For a community that endorses Bayesian epistemology we have had surprisingly few discussions about the most famous Bayesian contribution to epistemology: the Dutch Book arguments. In this post I present the arguments, but it is far from clear yet what the right way to interpret them is or even if they prove what they set out to. The Dutch Book arguments attempt to justify the Bayesian approach to science and belief; I will also suggest that any successful Dutch Book defense of Bayesianism cannot be disentangled from decision theory. But mostly this post is to introduce people to the argument and to get people thinking about a solution. The literature is scant enough that it is plausible people here could actually make genuine progress, especially since the problem is related to decision theory.1

Bayesianism fits together. Like a well-tailored jacket it feels comfortable and looks good. It's an appealing, functional aesthetic for those with cultivated epistemic taste. But sleekness is not a rigourous justification and so we should ask: why must the rational agent adopt the axioms of probability as conditions for her degrees of belief? Further, why should agents accept the principle conditionalization as a rule of inference? These are the questions the Dutch Book arguments try to answer.

The arguments begin with an assumption about the connection between degrees of belief and willingness to wager. An agent with degree of belief b in hypothesis h is assumed to be willing to buy wager up to and including $b in a unit wager on h and sell a unit wager on h down to and including $b. For example, if my degree of belief that I can drink ten eggnogs without passing out is .3 I am willing to bet $0.30 on the proposition that I can drink the nog without passing out when the stakes of the bet are $1. Call this the Will-to-wager Assumption. As we will see it is problematic.

The Synchronic Dutch Book Argument

Now consider what happens if my degree of belief that I can drink the eggnog is .3 and my degree of belief that I will pass out before I finish is .75. Given the Will-to-wager assumption my friend can construct a series of wagers that guarantee I will lose money.  My friend could offer me a wager on b where I pay $0.30 for $1.00 stakes if I finish the eggnog. He could simultaneously offer me a bet where I pay $0.75 for $1.00 stakes if pass out. Now if I down the eggnog I win $0.70 from the first bet but I lose $0.75 from the second bet, netting me -$0.05. If I pass out I lose the $0.30 from the first bet, but win $0.25 from the second bet, netting me -$0.05. In gambling terminology these lose-lose bets are called a Dutch book. What's cool about this is that violating the axioms of probability is a necessary and sufficient condition for degrees of belief to be susceptible to Dutch books, as in the above example. This is quite easy to see but the reader is welcome to pursue formal proofs: representing degrees of belief with only positive numbers, setting b(all outcomes)=1, and making b additive makes it impossible to construct a Dutch book. A violation of any axiom allows the sum of all b in the sample space to be greater than or less than 1, enabling a Dutch book.

The Diachronic Dutch Book Argument

What about conditionalization? Why must a rational agent believe h1 at b(h1|h2) once she learns h2? For this we update the Will-to-wager assumption to have it govern degrees of belief for hypothesis conditional on other hypotheses. An agent with degree of belief b in hypothesis h1|h2 is assumed to be willing to wager up to and including $b in a unit wager on h1 conditional on h2. This is a wager that is canceled if h2 turns out false but pays out if h2 turns out true. Say I believe with b=0.9 that I will finish ten drinks if we decide to drink cider instead of eggnog. Say I also believe with b=0.5 that we will drink cider and 0.5 that we drink eggnog. But say I *don't* update my beliefs according to the principle of conditionalization. Once I learn that we will drink cider my belief that I will finish the drinks is only b=0.7.  Given the Will-to-wager assumption I accept the following wagers.

(1) An unconditional wager on h2 (that we drink cider not eggnog) that pays $0.20 if h2 is true at b(h2)=0.5*$0.20= $0.10

(2) A unit wager on h1 (finishing ten drinks) conditional on h2 that pays $1.00 at b(h1|h2)=0.9*$1.00= $0.90

If h2 is false I lose $0.10 on wager (1). If h2 is true I win $0.10. But now I'm looking at all that cider and not feeling so good. I decide that my degree of belief that I will finish those ten ciders is only b=0.7. So my buys from me an unconditional wager (3) on h1 that pays $1.00 at b(h1)=0.7*$1.00=$0.7.

Then we start our drinking. If I finish the cider I gain $0.10 from wager (2) which puts me up $0.20, but then I lose $0.30 on wager (3) and I'm down $0.10 on the day. If I don't finish that cider I win $0.70 from wager (3) which puts me at $0.80 until I have to pay out on wager (2) and go down to -$0.10 on the day.

Note again that any update in degree of belief in any hypothesis h upon learning evidence e that doesn't equal b(h|e) is vulnerable to a Diachronic Dutch booking.

The Will-to-wager Assumption or Just What Does This Prove, Anyway?

We might want to take the above arguments literally and say they show not treating your degrees of belief like probabilities is liable to lead you into lose-lose wagers. But this would be a very dumb argument: there is no reason for anyone to actually make wagers in this manner. These are wagers which have zero expected gain and which presumably involve transaction costs. No rational person would make these wagers according to the Will-to-wager assumption. Second, the argument presented above uses money and as we are all familiar, money has diminishing return. You probably shouldn't bet $100 for a one in a million shot at $100,000,000 because a hundred million dollars is probably not a million times more useful than a hundred dollars. Third, the argument assumes a rational person must want to win bets. A person might enjoy the wager even if the odds aren't good or might prefer life without the money.

Nonetheless, the Will-to-wager Assumption doesn't feel arbitrary, it just isn't clear what it implies. There are a couple different strategies we might pursue to improve this argument. First, we can improve the Will-to-wager assumption and corresponding Dutch book theorems by making them about utility instead of money.

We start by defining a utility function, υ: XR where X is the set of outcomes and R is the set of real numbers. A rational agent is one that acts to maximize R according to their utility function. An agent with degree of belief b in hypothesis h is assumed to be willing to wager up to and including b(util) in a one unil wager on h. As a literal ascription of willingness to wager this interpretation still doesn't make sense. But we can think of the wagers here as general stand-ins for decisions made under uncertainty. The Will-to-Wager assumption fails to work when taken literally because in real life we can always decline wagers. But we can take every decision we make as a forced selection of a set of wagers from an imaginary bookie that doesn't charge a vig, pays out in utility whether you live or die. The Bookie sometimes offers a large, perhaps infinite selection of sets of wagers to pick from and sometimes offers only a handful. The agent can choose one and only one set at a time. Agents have little control over what wagers get offered to them but in many cases one set will clearly be better than the others. But the more an agent's treatment of her beliefs diverges from the laws of probability the more often she's going to get bilked by the imaginary bookie.  In other words, the key might be to transform the Dutch Book arguments into decision theory problems. These problems would hopefully demonstrate that non-Bayesian reasoning creates a class of decision problem which the agent always answers sub-optimally or inconsistently. 2

A possible downside to the above strategy is that it leaves rationality entangled with utility. There have been some attempts to rewrite the Dutch Book arguments to remove the aspects of utility and preference embedded in them. The main problem with these strategies is that they tend to either fail to remove all notions of preference3 or have to introduce some kind of apparatus that already resembles probability for no particular reason.4,5 Our conception of utility is in a Goldilocks spot- it has exactly what we need to make sense of probability while also being something we're familiar with, we don't have to invent it whole cloth. We might also ask a further question: why should beliefs come in degrees. The fact that our utility function (such as humans have one) seems to consist of real numbers and isn't binary (for example) might explain why. You don't need degrees of belief if all but one possible decision are always of value 0. In discussions here many of us have also been given to concluding that probability was epiphenomenal to optimum decision making. Obviously if we believe that we're going to want a Dutch book argument that includes utility. Moreover, any successful reduction of degrees of belief to some decision theoretic measure would benefit from a set of Dutch book arguments that left out degrees of belief altogether. 

As you can see, I think a successful Dutch book will probably keep probability intertwined with decision theory, but since this is our first encounter with the topic: have at it. Use this thread to generate some hypotheses, both for decision theoretic approaches and approaches that leave out utility.

1 This post can also be thought of as an introduction to basic material and a post accompanying "What is Bayesianism".

2 I have some more specific ideas for how to do this, but can't well present everything in this post and I'd like to see if others come up with the similar answers. Remember: discuss a problem exhaustively before coming to a conclusion. I hope people will try to work out their own versions, here in the comments or in new posts. It is also interesting to examine what kinds of utility functions can yield Dutch books- consider what happens for example when the utility function is strictly deontological where every decision consists of a 1 for one option and a 0 for all the others. I also worry that some of the novel decision theories suggested here might have some Dutch book issues. In cases like the Sleeping Beauty problem where the payoff structure is underdetermined things get weird. It looks like this is discussed in "When Betting Odds and Credences Come Apart" by Bradley and Leitgeb. I haven't read it yet though.

3 See Howson and Urbach, "Scientific Reasoning, the Bayesian Approach" as an example.

4 Helman, "Bayes and Beyond".

5 For a good summary of these problems see Maher, "Depragmatizing Dutch Book Arguments" where he refutes such attempts. Maher has his own justification for Bayesian Epistemology which isn't a Dutch Book argument (it uses Representation theory, which I don't really understand) and which isn't available online that I can find. This was published in his book "Betting on Theories" which I haven't read yet. This looks pretty important so I've reserved the book, if someone is looking for work to do, dig into this.

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I have a question about Dutch books. If I give you a finite table of probabilities, is there a polynomial time algorithm that will verify that it is or is not Dutch bookable? Or: help me make this question better-posed.

It reminds me of Boolean satisfiability, which is known to be NP complete, but maybe the similarity is superficial.

On the assumption that the known probability is the implied payoff (in reverse of betting, where the known payoff is assumed to be the implied probability of the bet) you can check for a dutch-book by summing the probabilities. Above one and it books the gambler (who will probably not buy it), below one and it books the bookmaker. This is because Dutch books have to be profitable over every possible outcome. There is a procedure called dutching which is very similar, except it doesn't guarantee a profit; it just forms a Dutch book over a restricted set of bets. This is no longer exhaustive, so there are outcomes in dutching where all of your bets fail and you lose money. I am not sure what changes if the payoff and the probability of paying off are not equivalent.
Yes. You can do the same for a diachronic Dutch book which takes the table of probabilities that describes an agents beliefs before the agent learns E and after the agent learns E. For all H in table two p(H) must = p(H|E) in table 1. If p(H) does not = p(H|E) the the agent these tables describe is Dutch bookable assuming she will wager at those probabilities.
It would seem that something is "Dutch Bookable" so long as the sum of probabilities doesn't add up to 1, which should not be a very difficult task at all. I'm hoping this helps you clarify the question, since I feel like this answer probably doesn't actually address your intent :)
Well, depends on if the probabilities overlap. So: P(A)=.5 P(A&B)=.1 P(&~B)=.2 is Dutch-Bookable It seems closer to the solvability of a system of linear equations. Depends on what kind of probabilities you get? Like if you have P(A), p(B), p(C), p(A&B)=P(B&C)=P(A&C)=0, it's trivial. But if you have P()=1/4 then you've got trouble.
Everyone's of course right. But it means I don't see a place for my train of thought to go.
Great question, I think I'll work on it. I agree with Eliezer - my intuition tells me that there exists some algorithm that does this in polynomial time in the size of the table... but I'm also curious how many bets the Dutch book has to have, in general (as a function of the number of branching points and the number of choices at each update).
If we let the probabilities go to 0 and 1, doesn't this problem just become SAT? EDIT: As it turns out, no, no it doesn't.
0Eliezer Yudkowsky13y
I expect an algorithm polynomial in the size of the table would exist.

It's NP-hard. Here's a reduction from the complement problem of 3SAT: let's say you have n clauses of the form (p and not-q and r), i.e., conjunctions of 3 positive or negated atoms. Offer bets on each clause that cost 1 and pay n+1. The whole book is Dutch iff the disjunction of all the clauses is a propositional tautology.

I've written some speculations about what this might mean. The tentative title is "Against the possibility of a formal account of rationality":

I really like the Less Wrong community's exposition of Bayesianism so I'd be delighted to have feedback!

3Eliezer Yudkowsky12y
Hm... if your probability assignments are conjunctions of that form, is it still true that finding a Dutch book is polynomial in the size of the probability table that would be required to store the entire joint probability distribution corresponding to every possible assignment of all atoms? I.e., NP-hard in the number of conjunctions, but polynomial in the size of the entire probability distribution?
Interesting. I'm actually not sure. The general result by Paris I cited is a little unclear. He proves CONSISTENCY (consistency of a set of personal probability statements) to be NP-complete. First he gets SAT \leq_P CONSISTENCY, but SAT is only O(2^n) in the number of atoms, not in the number of constraints. However, the corresponding positive result, that CONSISTENCY is in NP, is proven using an algorithm whose running time depends on the whole length of the input. It could be that if you have the whole table in front of you, checking consistency is just checking that all the rows and columns sum to 1. However, I don't think that looking at the complete joint distribution is the right formalization of the problem. For example, I have beliefs about 100 propositions, but it doesn't seem like I have 2^100 beliefs about the probabilities that they co-occur.
Yes, a complete probability table is coherent iff all entries sum to 1. But what do you mean by "the" complete probability table corresponding to a given set of constraints? There's often more than one such table.
Oh, thanks, you're completely right. To unify all the language and make things explicit: if you have n atoms, then there are 2^n possible states of the world (truth assignments to the atoms). Then, if you have a personal probability for each of the 2^n states ("complete joint distribution", "complete table"), you can check consistency by summing them and seeing that you get 1. This is O(n) in the size of the table. The question at stake seems to be something like this: does the agent legitimately have access to her (exponentially large) complete joint distribution? Or does she only have access to personal probabilities for a small number of statements (for example, a few conjunctions of atoms)? In the second case, there may be no complete joint distribution corresponding to her personal probabilities (if she's inconsistent), exactly one (if the joint distribution is completely specified, possibly implicitly via independence assumptions that uniquely determine it), or infinitely many.
Can I just take a second to boast about this.
Here's a paper that I think you'll find interesting.

I'm not very familiar with the literature, but just off the top my head:

It seems to me that the force of 'Dutch book' type arguments can easily be salvaged from criticisms such as these.

there is no reason for anyone to actually make wagers in this manner. These are wagers which have zero expected gain and which presumably involve transaction costs. No rational person would make these wagers according to the Will-to-wager assumption.

OK. Rather than ask the subject whether they would make this wager, ask them, "if you were forced to accept one side ... (read more)

I remembering that in another discussion where the diminshing return of money was given as an issue with a thought experiment, somebody suggested that you could eliminate the effect by just stipulating that (in the case of a wager) your winnings will be donated to a highly efficient charity that feeds starving children or something. My own thoughts: If the amount of money involved is so small that it would be worthless to any charity, just multiply everything by a sufficiently large constant. If you run out of starving children, start researching a cure for cancer, and once that's cured you can start in on another disease, etc. Once all problems are solved, we can assume that the standard of living can be improved indefinitely.

Hm. I'd been meaning to ask about this apparent circularity in the foundations for a bit, and now this tells me the answer is "we don't know yet".

(Specifically: VNM proves the analogue of the "will-to-wager" assumption, but of course it assumes our usual notion of probability. Meanwhile Dutch book argument proves that you need our usual notion of probability - assuming the notion of utility! I guess we can say these at least demonstrate the two are equivalent in some sense. :P )

Savage's representation theorem in Foundations of Statistics starts assuming neither. He just needs some axioms about preference over acts, some independence concepts and some pretty darn strong assumptions about the nature of events. So it's possible to do it without assuming a utility scale or a probability function.
I suppose this would be a good time to point anyone stumbling on this thread to my post that I later wrote on that theorem. :)

It seems like building a bet that only a Bayesian system wouldn't get Dutch-booked on might be possible.

Hmm. What exactly do you mean?
Nevermind. I had it in mind that un-dutch-bookability was a property of perfect Bayesianism alone - so that in the way that all other methods are approximations of Bayes, plus or minus some constant, or restricted in some way, these departures from Bayes would open up non-Bayes systems to some incorrect calculation of probabilities, and so it would be possible to set up a situation where their calculation departs from Bayes in a systematic way, and then set up a Dutch book diachronically. It seems like you might be able to do this for frequentism, but it turns out that there are other, non-Bayes systems that are immune to Dutch books as well.

[Late night paraphrasing deleted as more misunderstanding/derailing than helpful. Edit left for honesty purposes. Hopeful more useful comment later.]

If you weaken your will-to-wager assumption and effectively allow your agents to offer bid-ask spreads on bets (i'll buy bets on H for x, but sell them for y) then you get "Dutch book like" arguments that show that your beliefs conform to Dempster-Shafer belief functions, or Choquet capacities, depending on what other constraints you allow.

Or, if you allow that the world is non-classical – that the function that decides which propositions are true is not a classical logic valuation function – then you get similar results.

Other arguments for havin... (read more)

I don't share the intuition that our utility function "seems to consist of real numbers". It seems to consist of ordinal numbers, at best: this is better than that, which is better than the other. "At best" because it's not even clear that, for two outcomes neither of which I have ranked higher than the other, I'm generally able to say that I'm indifferent between them. Ambivalence is not necessarily indifference.

I think we should at least mention that there are other good arguments for why adopting the probability theory is a good idea. For example Cox's theorem.

This seems to be orthogonal to the current argument. The Dutch book argument says that your will-to-wager fair betting prices for dollar stakes had better conform to the axioms of probability. Cox's theorem says that your real-valued logic of plausible inference had better conform to the axioms of probability. So you need the extra step of saying that your betting behaviour should match up with your logic of plausible inference before the arguments support each other.

Rigorous formulation of dutch book arguments:

  • There is some set H of possible pre-existing states of the world, and the information contained within is hidden.
  • There is some set O of possible outcomes.
  • An action is a function from H to O.
  • A choice is a set of 2 or more actions, and an agent is a function from choices to actions within that choice. This is decision, stripped of any notion of probability or utility.

The dutch book arguments give you probability from utility. What we want to define is:

We choose a function $ from the reals to O (not the othe... (read more)

I wonder if this can stand in for the Maher?

Depragmatized Dutch Book Arguments

Yes, this is the article that covers several attempts to depragmatize the arguments. I highly recommend it. Unfortunately it doesn't explain his own approach in any detail. Edit: This contains a summary and a claimed refutation of Maher's theory, but it isn't complete enough to let me understand what Maher says.

How do you design a Dutch book to take advantage of someone whose estimations sum to less than one, instead of more?

Likewise, what does it look like to have negative b?

You buy the wagers from them instead of sell to them. So if you have -4(h) you will sell me a unit wager on h for negative $4 (in other words you will pay me to take the bet). Perhaps I need to rephrase the Will-to-wager assumption differently to make this possibility more explicit? (Edit: I have done so)
Thanks for the revision. The Will-to-wager assumption feels too strong for me. I would like, for instance, to be able to say "I will wager up to $0.30 on H, or up to $0.60 on ~H. Likewise, I will sell you a wager on H for $0.70 or more, and on ~H for $0.40 or more." Of course, that's effectively setting up my own Dutch book, but it feels very natural to me to associate uncertainty in an outcome with "gaps" that I don't want to commit to either hypothesis without more data. Then again, I'm a fan of DST, and that's sort of the point.
I would say that the reason for your intuition that uncertainty => gaps (which seems separate from risk-aversion-induced gaps) is that the person on the other end of the bet may have information you don't, and so them offering to bet you counts as Bayesian evidence that the side they're betting on is correct. However, e.g., a simple computer program can commit to not knowing anything about the world, and solve this problem.
Well, this is sound betting strategy. As I say, you shouldn't take bets with 0 expected return unless you just enjoy gambling; it's a waste of your time. The question we need to answer is whether or not this principle can be given a more abstract or idealized interpretation that says something important about why Bayesianism is rational- the argument certainly doesn't prove that non-Bayesians are going to get bilked all the time.
I think this misses the point, somewhat. There are important norms on rational action that don't apply only in the abstract case of the perfect bayesian reasoner. For example, some kinds of nonprobabilistic "bid/ask" betting strategies can be Dutch-booked and some can't. So even if we don't have point-valued will-to-wager values, there are still sensible and not sensible ways to decide what bets to take.
The question that needs answering isn't "What bets do I take?" but "What is the justification for Bayesian epistemology?".
What the Dutch book theorem gives you are restrictions on the kinds of will-to-wager numbers you can exhibit and still avoid sure loss. It's a big leap to claim that these numbers perfectly reflect what your degrees of belief ought to be. But that's not really what's at issue. The point I was making is that even among imperfect reasoners, there are better and worse ways to reason. We've sorted out the perfect case now. It's been done to death. Let's look at what kind of imperfect reasoning is best.
Yes. This was the subject of half the post. It actually is what was at issue in this year old post and ensuing discussion. There is no consensus justification for Bayesian epistemology. If you would rather talk about imperfect reasoning strategies than the philosophical foundations of ideal reasoning than you should go ahead and write a post about it. It isn't all that relevant as a reply to my comment.
I'd always thought "What bets do I take" was the justification for Bayesian epistemology. Every policy decision (every decision of any kind) is a statement of the form "I'm prepared to accept these costs to receive these outcomes given these events", this is a bet. If Bayesian epistemology lets you win bets then that's all the justification it could ever need.
The above discussion about "what bets do I take?" is about literal, monetary wager-making. The sense in which any decision can be described in a way that is equivalent to such a wager is precisely the question being discussed here.

But mostly this post is to introduce people to the argument and to get people thinking about a solution.

I'm afraid I don't understand what problem you are trying to solve here. How is what you want to accomplish different from what is done, for example, in Chapter 1 of Myerson?

Not having read that book I couldn't really tell you how or even if what I want to accomplish is different. I'm introducing people to the central arguments of Bayesian epistemology, the right way to interpret those arguments being a matter of controversy in the field. It seems unlikely the matter is conclusively settled in this book, but if it is the Myerson's point needs to be promoted and someone would do well to summarize it here. There are of course many books and articles that go into the matter deeper than I have here- if you are sufficiently familiar with the literature you may have been impressed with someone's treatment of it even though the field has not developed a consensus on the matter. Can you explain Myerson's? ETA: I just found in on Google. Give me a minute. Update: Myerson doesn't mention the Dutch book arguments in the pages I have access to. I've just skimmed the chapter and I don't see anything that obviously would provide a satisfactory interpretation of the Dutch book arguments. You'll have to make it more explicit or give me time to get the full book and read closely.
Myerson gives an argument justifying probability theory, Bayesian updating, and expected utility maximization based on some plausible axioms about rational decision making. As I understand it, Dutch book arguments are another way of justifying (some of) these results, but you are seeking ways of doing that justification without assuming that a rational decision maker has to function as a bookie - being willing to bet on either side of any question (receiving a small transaction fee). Decision theoretic arguments, which instead force the decision maker to choose one side or the other (while preserving transitivity of preferences), are an alternative to Dutch book arguments, are what Myerson provides, and are what I thought you were looking for. But apparently I was wrong. So I repeat: I don't understand what problem you are trying to solve here.
Again, I don't have the book! I realize there are many plausible ways of justifying these results, the vast majority of which I have never read and larges classes of which I may have never read. I was particularly interested in arguments in the Dutch book areaspace but I am of course interested in other ways of doing it. I'm trying to talk about the foundations of our epistemology, the most prominent of which appears to be these Dutch book arguments. I want to know if there is a good way to interpret them or revise them. If they are unsalvageable then I would like to know that. I am interested in alternative justifications and the degree to which they preserve the Dutch book argument's structure and the degree to which they don't. I haven't given a specification of the problem. I've picked a concept which has some problems and suggested we talk about it and work on it. So why don't you just explain how Myerson's argument works.
It is essentially the same as that of Anscombe and Aumann. Since that classic paper is available online, you can go straight to the source. But the basic idea is straightforward and has been covered by Ramsey, Savage, von Neumann, Luce and Raiffa, and many others. The central assumptions are that preferences are transitive, together with something variously called "The sure thing principle" (Savage) or the "Axiom of Independence" (von Neumann).
Thanks for the link. So that particular method (the one in the paper you link) has, to my mind, a rather troubling flaw: it bases subjective probability on so-called physical probability. I agree with what appears to be the dominant position here that all probabilities are subjective probabilities which makes the Anscombe and Aumann proof rather less interesting-- in fact it is question begging. (though it does work as a way of getting from more certain "objective" probabilities to less certain probabilities). They say that most of the other attempts have not relied on this, so I guess I'll have to look at some of those. I'm also not sure Anscombe and Aumann have in anyway motivated agents to treat degrees of belief as probability: they've just defined such a agent, not shown that such conditions are necessary and sufficient for that agent to be considered rational (I suppose an extended discussion of those central assumptions might do the the trick). But yes, these arguments are somewhat on topic.
0Wei Dai13y
Jack, you might be more interested in the paper linked to in this post.
This is not as clear as it could be in your original post. It might be helpful for others if you add an introduction that explicitly says what your aim is.