5Utility

5.1Two conceptions of utility

Daniel Bernoulli realized that rational agents don’t always maximize expected monetary payoff: £1000 has more utility for a pauper than for a rich man. But what is utility?

Until the early 20th century, utility was widely understood to be some kind of psychological quantity, often identified with degree of pleasure and absence of pain. On that account, an outcome has high utility for an agent to the extent that it increases the agent’s pleasure and/or decreases her pain.

Let’s assume for the sake of the argument that one can represent an agent’s total amount of pleasure and pain by a single number – the agent’s “degree of pleasure”. Can we understand utility as degree of pleasure? The answer depends on what role we want the concept of utility to play.

One such role lies in ethics. According to utilitarianism, an act is morally right just in case it would bring about the greatest total utility for all people. In this context, identifying utility with degree of pleasure implies that only pleasure and pain have intrinsic moral value; everything else – autonomy, integrity, respect of human rights, and so on – would be morally relevant only insofar as it causes pleasure or pain. This assumption is known as ethical hedonism. We will not pursue it any further.

Another role for a concept of utility lies in the theory of practical rationality. According to the MEU Principle, practically rational agents choose acts that maximize the credence-weighted average of the utility of the possible outcomes. If we identify utility with degree of pleasure, the MEU principle turns into what we might call the ‘MEP Principle’:

An act’s expected degree of pleasure is the probability-weighted average of the degree of pleasure that might result from the act.

The MEP Principle is a form of psychological hedonism. Psychological hedonism is the view that the only thing that ultimately motivates people is their own pleasure and pain.

The founding fathers of modern utilitarianism, Jeremy Bentham and John Stuart Mill, had sympathies for both ethical hedonism and psychological hedonism. As a consequence, the two conceptions of utility – the two roles associated with the word ‘utility’ – were not properly distinguished. Today, both kinds of hedonism have long fallen out of fashion, but the two conceptions are still often conflated.

For the most part, contemporary utilitarians hold that the standard of moral rightness is the total welfare or well-being produced by an act, which is not assumed to coincide with total degree of pleasure. Thus ‘utility’ is nowadays often used as a synonym for ‘welfare’ or ‘well-being’. But the word is also widely used in the other sense, to denote whatever motivates (rational) agents.

Some have argued that the two uses actually coincide: that the only thing that motivates rational agents is their own welfare or well-being. This may or may not be true. But it needs to be backed up by data and argument; it does not become true through sloppy use of language.

In these notes, ‘utility’ is only used in the second sense. The utility of a outcome measures the extent to which the agent in question wants the outcome to obtain. We do not assume that the only thing agents ultimately want is to increase their degree of pleasure, their welfare, their well-being, or anything like that.

Note that psychological hedonism, or the slightly more liberal claim that people only care about their welfare, is at most a contingent fact about humans. One can easily imagine agents who are motivated by other things. We can imagine a mother who knowingly takes on hardships for the benefit of her children, or a soldier who intentionally chooses a painful death in order to save her comrades. Psychological hedonists hold that humans would never consciously do such things: whenever an agent sacrifices her own good to benefit others, she mistakenly believes that her choice will actually make herself better off than the alternatives. Again, we don’t need to argue over whether this is true. The important point is that utility, as we use the term, does not mean the same as degree of pleasure or welfare or well-being.

A hedonist might object that while it is conceivable that an agent is motivated by things other than her personal pleasure, such agents would be irrational. After all, the MEP Principle only states that rational agents maximize their expected degree of pleasure; it doesn’t cover irrational agents.

This brings us to a tricky issue. What do we mean by ‘rational’? The label ‘rational’ is sometimes associated with cold-hearted selfishness. On this usage, a rational agent always looks out for her own advantage, with no concern for others. This idea of “economic rationality” has its use, but it is not our topic. The kind of rationality we’re interested in is a more minimal notion. Intuitively, it is the idea of “making sense”. If you want to reduce animal suffering, and you know you can achieve this by eating less meat, then it makes sense that you eat less meat. If you are sure that a picnic will be cancelled if it is raining, and you see that it is raining, then it doesn’t make sense to believe that the picnic will go ahead. The model we are studying is a model of agents who “make sense” in this kind of way.

Even if we were interested in the cold-hearted and selfish sense of rationality, we should not define utility as degree of pleasure or welfare. Consider a hypothetical agent who cares not just about herself, who sacrifices some of her own good to reduce the pain of others. The agent is “irrational” in the cold-hearted and selfish sense. But what is irrational about her? Does the fault lie in her beliefs, in her goals, or in the way she brings these together to make choices? Plausibly, the “fault” lies in her goals. Her concern for others is what goes against the standards of cold-hearted and selfish rationality. But if we were to define utility as degree of pleasure or welfare, we would have to say that the agent violates the basic norm of practical rationality, the MEU Principle.

The point generalizes. Consider a person in an abusive relationship who is manipulated into doing things that hurt or degrade her. We might reasonably think that the person shouldn’t do these things; it is against her interest to do them. But what is at fault? Arguably, the fault lies in her (manipulated) desires. What the person does may well be in line with what she wants to achieve – in particular, with her strong desire to please her partner. But a healthy, self-respecting person, we think, should have other desires.

By understanding utility as a measure of whatever the agent in question desires, we do not automatically sanction these desires as rational or praiseworthy. Our usage of ‘utility’ allows us to say that the person in the abusive relationship shouldn’t do what she is doing, because she should have different desires that would not support her actions.

5.2Sources of utility

An outcome’s utility measures the extent to which the agent is motivated to bring about the outcome. I will often say that this is the degree to which the agent desires the outcome, but we need to keep in mind that the word ‘desire’ can be misleading. For one thing, we need to cover “negative desire”. Being hungry might have greater utility for you than being dead, even though you do not desire either. More importantly, ‘desire’ is often associated with a particular type of motivational state. I might say that I got up early in the morning despite my strong desire to stay in bed; I got up not because I desired to get up, but because I had to. On this usage, my desires contrast with my sense of duty.

Utility comprises everything that motivates the agent, all the reasons she has for and against a particular action. As such, ‘utility’ is an umbrella term for a diverse set of psychological states or events. We can be motivated by bodily cravings, by moral commitments, by our image of the kind of person we want to be, by an overwhelming feeling of terror or love, and so on. These factors need not be conscious. There is good evidence that our true motives are often not what we believe or say they are. An agent’s utility function represents their true motives, and all of them.

Why should we believe that all the factors that motivate an agent can be amalgamated into a single numerical quantity? Would it not be better to allow for a whole range of utility functions: moral utility, emotional utility, and so on? We could certainly do that. But there are reasons to think that there must also be an amalgamated, all-things-considered utility (although the determinacy and numerical precision of utility functions is obviously an idealisation). When you face a decision, you have to make a single choice. You can’t choose one act on moral grounds and a different act on emotional grounds. Somehow, all your motives and reasons have to be weighed against each other to arrive at an overall ranking of your options.

We will have a brief look at the weighing of different considerations in chapter 7, but to a large extent this is really a topic for empirical psychology and neuroscience. If it turns out that there are 23 distinct factors that influence our motivation in an intricate network of inhibition and reinforcement, then so be it. We will model the whole network by a single utility function, staying neutral on “lower-level” details that can vary from agent to agent. But it’s important to keep in mind that a lot of interesting and complicated psychology is hiding in our seemingly simple concept of utility.

Consider the following scenario.

The kind of behaviour displayed by Emily is common. People tend to place a greater value on things they own than on things they don’t own. Initially, Emily considered the two mugs equally desirable. Having bought the red mug, Emily suddenly considers it better than the blue mug.

Psychologists have offered different explanations for this effect. Some say that forgoing an owned item feels like a loss, and we don’t like this feeling. Others have argued that we treat goods that we own as part of our identity; forgoing the good is thus perceived as a threat to our identity. We don’t need to adjudicate between these (and other) proposals. What’s important for us is that whichever explanation is correct, it should be reflected in Emily’s utility function. If Emily subconsciously regards her belongings as part of her identity, and she is subconsciously motivated to preserve her identity, then her utility for an outcome that involves giving up a previously owned good is comparatively low.

Outside philosophy – especially in economics – utility is often assumed to be a function of material goods (“commodity bundles”). On this usage, one can speak of the utility (for Emily) of the red mug, but one can’t distinguish between, for example, the utility of not getting the red mug and giving away the red mug. No matter what utility we then assign to the two mugs, Emily’s behaviour is found to violate the MEU Principle. If the red cup has greater utility than the blue cup, then Emily shouldn’t have been indifferent when she decided which cup to buy. If the two cups have equal utility for Emily, then Emily should be happy to swap the red cup for the blue cup.

On our usage of ‘utility’, Emily’s behaviour is perfectly compatible with the MEU Principle. Emily doesn’t just care about which material goods she owns. She also cares about changes to her possessions. If she is a real person, she will also care about other things that have little to do with material goods. If we want a general model of how beliefs and desires relate to choices, we need to make room for all the desires an agent might have. We could follow the economics tradition and restrict an agent’s utility function to material goods. But then we would have to add other elements to our model to account for desires that don’t pertain to the possession of material goods. We will choose the theoretically simpler option of widening the definition of ‘utility’, so that an agent’s utility function reflects everything the agent cares about. We are going to return to this theme in chapter 8.

Officially, we will use ‘utility’ to measure anything that motivates the relevant agent. It is worth pointing out, however, that our model can be usefully applied with other conceptions of utility. We might want to know, for example, what an agent should do, from a moral perspective, in a situation like the Miners Problem from chapter 1, where crucial information about the world is missing. A tempting idea is that the agent should maximize expected moral utility, where the moral utility of an outcome is defined by some ethical theory (utilitarianism, perhaps). Similarly, a corporation’s board of directors may want to know how to promote shareholder value in the light of such-and-such common information. Here the relevant utility function might be derived from the stipulated goal of promoting shareholder value, and the “credence” function might be derived from the shared information. Neither of these needs to match the beliefs and desires of any individual member of the board.

5.3The structure of utility

Now that we know what utility is, let’s have a closer look at its formal structure.

First of all, what are the bearers of utility? In ordinary language, we often say that people desire things: tea, cake, a concert ticket, a larger flat. As we saw in the previous section, we need a more general conception to capture an agent’s desire not to lose a previously owned good. We might also desire that our friends are happy, that it won’t rain tomorrow, that so-and-so will win the next elections. Here the object of desire isn’t a thing, but a possible state of the world. Even when we say that people desire things, plausibly the desire is really directed at a possible state of the world. When you desire tea, you desire to drink the tea. Your desire wouldn’t be satisfied if I gave you a certificate of ownership for a cup of tea that is locked away in a safe.

So we’ll assume that the objects of desire are the same kinds of things as the objects of belief: propositions, or possible states of the world. As in the case of belief, we don’t distinguish between logically equivalent states of the world. If you assign high utility to drinking tea then you also assign high utility to drinking tea or coffee but not coffee.

Let’s study how an agent’s desires towards logically related propositions are related to one another. Suppose you assign high utility to the proposition that it won’t rain tomorrow (perhaps because you want to go on a picnic). Then you should plausibly assign low utility to the proposition that it will rain. You can’t hope that it will rain and also that it won’t rain. In this respect, desire resembles belief: if you are confident that it will rain, you can’t also be confident that it won’t rain. The Negation Rule of probability captures the exact relationship between \(\Cr (A)\) and \(\Cr (\neg A)\), stating that \(\Cr (\neg A) = 1 - \Cr (A)\). Does the rule also hold for utility? More generally, do utilities satisfy the Kolmogorov axioms? It will be instructive to go through the three axioms.

Kolmogorov’s axiom (i) states that probabilities range from 0 to 1. If there are upper and lower bounds on utility, we could adopt axiom (i) for utilities as a convention of measurement: we simply use 1 for the upper bound and 0 for the lower bound. However, it is not obvious that there are such bounds. Couldn’t there be an infinite series \(A_1, A_2, A_3, \ldots \) of states of increasing utility in which the difference in utility between successive states is always the same? If there is such a series, then utility can’t be measured by numbers between 0 and 1. Philosophers are divided over the question. Some think utility must be bounded, others think it can be unbounded. There are arguments for both sides. We will not pause to look at them.

Kolmogorov’s axiom (ii) states that logically necessary propositions have probability 1. If utilities satisfy the probability axioms, this would mean that logically necessary propositions have maximal utility. However much you desire that it won’t rain tomorrow, your desire that it either will or won’t rain should be at least as great.

This does not look plausible. Intuitively, if something is certain to be the case, it makes no sense to desire it. But this could mean two things. It could mean that degrees of desire are not even defined for logically necessary propositions. Or it could mean that an agent should always be indifferent towards logically necessary propositions – neither wanting them to be the case nor wanting them to not be the case. Our common-sense conception of desire arguably sides with the first option: if you are certain of something, even asking how strongly you desire it seems odd. But the issue isn’t clear. For our purposes, it proves more convenient to go with the second option. We will say that even logically necessary propositions have well-defined utility, and that their utility measures the point between “positive” and “negative” desire. If you positively want something to be the case, the utility you assign to it is greater than the utility of a tautology. If you want something not to be the case, its utility is lower than that of a tautology. Some authors make this more concrete by adopting a convention that logically necessary propositions always have utility 0.

Axiom (iii) states that if \(A\) and \(B\) are logically incompatible, then the probability of \(A\lor B\) equals the sum of the probabilities of \(A\) and \(B\). To illustrate, suppose there are three possible locations for a picnic: Alder Park, Buckeye Park, and Cedar Park. Alder Park and Buckeye Park would be convenient for you; Cedar Park would not. Now how much do you desire that the picnic takes place in either Alder Park or Buckeye Park? If axiom (iii) holds for utilities, then if you desire Alder Park and Buckeye Park to equal degree \(x\), then your utility for the disjunction should be \(2x\): you should be more pleased to learn that the picnic takes place in either Alder Park or Buckeye Park than to learn that it takes place in Alder Park. That’s clearly wrong. Axiom (iii) also fails.

What is the true connection between the utility of \(A \lor B\) and the utilities of \(A\) and \(B\)? Intuitively, if \(A\) and \(B\) have equal utility \(x\), then the utility of \(A \lor B\) should also be \(x\). What if the utilities of \(A\) and \(B\) are not equal? What if, say, \(\U (A) > \U (B)\)? Then the utility of \(A \lor B\) should plausibly lie in between the utilities of \(A\) and \(B\): \[ \U (A) \geq \U (A \lor B) \geq \U (B). \] That is, if Alder Park is your first preference and Buckeye your second, then the disjunction either Alder Park or Buckeye Park can’t be worse than Buckeye Park or better than Alder Park. But where does \(\U (A \lor B)\) lie in between \(\U (A)\) and \(\U (B)\)? At the mid-point?

Suppose you prefer Alder Park to Buckeye Park, and Buckeye Park to Cedar Park. You think it is highly unlikely that the picnic will take place in Buckeye Park. Now how pleased would you be to learn the picnic won’t take place in Cedar Park – equivalently, that it will take place either in Alder Park or in Buckeye Park? You should be quite pleased. If you’re confident that \(B\) is false, then \(\U (A \lor B)\) should plausibly be close to \(\U (A)\). If you’re confident that \(A\) is false, then \(\U (A \lor B)\) should be near \(\U (B)\).

Your utilities depend on your beliefs! On reflection, this should not come as a surprise. A lot of the things we desire we only desire because we have certain beliefs. If you want to buy a hammer to hang up a picture, then your desire for the hammer is based (in part) on your belief that the hammer will allow you to hang up the picture.

Here is the general rule for \(\U (A \lor B)\), assuming \(A\) and \(B\) are incompatible. The rule was discovered by Richard Jeffrey in the 1960s and is our only basic rule of utility, apart from the assumption that logically equivalent propositions have the same utility.

In words: the utility of \(A \lor B\) is the weighted average of the utility of \(A\) and the utility of \(B\), weighted by the probability of the two disjuncts, conditional on \(A \lor B\).

Why ‘conditional on \(A \lor B\)’? Why don’t we simply weigh the utility of \(A\) and \(B\) by their unconditional probability? Because then highly unlikely propositions would automatically have a utility near 0. If you are almost certain that the picnic will take place in Cedar Park, both \(\Cr (\emph {Alder Park})\) and \(\Cr (\emph {Buckeye Park})\) will be close to 0. But the mere fact that a proposition is unlikely does not make it undesirable. To evaluate the desirability of a proposition, we should bracket its probability. That’s why Jeffrey’s axiom defines \(\U (A \lor B)\) as the probability-weighted average of \(\U (A)\) and \(\U (B)\) on the supposition that \(A \lor B\) is true.

The following consequence of Jeffrey’s axiom is often useful. Assume that \(S_{1},\ldots ,S_{n}\) are propositions for which it is guaranteed that exactly one of them is true. That is, any two propositions in \(S_{1},\ldots ,S_{n}\) are logically incompatible (no two of the propositions can be true together), and the disjunction \(S_{1}\lor \ldots \lor S_{n}\) is logically necessary (one of the propositions must be true). Such a collection of propositions is called a partition. Intuitively, a partition divides the space of possible worlds into disjoint regions.

Now, Jeffrey’s axiom entails that if \(S_{1},\ldots ,S_{n}\) is a partition, then for any proposition \(A\) with \(\Cr (A) > 0\), \begin {equation*} \U (A) = \U (A \land S_1)\cdot \Cr (S_{1}/A) + \ldots + \U (A \land S_{n})\cdot \Cr (S_{n}/A). \end {equation*} Let’s call this the partition formulation of Jeffrey’s axiom.

Think of \(A\) as a region in logical space. Each \(A \land S_{i}\) is a disjoint subregion of \(A\). The partition formulation says that the desirability of the whole region \(A\) is a weighted average of the desirability of the subregions, weighted by their probability conditional on \(A\).

5.4Basic desire

I have presented Jeffrey’s axiom as the sole formal requirement on rational utility. Even this much is controversial. Many philosophers hold that rationality imposes no constraints at all on an agent’s desires. (In a way, this is the opposite extreme of the hedonist doctrine that rational agents desire nothing but their own pleasure.) The idea was memorably expressed by David Hume in his Treatise of Human Nature:

’tis not contrary to reason to prefer the destruction of the whole world to the scratching of my finger. ’Tis not contrary to reason for me to chuse my total ruin, to prevent the least uneasiness of an Indian or person unknown to me.

Hume held that our basic desires are not responsive to evidence, reason, or argument. If your ultimate goal is to help some distant stranger, there is no non-circular argument that could prove your goal to be wrong, nor could we fault you for not taking into account any relevant evidence. Whatever facts you might find out about the world, you could coherently retain your ultimate goal of helping the stranger.

For Hume, beliefs and desires are in principle independent. What you believe is one thing, what you desire is another. Beliefs try to answer the question: what is the world like? Desires answer an entirely different question: what do you want the world to be like? On the face of it, these two questions really appear to be logically independent. Two agents could in principle give the same answer to the first question and different answers to second, or the other way around.

What we have seen in the previous section seems to contradict these intuitions. We have seen that an agent’s utilities are thoroughly entangled with her credences. Indeed, we can read off an agent’s credence in any proposition \(A\) from her utilities, assuming the utilities obey Jeffrey’s axiom, the credences obey the probability axioms, and the agent is not disinterested in \(A\). Here is how.

By Jeffrey’s axiom, \[ \U (A \lor \neg A) = \U (A)\cdot \Cr (A) + \U (\neg A)\cdot \Cr (\neg A). \] By the Negation Rule, we can replace \(\Cr (\neg A)\) by \(1 - \Cr (A)\). Multiplying out, we get \[ \U (A \lor \neg A) = \U (A)\cdot \Cr (A) + \U (\neg A) - \U (\neg A)\cdot \Cr (A). \] Now we solve for \(\Cr (A)\): \[ \Cr (A) = \frac {\U (A \lor \neg A) - \U (\neg A)}{\U (A) - \U (\neg A)}. \] The ratio on the right-hand side is defined whenever \(\U (A) \not = \U (\neg A)\), which I meant when I said that the agent is “not disinterested” in \(A\).

What is going on here? Have we refuted Hume? Have we shown that an agent’s beliefs are part of her desires?

Of course not – or not in any interesting sense. We need to distinguish basic desires from derived desires. If you are looking for a hammer to hang up a picture, your desire to find the hammer is not a basic desire. It is derived from your desire to hang up the picture and your belief that you need a hammer to achieve that goal. By contrast, a desire to be free from pain is typically basic. If you want a headache to go away, this is usually not (or not only) because you think having no headache is associated with other things you desire. You simply don’t want to have a headache, and that’s the end of the story.

When Hume claimed that desires are independent of beliefs, he was talking about basic desires.

How are basic desires related to an agent’s utility function?

Let’s pretend that you have only one basic desire: to be free from pain. Let’s also pretend that this is an all-or-nothing matter. By your lights, all possible worlds in which you are free from pain are then equally good, equally desirable. In each of them, you have everything you want. The worlds in which you are not free from pain are also equally good. In each of them, you do not have what you want.

Let’s say that a proposition has uniform utility for an agent if the agent does not care how the proposition is realized: all subsets of the proposition (understood as a set of possible worlds) have equal utility. In our example, being pain-free and being in pain have uniform utility.

Let’s change the scenario so that you have two basic desires: being free from pain and being admired by other people. These are logically independent, so there are four combinations: (1) being pain-free and admired, (2) being pain-free and not admired, (3) being in pain and admired, and (4) being in pain and not admired. Note that these form a partition.

Being in pain no longer has uniform utility. The worlds where you are in pain divide into (better) worlds where you are in pain and admired and (worse) worlds where you are in pain and not admired. As a consequence, the utility of being in pain now depends on your beliefs: the stronger you believe that you are admired if you are in pain, the more you desire being in pain.

The four combinations of being pain-free and admired, however, have uniform utility. All worlds in which you are, say, in pain and admired are equally desirable (still pretending these are all-or-nothing matters). I will say that these combinations are your concerns. Intuitively, a concern is a proposition that settles everything the agent ultimately cares about. An agent’s concerns always form a partition.

Remember that an outcome in a decision matrix must settle everything the agent cares about. Every outcome in every decision problem is therefore a concern. Many decision theorists use the word ‘outcome’ where I use ‘concern’. I prefer a different label, if only because some of an agent’s concern may never figure as outcomes in a decision situation.

It will be useful to have a name for an agent’s utility function restricted to their concerns. I’ll call it the agent’s intrinsic utility function. (Some people say ‘value function’; many just say ‘utility function’ and never consider the wider function we call the agent’s ‘utility function’.)

Formally, an intrinsic utility function assigns numbers to some partition of propositions. Intuitively, each of these propositions settles everything the agent cares about, and the numbers tell us how strongly the agent desires any particular way of settling the things they care about. In the above example, your intrinsic utility function might by fully given as follows: \begin {gather*} \U (\emph {Pain} \land \emph {Admired}) = 1,\\[0.5em] \U (\neg \emph {Pain} \land \emph {Admired}) = 5,\\[0.5em] \U (\emph {Pain} \land \neg \emph {Admired}) = -5,\\[0.5em] \U (\neg \emph {Pain} \land \neg \emph {Admired}) = 0. \end {gather*}

An agent’s intrinsic utility function represents the belief-independent aspect of their goals or desires. Every possible credence function is compatible with every possible intrinsic utility function.

Since no concern is ever a disjunction of other concerns, Jeffrey’s axiom imposes no constraint on intrinsic utility functions. It only enters the picture when we look at the utility of propositions that aren’t concerns. In effect, the axiom tells us how to derive an agent’s utility for non-concerns from the agent’s intrinsic utility function and their credence function. (The axiom’s partition formulation makes the derivation transparent.)

In chapter 7, we will look at how an agent’s intrinsic utility function might be determined by less specific desires – by a desire to be free from pain, for example, and a desire to be admired. Before we do this, we need to say more about what the utility numbers are supposed to represent. What, exactly, does it mean that a proposition has utility 5, as opposed to -5 or 27?