*Author’s Note: Please don’t get scared off by the first sentence. I promise it's not as bad as it sounds.*

There’s a theorem from the early days of group theory which says that any continuous, monotonic function which does not depend on the order of its inputs can be transformed to addition. A good example is multiplication of positive numbers: f(x, y, z) = x*y*z. It’s continuous, it’s monotonic (increasing any of x, y, or z increases f), and we can change around the order of inputs without changing the result. In this case, f is transformed to addition using a logarithm: log(f(x, y, z)) = log(x) + log(y) + log(z).

Now, at first glance, we might say this is a very specialized theorem. “Continuous” and “monotonic” are very strong conditions; they’re not both going to apply very often. But if we actually look through the proof, it becomes clear that these assumptions aren’t as important as they look. Weakening them does change the theorem, but the core idea remains. For instance, if we remove monotonicity, then our function can still be written in terms of *vector* addition.

Many theorems/proofs contain pieces which are really just there for modelling purposes. The central idea of the proof can apply in many different settings, but we need to pick one of those settings in order to formalize it. This creates some mathematical boilerplate. Typically, we pick a setting which keeps the theorem simple - but that may involve stronger boilerplate assumptions than are strictly necessary for the main idea.

In such cases, we can usually relax the boilerplate assumptions and end up with slightly weaker forms of the theorem, which nonetheless maintain the central concepts.

Unfortunately, the boilerplate occasionally distracts people who aren’t familiar with the full idea underlying the proof. For some reason, I see this problem most with theorems in economics, game theory and decision theory - the sort of theorems which say “either X, or somebody is giving away free money”. People will come along and say “but wait, the theorem assumes Y, which is completely unrealistic!” But really, Y is often just boilerplate, and the core ideas still apply even if Y is relaxed to something more realistic. In fact, in many cases, the confusion is over the *wording* of the boilerplate! Just because we use the word “bet”, doesn’t mean people need to be at a casino for the theorem to apply.

A few examples:

- “VNM utility theorem is unrealistic! It requires that we have preferences over every possible state of the universe.” Response: Completeness is really just there to keep the math clean. The core ideas of the proof still show that, if we don’t have a utility function over some neighborhood of world-states, then we can be exploited using only those world-states.
- “All these rationality theorems are unrealistic! They’re only relevant to worlds where evil agents are constantly running around looking to exploit us.” Response: We face trade-offs in the real world, and if we don’t choose between the options consistently, we’ll end up throwing away resources unnecessarily. Whether an evil person manipulates us into it, or we stumble into it, isn’t really relevant. The more situations where we locally violate VNM utility, the more situations where we’ll lose resources.
- “VNM utility theorem is unrealistic! It assumes we’re willing to accept either a trade or its opposite (or both) - rather than just ignoring offers.” Response: We face trade-offs in the real world where “ignore” is not an option, and if we don’t choose between the options consistently, we’ll end up throwing away resources unnecessarily.
- “Dutch Book theorems are unrealistic! They assume we’re willing to accept either a bet or it’s opposite, rather than ignoring both.” Response: same as previous. Alternatively, we can build bid-ask spreads into the model, and most of the structure remains.
- “Dutch Book Theorems are unrealistic! They assume we’re constantly making bets on everything possible.” Response: every time we make a decision under uncertainty, we make a bet. Do so inconsistently, and we throw away resources unnecessarily.

In closing, one important note: I definitely do not want to claim that *all* objections to the use of VNM utility theorem, Dutch Book theorems, etc make this kind of mistake.

I think your post needs a counterpoint: to deserve that kind of trust, a result needs to also have good empirical reputation. Not all theoretical results are like that. For example, Aumann agreement makes perfect sense in theory and is robust to small changes, but doesn't happen in reality. A big part of studying econ is figuring out which parts have empirical backing and how much.

Is Aumann robust to untrustworthiness ?

I don’t think this post passes the Intellectual Turing Test for people (like me) who object to the sorts of theorems you cite.

You say:

But in most such cases, whether the “weaker forms” of the theorems do, in fact, “maintain the central concepts”, is

exactly what is at issue.Let’s go through a couple of examples:

This one is not a central example, since I’ve not seen any VNM-proponent put it in quite these terms. A citation for this would be nice. In any case, the sort of thing you cite is not really

myprimary objection to VNM (insofar as I even have “objections” to the theorem itself rather than to the irresponsible way in which it’s often used), so we can let this pass.Yes, this is exactly the claim under dispute. This is the one you need to be defending, seriously and in detail.

Ditto.

Ditto again. I have asked for a demonstration of this claim many times, when I’ve seen Dutch Books brought up on Less Wrong and in related contexts. I’ve never gotten so much as a serious attempt at a response. I ask you the same: demonstrate, please, and with (real-world!) examples.

Once again, please provide some

real-worldexamples of when this applies.In summary: you seem to think, and claim, that people simply

aren’t awarethat there’s a weaker form of the theorem, which is still claimed to be true. I submit to you that if your interlocutor is intelligent and informed, then this is almost alwaysnotthe case. Rather, peopleareaware of the “weaker form”,but do not accept it as true!(After all, the “strong form” has a

proof, which we can, like, look up on the internet and so on. The “weak form” has… what? Usually, nothing but hand-waving… or that’s how it seems, anyway! In any case, making a serious, convincing case for the “weak form”,with real-world examples, that engages with doubters and addresses objections, etc., is where the meat of this sort of argument has to be.)VNM is used to show why you need to have utility functions if you don't want to get Dutch-booked. It's not something the OP invented, it's the whole point of VNM. One wonder what you thought VNM was about.

That we face trade-offs in the real world is a claim under dispute ?

Another way of phrasing it is that we can model "ignore" as a choice, and derive the VNM theorem just as usual.

Ditto.

OP said it: every time we make a decision under uncertainty. Every decision under uncertainty can be modeled as a bet, and Dutch book theorems are derived as usual.

This is a confused and inaccurate comment.

The von Neumann-Morgenstern utility theorem states that

ifan agent’s preferences conform to the given axioms,thenthere exists a “utility function” that will correspond to the agent’s preferences (and so that agent can be said to behave as if maximizing a “utility function”).We may then ask whether there is any

normativereason for our preferences to conform to the given axioms (or, in other words, whether the axioms are justified by anything).If the answer to this latter question turned out to be “no”, the VNM theorem would continue to hold. The theorem is entirely agnostic about whether any agent “should” hold the given axioms; it only tells us a certain mathematical fact about agents that

dohold said axioms.It so happens to be the case that for at least some[1] of the axioms, an agent that violates that axiom will agree to a Dutch book. Note, however, that the truth of this fact is independent of the truth of the VNM theorem.

Once again: if the VNM theorem were false, it could still be the case that an agent that violated one or more of the given axioms would agree to a Dutch book; and, conversely, if the latter were not the case, the VNM theorem would remain as true as ever.

[1] It would be rather audacious to claim that this is true for

eachof the four axioms. For instance, do please demonstrate how you would Dutch-book an agent that does not conform to the completeness axiom!Your questions give the impression that you’re being deliberately dense.

Obviously it’s true that we face trade-offs. What is not so obvious is

literally the entire rest of the section I quoted.As I explained above, the VNM theorem is orthogonal to Dutch book theorems, so this response is a

non sequitur.More generally, however… I have heard glib responses such as “Every decision under uncertainty can be modeled as a bet” many times. Yet if the applicability of Dutch book theorems is so ubiquitous, why do you (and others who say similar things) seem to find it so difficult to provide an actual, concrete, real-world example of any of the claims in the OP? Not a class of examples; not an analogy; not even a formal proof that examples exist; but an

actual example. In fact, it should not be onerous to provide—let’s say—threeexamples, yes? Please bespecific.How can an agent not conform the completeness axiom ? It literally just say "either the agent prefer A to B, or B to A, or don't prefer anything". Offer me an example of an agent that don't conform to the completeness axiom.

The entire rest of the section is a straightforward application of the theorem. The objection is that X don't happen in real life, and the counter-objection is that something like X do happen in real life, meaning the theorem do apply.

Yeah, sorry for being imprecise in my language. Can you just be charitable and see that my statement make sense if you replace "VNM" by "Dutch book" ? Your behavior does not really send the vibe of someone who want to approach this complicated issue honestly, and more send the vibe of someone looking for Internet debate points.

(Note: I ask that you not take this as an invitation to continue arguing the primary topic of this thread; however, one of the points you made is interesting enough on its own, and tangential enough from the main dispute, that I wanted to address it for the benefits of anyone reading this.)This turns out to be an interesting question.

One obvious counterexample is simply an agent whose preferences are not totally deterministic; suppose that when choosing between A and B (though not necessarily in other cases involving other choices), the agent flips a coin, preferring A if heads, B otherwise (and thenceforth behaves according to this coin flip). However, until they actually have to make the choice, they have no preference. How do you propose to construct a Dutch book for this agent? Remember, the agent will only determine their preference

afterbeing provided with your offered bets.A less trivial example is the case of bounded rationality. Suppose you want to know if I prefer A to B. However, either or both of A/B are outcomes that I have not considered yet. Suppose also (as is often the case in reality) that whenever I do encounter this choice, I will at once perceive that to fully evaluate it would be computationally (or otherwise cognitively) intractable given the limitations of time and other resources that I am willing to spend on making this decision. I will therefore rely on certain heuristics (which I have inherited from evolution, from my life experiences, or from god knows where else), I will consider certain previously known data, I will perhaps spend some small amount of time/effort on acquiring information to improve my understanding of A and B, and then form a preference.

My preference will thus depend on various contingent factors (what heuristics I can readily call to mind, what information is easily available for me to use in deciding, what has taken place in my life up to the point when I have to decide, etc.). Many, if not most, of these contingent factors, are not known to you; and even were they known to you, their effects on my preference are likely to be intractable to determine. You therefore are not able to model me as an agent whose preferences are complete. (We might, at most, be able to say something like “Omega, who can see the entire manifold of existence in all dimensions and time directions, can model me as an agent with complete preferences”, but certainly not that

you, nor any other realistic agent, can do so.)Finally, “Expected Utility Theory without the Completeness Axiom” (Dubra et. al., 2001) is a fascinating paper that explores some of the implications of completeness axiom violation in some detail. Key quote:

I encourage you to read the whole thing (it’s a mere 13 pages long).

P.S.Here’s the aforementioned “Aumann (1962)” (yes, that very same Robert J. Aumann)—a paper called “Utility Theory without the Completeness Axiom”. Aumann writes in plain language wherever possible, and the paper isveryreadable. It includes this line:The full elaboration for this (perhaps quite shocking) comment is too long to quote; I encourage anyone who’s at all interested in utility theory to read the paper.

I happened upon this old thread, and found the discussion intriguing. Thanks for posting these references! Unless I'm mistaken, it sounds like you've discussed this topic a lot on LW but have never made a big post detailing your whole perspective. Maybe that would be useful! At least I personally find discussions of applicability/generalizability of VNM and other rationality axioms quite interesting.

Indeed, I think I recently ran into

anotherold comment of yours in which you made a remark about how Dutch Books only hold for repeated games? I don't recall the details now.I have some comments on the preceding discussion. You said:

For me, it seems that transitivity and completeness are on an equally justified footing, based on the classic money-pump argument.

Just to keep things clear, here is how I think about the details. There are

outcomes. Then there aregambles,which we will define recursively. An outcome counts as a gamble for the sake of the base case of our recursion. For gambles A and B, pA+(1-p)B also counts as a gamble, where p is a real number in the range [0,1].Now we have a preference relation > on our gambles. I understand its negation to be ≤; saying ¬(A>B) is the same thing as A≤B. The indifference relation, A∼B is just the same thing as (A≤B)&(B≤A).

This is different than the development on wikipedia, where ~ is defined separately. But I think it makes more sense to define > and then define ~ from that. A>B can be understood as "definitely choose A when given the choice between A and B". ~ then represents indifference as well as uncertainty like the kind you describe when you discuss bounded rationality.

From this starting point, it's clear that

either A<B, or B<A, or A~B. This is just a way of saying "either A<B or B<A or neither". What's important about the completeness axiom is the assumption thatexactlyone of these hold; this tells us that we cannot have both A<B and B<A.But this is practically the same as circular preferences A<B<C<A, which transitivity outlaws. It's just a circle of length 2.

The classic money-pump against circularity is that if we have circular preferences, someone can charge us for making a round trip around the circle, swapping A for B for C for A again. They leave us in the same position we started, less some money. They can then do this again and again, "pumping" all the money out of us.

Personally I find the this argument extremely metaphysically weird, for several reasons.

possibility ofB, but also a new offer from God, namely, to move on to C. So God is actually fooling the agent, making an offer of B but really giving the agent something different than B. Bad decision-making should not count against the agent if the agent was mislead in such a manner!I have similar (but less severe) objections to Dutch-book arguments.

However,

I also find the argument extremely practically applicable, so much so that I can excuse the metaphysical weirdness. I have come to think of Dutch-book and money-pump arguments asillustrative of important types of (in)consistencyrather than literal arguments.OK, why do I find money-pumps practical?

Simply put, if I have a loop in my preferences, then I will waste a lot of time deliberating. The real money-pump isn't someone taking advantage of me, but rather, time itself passing.

What I find is that I get stuck deliberating until I can find a way to get rid of the loop. Or, if I "just choose randomly", I'm stuck with a yucky dissatisfied feeling (I have regret, because I see another option as better than the one I chose).

This is equally true of three-choice loops and two-choice loops. So, transitivity and completeness seem equally well-justified to me.

Stuart Armstrong argues that there is a weak money pump for the independence axiom. I made a very technical post (not all of which seems to render correctly on LessWrong :/) justifying as much as I could with money-pump/dutch-book arguments, and similarly got everything except continuity.

I regard continuity as not very theoretically important, but highly applicable in practice. IE, I think the pure theory of rationality should exclude continuity, but a realistic agent will usually have continuous values. The reason for this is again because of

deliberation time.If we drop continuity, we get a version of utility theory with infinite and infinitesimal values. This is perfectly fine, has the advantage of being more general, and is in some sense more elegant. To reference the OP, continuity is definitely just boilerplate; we get a nice generalization if we want to drop it.

However, a real agent will ignore its own infinitesimal preferences, because

it's not worth spending time thinking about that. Indeed, it will almost always just think about the largest infinity in its preferences. This is especially true if we assume that the agent places positive probability on a really broad class of things, which again seems true of capable agents in practice. (IE, if you have infinities in your values, and a broad probability distribution, you'll be Pascal-mugged -- you'll only think of the infinite payoffs, neglecting finite payoffs.)So all of the axioms except independence have what appear to me to be rather practical justifications, and independence has a weak money-pump justification (which may or may not translate to anything practical).

Correction: I now see that my formulation turns the question ofcompletenessinto a question oftransitivity of indifference. An "incomplete" preference relation should not be understood as one in which allows strict preferences to go in both directions (which is what I interpret them as, above) but rather, a preference relation in which the ≤ relation (and hence the ∼ relation) is not transitive.In this case, we can distinguish between ~ and "gaps", IE, incomparable A and B. ~ might be transitive, but this doesn't bridge across the gaps. So we might have a preference chain A>B>C and a chain X>Y>Z, but not have any way to compare between the two chains.

In my formulation, which lumps together indifference and gaps, we can't have this two-chain situation. If A~X, then we must have A>Y, since X>Y, by transitivity of ≥.

So what would be a completeness violation in the wikipedia formulation becomes a transitivity violation in mine.

But notice that I never argued for the transitivity of ~ or ≥ in my comment; I only argued for the transitivity of >.

I don't think a money-pump argument can be offered for transitivity here.

However, I took a look at the paper by Aumann which you cited, and I'm fairly happy with the generalization of VNM therein! Dropping uniqueness does not seem like a big cost. This seems like more of an example of John Wentworth's "boilerplate" point, rather than a counterexample.

Though there’s a great deal more I could say here, I think that when accusations of “looking for Internet debate points” start to fly, that’s the point at which it’s best to bow out of the conversation.

I really liked this post, because I have been on both sides of the coin here: that is to say, I have been the person who thought a theory was irrelevant because its assumptions were too extreme, and I have been the person trying to apply the core insights of the theory, and been criticized because the situation to which I was applying it did not meet various assumptions. I was confused each time, and I am pretty sure I have even been on both sides of the

same theoryat least once.It is practically inevitable that either side is the correct answer depending on the situation, and possible that I was closer to correct than the person I was disagreeing with. But then, I was confused each time; the simpler explanation by far is that I was confused about the theories under discussion.

When I am trying to learn about a particular theory or field, I now set as the first priority the historical context for its development. This is very reliable for communicating the underlying intuitions, and also can be counted on to describe the real-life situations that inspired them or to which they were applied.