Dutch Books, Money Pumps, and 'By Their Fruits' Reasoning

There is a species of reasoning deployed in some of the central arguments of formal epistemology and decision theory that we might call 'by their fruits' reasoning. It seeks to establish certain norms of rationality that govern our mental states by showing that, if your mental states fail to satisfy those norms, they lead you to make choices that have some undesirable feature. Thus, just as we might know false prophets by their behaviour, and corrupt trees by their evil fruit, so can we know that certain attitudes are irrational by looking not to them directly but to their consequences. For instance, the Dutch Book argument seeks to establish the norm of Probabilism for credences, which says that your credences should satisfy the axioms of the probability calculus. And it does this by showing that, if your credences do not satisfy those axioms, they will lead you to enter into a series of bets that, taken together, lose you money for sure (Ramsey 1931, de Finetti 1937). The Money Pump argument seeks to establish, among other norms, the norm of Transitivity for preferences, which says that if you prefer one option to another and that other to a third, you should prefer the first option to the third. And it does this by showing that, if your preferences are not transitive, they will lead you, again, to make a series of choices that loses you money for sure (Davidson, et al. 1955). Both of these arguments use 'by their fruits' reasoning. In this paper, I will argue that such arguments fail. I will focus particularly on the Dutch Book argument so that I can illustrate the points with examples. But the objections I raise apply equally to Money Pump arguments.

The Dutch Book argument: an example

Joachim is more confident that Sarah is an astrophysicist and a climate activist (proposition $A\ \&\ B$) than he is that she is an astrophysicist (proposition $A$). He is 60% confident in $A\ \&\ B$ and only 30% confident in $A$. But $A\ \&\ B$ entails $A$. So, intuitively, Joachim's credences are irrational.

How can we establish this? According to the Dutch Book argument, we look to the choices that Joachim's credences will lead him to make. The first premise of that argument posits a connection between credences and betting behaviour. Suppose $X$ is a proposition and $S$ is a number $S$, positive, negative, or zero. Then a £$S$ bet on $X$ is a bet that pays £$S$ if $X$ is true and £$0$ if $X$ is false. £$S$ is the stake of the bet. The first premise of the Dutch Book argument says that, if you have credence $p$ in $X$, you will buy a £$S$ bet on $X$ for anything less than £$pS$. That is, the more confident you are in a proposition, the greater a proportion of the stake you are prepared to pay to buy it. Thus, in particular:

• Bet 1: Joachim will buy a £$100$ bet on $A\ \&\ B$ for £$50$;
• Bet 2: Joachim will sell a £$100$ bet on $A$ for £$40$.

The total net gain of these bets, taken together, is guaranteed to be negative. Thus, his credences will lead him to a perform a pair of actions that, taken together, loses him money for sure. This is the second premise of the Dutch Book argument against Joachim. We say that this pair of actions (buy the first bet for £$40$; sell the second for £$50$) is dominated by the pair of actions in which he refuses to enter into each bets (refuse the first bet; refuse the second). The latter pair is guaranteed to result in greater total value than the former pair; the latter pair results in no loss and no gain, while the former results in a loss for sure. The third premise of the Dutch Book argument contends that, since it is undesirable to choose a pair of dominated options, it is irrational to have credences that lead you to do this. Ye shall know them by their fruits.

Thus, a Dutch Book argument has three premises. The first premise posits a connection between having a particular credence in a proposition and accepting certain bets on that proposition. The second is a mathematical theorem that shows that, if the first premise is true, and if your credences do not satisfy the probability axioms, they will lead you to make a series of choices that is dominated by some alternative series of choices you might have made instead; the third premise says that your credences are irrational if, together with the connection posited in the first premise, they lead you to choose a dominated series of options. My objection is this: there is no account of the connection between credences and betting behaviour that makes both the first and third premise plausible; those accounts strong enough to make the third premise plausible are too strong to make the first premise plausible. Our strategy will be to enumerate the possible putative accounts of that connection and show either that either the first or the third premise is false when we adopt that account.

Let $C(p, X)$ be the proposition that you have credence $p$ in proposition $X$;  and let $B(x, S, X)$ be the proposition that you pay £$x$ for a £$S$ bet on $X$. Then the first premise of the Dutch Book argument has the following form:

For all credences $p$, propositions $X$, prices $x$, and stakes $S$, if $x < pS$
$$O(C(p, X) \rightarrow B(x, S, X))$$where $O$ is a modal operator. But which modal operator? Different answers to this constitute different versions of the connection between credences and betting behaviour that appears in the first and third premise of the Dutch Book argument. We will consider six different candidate operators and argue that none makes the first and third premises both true. The six candidates are: metaphysical necessity; nomological necessity; nomological high probability; deontic necessity; deontic possibility; and the modality of defeasible reasons.

$O$ is metaphysical necessity

We begin with metaphysical modality. According to this account, the first premise of the Dutch Book argument says that it is metaphysically impossible to have a credence of $p$ in $X$ while refusing to pay £$x$ for a £$S$ bet on $X$ (for $x < pS$). If you were to refuse such a bet, that would simply mean that you do not have that credence. This sort of account would be appealing to a behaviourist, who seeks an operational definition of what it means to have a particular precise credence in a proposition---a definition in terms of betting behaviour might well satisfy them.

If this account were true, the third premise of the Dutch Book argument would be plausible. If having a set of mental states were to guarantee as a matter of metaphysical necessity that you'd make a dominated series of choices when faced with a particular series of decisions, that seems sufficient to show that those credences are irrational. The problem is that, as David Christensen (1996) shows, the account itself cannot be true. Christensen's central point is this: credences are often and perhaps typically connected to betting behaviour and decision-making more generally; but they are often and perhaps typically connected to other things as well, such as emotional states, conative states, and other doxastic states. If I have a high credence that my partner loves me, I'm likely to pay a high proportion of the stake for a bet on it; but I'm also likely to feel joy, plan to spend more time with him, hope that his love continues, and believe that we will still be together in five years' time. What's more, none of these connections is obviously more important than any other in determining that a mental state is a credence. And each might fail while the others hold. Indeed, as Christensen notes, in Dutch Book arguments, we are concerned precisely with those cases in which there is a breakdown of the rationally required connections between credences, namely, the connections described by the probability axioms. Having a credence in one proposition usually leads you to have at least as high a credence in another proposition it entails. But, as we saw in Joachim's case, this connection can break down. So, just as Joachim's case shows that it is metaphysically possible to have a particular credence  that has all the other connections that we typically associate with it except the connection to other credences, so it must be at least metaphysically possible to have a credence has all the other connections that we associate with it but not the connection to betting behaviour posited by the first premise. Such a mental state would still count as the credence in question because of all the other connections; but it wouldn't give rise to the apparently characteristic betting behaviour that is required to run the Dutch Book argument. Moreover, note that we need not assume that the credence has none of the usual connections to betting behaviour. Consider Joachim again. Every Dutch Book to which he is vulnerable involves him buying a bet on $A\ \&\ B$ and selling a bet on $A$. That is, it involves him buying a bet on $A\ \&\ B$ with a positive stake and buying a bet on $A$ with a negative stake. So he would evade the argument if his credence in $A\ \&\ B$ were to lead him to buy the bets with any stake that the first premise says they will, while his credence in $A$ were only to lead him to buy the bets with positive stake that the first premise says they will. In this case, we'd surely say he has the credences we assign to him. But he would not be vulnerable to a Dutch Book argument.

Thus, if $O$ is metaphysical necessity, the third premise might well be true; but the first premise is false.

$O$ is nomological necessity

Learning from the problems with the previous proposal, we might retreat to a weaker modality. For instance, we might suggest that $O$ is a nomological modality. There are two that it might be. We might say that the connection between credences and betting behaviour posited by the first premise is nomologically necessary---that is, it is entailed by the laws of nature. Or, we might say that it is nomologically highly probable---that is, the objective chance of the consequent given the antecedent is high. Let's take them in turn.

First, $O$ is nomological necessity. The problem with this is the same as the problem with the suggestion from the previous section that $O$ is metaphysical necessity. Above, we imagined a mental state that had all the other features we'd typically expect of a particular credence in a proposition, except some range of connections to betting behaviour that was crucial for the Dutch Book argument. We noted that this would still count as the credence in question. All that needs to be added here is that the example we considered is not only metaphysically possible, but also nomologically possible. That is, this is not akin to an example in which the fine structure constant is different from what it actually is---in that case, it would be metaphysically possible, but nomologically impossible. There is no law of nature that entails that your credence will lead to particular betting behaviour.

Thus, again, the first premise is false.

$O$ is nomological high probability

Nonetheless, while it is not guaranteed by the laws of nature that an individual with a particular credence in a proposition will engage in the betting behaviour posited by the first premise, it does seem plausible that they are very likely to do so---that is, the objective chance that they will display the behaviour given that they have the credence is high. In other words, while weakening from metaphysical to nomological necessity doesn't make the first premise plausible, weakening further to nomological high probability does. So let's suppose, then, that $O$ is nomological high probability. Unfortunately, this causes two problems for the third premise.

Here's the first. Suppose I have credences in 1,000 mutually exclusive and exhaustive propositions. And suppose each credence is $\frac{1}{1,001}$. So they violate Probabilism. Suppose further that each credence is 99% likely to give rise to the betting behaviour mentioned in the first premise of the Dutch Book argument; and suppose that whether one of the credences does or not is independent of whether any of the others does or not. Then the objective chance that the set of 1,000 credences will give rise to the betting behaviour that will lose me money for sure is $0.99^{1,000} = 0.00004 \approx \frac{1}{23,163}$. And this tells against the third premise. After all, what is so irrational about a set of credences that will lead to a dominated series of choices less than once in every 20,000 times I face the bets described in the Dutch Book argument against me?

Here's the second problem. On the account we are considering, having a particular credence in a proposition makes it highly likely that you'll bet in a particular way. Let's say, then, that you violate Probabilism, and your credences do indeed result in you making a dominated series of choices. The third premise infers from this that your credences are irrational. But why lay the blame at the credences' door? After all, there is another possible culprit, namely, the probabilistic connection between the credence and the betting behaviour. Consider an analogy. Suppose that, as the result of some bizarre causal pathway, when I fall in love, it is very likely that I will feed myself a diet of nothing but mud and leaves for a week. I hate the taste of the mud and the leaves make me very sick, and so I lower my utility considerably by responding in this way. But I do it anyway. In this case, we would not, I think, say that it is irrational to fall in love. Rather, we'd say that what is irrational is my response to falling in love. Similarly, suppose I make a dominated series of choices and thus reveal some irrationality in myself. Then, for all the Dutch Book argument says, it might be that the irrationality lies not in the credences, but rather in my response to having those credences.

Thus, on this account, the first premise is plausible, but the third premise is unmotivated, for it imputes irrationality to my credences when it might instead lie in my response to having those credences.

$O$ is deontic necessity

A natural response to the argument of the previous section is that the analogy between the credence-betting connection and the love-diet connection fails because the first is a rationally appropriate connection, while the latter is not. This leads us to suggest, along with Christensen (1996), that the connection between credences and betting behaviour at the heart of the Dutch Book argument is not governed by a descriptive modality, such as metaphysical or nomological modality, but rather by a prescriptive modality, such as deontic modality. In particular, it suggests that what the first premise says is not that someone with a particular credence in a proposition will or might or probably will accept certain bets on that proposition; but rather that they should or may or have good but defeasible reason to do so.

Let's begin with deontic necessity. Here, my objection is that, if this is the modality at play in the first and third premise, then the argument is self-defeating. To see why, consider Joachim again. Suppose the modality is deontic necessity, and suppose that the first premise is true. So Joachim is rationally required to make a dominated series of choices---buy the £$100$ bet on $A\ \&\ B$ for £$50$; sell the £$100$ bet on $A$ for £$40$. Now suppose further that the third premise is true as well---it does, after all, seem plausible on this account of the modality involved. Then we conclude that Joachim's credences are irrational. But surely it is not rationally required to choose in line with irrational credences. Surely what is rationally required of Joachim instead is that he should correct his irrational credences so that they are now rational, and he should then choose in line with his new rational credences. Now, whatever other features they have, his new rational credences must obey Probabilism. If not, they will be vulnerable to the Dutch Book argument and thus irrational. But the Converse Dutch Theorem shows that, if they obey Probabilism, they will not rationally require or even permit Joachim to make a dominated series of choices. And, in particular, they neither require nor permit him to accept both of the bets described in the original argument. But from this we can conclude that the first premise is false. Joachim's original credences do not rationally require him to accept both of the bets; instead, rationality requires him to fix up those credences and choose in line with the credences that result. But those new fixed up credences do not require what the first premise says they require. Indeed, they don't even permit what the first premise says they require. So, if the premises of the Dutch Book argument are true, Joachim's credences are irrational, and thus the first premise of the argument is false.

Thus, on this account, the Dutch Book argument is self-defeating: if it succeeds, its first premise is false.

$O$ is deontic possibility

A similar problem arises if we take the modality to be deontic possibility, rather than necessity. On this account, the first premise says not that Joachim is required to make each of the choices in the dominated series of choices, but rather that he is permitted to do so. The third premise must then judge a person irrational if they are permitted to accept each choice in a dominated series of choices. If we grant that, we can conclude that Joachim's credences are irrational. And again, we note that rationality therefore requires him to fix up those credences first and then to choose in line with the fixed up credences. But just as those fixed up credences don't require him to make each of the choices in the dominated series, so they don't permit him to make them either. So the Dutch Book argument, if successful, undermines its first premise again.

Again, the Dutch Book argument is self-defeating.

$O$ is the modality of defeasible reasons

The final possibility we will consider: Joachim's credences neither rationally require nor rationally permit him to make each of the choices in the dominated series; but perhaps we might say that each credence gives him a pro tanto or defeasible reason to accept the corresponding bet. That is, we might say that Joachim's credence of 60% in $A\ \&\ B$ gives him a pro tanto or defeasible reason to buy a £$100$ bet on $A\ \&\ B$ for £$50$, while his credence of 30% in $A$ gives him a pro tanto or defeasible reason to sell a £$100$ bet on $A$ for £$40$. As we saw above, those reasons must be defeasible, since they will be defeated by the fact that Joachim's credences, taken together, are irrational. Since they are irrational, he has stronger reason to fix up those credences and choose in line with the fixed up one than he has to choose in line with his original credences. But his original credences nonetheless still provide some reason in favour of accepting the bets.*

Rendered thus, I think the first premise is quite plausible. The problem is that the third premise is not. It must say that it is irrational to have any set of mental states where (i) each state in the set gives pro tanto reason to make a particular choice and (ii) taken together, that series of choices is dominated by another series of choices. But that is surely false. Suppose I believe this car in front of me is two years old and I also believe it's done 200,000 miles. The first belief gives me pro tanto or defeasible reason to pay £$5,000$ for it. The second gives me pro tanto reason to sell it for £$500$ as soon as I own it. Doing both of these things will lose me £$4,500$ for sure. But there is nothing irrational about my two beliefs. The problem arises only if I make decisions in line with the reasons given by just one of the beliefs, rather than taking into account my whole doxastic state. If I were to attend to my whole doxastic state, I'd never pay £$5,000$ for the car in the first place.  And the same might be said of Joachim. If he pays attention only to the reasons given by his credence in $A\ \&\ B$ when he considers the bet on that proposition, and pays attention only to the reasons given by his credence in $A$ when he considers the bet on that proposition, he will choose a dominated series of options. But if he looks to the whole credal state, and if the Dutch Book argument succeeds, he will see that its irrationality defeats those reasons and gives him stronger reason to fix up his credences and act in line with those. In sum, there is nothing irrational about a set of mental states each of which individually gives you pro tanto or defeasible reason to choose an option in a dominated series of options.

On this account, the first premise may be true, but the third is false.

Conclusion

In conclusion, there is no account of the modality involved in the first and third premises of the Dutch Book argument that can make both premises true. Metaphysical and nomological necessity are too strong to make the first premise true. Nomological high probability is not, but it does not make the third premise true. Deontic necessity and possibility render the argument self-defeating, for if the arguments succeeds, the first premise must be false. Finally, the modality of defeasible reasons, like nomological high probability, renders the first premise plausible. But it is not sufficient to secure to the third premise.

Before we conclude, let's consider briefly how these considerations affect money pump arguments. The first premise of a money pump argument does not posit a connection between credences and betting behaviour, but between preferences and betting behaviour. In particular: if I prefer one option to another, there will be some small amount of money I'll be prepared to pay to receive the first option rather than the second. As with the Dutch Book argument, the question arises what the modal force of this connection is. And indeed the same candidates are available. What's more, the same considerations tell against each of those candidates. Just as credences are typically connected not only to betting behaviour but also to emotional states, intentional states, and other doxastic states, so preferences are typically connected to emotional states, intentional states, and other preferences. If I prefer one option to another, then this might typically lead me to pay a little to receive the first rather than the second; but it will also typically lead me to hope that I will receive the first rather than the second, to fear that I'll receive the second, to intend to choose the first over the second when faced with such a choice, and to have a further preference for the first and a small loss of money over the second. And again the connections to behaviour are no more central to this preference than the connections to the emotional states of hope and fear, the intentions to choose, and the other preferences. So the modal force of the connection posited by the first premise cannot be metaphysical or nomological necessity. And for the same reasons as above, it cannot be nomological high probability, deontic necessity or possibility, or the modality of defeasible reasons. In each case, the same objections hold.

So these two central instances of 'by their fruits' reasoning fail. We cannot give an account of the connection between the mental states and their evil fruit that renders the argument successful.

[* Thanks to Jason Konek for pushing me to consider this account.]

References

• Christensen, D. (1996). Dutch-Book Arguments Depragmatized: Epistemic Consistency for Partial Believers. The Journal of Philosophy, 93(9), 450–479. 
• Davidson, D., McKinsey, J. C. C., & Suppes, P. (1955). Outlines of a Formal Theory of Value, I. Philosophy of Science, 22(2), 140–60.
• de Finetti, B. (1937). Foresight: Its Logical Laws, Its Subjective Sources. In H. E. Kyburg, & H. E. K. Smokler (Eds.) Studies in Subjective Probability. Huntingdon, N. Y.: Robert E. Kreiger Publishing Co.
• Ramsey, F. P. (1931). Truth and Probability. The Foundations of Mathematics and Other Logical Essays, (pp. 156–198).