# Multidimensional utility

I'm an outsider to the philosophy community (I'm a mathematics PhD student), and I'm curious of whether the following critique has been addressed. During some superficial discussions I had on utilitarianism, I frequently encountered the assumption that the total utility of an (intelligent) agent is 1. additive and 2. one-dimensional. To be precise, if an agent undergoes experiences E1, ... En, then each unit of experience has the associated utilities u1, ... un, which are real numbers, and the agent's total utility is u1 + ... + un (to account for infinitely many experiences, use an integral instead). At the very least, this was what I understood to be assumed in the original proposal of utilitarianism and its elementary criticisms.

I have two objections regarding this model of utilitarianism. I think that the assumption of regarding u1, ... un as real numbers is too restrictive, and I also think that using the total sum u1 + ... + un as the total utility is too restrictive.

1. A simple proposal is to model u1, ... un as vectors in higher dimensional spaces, possibly infinite. This will allow capturing various aspects of experiences in many coordinates of each experience. For example, one can think of each coordinate as "joy", "sadness", "familial bond", and so on. Note that even sadness is seen as good in the right context, for example in viewing a masterfully created film of high emotional value.

2. Instead of directly taking u1 + ... + un, the framework of utilitarianism should allow history-dependence, such as diminishing return. We are well aware that repeating the same type of experience repeatedly makes the experience have less value on us.

3. Utility should also allow perspective-dependence, which can change wildly through our lives. Depending on our value system, pain and suffering can have great utility, which can encapsulate anything from sensual pleasure to martyrdom. This value system can very well change through our lives, and thus I think that an agent's aggregate experience should be able to be filtered through a 'perspective function', to yield lower-dimensional (possibly 1-dimensional) utility.

Thus a simple extended model of utility I propose would be equipped with the following:

1. A real vector space H, whose elements represent events in an agent's life.
2. A sequence An = (E1, E2, ..., En) of elements of H, representing an agent's life.
3. A set P of perspective functions H^k -> R, ranging over all natural numbers k
4. A model of utility f(An) that depends on the timestep n and the perspective function f.

It seems likely that these concerns have been addressed in the philosophy community one way or another. Some simple Google searches at least reveal some papers in economics that deal with higher dimensional models of utility. Are there existing references to my concerns? Thank you in advance.

• I guess a basic concern is that at some point one may have to make a choice across dimensions. It's not clear how that is to be handled in such a vectoral context. There are lots of papers in psychology on this; see e.g. onlinelibrary.wiley.com/doi/abs/10.1111/spc3.12509 Commented Apr 21, 2021 at 10:38
• The classical utilitarianism of Bentham was arguably multi-dimensional along your lines. Warke explains in Classical Utilitarianism and the methodology of determinate choice how and why this conception was gradually abandoned because it was (deemed) unworkable for "unavoidable ambiguity in determining optimality, due to the multi-dimensionality of its optimality criterion", among other things. It is fundamentally unclear how to set up "perspective functions" that produce determinate choices. Commented Apr 21, 2021 at 12:32
• partly as a self-reference, I found this question to be relevant as well: philosophy.stackexchange.com/questions/407/… Commented Apr 21, 2021 at 15:43
• @Conifold That seems to be a problem of giving a total order on a real vector space. I'd argue that we don't always need or have a total order of preferences. Also, a simple 1-dimensional projection along any vector produces a perspective function. Commented Apr 22, 2021 at 8:27
• @FinnLim The central purpose of ethics is deciding how to act, a method that does not reliably select a top preference is of limited use there. Perspective functions also have to be ethically meaningful, simply picking one of the "pleasures" and acting on it alone is not a sound "perspective" most of the time, however nice it may be mathematically. Generally, multi-objective optimization in ethics raises the issue of moral dilemmas, which is very thorny and often drives people away from any form of utilitarianism in deciding action. Commented Apr 22, 2021 at 17:51

It turns out that if a rational agent has a complete and consistent preference relation among possible outcomes, and uses this relation to choose between probabilistic outcomes, then the agent must act as if they are assigning a single real-numbered utility to each different (probabilistic) outcome.

This is the theoretical justification for utility always being a one-dimensional real number.

However, note that this theorem assumes the agent is "rational." For instance, if the agent prefers A to B and B to C, then the agent must prefer A to C. In real life, we do not have time or cognitive ability to arrive at a complete and consistent set of preferences. We are only approximately rational, which may leave room for a more approximate and complicated notion of utility.

• Philosopher Frank Ramsey pioneered the idea of determining a preference function through choices between different probabilistic outcomes (like 'would you prefer a 50% chance of ice cream or an 80% chance of a popsicle'), see section 6 of his SEP article and section 2.2 of the decision theory article, but section 3.1 of the latter says 'Ramsey neither gave a full proof of his result nor provided much detail of how it would go' Commented Apr 21, 2021 at 23:41
• "However, note that this theorem assumes the agent is "rational." For instance, if the agent prefers A to B and B to C, then the agent must prefer A to C." That isn't always true, though? Take the example of intransitive dice (en.wikipedia.org/wiki/Intransitive_dice) or Paper-Scissors-Rock, for instance. The utility of Rock might beat the utility of Scissors, and the utility of Scissors might beat that of Paper, but the same doesn't hold true for Paper and Rock. Commented Apr 22, 2021 at 4:40
• @nick012000 intransitive dice are not a rational agent. An agent who prefers A to B, and B to C, and C to A, would behave irrationally. Suppose he has A; then we offer him C in exchange for A and a penny, and he agrees. Now he has C and we offer him B in exchange for C and a penny, and he agrees. Now he has B and we offer A in exchange for B and a penny, and he agrees. Now he's down three pennies and he has A again, and we can repeat the cycle indefinitely, and he will give us all he has. Commented Apr 22, 2021 at 4:50
• Thank you for the reference; it's indeed a delightful theorem. However, I think this answer may be missing the point. The von Neumann-Morgenstein setup doesn't account for the possibility that people's preferences could change over time, which is where my point 2 (diminishing returns) and 3 (perspective dependence) enter. Commented Apr 22, 2021 at 8:17
• "The von Neumann-Morgenstein setup doesn't account for the possibility that people's preferences could change over time" Why would preference changing over time imply a multidimensional utility function, as opposed to a time-varying real-valued utility function? At any given moment an agent's preferences might still satisfy the vN-M axioms, i.e. if you could somehow make a large number of copies of the agent's state of mind at a particular moment and immediately ask each copy about their preferences in gambling games with different outcomes, their collective answers could satisfy the axioms. Commented Apr 22, 2021 at 15:41

There's actually a bit more to be said here and vN-M actually anticipated some of this; Dubra, Maccheroni and Ok (2003):

Curiously, the basic idea has already been suggested, albeit elusively, by von Neumann and Morgenstern [1944, pp. 19–20]:

We have conceded that one may doubt whether a person can always decide which of two alternatives he prefers. If the general comparability assumption is not made, a mathematical theory is still possible. It leads to what may be described as a many-dimensional vector concept of utility. This is a more complicated and less satisfactory set-up, but we do not propose to treat it systematically at this time.

In evaluation of this statement, Aumann [1962, p. 449] notes that ‘‘Details were never published. What they probably had in mind was some kind of mapping from the space of lotteries to a canonical partially ordered euclidean space, but it is not clear to me how this approach can be worked out.’’ Our objective here is actually nothing other than formalizing Aumann’s interpretation of the von Neumann– Morgenstern suggestion. [...]

Given the well-known characterization of the stochastic dominance orderings in terms of linear functionals that possess an expected utility form, we would like to propose here a multi-utility representation for such a preorder. [...]

The main result of this paper states that any preference relation that satisfies the independence and continuity axioms admits an expected multi-utility representation, provided that the prize space X is compact.

(I've omitted any math formulae since they're almost impossible write here.)

Then Danan, Gajdos and Tallon (2015) prove a generalization of Harsanyi's (1955) [social] aggregation theorem for such incomplete preferences, relying on the multi-utility basically in the sense of afore-quoted paper. Basically their setup

allows preference incompleteness at boht the individual and social level.

Interestingly, Danan et al. they note that in their setup the resulting social preference can be complete even when all individual preferences are incomplete, but also vice-versa.

in the expected multi-utility setting, Pareto indifference (resp. preference) is necessary and sufficient for the set of social utility function to consist of a set of bi-utilitarian (resp. utilitarian) aggregations of individual utility functions.

Bi-utilitarianism cannot in general be reduced to signed utilitarianism [...]

And that it's an

open problem to find weaker conditions allowing society to make a selection within the individual sets of utility functions (thereby reducing social incompleteness) while retaining the separation between weights and utilities.

As Wikipedia doesn't seem to mention this at all, the (original) vNM result has restated been restated in the slightly more general context of mixture sets and mixture-preserving functions, which don't explicitly reference lotteries anymore; in particular the monograph of Fishburn (1982) deals with this MS-MP form extensively; for a precis see Mongin (2001). Basically, due to axiom 4 of VNM (reduction of compound lotteries), which is incorporated in the definition of mixture sets, the "non-degenerate" mixture sets [I'm punting here on the exact def of non-degenerate; see Mongin] can always be embedded into a convex subset of a vector space; so in a sense, the mixture sets are a "slight" generalization of convex sets.

Philosophically however, the point of mixture sets is precisely that that you can have a mathematical notion of what it means to mix utilities of "arbitrary kinds".

• SE is very capable of displaying equations. This is probably the simplest way, for when it is worth it in an answer meta.stackexchange.com/a/76944 Commented Apr 21, 2021 at 21:56
• @CriglCragl: you mean it's very capable of displaying images (when LaTeX is not enabled.) Commented Apr 21, 2021 at 21:58

You raise important questions. All 3 of your points are captured in known theories, but I only have good reference for 1. 2 and 3 are captured within the utility calculation itself. Each experience is not assigned the same utility due to diminishing returns, but this doesn't require weighting the utilities differently that are summed, just decreasing the utility assignment from that experience. Since the utility function is normally subject dependent, utility can be calculated differently based on each subject (e.g. animals vs humans) or consider the time period of their life, but you have an epistemic limitation here. Some would say it is better if your utility has a positive slope, and the utility calculation can get a boost from that.

For 1, the reason why truly multidimensional utilitarianism is rare is because even on value pluralism (e.g. objective list theories), one usually takes some linear (can be nonlinear but...) combinations of these things of value, so utility is still understood as reducible along 1 dimension as a combination of multiple things that contribute to utility. Because there are concerns about comparability mentioned above.

However, my professor Martin Peterson wrote a book, The Dimensions of Consequentialism, on a truly irreducible version he calls Multi-Dimensional Consequentialism that seems to be exactly what you are looking for (for #1). He has some papers on it and there is a special issue of a journal dedicated to it.

• That is quite delightful! Looks like there are people who indeed addressed my concerns to an extent. I personally think that nonlinear operations for considering aggregate utility is actually very important. For example, I doubt that our self evaluation of life is truly additive; perspectives can highlight different parts of our memories and weave them into new narratives with radically different value, and that operation is very nonlinear. Commented Apr 22, 2021 at 8:20
• I think there's merit in the type of critique that goes "in the end, you have to make a 1-dimensional evaluation", but perhaps sweeps under the rug how exactly we assign utilities to different events. My proposal was that this underlying mechanism can be elucidated with "latent higher dimensional vectors" and "perspective functions". Commented Apr 22, 2021 at 8:24
• @FinnLim I can definitely see that! Being a monist about value (e.g. only pleasure or desire-satisfaction has intrinsic value) certainly makes things easier for reducing to 1D, but probably misses out on accuracy like you suggest Commented Apr 22, 2021 at 16:26

To refute the specific point that utilities must be always additive, imagine an agent having the same experience over and over, or with trivial variations. A utility that consider their net value additively would not be a very smart utility function for that agent to have.

To refute the point that utilities are always uni-dimensional requires probably a more nuanced argument that makes some assumptions about the values that are driving a specific agent in question. But one can certainly see that an experience has utility to an agent only in the measure that it supports or helps the advance of that agent particular agenda. Hence the value or utility of an experience can vary wildly across agents with different agendas