Multidimensional utility

Question

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.

Answer 1

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.

See the Von Neumann-Morgenstern utility theorem.

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.

Answer 2

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.

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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".

Answer 3

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.

Answer 4

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