Dutch Book Argument Research Articles

Just As Planned: Bayesianism, Externalism, and Plan Coherence

Two of the most influential arguments for Bayesian updating ("Conditionalization") -- Hilary Greaves' and David Wallace's Accuracy Argument and David Lewis' Diachronic Dutch Book Argument-- turn out to impose a strong and surprising limitation on rational uncertainty: that one can never be rationally uncertain of what one's evidence is. Many philosophers ("externalists") reject that claim, and now seem to face a difficult choice: either to endorse the arguments and give up Externalism, or to reject the arguments and lose some of the best justifications of Bayesianism. The author argues that the key to resolving this conflict lies in recognizing that both arguments are plan-based, in that they argue for Conditionalization by first arguing that one should planto conditionalize. With this in view, we can identify the culprit common to both arguments: for an externalist, they misconceive the requirement to carry out a plan made at an earlier time. They should therefore not persuade us to reject Externalism. Furthermore, rethinking the nature of this requirement allows us to give two new arguments for Conditionalization that do not rule out rational uncertainty about one's evidence and that can thus serve as common ground in the debate between externalists and their opponents.

Epistemic confidence in the observed confidence interval

AbstractWe define confidence to be epistemic if it applies to an observed confidence interval. Epistemic confidence is unavailable—or even denied—in orthodox frequentist inference, as the confidence level is understood to apply to the procedure. Yet there are obvious practical and psychological needs to think about the uncertainty in the observed interval. We extend the Dutch Book argument used in the classical Bayesian justification of subjective probability to a stronger market‐based version, which prevents external agents from exploiting unused information in any relevant subset. We previously showed that confidence is an extended likelihood, and the likelihood principle states that the likelihood contains all the information in the data, hence leaving no relevant subset. Intuitively, this implies that confidence associated with the full likelihood is protected from the Dutch Book, and hence is epistemic. Our goal is to validate this intuitive notion through theoretical backing and practical illustrations.

Formal Epistemology Meets Mechanism Design

This article connects recent work in formal epistemology to work in economics and computer science. Analysing the Dutch Book Arguments, Epistemic Utility Theory and Objective Bayesian Epistemology we discover that formal epistemologists employ the same argument structure as economists and computer scientists. Since similar approaches often have similar problems and have shared solutions, opportunities for cross-fertilisation abound.

Reflection and Conditionalization: Comments on Michael Rescorla

AbstractRescorla explores the relation between Reflection, Conditionalization, and Dutch book arguments in the presence of a weakened concept of sure loss and weakened conditions of self‐transparency for doxastic agents. The literature about Reflection and about Dutch Book arguments, though overlapping, are distinct, and its history illuminates the import of Rescorla's investigation. With examples from a previous debate in the 70s and results about Reflection and Conditionalization in the 80s, I propose a way of seeing the epistemic enterprise in the light of practical requirements to be met by demands for synchronic coherence and probability updating policies. This includes a defense of principles rejected by Rescorla, while allowing for the value of his results in the borderland between theories of cognition and formal epistemology.

Revisiting McGee’s Probabilistic Analysis of Conditionals

This paper calls for a re-appraisal of McGee’s analysis of the semantics, logic and probabilities of indicative conditionals presented in his 1989 paper Conditional probabilities and compounds of conditionals. The probabilistic measures introduced by McGee are given a new axiomatisation—built on the principle that the antecedent of a conditional is probabilistically independent of the conditional—and a more transparent method of constructing such measures is provided. McGee’s Dutch book argument is restructured to more clearly reveal that it introduces a novel contribution to the epistemology of semantic indeterminacy, and shows that its more controversial implications are unavoidable if we want to maintain the Ramsey Test along with the standard laws of probability. Importantly, it is shown that the counterexamples that have been levelled at McGee’s analysis—generating a rather wide consensus that it yields ‘unintuitive’ or ‘wrong’ probabilities for compounds —fail to strike at their intended target; for to honour the intuitions of the counterexamples one must either give up the Ramsey Test or the standard laws of probability. It will be argued that we need to give up neither if we take the counterexamples as further evidence that the indicative conditional sometimes allows for a non-epistemic ‘causal’ interpretation alongside its usual epistemic interpretation.

Reflecting on diachronic Dutch books

AbstractConditionalization governs how to reallocate credence in light of new evidence. One prominent argument in favor of Conditionalization holds that an agent who violates it is vulnerable to a diachronic Dutch book: a series of acceptable bets offered at multiple times that inflict a sure loss. van Fraassen argues that an agent who violates the Principle of Reflection is likewise vulnerable to a diachronic Dutch book. He concludes that agents should conform to both Conditionalization and Reflection. Some authors reply that Reflection is implausible and hence that there must be something wrong with diachronic Dutch book arguments. Other authors try to isolate a principled difference between the Dutch book argument for Conditionalization and the Dutch book argument for Reflection, such that the former argument may succeed even though the latter fails. I pursue a version of this strategy. I contend that, once we properly elucidate the notion of sure loss, non‐reflectors are not vulnerable to a sure loss. An agent who violates Reflection is not thereby subject to a diachronic Dutch book. Appearances to the contrary result from an unmotivated focus upon an overly narrow set of gambling scenarios.

Rational Degree of Belief Ceases to Be Additive When the Dutch Book Argument Is Transported to a New Environment

The strength of a person’s beliefs can be measured by the buying and selling prices they offer on contingent promissory notes. Consider a promissory note contingent on a proposition; it pays off one unit of money if the proposition is true and nothing otherwise. The more strongly a person believes the proposition, the higher the minimum price would be at which they would sell it. The same would apply to the maximum purchase price. The well-known Dutch Book Argument claims that, if the person’s beliefs are rational, their buying/selling prices should combine additively, meaning that the price of a promissory note contingent on the disjunction of two incompatible propositions should be the sum of the prices of the promissory notes contingent on the individual incompatible propositions. This paper shows that the essence of the Dutch Book Argument is that rational belief is additive because money is additive. It is proved that, if the structure of the Dutch Book Argument is kept, but a nonadditive resource is substituted for money, then rational belief will follow a nonadditive combining rule. It is also shown how rational buying/selling prices behave when the pay-off amount of a contingent note changes.

Dutch Books and Logical Form

Dutch Book Arguments (DBAs) have been invoked to support various alleged requirements of rationality. Some alleged requirements, such as probabilistic coherence and conditionalization, are plausible. Others, such as credal transparency and reflection, may be less so. Anna Mahtani has argued for a new understanding of DBAs that, she claims, allows us to keep the DBAs for the plausible requirements while rejecting those for the implausible ones. I argue that Mahtani’s new account fails, as (a) it does not support highly plausible requirements of rational coherence and (b) it does not succeed in undermining the DBAs for credal transparency or reflection.

Socratic Irony and Argumentation

Socratic irony can be understood independently of the immortal heroics of Plato’s Socrates. We need a systematic account and criticism of it both as a debate-winning strategy of argumentation and teaching method. The Speaker introduces an issue pretending to be at a lower intellectual level than her co-debaters, or Participants. An Audience looks over and evaluates the results. How is it possible that the Speaker like Socrates is, consistently, in the winning position? The situation is ironic because the Participants fight from a losing position but realize it too late. Socratic irony compares with divine irony: divine irony is a subtype of Socratic irony since you lose when you challenge gods. Socratic irony is also, prima facie, a subtype of dramatic irony when the Audience knows more than the Participants on the stage. We must distinguish between the ideal and realistic elements of Socratic Irony. The very idea of Socratic irony looks idealized, or it is an ideal case, which explains the Speaker’s consistently winning position. In real life, the debate must be rigged, or the Dutch Book argument applies to the Participants, if the Speaker is so successful.

An Application of Quantum Logic to Experimental Behavioral Science

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Bayesian Updating When What You Learn Might Be False (Forthcoming in Erkenntnis)

Rescorla (Erkenntnis, 2020) has recently pointed out that the standard arguments for Bayesian Conditionalization assume that whenever I become certain of something, it is true. Most people would reject this assumption. In response, Rescorla offers an improved Dutch Book argument for Bayesian Conditionalization that does not make this assumption. My purpose in this paper is two-fold. First, I want to illuminate Rescorla’s new argument by giving a very general Dutch Book argument that applies to many cases of updating beyond those covered by Conditionalization, and then showing how Rescorla’s version follows as a special case of that. Second, I want to show how to generalise R. A. Briggs and Richard Pettigrew’s Accuracy Dominance argument to avoid the assumption that Rescorla has identified (Briggs and Pettigrew in Noûs, 2018). In both cases, these arguments proceed by first establishing a very general reflection principle.

Two Forms of Inconsistency in Quantum Foundations

Recently, there has been some discussion of how Dutch Book arguments might be used to demonstrate the rational incoherence of certain hidden variable models of quantum theory (Feintzeig and Fletcher 2017). In this paper, we argue that the 'form of inconsistency' underlying this alleged irrationality is deeply and comprehensively related to the more familiar 'inconsistency' phenomenon of contextuality. Our main result is that the hierarchy of contextuality due to Abramsky and Brandenburger (2011) corresponds to a hierarchy of additivity/convexity-violations which yields formal Dutch Books of different strengths. We then use this result to provide a partial assessment of whether these formal Dutch Books can be interpreted normatively.

Externalism and exploitability

AbstractAccording to Bayesian orthodoxy, an agent should update—or at least should plan to update—her credences by conditionalization. Some have defended this claim by means of a diachronic Dutch book argument. They say: an agent who does not plan to update her credences by conditionalization is vulnerable (by her own lights) to a diachronic Dutch book, i.e., a sequence of bets which, when accepted, guarantee loss of utility. Here, I show that this argument is in tension with evidence externalism, i.e., the view that an agent's evidence can entail non‐trivial propositions about the external world. I argue that this tension casts doubt on the idea that diachronic Dutch books can be used to justify or vindicate updating plans.

A Dutch book for CDT thirders

I give a Dutch book argument for CDT thirders in the context of a generalized Sleeping Beauty scenario. In combination with the Briggs (2010) Dutch book for EDT thirders, this amounts to a purely decision-theoretic argument for halfing in Sleeping Beauty. In combination with the Hitchcock (2004) Dutch book for CDT halfers, this amounts to a Dutch book argument against CDT. The combined Dutch book against CDT invites a plausible diagnosis of the reasons for CDT’s failure: CDT is incompatible with the Dutch book requirement because it allows the values of epistemically impossible outcomes influence one’s decision.

One Hundred Years After Keynes’s a Treatise on Probability Appeared, There Is Still Pervasive Confusion Among Academics About the Logical Theory of Probability: Keynes’s Theory of Logical Probability Is an Interval Valued Theory of Probability Based On Boole That the 18 Year Old Ramsey Never Understood in His Life

The very recent publication of C. Misak’s autobiography of F P Ramsey in 2020, as well as a number of book reviews made by different reviewers of that book, have resulted in the resurrection of highly misleading claims made by F. Ramsey, when he was 18 years old, that can only be characterized as being silly -to someone who has actually read the entire book. Therefore, individuals, who have only read Part I of Keynes’s book, are highly susceptible to claims made when reading book reviews of academics who never read the book they claimed to be reviewing. Such is the case with Frank Ramsey. The Keynes -Townshend correspondence in 1937 – 1938 over the connections between the A Treatise on Probability and the General Theory, as well as the Keynes-Tinbergen exchanges of 1938-1940 concerning Keynes’s critique or Tinbergen’s use of precise probability and the Normal distribution to model the business cycle, make it crystal clear that Keynes never accepted Ramsey’s critique in his lifetime and continued to use his own logical theory of imprecise probability .Ramsey’s subjective theory of probability is a theory of precise probability that can’t deal with overlapping evidence or conflicting evidence. Of course, Keynes did believe that Ramsey had strengthened the logical foundations for PRECISE probability with his betting quotient approach combined with his Dutch book argument regarding degrees of belief. However, Keynes’s logical theory is one that is based on rational degrees of belief and imprecise probability. Only in a very special case, where Keynes’s degree of the completeness of the relevant evidence, w, equaled 1 on the unit interval would Keynes have considered using Ramsey’s theory. A similar fate has occurred regarding Keynes’s separate theory of the Evidential Weight of the Argument. Here we will find a plethora of mathematical and logical errors littering the academic journals in economics and philosophy.

Avoiding Dutch Books despite inconsistent credences

It is often loosely said that Ramsey (in: Braithwaite (ed) The foundations of mathematics and other logical essays, Routledge and Kegan Paul, Abingdon, pp 156–198, 1931) and de Finetti (in: Kyburg, Smokler (eds) Studies in subjective probability, Kreiger Publishing, Huntington, 1937) proved that if your credences are inconsistent, then you will be willing to accept a Dutch Book, a wager portfolio that is sure to result in a loss. Of course, their theorems are true, but the claim about acceptance of Dutch Books assumes a particular method of calculating expected utilities given the inconsistent credences. I will argue that there are better ways of calculating expected utilities given a potentially inconsistent credence assignment, and that for a large class of credences—a class that includes many inconsistent examples—these ways are immune to Dutch Books and single-shot domination failures. The crucial move is to replace Finite Additivity with Monotonicity (if $$A\subseteq B$$ , then $$P(A)\le P(B)$$ ) and then calculate expected utilities for positive U via the formula $$\int _0^\infty P(U>y)\, dy$$ . This shows that Dutch Book arguments for probabilism, the thesis that one’s credences should be consistent, do not establish their conclusion. Finally, I will consider a modified argument based on multi-step domination failure that does better, but nonetheless is not as compelling as the Dutch Book arguments appeared to be.

Keynes Rejected the Concepts of Probabilistic Truth, True Expected Values, True Expectations, True Probability Distributions and True Probabilities: ‘Probability Begins and Ends With Probability’ (Keynes, 1921)

Keynes Rejected the Concepts of Probabilistic Truth, True Expected Values, True Expectations, True Probability Distributions and True Probabilities: ‘Probability Begins and Ends With Probability’ (Keynes, 1921)

The influence of financial practice in developing mathematical probability

The purpose of this paper is to discuss the role of financial practice in the development of mathematics as applied in human judgement. The basis of the paper is in historical research from the 1990s that argues that the monetisation of western commerce, which abstracted value into quantified price, was synthesised with scholastic analysis resulting in a “mathematical mechanistic world picture” that led to the widespread use of mathematics in science from the seventeenth century. An aspect of this process was the quantification of chance that led to the development of mathematical probability, the branch of mathematics most relevant to judgement and the focus of this paper. Ideas from this historical research are related to the fundamental theorem of asset pricing (FTAP), the foundational theory of contemporary financial mathematics. The paper observes that vestiges of medieval scholastic attitudes to financial ethics can be discerned in the FTAP, offering a novel interpretation of the mathematical theorem. The paper then considers the Dutch book argument (DBA), the most popular justification for subjective probability. The paper’s main contribution is in describing the significance of financial practice in validating the DBA and the paper explains how the FTAP addresses some criticisms of how the DBA represents beliefs. The conclusions emphasise the distinction between pure and practical reasoning and that this should be mirrored in a distinction between the mathematics of physica and practica. This point is important as mathematics is becoming more widespread in modelling, and directing, social systems.

Bayesian Beauty

The Sleeping Beauty problem has attracted considerable attention in the literature as a paradigmatic example of how self-locating uncertainty creates problems for the Bayesian principles of Conditionalization and Reflection. Furthermore, it is also thought to raise serious issues for diachronic Dutch Book arguments. I show that, contrary to what is commonly accepted, it is possible to represent the Sleeping Beauty problem within a standard Bayesian framework. Once the problem is correctly represented, the ‘thirder’ solution satisfies standard rationality principles, vindicating why it is not vulnerable to diachronic Dutch Book arguments. Moreover, the diachronic Dutch Books against the ‘halfer’ solutions fail to undermine the standard arguments for Conditionalization. The main upshot that emerges from my discussion is that the disagreement between different solutions does not challenge the applicability of Bayesian reasoning to centered settings, nor the commitment to Conditionalization, but is instead an instance of the familiar problem of choosing the priors.

Jeffrey conditionalization: proceed with caution

It has been argued that if the rigidity condition is satisfied, a rational agent operating with uncertain evidence should update her subjective probabilities by Jeffrey conditionalization (JC) or else a series of bets resulting in a sure loss could be made against her (the Dynamic Dutch Book Argument). We show, however, that even if the rigidity condition is satisfied, it is not always safe to update probability distributions by JC because there exist such sequences of non-misleading uncertain observations where it may be foreseen that an agent who updates her subjective probabilities by JC will end up nearly certain that a false hypothesis is true. We analyze the features of JC that lead to this problem, specify the conditions in which it arises and respond to potential objections.