Consequentialist Foundations for Expected Utility

This is a linkpost for Consequentialist Foundations for Expected Utility by Peter J. Hammond, which was published in July 1988 in Theory and Decision. Below are the setup, premises, and conclusions of the theorems presented there as summarised by Claude Opus 5 in plain language. Peter said "It Is a very good summary of the most important parts". Relatedly, you may be interested in Peter's article Prerationality as avoiding predictably regrettable consequences, which Peter "regard[s] as an advance on that earlier paper [from 1988]", and was published in 2022 in Revue Éonomique.

Setup

Hammond models a decision as a decision tree: a branching diagram in which some branch points are choices the agent makes, some are chance events with known probabilities, and some are points where an uncertain state of the world gets revealed (with no probability attached — those probabilities are derived later, not assumed). The tips of the branches carry consequences.

What the theorems are about is a decision rule: a policy that says, at every choice point of every such tree, which of the available options are acceptable. Two features of this are built into the framework rather than argued for, and both do real work below. The rule may name more than one option as acceptable. And it always names at least one — it never returns "nothing here is acceptable".

Premises

1. Consequentialism. Only consequences matter. If two decision trees make the same set of consequences achievable, then acceptable behaviour must lead to the same consequences in both, however differently the two trees are shaped. So the rule boils down to a way of picking, from any menu of achievable consequences, which ones it is acceptable to end up with. The substantive content lives in what counts as a consequence — anything that ought to affect the decision (regret, sunk costs, even the decision process itself) has to be built into the consequence domain, not left outside it.

2. Consistency (dynamic consistency). What you do at a choice point doesn't depend on how you got there. Take the part of the tree that still lies ahead of you, treat it as a decision problem in its own right, and the rule must recommend the same things it recommended at that point in the original tree.

3. Unrestricted domain. The rule applies to every logically possible finite decision tree with consequences in the given domain. This is what lets the proofs construct whatever tree is needed to pin preferences down.

4. Continuity. As the probabilities of the chance events vary, the set of acceptable choices varies continuously — a small change in the odds can't make an acceptable option abruptly unacceptable. This excludes discontinuous preferences.

5. State-independence. If the state of the world has no bearing on what you actually receive, it should have no bearing on your choice either.

Conclusions

Premises 1–3 hold if and only if behaviour can be described as maximising a family of preference rankings (one per event) with the following four properties:

• Completeness. Any two prospects can be compared: for every pair, the ranking says one is at least as good as the other, or that they are equally good. There are no gaps.
• Transitivity. Preferences don't cycle: if λ is at least as good as µ, and µ is at least as good as ν, then λ is at least as good as ν. (Completeness and transitivity together are what make the rankings preference orderings.)
• Independence. Mixing two prospects with a common third prospect, in the same proportion, never reverses the ranking: λ is preferred to µ exactly when "α-chance-of-λ-else-ν" is preferred to "α-chance-of-µ-else-ν".
• Sure-thing principle for independent probabilities. A weakened version of Savage's sure-thing principle: an outcome shared across some states can be ignored when ranking prospects that agree there — but only when the prospects across the different events are probabilistically independent.

Premises 1–4 hold if and only if behaviour maximises expected utility. Adding continuity (premise 4) rules out discontinuous preferences and delivers a utility function over consequences, unique up to a positive affine transformation, whose expected value the agent maximises at every choice point. "Unique up to a positive affine transformation" means the utility scale's zero point and unit are arbitrary — much like temperature, where Celsius and Fahrenheit encode the same facts despite different numbers — so any two utility functions representing the same behaviour are related by u′ = a + bu with b > 0.

Premises 1–5 hold if and only if behaviour maximises Bayesian (subjective) expected utility — provided two further standard assumptions hold: the same menu of consequences is available in every state of the world (although different ones can be realised depending on the state), and that menu contains at least three options the agent ranks differently.