central}, \text {£400}} \end {gather*} Among flats that cost £500, you prefer central 50 m\(^2\) flats to 40 \(\m ^2\) flats at the beach. But among flats that cost £400, your preferences are reversed: you prefer 40 \(\m ^2\) beach flats to 50 \(\m ^2\) central flats. In a sense, your preferences for size and location depend on price. But we don’t have a violation of weak separability, simply because the relevant attribute lists differ in more than one position.

That’s why weak separability is called ‘weak’. To rule out the present kind of dependence, we need to strengthen the concept of separability. Preferences are called strongly separable if the ranking of lists that differ in one or more positions does not depend on the attributes in the remaining positions, in which they do not differ. In the example, your ranking of \(\t {50 \m ^2, \text {central}, -}\) and \(\t {40 \m ^2, \text {beach}, -}\) depends on how the blank (‘\(-\)’) is filled in. Your preferences aren’t strongly separable.

(Are they weakly separable? We can’t say. I have only specified how you rank two pairs of lists. Your preferences are presumably defined for many other combinations of flat size, location, and costs. There’s no violation of weak separability in the two data points I have given. But there might be a violation elsewhere.)

In 1960, Gérard Debreu proved that strong separability is exactly what is needed to ensure additivity.

To state Debreu’s result, let’s say that an agent’s preferences over attribute lists have an additive representation if there are a function \(\U \), assigning numbers to the lists, and subvalue functions \(V_1, V_2, \ldots , V_n\), assigning numbers to the items on the lists, such that the following two conditions are satisfied. First, the preferences are represented by \(\U \). That is, for any two lists \(A\) and \(B\), \begin {gather*} A \pref B \text { iff } \U (A) > \U (B), \text { and }\\[0.5em] A \sim B \text { iff }\U (A) = \U (B). \end {gather*} Second, the \(\U \)-value assigned to any list \(\t {A_1,A_2,\ldots ,A_n}\) equals the sum of the subvalues assigned to the items on the list: \[ \U (\t {A_1,A_2,\ldots ,A_n}) = V_1(A_1) + V_2(A_2) + \ldots + V_n(A_n). \]

Now, in essence, Debreu’s theorem states that if preferences over attribute lists are complete and transitive, then they have an additive representation if and only if they are strongly separable.

A technical further condition is needed if the number of attribute combinations is uncountably infinite; we’ll ignore that. Curiously, the result also requires that there are at least three attributes that matter to the agent. For two attributes, a stronger condition called ‘double-cancellation’ is required. Double-cancellation says that if \(\t {A_1,B_1} \succsim \t {A_2,B_2}\) and \(\t {A_2,B_3} \succsim \t {A_3,B_1}\) then \(\t {A_2,B_3} \succsim \t {A_3,B_2}\). But let’s just focus on cases with at least three relevant attributes.

Debreu’s theorem has an interesting corollary. Suppose a utility function \(\U \) has an additive representation in terms of certain attributes. One can show that if the attributes are sufficiently fine-grained, and small differences to the attributes make for small difference in overall utility, then every utility function \(\U '\) that has an additive representation in terms of the relevant attributes differs from \(\U \) at most in the choice of unit and zero.

This suggests a new response to the ordinalist challenge. The ordinalists claimed that utility assignments are arbitrary as long as they respect the agent’s preference order. In response, one might argue that rational (intrinsic) preferences should be strongly separable and that an adequate representation of such preferences should involve an additive utility function. The only arbitrary aspect of a utility representation would then be the choice of unit and zero.

Why might one think that rational preferences should be separable? Remember that we are talking about preferences over “attribute lists” that settle everything the agent ultimately cares about, with each position in a list settling one question that intrinsically matters to the agent. In our toy example, these were the size, location, and costs of their flat. More realistically, items in the attribute list might be the agent’s level of happiness, their social standing, the well-being of their relatives, etc. Now, if an agent has a basic desire for, say, happiness, then we would expect that increasing the level of happiness, while holding fixed everything else the agent cares about, always is a change for the better. That is, if two worlds \(w_{1}\) and \(w_{2}\) agree in all respects that matter to the agent except that the agent is happier in \(w_{1}\) than in \(w_{2}\), then we would expect the agent to prefer \(w_{1}\) over \(w_{2}\). From this perspective, separability might be understood as a condition on how to identify basic desires: if an agent’s preferences over some attribute lists are not separable, then the attributes don’t represent (all) the agent’s basic (intrinsic) desires.

7.4Separability across time

According to psychological hedonism, the only thing people ultimately care about is their personal pleasure. But pleasure isn’t constant. The hedonist conjecture leaves open how people rank different ways pleasure can be distributed over a lifetime. Unless an agent just cares about their pleasure at a single point in time, a basic desire for pleasure is really a concern for a lot of things: pleasure now, pleasure tomorrow, pleasure the day after, and so on. We can think of these as the “attributes” in the agent’s intrinsic utility function. The hedonist’s intrinsic utility function somehow aggregates the value of pleasure experienced at different times.

To keep things simple, let’s pretend that pleasure does not vary within any given day. We might then model a hedonist utility function as a function that assigns numbers to lists like \(\t {1,10,-1,2,\ldots }\), where the elements in the list specify the agent’s degree of pleasure today (1), tomorrow (10), the day after (-1), and so on. Such attribute lists, in which successive positions correspond to successive points in time, are called time streams.

A hedonist agent would plausibly prefer more pleasure to less at any point in time, no matter how much pleasure there is before or afterwards. If so, their preferences between time streams are weakly separable. Strong separability is also plausible: whether the agent prefers a certain amount of pleasure on some days to a different amount of pleasure on these days should not depend on how much pleasure the agent has on other days. It follows by Debreu’s theorem that the utility the agent assigns to a time stream can be determined as the sum of the subvalues she assign to the individual parts of the stream. That is, if \(p_1\), \(p_2\), …, \(p_n\) are the agent’s degrees of pleasure on days \(1, 2, \ldots , n\) respectively, then there are subvalue functions \(V_1,V_2,\ldots ,V_n\) such that \[ V(\t {p_1,p_2,\ldots ,p_n}) = V_1(p_1) + V_2(p_2) + \ldots + V_n(p_n). \]

We can say more if we make one further assumption. Suppose an agent prefers stream \(\t {p_1,p_2,\ldots ,p_n}\) to an alternative \(\t {p_1',p_2',\ldots ,p_n'}\). Now consider the same streams with all entries pushed one day into the future, and prefixed with the same degree of pleasure \(p_0\). So the first stream turns into \(\t {p_0, p_1,p_2,\ldots ,p_n}\) and the second into \(\t {p_0, p_1',p_2',\ldots ,p_n'}\). Will the agent prefer the modified first stream to the modified second stream, given that she preferred the original first stream? If the answer is yes, then her preferences are called stationary. From a hedonist perspective, stationarity seems plausible: if there’s more aggregated pleasure in \(\t {p_1,p_2,\ldots ,p_n}\) than in \(\t {p_1',p_2',\ldots ,p_n'}\), then there is also more pleasure in \(\t {p_0,p_1,p_2,\ldots ,p_n}\) than in \(\t {p_0,p_1',p_2',\ldots ,p_n'}\).

It is not hard to show that if preferences over time streams are separable and stationary (as well as transitive and complete), then they can be represented by a function of the form \[ \U (\t {A_1,\ldots ,A_n}) = V_1(A_1) + \delta \cdot V_1(A_2) + \delta ^2 \cdot V_1(A_3) \ldots + \delta ^{n-1} \cdot V_1(A_n), \] where \(\delta \) is a fixed number greater than 1. The interesting thing here is that the subvalue function for any time equals the subvalue function \(V_1\) for the first time, scaled by an exponential discounting factor \(\delta ^i\).

If a hedonist has strongly separable and stationary preferences, then her preferences over time streams are fixed by two things: how much she values present pleasure, and how much she discounts the future. If \(\delta = 1\), the agent values pleasure equally, no matter when it occurs. If \(\delta = \nicefrac {1}{2}\), then one unit of pleasure tomorrow is worth half as much as to the agent as one unit today; the day after tomorrow it is worth a quarter; and so on.

Even if you’re not a hedonist, you probably care about some things that can occur (and re-occur) at different times: talking to friends, going to concerts, having a glass of wine, etc. The formal results still apply. If your preferences over the relevant time streams are separable and stationary, then they are fixed by your subvalue function for the relevant events (talking to friends, etc.) right now and by a discounting parameter \(\delta \).

Some have argued that stationarity and separability across times are requirements of rationality. Some have even suggested that the only rationally defensible discounting factor is 1, on the ground that we should be impartial with respect to different parts of our life.

An argument in favour of stationarity is that it is often thought to be required to protect the agent from a kind of disagreement with her future self. To illustrate, suppose you prefer \(\t {10, 0, 0, 0, \ldots }\) to \(\t {0, 11, 0, 0, \ldots }\) because you care more about today’s pleasure than about tomorrow’s. You care less about the difference between getting pleasure in four days and getting it in five days, so you prefer \(\t {0, 0, 0, 0, 11, 0, 0, \ldots }\) to \(\t {0, 0, 0, 11, 10, 0, 0, \ldots }\). These preferences violate stationarity. Stationarity would imply that if you prefer \(\t {10, 0, 0, 0, \ldots }\) to \(\t {0, 11, 0, 0, \ldots }\) then you also prefer the first stream to the second if both are prefixed with 0, and therefore also if both are prefixed with two 0s, and with three 0s. Now suppose your (non-stationary) preferences remain the same for the next 4 days. At the end of this time, you’d still rather have 10 units of pleasure today than 11 tomorrow: you still prefer \(\t {10, 0, 0, 0, \ldots }\) to \(\t {0, 11, 0, 0, \ldots }\). But your “today” is what used to be “in 4 days”. Your new preferences disagree with those of your earlier self, in the sense that the worlds your former self regarded as better you now regard as worse. This kind of disagreement is called time inconsistency.

Empirical studies suggest that time inconsistency is pervasive. People often prefer their future selves to study, eat well, and exercise, but choose burgers and TV for today.

These preferences do look problematic. Other apparent violations of stationarity, and even separability across time, however, look OK. Suppose you like to have a glass of wine every now and then. But only now and then; you don’t want to have wine every day. It seems to follow that your preferences violate both separability and stationarity. You violate stationarity because even though you might prefer a stream \(\t {\text {wine}, \text {no wine}, \text {no wine}, \ldots }\) to \(\t {\text {no wine}, \text {no wine}, \text {no wine}, \ldots }\), your preference reverses if both streams are prefixed with wine (or many instances of wine). You violate separability because whether you regard having wine in \(n\) days as desirable depends on whether you will have wine right before or after these days.

Even if an agent only cares about pleasure, it is not obvious why a rational agent might not (say) prefer relatively constant levels of pleasure over wildly fluctuating levels, or the other way round.

One might argue, however, that in these cases the items in the time streams do not represent you basic desires, or not all of them. If, for example, you have a preference for constant levels of pleasure, then your basic desires don’t just pertain to how much pleasure you have today, how much pleasure you have tomorrow, and so on. You have a further basic desire: that your pleasure be constant from day to day.

Let’s briefly return to the problematic kind of time-inconsistency, manifested by the common desire for vice today and virtue tomorrow. What could explain this phenomenon?

Part of the explanation might be that our preferences have different sources (as I emphasized in chapter 5). When we reflect on having fries or salad now, we are more influenced by spontaneous cravings than when we consider the same options for tomorrow.

We could represent different sources of value by different subvalue functions. We might, for example, have a subvalue function \(V_{c}\) that measures the extent to which a proposition satisfies you present cravings, and another subvalue function \(V_{m}\) that measures to what extent it matches your moral convictions. Your intrinsic utility function is some kind of aggregate of these components. Here, too, separability is plausible. If, for example, you think that one world is morally better than another, and the two worlds are equally good with respect to all your other motives (your cravings are equally satisfied in either, etc.), then you plausibly prefer the first world to the second. This suggests that different sources of intrinsic utility combine in an additive manner.

7.5Harsanyi’s “proof of utilitarianism”

The ordinalist movement posed a challenge not only to the MEU Principle, but also to utilitarianism in ethics. Utilitarianism is a combination of two claims. The first says that an act is right iff it brings about the best available state of the world. The second says that the “goodness” of a state is the sum of the utility of all people. Without a numerical (and not just ordinal) measure of personal utility, this second claim makes no sense. We would need a new criterion for ranking states of the world.

One such criterion was proposed by Pareto. Recall that Pareto did not deny that people have preferences. If we want to know which of two states is better, we can still ask which of them people prefer. This allows us to define at least a partial order on the possible states:

Unlike classical utilitarianism, however, the Pareto Condition offers little moral guidance. For instance, while classical utilitarianism suggests that one should harvest the organs of an innocent person in order to save ten others, the Pareto Condition does not settle whether it would be better or worse to harvest the organs, given that the person to be sacrificed ranks the options differently than those who would be saved.

In 1955, John Harsanyi proved a remarkable theorem that seemed to rescue, and indeed vindicate, classical utilitarianism.

As a first step, Harsanyi adopts von Neumann’s response to the ordinalist challenge. He assumes that each individual has preferences not only among the relevant states, but also among lotteries involving the states, and that their preferences conform to the von Neumann and Morgenstern axioms. We can then represent their preferences by personal utility functions \(\U _{1},\ldots ,\U _{n}\) (one for each individual) that are unique up to the choice of unit and zero.

Our goal is to derive a “social preference” relation between states that settles whether a state is overall better than another. Harsanyi assumes that this social preference relation can be extended to lotteries in a way that conforms to the von Neumann and Morgenstern axioms. It follows that social preference is also represented by a (“social”) utility function \(\U _{s}\) that is unique up to the choice of unit and zero.

Harsanyi now showed that if we add the Pareto condition (for both states and lotteries), then the individual and social preferences are represented by utility functions \(\U _1,\ldots ,\U _n\) and \(\U _s\) in such a way that social utility is simply the sum of the individual utilities: for any state \(A\), \[ \U _s(A) = \U _1(A) + \ldots + \U _n(A). \] Once we have allowed lotteries into the picture, the Pareto condition entails full-blown utilitarianism! How is this possible?

The Pareto condition implies that the social utility of any state is determined by the personal utility each individual assigns to the state. For suppose the social utility of some state \(A\) depends on an aspect of \(A\) that doesn’t affect the personal utilities. Then there is an alternative \(B\) to \(A\) (that differs from \(A\) in this aspect) for which \(\U _{s}(B) \not = \U _{s}(A)\) even though every individual assigns the same utility to \(A\) and \(B\). This contradicts the Pareto condition.

So the only “attributes” of a state that are relevant to its social utility are its personal utility scores. We can represent a state by a list of numbers \(\t {u_{1}, \ldots , u_{n} }\), each of which specifies how desirable the state is for a particular individual.

Most non-utilitarians would disagree on this point. They would hold that even if everyone is indifferent between two states \(A\) and \(B\), \(A\) might still be worse than \(B\), if it involves gratuitous human rights violations, animal suffering, sin, or whatever.

The really surprising part of Harsanyi’s theorem is that the social utility of a state is simply the sum of its personal utility scores \(u_{1} + \ldots + u_{n}\). This tells us that social preference is separable across the personal utilities, and that each personal utility (each attribute) simply contributes its value to social utility. How does this come about? Couldn’t an even distribution \(\t {10,10,10,10,\ldots }\) be better than an uneven distribution \(\t {0,20,0,20,\ldots }\)? Relatedly, couldn’t personal utility have “declining social value”, so that adding 1 unit of personal utility to an individual whose utility is already at 1000 contributes less to social utility than adding 1 unit to an individual who stands at 0?

These possibilities are ruled out by three assumptions that look harmless in isolation, but have great power when combined.

One is the assumption that the Pareto condition holds for both lotteries and states. This implies that if every individual is indifferent between some lottery \(L\) and some state \(A\), then the social preference relation is indifferent between \(L\) and \(A\).

The second assumption is that each individual evaluates lotteries by their expected (personal) utility. Let \(L\) be a fair lottery between \(\t {0,20,0,20,\ldots }\) and \(\t {20,0,20,0\ldots }\). The expected personal utility for each individual is 10. If everyone evaluates the lottery by its expected personal utility, then everyone is indifferent between \(L\) and \(\t {10,10,10,10,\ldots }\). By the first assumption, it follows that the social preference order is indifferent between \(L\) and \(\t {10,10,10,10,\ldots }\).

Finally, we have assumed that the social preference order ranks lotteries by their expected social utility. Assuming that the number of individuals is even, the states \(\t {20,0,20,0,\ldots }\) and \(\t {0,20,0,20,\ldots }\) plausibly have the same social utility. It follows that the social preference order is indifferent between either of these states and \(L\). (If \(A\) and \(B\) have equal utility, then the expected utility of a lottery between \(A\) and \(B\) must equal the utility of \(A\) and \(B\).) But we’ve just seen that the social preference order is indifferent between \(L\) and \(\t {10,10,10,10,\ldots }\). It follows that \(\t {0,20,0,20,\ldots }\) and \(\t {10,10,10,10,\ldots }\) have equal social utility.

If we think that even distributions of utility are better than uneven distributions, we have to reject at least one of the three assumptions. If we also accept that the right way to evaluate lotteries is by expected utility, it looks like the first assumption has to go. \(L\) is worse than \(\t {10,10,10,10,\ldots }\) even though each individual is indifferent between the two.

But should we accept that the right way to evaluate lotteries is by expected utility? This is the question to which we turn next.