--- title: "Justifying expected utility maximization from first principles (Von Neumann–Morgenstern)." date: 2025-11-02 tags: - machine learning - statistics - opinion - _recent-highlight has_math: false --- I've recently found myself arguing for expected utility maximization as an approach to practical decision making problems at work (mostly regarding molecule selection). This post is my attempt to write out an argument for using expected utility, targeted at non-mathematical (but still technical) readers.

Disclaimer: No technical content in this post is original (I bet LLMs can explain the same content just as well).

## Starter: the Von Neumann–Morgenstern utility theorem I think the simplest path to expected utility is the Von Neumann–Morgenstern (VNM) utility theorem. The VNM utility theorem essentially states that if you have _preferences_ over uncertain outcomes, and these preferences obey a set of axioms (aka rules), then there exists a utility function where maximizing expected utility will always choose your most preferred option. My argument will be to go through the axioms, and if you accept the axioms then expected utility is the natural conclusion. VNM operates specifically on preferences over "lotteries": essentially a distribution over outcomes with known probabilities. Think of it like a "mystery box" from a video game: you are given the odds of different boxes, but must choose a box _without_ knowing exactly which outcome you will get. Examples of a lunch lottery could be: - A: 50% chance of hamburger, 50% pizza - B: 25% dumplings, 25% bread, 50% sushi We'll come back to framing other things as lotteries at the end of the post. ## The VNM axioms The first axiom is **completeness**: for any lotteries A and B, you will either prefer A, prefer B, or be indifferent between the options. I.e.: you can always make a choice. This seems reasonable. The second axiom of **transitivity**: if you prefer A to B, and B to C, then you also prefer A to C. Hopefully this also seems reasonable. The third axiom is **continuity**: if you prefer A > B > C, there is some probability p (between 0 and 1) where the "nested" lottery "A with probability p, C with probability (1-p)" is equally good as lottery B. This essentially says that you are always willing to risk getting a worse option in exchange for a chance at the best option at _some_ level of risk. For example, if you like hamburger > pizza > sushi, for some level of risk p you would think "p chance of hamburger, (1-p) chance of sushi)" is as good as 100% chance of pizza. Hopefully this also seems reasonable. The final axiom is **independence**: if you prefer A > B, then adding an "irrelevant" nested lottery C should not change this preference, so pC + (1-p)A \> pC + (1-p)B. For example, if you prefer hamburgers to pizza, then you should prefer "95% chance getting hamburger, 5% salad" to "95% of getting pizza, 5% salad" because the "5% salad" is the same on both sides. Hopefully this seems reasonable. ## Possible objections to these axioms I am the type of person who reads these axioms and happily accepts them all as intuitively. For readers who don't have this reaction, or who have some objections in mind, let me try to go through common critiques. One possible critique comes from over-interpreting the goals or scope of these axioms. These axioms are: - **NOT** claims about how humans make decisions in the real world (humans can violate these, although it isn't a _terrible_ model either) - **NOT** claiming that you must have a fixed set of preferences which does not change with time (a common but oversimplified model used in economics) - **NOT** claiming that all _computer_ algorithms must follow these guidelines All these axioms **are** is a description of preferences which (hopefully) seems "reasonable". Here are responses to some individual sets of critiques (click to open).
Objections to completeness
Objections to transitivity This one is usually the least objectionable. The most common objection is probably something like "in some situations I prefer A, in some situations I prefer B", but this "objection" is just misinterpreting the axioms as assuming preferences don't vary with time or context (see bullet point in previous box).
Objections to continuity This axiom probably feels unintuitive because we aren't usually presented with choices like this in real life. However, risking worse outcomes for a chance at better outcomes is something people do all the time. Here are some scenarios/responses:
Objections to independence Historically this axiom has received a lot of criticism and debate in the academic literature, but as far as I can tell most of this criticism is directed at utility maximization as a description of human decision making. Famous counter-examples like the Allais paradox show that humans can violate independence in real-world gambling contexts. To resolve this, remember that these axioms are not meant to be a description of real human decision-making: instead they are descriptions of rational decision making. I simply don't see how a decision maker violating independence could be seen as an "intelligent" choice.


## Accepting the axioms leads to expected utility I will not write out a proof here ([Wikipedia has a pretty good one](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem#Proof_sketch)), but accepting these axioms directly implies the existence of a utility function such that decisions for any lottery are made by maximizing expected utility. Believing in these axioms _is_ believing in expected utility. Of course, the VNM theorem is not stating that decision-makers operate by _explicitly calculating_ expected utility (clearly they do not need to). It only says that an expected-utility agent can _replicate_ the behaviour of another agent by doing explicit expected utility calculations. ## Does this mean AI algorithms should calculate and maximize expected utility? Sort of: it says it is a viable option. If decisions are made through a complex process (eg human thought) but satisfy the VNM axioms, it means that direct calculation is a viable alternative. However, there are many situations where this is not possible: - The odds of different outcomes are unknown (or hard/impossible to specify numerically) - In a high-dimensional space it might be difficult to directly construct a utility function which represents a user's preferences (eg, think of learning art preferences directly in pixel space). - In repeated rounds of decision making there is an exploration-exploitation trade-off which expected utility does not address That being said, I am convinced that expected utility is a reasonable first approach for many problems. ## Epilogue: how does this apply to molecules and other scientific problems? In the context of such _scientific design_ problems (like molecule design in drug discovery), scientists design high-dimensional objects (ie molecules) with multiple (conflicting) objectives (eg potency, toxicity, solubility), each of which is difficult to predict (and also difficult to measure experimentally). I think each molecule can be thought of as a _lottery_, and its _properties_ are the potentially uncertain outcomes. The advantage of expected utility in this context is that it allows you to decouple _uncertainty in outcomes_ from _preferences across multiple objectives_, specify each one separately, then bring them together into a principled decision-making algorithm. This is possible _if_ the chemists' preferences satisfy the VNM axioms. Being humans, their past decisions might not always follow it, but if the chemists agree that these axioms are reasonable then it means constructing a utility function _is_ possible (at least in theory). Also, elucidating chemists preferences for hypothetical molecules with _known_ properties is much easier than for real molecules with unknown properties, because their decisions will be a mix of preferences and property predictions. I am personally hopeful that this approach (or an approach with this general flavour) can be useful for getting AI to "work" for drug discovery, at least to a degree that medicinal chemists are happy with! --- ## References For this post I mostly referred to Wikipedia's page on this, which is very good: .