Cooking the Dutch Books

Jake Metzger

In today’s post, we’re going to talk a bit about about Dutch Books and do a bit of analysis of a post from a self-described “socratic pedagogue to the statistically minded” on LinkedIn, because of course that’s going to be an accurate moniker. More often than not, I find that these statisticians-with-opinions want to express their philosophical opinions rather than opine on statistics that they’re actually trained with, leading to saying silly things. This is one of those cases, but it’s hopefully instructive.[$]

Dutch Books and Bayesian Justification

When it comes to establishing the rationality of Bayesianism, there are typically two approaches: an axiomatic approach (following the footsteps of folks like Cox and his theorem) and a decision-theoretic approach. Dutch Books are in this second category of justifications. Dutch Books are thought experiments which construct betting scenarios which demonstrate that failing to make decisions according to the laws of probability can lead to sure loss. That is, in such scenarios, non-probabilistic (i.e. non-Bayesian) reasoners are guaranteed to lose cash, regardless of how they bet.

Axioms of Probability

First, let us recall the core axioms that govern probability [@].

Let X be a set of propositions (or events, depending on how you choose your foundations). Let A and B be arbitrary members of X. Let p be a probability assignment function on X.

  1. \(p(A) ≥ 0\) (Non-negativity)
  2. If \(A\) is a tautology, then \(p(A) = 1\) (Normalization)
  3. If \(A\) and \(B\) are independent, then \(p(A or B) = p(A) + p(B)\) (Finite additivity)

Note that in more mathematical contexts, finite additivity is often replaced with countable additivity. In such contexts, (merely) finitely-additive probabilities might be called quasi-probabilities. Such details are not important here; finite additivity is sufficient and is implied by countable additivity.

Dutch Book Theorems

There are two key theorems related to Dutch Books. The (Forward) Dutch Book Theorem states that for any probabilistically incoherent decision making procedure, there exists a Dutch Book which leads to a sure loss. The Converse Dutch Book Theorem states that if a reasoner is probabilistically coherent (i.e. Bayesian), there does not exist any such Dutch Book. These theorems together are intended to be justification for Bayesian decision procedures over alternatives. It should be noted that these theorems do not describe how bad these losses will be, only that these losses are guaranteed to be positive.

Rather than spending time on preliminaries, let’s jump directly into (informal) proofs of these Dutch Book theorems, because they’ll also serve as examples of how Dutch Books function. Let’s, however, introduce one definition:

As I’m going to use the term here, a fair bet is a bet where the cost of placing the bet exactly equals your expected value of the bet. That is, if you assign event A probability \(p(A) = q\), then a bet with potential payoff \(S\) is fair when it costs \(qS\) to place the bet. In such a case, you are indifferent about placing the bet.

Note here that we’re not necessarily talking about the betting behavior of an actual person but a kind of idealized betting behavior. Because Dutch Books, as discussed here, are intended as thought experiments about idealized practical concepts, this isn’t a problem. This is a problem, however, in more applied settings like behavioral economics or psychology, which are far beyond our scope. Continuing on… (For those who don’t like the typography below, I refer to Williamson 2010, p. 35-37)

Axiom 1: Non-negativity

Suppose that, with regard to a proposition (or event) \(A\), a reasoner assigns some probability \(p(A) = q < 0\). Knowing this, a bookie can ensure that this reasoner willingly lose money via fair bets.

Note that the agent’s loss about \(A\) is \(L_A = (q-I_A)S_A\), where \(I_A\) is the indicator function for the truth of \(A\) and \(S_A\) is the yet-to-be-defined stakes posted for the bookie. Then consider that the bookie defines \(S_A\) to be \(-1\) at \(A\) and \(0\) elsewhere (over the events in \(X\)). That is, essentially, \(S_A = -1\) for \(A\) and \(0\) for \(~A\).

Then, by substitution, the agent’s loss is \(L_A = I_A - q\), which it positive regardless of the value of \(I_A\). That is, regardless of the truth of \(A\), the reasoner is guaranteed to lose money.

Axiom 2: Normalization

Suppose that, with regard to a proposition (or event) \(A\), a reasoner assigns \(p(A) = q\) and yet assigns \(p(~A) = r\) such that \(q+r\) does not equal \(1\). That is, he assigns \(p(A or ~A)\) something other than \(1\). Knowing this, a bookie can ensure that reasoner willingly lose money via fair bets.

Let us define \(A’\) as the proposition \((A or ~A)\).

Without loss of generality, assume that \(p(A’) = q < 1\). Then consider that the bookie define \(S_A’\) similarly as before: \(-1\) at \(A\) and \(0\) elsewhere (that is, \(-1\) for \(A\) and \(0\) for \(~A\)). Then, by substitution, the agent’s loss is \(L_A’ = I_A’-q = 1-q\), which is positive (note that \(I_A’=1\) for tautology \(A’\)). [To get the case where \(q >1\), just let \(S_A’\) be positive rather than negative at \(A\).]

Axiom 3: Finite Additivity

Suppose that, with regard to a proposition (or event) A, a reasoner assigns \(p(A) = q\) and \(p(B) = r\) to mutually exclusive events but yet assigns \(p(A or B)\) to something other than \(q+r\). Knowing this, a bookie can ensure the reasoner loses money via fair bets.

Let us define \(A’’\) as the proposition \((A or B)\) for the mutually exclusive \(A and B\).

Without loss of generality, suppose that \(p(A or B) < q+r\). Then suppose the bookie defines the stakes \(S_A’’\) as \(-1\) at \(A’’\), \(1\) at \(A\), \(1\) at \(B\), and \(0\) elsewhere. (Strictly speaking, we should assign \(1\) to any proposition that implies \(A\) and \(1\) to any proposition that implies B, but this sketch is sufficient).

The agent’s loss in this scenario looks like the following:

\(L_A’’ = I_A’’ - p(A’’) + (p(A) - I_A) + (p(B) - I_B)\)

But note that \(I_A’’ = I_A + I_B\) (because \(A\) and \(B\) are mutually exclusive), so \(L_A’’\) simplifies to \(p(A) + p(B) - p(A’’)\), which is positive. That is, the agent is sure to lose cash if the reasoner assigns values that are not (at least) finitely additive. (To get the case where \(p(A or B) > q+r\), flip the signs in \(S_A’’\).)

With these three subcases done, we’ve (informally) established the (Forward) Dutch Book Theorem, as Ramsey and DeFinetti had done a century ago. But we’re not done. Let’s also take up the Converse Dutch Book Theorem!

Converse Dutch Book Theorem

Recall that the Converse Dutch Book Theorem essentially states that if a reasoner’s beliefs follow the three axioms of probability discussed above that they are not subject to the kinds of sure losses in the above Dutch Books.

Let \(W\) be the set of atomic propositions (or elementary events) in \(X\) and suppose that the reasoner assigns probabilities to \(X\) in accordance with the axioms of probability. Let \(A\) be one of these atomic propositions in \(W\). Then if \(A\) is true, the agent’s loss is the sum of terms of the form \((p(u)-1)S_u\) where \(u\) covers all propositions in \(X\) implying \(A\) as well as of the form \(p(v)S_v\) for \(v\) covers all propositions in \(X\) that do not imply \(A\).

Rearranging, we get that this loss \(L_A\) is equal to the sum of terms of the form \(p(x)S_x\) over all sentences in \(X\) minus the sum of the bookie’s stakes \(S_u\), where \(u\) covers all propositions in \(X\) implying \(A\).

If we then calculate the expected loss of the reasoner over the atomic sentences in \(W\), we get a sum of the form \(p(w)L_w\). Sparing the rote arithmetic here, this loss is identically zero. A probabilistically-consistent reasoner cannot be Dutch Booked for a sure loss.

Now that we’ve walked through the Dutch Book Theorems, let’s discuss what happens when using Dutch Books to justify Bayesianism.

Rejecting Dutch Books?

These two theorems are not in question when frequentists object to the concerns raised by Dutch Books, nor can they be without raising far more questions – if a frequentist objection to Dutch Book arguments rested on rejecting more fundamental ideas in decision theory, the worse for frequentism. More usual responses fall into common categories which can be summarized as follows (hardly exhaustively or exclusively):

“Guarding against sure loss just isn’t something worth striving for in the first place.”

“Sure loss isn’t great, but if I have to open up myself to sure loss scenarios in order to control my error, then so be it. That is, error control is more important than coherence, and if I have to choose between them, I choose error control.”

“Dutch Book sure loss scenarios are too rare to matter in practical statistics, and even when they exist, a nonzero loss is very often still a very small loss. So, not much is lost by rejecting probabilistic coherence.”

“Sure, we can be Dutch Booked, but there are other betting situations that are bad for Bayesians.”

The first type of rejection is justified, if it is justified, by philosophical commitments elsewhere. For example, one might invoke commitment to Popperian Falsificationism to support their rejection of Dutch Book considerations, at least in the sciences. This leads down a rabbit hole of considerations further afield from this discussion, but it should be obvious that such philosophical presuppositions must themselves have sufficient, articulable support to make sense as a defeater against Dutch Books. Typically, though, discussions in this vein end in gestural appeals to the dispositions of scientists rather than actual scientific practice or argument. Needless to say, these dismissals can usually just be themselves dismissed.

The second type of rejection recognizes that, ceteris paribus, it would be nice to guard against sure loss scenarios, but they value empirical calibration more, so they’re willing to bite such a bullet. The issue with these kinds of rejections is that it frames probabilism as being incompatible with calibration, Bayesianism as incompatible with empirical error control. This is a false dichotomy, and the objective Bayesian position represents a splitting of the horns of that dilemma. The objective Bayesian utilizes empirical frequency information to bound rational inference to context-specific error rates. (See previous posts). These sorts of frequentists should welcome an objective Bayesian approach.

The third sort of rejection demonstrates a misunderstanding of the point of Dutch Book scenarios. As mentioned before, the Dutch Book Theorems demonstrate that any non-probabilistic decision-making rules imply the existence of sure loss scenarios, but they do not imply that these losses are large nor that they outweigh other sources of loss (as we’ll see later). Indeed, they would be failures if the “failure” were measured by certain empirical expectations. But that’s not what Dutch Books do. Rather, Dutch Books demonstrate an incoherence in a critical practical concept, namely putting one’s ideal decision procedure to use. Dutch Books show that any non-Bayesian betting procedure, no matter what else it has going for it, admits of betting scenarios where it is guaranteed to lose money on a bet. Given that the concepts of probability were literally born out of the practice of wagering, winning, and losing, it seems incoherent to choose a decision-making procedure that can be gerrymandered to turn bettors into sure losers on a bet, especially since Bayesian procedures cannot be so gerrymandered. Indeed, if a bettor learned that they were in a Dutch Book for their next bet, they had better choose to convert or potentially lose their shirt!

The fourth rejection type could be seen as a version of the second, but the valence is a bit different in the sense that they typically assume that they have some larger incoherence problem with Bayesianism specifically rather than an unwarranted assumption that they must choose between coherence and error control. We’ll analyze a concrete example of this below. Note, however, that in order for positive counterarguments to be successful against Bayesianism in general, they must target probabilism. If it does not, then it, at best, can only target certain kinds of Bayesianism. A partial case for objective Bayesianism is, for example, built on Dutch Books against frequentists as well as exploitation of the uncalibrated bets of the subjectivist Bayesians, so targeting subjectivists won’t quite do it.

Cooking the Dutch Books

Now let’s turn all of this high-level discussion to a critical reading of a LinkedIn post that tries to create a rather convoluted Dutch Book scenario[!]. The goal the scenario is, apparently, to make a point that, “[R]egardless of paradigm, a savvy gambler can turn a profit by exploiting differences in the amount of information used by others.” Basically, this purported Dutch Book scenario aims to, in some sense, Dutch Book everyone.

Now, how does this fare with the Dutch Book Theorems? Recall that the Converse Dutch Book Theorem establishes that probabilistically-consistent bettors are not Dutch Book-able. So, what, exactly is going on here?

The setup of the scenario has a few layers. In the first layer, we are to consider the problem of estimating the bias of the 15th coin from a bag of unknown size where all that’s known is that, in general, the biases of the coins in the bag follow some unknown Beta distribution. The question on this layer is, “How much would you be willing to pay for a chance of a $100 return that the coin’s bias is greater than 0.80?” So, we’re being asked to identify our fair bet prices, using our inference method of choice. This bet is to be used in the second layer.

The second layer of this scenario involves situating the various answers to the first question in a larger betting situation. We can recognize that different statistical inference procedures can produce different fair bet estimates. In this second layer, you no longer operate as an individual bettor but as a kind of free-roaming bookie in a sea of bettors from the first layer of the scenario. You, and only you, are able to approach pairs of bettors and have them willingly take opposite bets. The catch is that you, as the bookie, are able to form something that looks quite Dutch Book-like in this situation: if you approach two bettors with opposite bets that form a probabilistically incoherent pair, then you, as the bookie, get to walk away with a sure gain!

Not a Dutch Book, but Still Bad

As mentioned above, Dutch Books are typically understood to be betting scenarios which lead to sure loss. In this scenario, the bookie has a sure gain, but who has the sure loss? Not to spoil too much, but no one has a sure loss in this scenario. The author recognizes this point by instead focusing on long run expected loss with statements like:

“Typically, seasoned gamblers are concerned with the long-run performance of their betting strategies as this will determine whether they will profit or at least break even.” [!]

This is a bit of a sleight of hand but not totally wild. Dutch Books, at least as I think they’re best understood, target the internal coherence of our practical concepts, and there is something quite fishy about being able to be assuredly bilked at once using an ideal decision procedure, no matter how large or small the bilking turns out to be. In this case, however, we’re not considering the internal coherence of anything but rather a performance desideratum: following an ideal decision procedure should not lead to expected long-run losses, especially when there exist decision procedures that do not.

This desideratum is imminently reasonable, especially in the sciences, and one that I generally agree with as an objective Bayesian. But what does the author say on how well frequentism holds up here? It turns out not to be very much. In the “Morals Gleaned” section, the the author states, “[I]t’s interesting to see […] how a relevant willingness to bet comes from an understanding of long-run performance.” This latter clause is true, but what is the moral gleaned? If we’re to take away that long-run performance is relevant for betting, then I have no disagreement. But if we’re to take away that long-run performance is the only consideration relevant for betting, then that’s reaching too far. It turns out that investigating deeper gives a more satisfying answer.

Who Pays the Bookie?

In the author’s scenario, the bookie walks away with a sure gain, but, as we’ve said, no one has a sure loss. Indeed, if we were to consider the body of bettors as, in some sense, a single betting entity, then that entity would be Dutch Booked and be experiencing a sure loss, but this isn’t the case. Nonetheless, the sure gain by the bookie must be funded from somewhere. Are the losses that fund the bookie’s gain equal between all parties or are there some bettors that are being exploited more than others?

The author, not asking this question himself, incorrectly describes the situation as a savvy bookie being able to win regardless of inferential paradigm.[!] The actual answer is that there are some bettors being exploited more than others in this scenario, namely the uncalibrated bettors. Bettors whose betting probabilities systematically differ from “true” data-generating probabilities are exploitable over the long run.

Let’s see what objective Bayesian Jon Williamson said about scenarios like this in 2013 [*]:

Betting according to physical probabilities is also justifiable worst-case expected loss. Suppose evidence says that \(\theta \in S\mathcal{L}, P^*(\theta)=x\) and that the agent bets according to \(P_E(\theta)=q\)

Such a bet is interpreted as a payment of \(qS\) for a return of \(S\) if theta turns out to be true, where stake \(S\) is chosen by a stake-maker and can be positive or negative. Expected loss is then \(x(q-1) + (1-x)qS = (q-x)S\).

lf \(q > x\) then a stake-maker can (in the worst case, will) choose \(S > 0\) to ensure that the expected loss is positive. Similarly if \(q<x\) and \(S\) is chosen to be negative. Only if \(q = x\) is this kind of exploitation of betting commitments not possible. More generally, if \(E\) determines some \(P^*_L\) then exploitation is only possible if \(P_E\) lies outside the convex hull \(<P^*_L>\).

That is, if people bet with uncalibrated probabilities, then they can be exploited over the long run with an expected gain proportional to the degree of miscalibration. Hence why objective Bayesians argue for calibration against subjectivists.

How does this exploitation show up in the author’s scenario? Well, we can consider the fact that all of the Bayesian solutions considered make no effort at calibrating with empirical frequency information (a la Williamson, Gelman, Bernardo, or contemporary objective Bayesians in general). You, as the bookie in the author’s scenario, will find it far more advantageous to choose an uncalibrated bettor to be at least one of the bettors in your chosen betting pairs:

Suppose for the sake of argument that you’re forced to always include a bettor that has correctly identified the “true” data-generating probability \(q\) (or \(1-q\), as necessary) but have the option of taking a calibrated bettor or an uncalibrated one for the second. It follows immediately that, over the long run, it is cheaper to take bets with an uncalibrated second since you get to choose which side of the bet they’re on, and your expected long run gain is directly proportional to just how uncalibrated they are.

Relaxing the above assumption, the situation for you as the bookie is only better if you get to choose two particularly uncalibrated bettors instead of just one, given that you do not have to take bets with pairs of people whose uncalibrated fair prices are contingently unfavorable.

To sum, the author’s bookie walks away with a sure gain that is disproportionately funded (over the long run) by the long run losses of uncalibrated reasoners.

Another question immediately arises: What if, say, only calibrated reasoners are allowed to bet? That is, let’s assume that the room is filled with frequentists who, rather than using different inference procedures just use different methods of obtaining their confidence intervals. Maybe some appeal to the CLT, some use a bootstrap method, some use a conformal method, etc. What happens then? And what happens if objective Bayesians are added to the mix?

In the first situation, we should ideally expect that, since there is no long run expected loss to fund the bookie’s gain, the bookie fails to get his cash. However, due to the positive selection effect of the bookie’s choice as well as inherent stochasticity in some of these methods, some cash may be able to be siphoned off by the bookie on occasion. But 1) we should expect that the proportion of exploitable pairs of bettors to essentially vanish and 2) we should expect that the bookie’s gain be small in the long run. And, as it turns out, this is the same when objective Bayesians are added to the mix – we generate CI’s using frequentist methods, after all (see [*]).

Appropriate Morals Gleaned

So what are we to take from all this?

The scenario’s author, in his section about “Morals gleaned”, draws some themes about hierarchical modeling that seem largely irrelevant given the above analysis. He makes his comment about the relevance of long-run performance for bets, which I already discussed above. But then he goes on to make general comments about hierarchical modeling, which are mostly relevant for the first “layer” to his scenario. However, as I think should be rather obvious through the analysis, the first layer is practically irrelevant for the operation of his exploitative scenario – all that is needed is for there to be a pool of bettors of different degrees of calibration, the particular problem they’re solving is inconsequential. Despite this irrelevance, he goes on to wax about how “lack of certainty” isn’t a random variable, which is just incorrect – he may not find degree-of-belief probability to be useful, but there is no doubt that it’s a coherent interpretation of probability and, thus, forms random variables. What is correct is that doing inference with uncalibrated methods can be a problem and that probabilistic consistency alone doesn’t buy you calibration. He’s correct in lambasting subjectivists about this problem but misses the mark entirely when waxing about Bayesianism in general, precisely because to target Bayesianism you need to target probabilism, which he doesn’t.

What we should walk away from the author’s scenario recognizing is that frequentism has no guard against sure loss as they can be Dutch Booked and subjectivist Bayesians have no guard against long run expected loss, as the author demonstrates himself. This agrees with objective Bayesianism. What I find puzzling, though, is a frequentist who finds long-run exploitation problematic because of such-and-so pragmatic arguments but who, seemingly ad hoc-ly, rejects sure loss exploitation which also relies on similar pragmatic argumentation. These considerations are complimentary, not contradictory, and objective Bayesians have the high ground as the apparently sole contender compatible with both.

References and Notes: