I burned some tokens iterating with Claude fable and ended up learning quite a bit here:

**The background: two ways of describing one problem**

A sequential decision problem can be written in *extensive form* — a tree with choice nodes (where you act) and chance nodes (where nature acts), with consequences at the leaves — or in *normal form*, where you list every complete contingency plan ("strategy") and treat the problem as a single one-shot choice among those plans. Elga's two-bet setup in extensive form is: choose accept/reject A, then choose accept/reject B. In normal form it's one choice among four strategies: {BOTH, A-only, B-only, NEITHER}.

The anti-sequentialist move you were making in this thread is: since nothing happens between Elga's nodes, the extensive form is a misleading picture and the normal form is the *real* problem. Call this **normal-form reduction**: rational choice in a tree must agree with rational choice among the tree's strategies. It feels like a free, innocent principle — surely how a problem is *typeset* can't matter.

Hammond's theorem is the demonstration that it is not free. It is roughly the most expensive principle in decision theory.

**Hammond's conditions**

In "Consequentialist Foundations for Expected Utility" (*Theory and Decision*, 1988), Peter Hammond starts not from preference axioms but from *behavior*: a "behavior norm" that specifies, for every finite decision tree, which moves are acceptable at each node. He then imposes three structural conditions:

*Consequentialism.* Acceptable behavior in a tree depends only on the consequences the available strategies deliver — not on the tree's shape. Two trees whose strategies map to the same consequences must license the same consequences. This *is* normal-form reduction, stated as an axiom.

*Dynamic consistency.* The plan you'd endorse at the start is the plan your later selves actually continue; there is no predictable defection from your own strategy.

*Separability.* Behavior at a node in the middle of a tree is the same as behavior in the "snipped-off" subtree treated as a standalone problem starting there. Your past — the branch you traveled to get here — is normatively inert.

Plus an *unrestricted domain* assumption: the norm must deliver verdicts for every finite tree, not just convenient ones.

**The theorem, and why completeness falls out**

Hammond proves that any behavior norm satisfying these conditions is representable as maximization of expected utility: a complete, transitive preference ordering satisfying the independence axiom and — once chance nodes with subjective uncertainty enter — a single, precise probability function. Sharp credences, derived, not assumed.

The philosophically interesting part is where *completeness* comes from, since that's what the imprecise probabilist denies. It comes almost embarrassingly cheaply, and seeing why is the heart of the counterattack. Behavior, unlike preference, is always decisive: put an agent in a tree offering x or y and she leaves the room having done *something*. Define "x is weakly preferred to y" as "x is choosable from some tree offering exactly {x, y}." Since something is always choosable, this revealed relation is complete *by construction*. That's trivial so far — a genuinely conflicted agent also picks something, perhaps arbitrarily, and we shouldn't read a considered ranking off one forced pick.

Here's the catch: incompleteness that can't show up in any single choice can only ever show up as a *pattern across contexts* — choosing x over y inside one tree but y over x inside another, or refusing at a later node to complete a sequence your earlier choice began. Those context-sensitive patterns are precisely what consequentialism and separability forbid. Consequentialism says the embedding tree can't matter; separability says your history can't matter; dynamic consistency says your plan can't come apart from your conduct. Jointly they seal every exit through which incomparability could behaviorally *express itself*. Whatever tie-breaking the agent does under conflict gets laminated, by the consistency conditions, into a single coherent ordering across all trees — and Hammond shows the standard money-pump/Dutch-book tree constructions then force transitivity, independence, and precise probabilities. An agent who satisfies all of Hammond's conditions is *behaviorally indistinguishable from a sharp expected-utility maximizer*, whatever fuzzy inner life she reports.

**Why this is a counterattack on the anti-sequential polemic specifically**

Recall your polemic's shape: Elga's tree has no decision-relevant events between nodes, so reduce it to the normal form, pick from {BOTH, A-only, B-only, NEITHER} by policy-level maximality, exclude only NEITHER, and declare that imprecision survives with exactly the permissiveness Elga demanded.

Hammond's theorem says: you just endorsed consequentialism (the reduction) and dynamic consistency (your policy-follow-through). If you *also* accept separability and unrestricted domain, the theorem grinds forward and hands you back completeness — and with it sharp probabilities. Your hammer, swung with full force, rebuilds Elga's conclusion from the opposite direction. The reduction move is therefore not a safe harbor; it's the first premise of a proof of SHARP.

So the anti-sequentialist *must* locate a Hammond condition to reject, and the only live candidate is **separability** — she must say that what's rationally choosable at the B-node genuinely depends on the tree it sits in (whether an A-node preceded it). And now look at what Elga's Sally argument actually is: it is a direct intuition pump *for separability*. Sally at the B-node after rejecting A, and Sally offered B alone, are stipulated identical in beliefs, options, and everything she cares about; Elga insists rationality must treat them alike. That is separability, stated in a vignette. The dialectic thus closes into a perfect circle: the polemic escapes Sally by reduction, Hammond shows reduction-plus-separability yields sharpness, so the escape requires denying separability, and Sally is the argument for separability. The whole debate was never about probabilities at all. It is a debate about one axiom of dynamic choice, and both sides' machinery is downstream of it.

This is exactly the lesson of Teddy Seidenfeld's companion paper from the same year ("Decision Theory Without 'Independence' or Without 'Ordering'," *Economics and Philosophy* 1988): once you take dynamic coherence seriously, you face a forced trilemma — give up completeness/ordering, give up independence, or give up one of the dynamic conditions (separability or consistency or reduction). You cannot keep everything, and no position gets to win by framing. McClennen's resolute choice keeps ordering and reduction but drops separability; Levi and Seidenfeld keep separability-ish structure but drop ordering and accept lumpy behavior; Hammond keeps all the dynamic conditions and is thereby *committed* to sharp EU.

**The honest bottom line**

Hammond doesn't refute the anti-sequentialist. His conditions are premises, and separability in particular is deniable — plenty of serious people deny it. What the theorem does is strip the polemic of its air of costlessness. "It's obviously one decision typeset on two lines" presents normal-form reduction as bookkeeping; Hammond shows it is one third of an engine that manufactures completeness, so anyone wielding it owes an account of which sibling condition they're rejecting and why the money-pump constructions that enforce it are toothless against them. That's a real debt, payable in the separability literature — Seidenfeld, McClennen, Rabinowicz, Steele — rather than in polemic. The promotion I mentioned before stands, though: "which of Hammond's axioms is false?" is a far better fight for the imprecise probabilist than "why did you burn $5?"

Yeah good point I should've asked Claude, too!

My sense is that (perhaps similar to others in the comment section?) I don't buy that this problem set up is a sequential problem at all. 

from the paper:

Any perfectly rational agent who is sequentially offered
bets A and B in the above circumstances (full disclosure in
advance about the whole setup, no change of belief in H
during the whole process, utilities linear in dollars) will ac-
cept at least one of the bets.

I kind of struggle to understand what "full disclosure in advance" really means and how it does not invalidate the sequential structure itself. Perhaps I'm missing something.

Any agent who knows the entire set up in advance could just interpret this as "if you take both bets you will get $5 regardless of your probability estimates of anything or lack thereof" so it would be irrational not to do so. Is this in the "Plan" or "Sequence" category?
I suspect "full disclosure" is less strict than what I'm interpreting it to be here?

Yet there’s been surprisingly little comparison between these cause areas.

I’m not sure this is a widely shared observation? I think there’s been a lot of work in this direction, but it’s just a very hard problem?

(Eg Rethink Priorities)

There’s a descriptive and normative component to your observation. 
I’d argue that many donors are surprisingly diversified given direct application of philosophical theory, yes. I’d say the ultimate decision makers are donors, not eg evaluators. 
Does said theory predict well what they do? No. Should it? I’m not sure, this is not a theory designed to describe what actors that call themselves altruistic do.
Does it prescribe what they should do? Possibly, but I’d argue diversification is both good and you need another mechanism than diminishing returns to justify it. 

(Tbf the animal space is probably where you are most likely to hit diminishing returns among EA cause areas right now. (If you’re a ~ billionaire))

“Maximising expected choiceworthiness involves diversification because the cost-effectiveness of each intervention decreases with spending. Going "all in" on the option with the highest marginal cost-effectiveness would only make sense for a very limited budget”

I’d argue almost all decision makers in the EA space operate in the linear regime where cost effectiveness does not meaningfully decrease. 

 

Yeah I agree - expected utility maximisation really starts to fall apart in this existential risk regime, even over trajectories rather than applied statically, and it only makes sense “locally” and at the margin. 

Personally I’m very happy to bite the bullet and not be rigorously utilitarian, but I’m also a global health focussed “old school EA” thinking about how much to diversify donations across charities ;)
 

I guess the obvious question is: Where do you draw the line regarding who to include in "the community", why, and how does it affect the results? 

An explicit swap is very clean and the counterfactual is known. Could they make a non-official, un-enforcable deal (I am very much not an expert on this at all and much less a lawyer!)?

Thinking about how ones situation impacts effective opportunities is also reasonable [1], but who is to say that they have not already assessed their specific viable options and are still convinced that their preferred choice is the best? People want to use their donations to "vote" and express an opinion, and that's probably valuable?

[1] I suspect this is relatively common among people donating to underfunded areas in Animal welfare.

While this sounds intuitively right, I think in the simplest utility maximising setting (iid additive errors with mean zero) your first claim does not seem true? The best looking noisy option is still most likely to be the best?

(I need to think more about the maths, but at least you need some kind of shrinkage to a prior that can change the ranking, which you’re unlikely to get, and if you’re maximising utility the solution is always fully concentrated?)

To be clear, it need not be deliberate and they need not benefit personally!