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Introduction

Dutch Book arguments have been invoked to defend various probabilistic norms of rationality. The idea originates with Ramsey, who observed that an agent whose degrees of belief (or credences) fail to satisfy the basic axioms of probability would be vulnerable to a sure betting loss (i.e., a Dutch Book). Here probabilities are understood as functions that attach values to sentences (or propositions), with the basic axioms requiring: (1) For each A, 0 ≤ pr (A) [non-negativity] (2) if A is a tautology, pr (A) = 1 [normalization] (sometimes this is strengthened to require that all logical or necessary truths are assigned the maximum value), and (3) if A and B are mutually exclusive, pr (A or B) = pr (A) + pr (B) [finite additivity]. Intuitively, credences that are associated with sure losses are rationally defective. But there has been considerable discussion over how to fill in the details of the argument, which Ramsey sketched, and over its overall cogency in defending the probability axioms as a constraint on rational degrees of belief (probabilism). Beyond defending the probability axioms as a constraint on rational credences, Dutch Book arguments have also been given to defend a variety of other norms including conditionalization principles that purport to govern belief change over time. The first few sections of this article focus on works that consider the premises and structure of the basic Dutch Book argument, its interpretation, and its connection with decision theory. The remaining sections are devoted to extensions of the basic Dutch Book argument for the probability axioms to arguments for additional constraints on rational credences.

General Overviews

There are a number of survey articles on Bayesianism and Bayesian epistemology that include a brief discussion of Dutch Book arguments, including Easwaran 2013 and Talbott 2015, as well as some devoted exclusively to Dutch Book arguments, such as Hájek 2008 and Vineberg 2011 that provide more thorough coverage, including some of the nuances of their interpretation along with criticism.

Easwaran, Kenny. “Bayesianism.” In Oxford Bibliographies. 2013.
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Provides a brief but useful introduction to Bayesianism. Dutch Book arguments, accuracy arguments for probabilism, decision theory, conditionals, Bayesian philosophy of science and Bayesian epistemology are among the main topics.
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Hájek, Alan. “Dutch Book Arguments.” In Oxford Handbook of Rational and Social Choice. Edited by Paul Anand, Prasanta Pattanaik, and Clemens Puppe, 173–195. Oxford: Oxford University Press, 2008.
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A well-written introduction covering Dutch Book arguments for the probability axioms and various other norms. The treatment includes just the minimum technical detail needed to clearly present the relevant points about (and criticisms of) the arguments.
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Talbott, William. “Bayesian Epistemology.” In Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2015.
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Presents the fundamentals of Bayesian epistemology, including a short section on Dutch Book arguments.
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Vineberg, Susan. “Dutch Book Arguments.” In Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.
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Gives broad coverage of Dutch Book arguments for the probability axioms, conditionalization, and other norms. Provides the basic mechanics of these arguments and discusses their interpretation in considerable detail.
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Textbooks

There are a couple of older textbooks, Skyrms 1986 and Howson and Urbach 1993, with introductory material on probability that cover Dutch Book arguments.

Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. 2d ed. Chicago: Open Court, 1993.
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The second edition of this text combines a lengthy introduction to probability and Dutch Book arguments, as well as to Bayesian statistical reasoning, with a spirited philosophical defense of Bayesian methodology within the philosophy of science. Numerous changes have been made between the editions, with the discussion of Dutch Book arguments largely removed in the third.
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Skyrms, Brian. Choice and Chance: An Introduction to Inductive Logic. 3d ed. Belmont, CA: Wadsworth, 1986.
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Provides an elementary introduction to probability and decision, including a discussion of betting terminology and the Dutch Book argument. The third edition has the most thorough coverage, although the early and later editions cover much of the same material.
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The Basic Dutch Book Argument for Coherence

A set of credences that does not satisfy the axioms of probability is termed incoherent. The Dutch Book argument claims that having incoherent credences is irrational. It does so by linking credences with betting quotients, where the betting quotient for a bet is defined in terms of the amount paid if the bet is won and the amount forfeited if it is lost [if the amount won is W and the amount lost is L, the betting quotient is L / (W + L)]. A set of betting quotients is also said to be incoherent if it violates the probability axioms. A key assumption is that an agent’s credences determine his or her fair betting quotients, with incoherent credences yielding incoherent betting quotients. Also central to the argument is a mathematical result, the Dutch Book theorem, stating that for any set of betting quotients that do not satisfy the axioms of probability, there is a series of bets with those quotients that together produce a sure loss to one side, regardless of the outcome of the individual bets. The argument then is that incoherent credences are irrational because they lead, or can lead, to sure losses. There is also a Converse Dutch Book theorem, which shows that, subject to certain constraints, betting quotients that do satisfy the probability axioms are not subject to a Dutch Book. This result is then used along with the Dutch Book theorem to argue for probabilism, that is, that the probability axioms are a norm of rational credence. Whereas both the Dutch Book theorem and its converse are uncontroversial mathematical theorems, many questions have been raised about the connection between credences and betting quotients, what it means to be vulnerable to a sure loss, and the significance of being subject to such a loss.

Technical Results

Central to the Dutch Book argument for coherence are various technical results concerning betting quotients and the conditions under which a Dutch Book can be constructed. Ramsey 1926 first proposed investigating credences by identifying them with betting quotients and provided a detailed theory of credences and their measurement using this idea. Working independently, de Finetti 1937 rejected objective interpretations of probability, worked out in greater mathematical detail a subjective interpretation in terms of degrees of belief through the theory of betting quotients. Whereas these preliminary works gave a general theoretical underpinning to the subjective theory of probability in terms of betting quotients, a number of other papers have focused on specific features of betting systems of importance in Dutch Book arguments in order to provide a foundation for the theories of probability and confirmation. Kemeny 1955 and Lehman 1955 each independently establish the Converse Dutch Book theorem showing that, subject to certain assumptions on bets, a Dutch Book cannot be made against someone who satisfies the probability axioms. Shimony 1955 considers conditional probabilities used to characterize confirmation and uses betting quotients to argue for associated axioms of confirmation on conditional probabilities. Adams 1962 generalizes the results of Kemeny, Lehman, and Shimony by relaxing some of their assumptions on bets (including the claim that a set of bets is finite) to allow a Dutch Book to be constructed when the axiom of countable additivity is violated. Williamson 1999 discusses countable additivity and the Dutch Book argument in some detail. Whereas conditional probabilities are sometimes taken as defined, Popper gave an axiomatization of conditional probability directly, which has been of considerable interest. Stalnaker 1970 gives a Dutch Book argument for this axiomatization.

Adams, Ernest W. “On Rational Betting Systems.” Archiv für Mathematische Logik and Grundlagenforschung 6 (1962): 7–29.
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This paper generalizes the results of Kemeny, Lehman, and Shimony, first by placing assumptions on allowable bets that are more consistent with bookmaking practice, and second by allowing for infinite betting systems in order to justify countable additivity.
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de Finetti, Bruno. “Foresight: Its Logical Laws, Its Subjective Sources.” In Studies in Subjective Probability. Edited by Henry E. Kyburg and Howard E. K. Smokler, 53–118. Huntington, NY: Robert E. Kreiger, 1937.
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A fundamental paper in the development of the theory of subjective probability that, independently of Ramsey’s earlier work, stems from the consideration of betting behavior.
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de Finetti, Bruno. Probability, Induction and Statistics. New York: Wiley, 1972.
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An opinionated textbook on the theory of (subjective) probability.
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Kadane, Joseph, Mark Schervish, and Teddy Seidenfeld. Rethinking the Foundations of Statistics. Cambridge, UK: Cambridge University Press, 1999.
DOI: 10.1017/CBO9781139173230 Save Citation »Export Citation »E-mail Citation »
Contains a number of papers by the authors in which technical results concerning Dutch Books are presented. One of particular interest here is “A Conflict Between Finite Additivity and Avoiding Dutch Book” by Seidenfeld and Schervish, pp. 194–210.
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Kemeny, John. “Fair Bets and Inductive Probabilities.” Journal of Symbolic Logic 20.3 (1955): 263–273.
DOI: 10.2307/2268222 Save Citation »Export Citation »E-mail Citation »
Presents and proves a converse Dutch Book theorem for probabilism.
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Lehman, R. Sherman. “On Confirmation and Rational Betting.” Journal of Symbolic Logic 20.3 (1955): 251–262.
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Proves a number of technical results concerning probabilities including a Converse Dutch Book theorem for probabilism.
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Ramsey, F. P. “Truth and Probability.” In Studies in Subjective Probability. Edited by Henry E. Kyburg and Howard E. Smokler, 23–52. Huntington, NY: Robert E. Kreiger, 1926.
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The first paper to give a rigorous treatment of partial beliefs by identifying them with betting quotients. Provides a theory of subjective probability in terms of such beliefs and an investigation of their logic. It not only breaks new ground and offers a sophisticated technical development but also contains a particularly rich philosophical discussion of great breadth and depth. It is essential reading for anyone interested in the philosophical aspects of probability.
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Shimony, Abner. “Coherence and the Axioms of Confirmation.” Journal of Symbolic Logic 20.3 (1955): 1–28.
DOI: 10.2307/2268039 Save Citation »Export Citation »E-mail Citation »
Proves and discusses a number of results concerning coherence for conditional probabilities.
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Stalnaker, Robert. “Probability and Conditionals.” Philosophy of Science 37.1 (1970): 64–80.
DOI: 10.1086/288280 Save Citation »Export Citation »E-mail Citation »
Gives a Dutch Book argument for Popper’s axiomatization of conditional probability.
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Williamson, Jon. “Countable Additivity and Subjective Probability.” British Journal for the Philosophy of Science 50 (1999): 401–416.
DOI: 10.1093/bjps/50.3.401 Save Citation »Export Citation »E-mail Citation »
Gives a Dutch Book argument for countable additivity, with considerable discussion of de Finetti’s objections to the axiom.
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Interpretations and Criticisms of Dutch Book Arguments

Ramsey 1926 (cited under Incoherence as a Form of Inconsistency) mentioned that someone whose degrees of belief violate the axioms of probability is vulnerable to a Dutch Book; however, Ramsey did not spell its form out in detail, leaving questions about whether and how the argument may cogently be formulated. The argument presumes that degrees of belief are associated with betting quotients and that having incoherent betting quotients, which are associated with a Dutch Book, is irrational; however, the precise interpretation of these claims, and thus whether or not they are correct, has been subject to tremendous debate.

Incoherence as a Pragmatic Defect

A common understanding is that the irrationality of being Dutch Booked, or the mere vulnerability to such a loss, stems from the guaranteed loss or potential loss of value. Jackson and Pargetter 1976 views the argument in this way, providing a formulation in terms of forced betting situations aimed at resolving some issues raised in Baillie 1973 concerning the connections between credences and betting quotients and between having betting quotients that permit a Dutch Book and irrationality. Their “modified” Dutch Book argument is criticized in Kennedy and Chihara 1979. Davidson and Pargetter 1985 offers a defense of the Jackson and Pargetter version of the argument against these criticisms. Whether or not an agent is forced to bet, it has been observed in Ramsey 1926 (cited under Incoherence as a Form of Inconsistency), Kaplan 1993, Kyburg 1978, and others, that her credence need not be related straightforwardly to her betting quotients. To obtain the additivity axiom, the Dutch Book argument requires what is often called the package principle, namely that an agent who finds each of two bets acceptable will find them jointly acceptable. Kyburg 1978, Schick 1986, and Kaplan 1993 are among the many noting counterexamples to the principle. Hájek 2005 observes that in taking the irrationality to stem from sure losses, the good outcome of sure gains from incoherence is ignored. Finite additivity and normalization have the consequence that the probability of a proposition and the probability of its negation sum to 1. Hedden 2013 observes that this consequence is presupposed, rather than proven, by the Dutch Book argument, which may be seen as a limitation of the argument. This fact leaves open the possibility of satisfactory accounts of epistemic probability, such as that of Dempster-Shafer theory, in which this consequence fails.

Baillie, Patricia. “Confirmation and the Dutch Book Argument.” British Journal for the Philosophy of Science 24.4 (1973): 393–397.
DOI: 10.1093/bjps/24.4.393 Save Citation »Export Citation »E-mail Citation »
Discusses a number of objections to the Dutch Book argument as presented in Kemeny 1955, Lehman 1955, and Shimony 1955 (all cited under Basic Dutch Book Argument for Coherence: Technical Results).
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Davidson, Barbara, and Robert Pargetter. “In Defense of the Dutch Book Argument.” Canadian Journal of Philosophy 15.3 (1985): 405–424.
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Offers a defense of the Dutch Book argument presented in Jackson and Pargetter 1976 against various criticisms of Kennedy and Chihara 1979.
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Hájek, Alan. “Scotching Dutch Books?” Edited by John Hawthorne. Philosophical Perspectives 19 (2005): 139–151.
DOI: 10.1111/j.1520-8583.2005.00057.x Save Citation »Export Citation »E-mail Citation »
Observes that there is a “Good Book” argument for the claim that one ought to violate the probability axioms, which yields a new criticism of many versions of the Dutch Book argument, but to which Ramsey’s original argument is impervious.
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Hedden, Brian. “Incoherence without Exploitability.” Nous 47.3 (2013): 482–495.
DOI: 10.1111/j.1468-0068.2011.00842.x Save Citation »Export Citation »E-mail Citation »
Discusses the fact that the Dutch Book argument presupposes a consequence of finite additivity and normalization, namely negation coherence (i.e., the claim that the probability of a proposition and the probability of its negation sum to 1).
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Jackson, Frank, and Robert Pargetter. “A Modified Dutch Book Argument.” Philosophical Studies 29 (1976): 403–407.
DOI: 10.1007/BF00646318 Save Citation »Export Citation »E-mail Citation »
Offers a version of the Dutch Book argument intended to clarify and bolster the connection between having incoherent credences and being irrational. They identify a person’s credences with his or her betting quotients in a forced betting situation, and then appeal to the Dutch Book theorem to claim that such credences are irrational in such situations. They then argue that credences that are irrational in forced betting situations are irrational generally.
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Kaplan, Mark. “Not by the Book.” Philosophical Topics 21.1 (1993): 153–171.
DOI: 10.5840/philtopics19932116 Save Citation »Export Citation »E-mail Citation »
Rejects the assumption of the Dutch Book argument that credences involve a disposition to accept bets as corresponding odds, as well as the package principle. Offers an alternative argument for probabilism by making some relatively simple assumptions about preferences.
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Kennedy, Ralph, and Charles Chihara. “The Dutch Book Argument: Its Logical Flaws, Its Subjective Sources.” Philosophical Studies 36 (1979): 19–33.
DOI: 10.1007/BF00354378 Save Citation »Export Citation »E-mail Citation »
Gives Jackson and Pargetter’s modified Dutch Book argument a thorough critique, questioning their claims that degrees of belief match betting quotients in a forced betting situation, that incoherent credences are irrational in such scenarios, and their claim that this can be used to show that such credences are generally irrational.
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Kyburg, Henry. “Subjective Probability: Criticisms, Reflections, and Problems.” Journal of Philosophical Logic 7.1 (1978): 157–180.
DOI: 10.1007/BF00245926 Save Citation »Export Citation »E-mail Citation »
Offers numerous criticisms of the Dutch Book argument concerning its assumptions about the relationship between degrees of belief and betting quotients, as well as its assumption concerning an incoherent agent’s willingness to accept the bets required for a Dutch Book. It is noted that outside of the forced betting scenarios described by Jackson and Pargetter, there is little real threat of loss since one can generally avoid Dutch Books if confronted by a Dutch Bookie by refusing to bet.
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Schick, Frederick. “Dutch Books and Money Pumps.” Journal of Philosophy 83 (1986): 112–119.
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Argues against the package principle (i.e., the claim that an agent who finds each of two bets acceptable will find them jointly acceptable), which is crucial in the Dutch Book argument to establishing additivity. Also argues against the assumption of the money pump arguments for the axioms of rational preference that bets that are individually acceptable are sequentially acceptable.
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Incoherence as a Form of Inconsistency

Various authors argue that the Dutch Book theorem exposes a kind of inconsistency in degrees of belief that violate the axioms of probability, and that it is from this that the irrationality of such credences derives rather than the practical threat of monetary loss. This reading is suggested by a remark in Ramsey 1926 that Dutch Book vulnerability involves inconsistent preferences, and is emphasized in Skyrms 1987. There is a question here though about how such inconsistency is to be understood, and whether the argument works as so interpreted. Some suggestions for what such inconsistency amounts to have been given in Armendt 1993, as well as in Howson and Urbach 1993 and are analyzed in Vineberg 2001. Hájek 2008 notes that the move toward understanding the defect of incoherence in terms of inconsistency does not suffice to make the Dutch Book argument a convincing one for probabilism. Briggs 2009 (cited under Diachronic Dutch Book Arguments: Dutch Strategy Argument for Reflection) argues that vulnerability to a Dutch Book need not stem from inconsistent credences, with a different take on inconsistency than previously suggested. A contrasting account of Dutch Book arguments and inconsistency is given in Mahtani 2015.

Armendt, Brad. “Dutch Books, Additivity and Utility Theory.” Philosophical Topics 21.1 (1993): 1–20.
DOI: 10.5840/philtopics19932111 Save Citation »Export Citation »E-mail Citation »
Claims that incoherence is irrational in the sense that it involves what he terms “divided-mind” inconsistency, which involves attaching two different choice-guiding evaluations to the same thing.
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Hájek, Alan. “Arguments For—or Against—Probabilism?” British Journal for the Philosophy of Science 59 (2008): 793–819.
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Argues that violating the probability axioms involves having inconsistent attitudes toward a set of bets, and that there is no converse Dutch Book theorem to the effect that one will not have inconsistent attitudes if the axioms are satisfied.
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Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. 2d ed. Chicago: Open Court, 1993.
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Takes having a degree of belief in H to involve evaluating certain bets on H as fair, and then argues that the Dutch Book theorem establishes that violations of the probability axioms involve regarding a set of bets as fair that in fact cannot be, which involves a form of inconsistency.
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Mahtani, Anna. “Dutch Books, Coherence, and Logical Consistency.” Nous 49 (2015): 522–537.
DOI: 10.1111/nous.12070 Save Citation »Export Citation »E-mail Citation »
Attempts to draw the analogy between inconsistent full and partial beliefs in a new way, by focusing on the idea that consistent sentences are those that are true under every interpretation. Takes incoherence to involve the acceptance as fair of a set of bets that would result in a loss under any interpretation.
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Ramsey, F. P. “Truth and Probability.” In Studies in Subjective Probability. Edited by Henry E. Kyburg and Howard E. Smokler, 25–52. Huntington, NY: Robert E. Kreiger, 1926.
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Undertakes to investigate the logic of partial belief by associating degrees of belief with dispositions to bet. The possibility of making a Dutch Book against a person whose degrees of belief violate the axioms of probability is noted in the course of developing a theory of partial beliefs and their connection with rational action, with such Dutch Book vulnerability described as a form of inconsistency.
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Skyrms, Brian. “Coherence.” In Scientific Inquiry in Philosophical Perspective. Edited by N. Rescher, 225–242. Pittsburgh, PA: University of Pittsburgh Press, 1987.
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Notes that Ramsey saw Dutch Book vulnerability as a form of inconsistency and examines the source of such vulnerability in the theory of rational preference.
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Vineberg, Susan. “The Notion of Consistency for Partial Belief.” Philosophical Studies 102 (2001): 281–296.
DOI: 10.1023/A:1010309526393 Save Citation »Export Citation »E-mail Citation »
Investigates the claim that Dutch Book vulnerability involves a form of inconsistency and discusses why a variety of proposals for understanding what such inconsistency amounts to fails to yield a successful analysis of the Dutch Book argument.
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Dutch Book Arguments and Decision Theory

Dutch Book arguments are closely tied to decision theory. Ramsey 1926 (cited under Basic Dutch Book Argument for Coherence: Technical Results) identified degrees of belief with betting preferences, and took the inconsistency in being incoherent to be associated with violating the axioms of rational preference, upon which the theory of decision rests. Ramsey’s paper outlines the idea that an agent who satisfies the axioms of rational preference can be represented as having degrees of belief that satisfy the axioms of probability. Moreover, starting with the agent’s preferences, one can assign utilities to outcomes, which are additive. Where the payoffs for bets are given in utilities, the problem that the value of monetary bets need not be additive, which is needed for the Dutch Book argument, vanishes. Skyrms 1987 reads these results as showing that the Dutch Book argument itself has its roots in decision theory and that the objections to the argument can be responded to by appealing to its decision-theoretic underpinnings to remove the simplifications made by the argument in appealing to monetary bets. On the other hand, Kaplan 1996 and Maher 1993 instead think that the Dutch Book argument should be jettisoned in favor of one that comes directly from decision theory. Armendt 1993 observes that there are assumptions of both the Dutch Book argument and the decision-theoretic argument for probabilism that call for support and thinks the best route to probabilism involves the two arguments working together in tandem. However, for some purposes, such as investigating rational betting behavior, it is useful to begin with decision theory. This approach is put to work in Adams and Rosenkrantz 1980 in examining the Dutch Book argument’s assumptions about rational betting quotients.

Adams, Ernest W., and Roger D. Rosenkrantz. “Applying the Jeffrey Decision Model to Rational Betting and Information Acquisition.” Theory and Decision 12 (1980): 1–20.
DOI: 10.1007/BF00154655 Save Citation »Export Citation »E-mail Citation »
Uses Jeffrey’s decision theory to investigate the problem of the bookmaker in deciding upon the odds at which to offer bets that he will accept and the conditions under which it may be rational to post incoherent betting quotients.
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Armendt, Brad. “Dutch Books, Additivity and Utility Theory.” Philosophical Topics 21.1 (1993): 1–20.
DOI: 10.5840/philtopics19932111 Save Citation »Export Citation »E-mail Citation »
Observes that the Dutch Book justification of the probability axioms is closely allied with the assumptions of utility and preference. Argues that neither the Dutch Book argument nor a decision-theoretic one can provide an unassailable foundation for taking credences as probabilities but that both arguments play a role in motivating the Bayesian program.
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Kaplan, Mark. Decision Theory as Philosophy. Cambridge, UK: Cambridge University Press, 1996.
DOI: 10.1017/CBO9780511804847 Save Citation »Export Citation »E-mail Citation »
Defends a version of probabilism from assumptions about rational preference but argues against defending the view with the Dutch Book argument. Discusses the general role of credence and belief, along with the place of decision theory, in epistemology.
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Maher, Patrick. Betting on Theories. Cambridge, UK: Cambridge University Press, 1993.
DOI: 10.1017/CBO9780511527326 Save Citation »Export Citation »E-mail Citation »
Develops a version of Bayesianism in the philosophy of science stemming from the basic principles of decision theory. Includes arguments that the Dutch Book argument for probabilism, the principle of conditionalization, and the principle of reflection are fallacious.
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Skyrms, Brian. “Coherence.” In Scientific Inquiry in Philosophical Perspective. Edited by N. Rescher, 225–242. Pittsburgh, PA: University of Pittsburgh Press, 1987.
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Investigates the decision-theoretic underpinning of the Dutch Book argument and the source of the inconsistency involved in violating the axioms of probability.
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Depragmatizing Dutch Book Arguments

Some philosophers who, with Ramsey and others, read violation of the probability axioms as a form of inconsistency are nevertheless dissatisfied with the way in which the Dutch Book argument depends upon his assumed connection between credence and action. Part of the trouble concerns the difficulty in spelling out this connection in an accurate and convincing way. Additionally, there is a concern, expressed in Christensen 1996 and Christensen 2004, that if credences are at bottom to be understood in terms of preference, it is unclear why violating the probability axioms is a logical or epistemic defect. In response, Christensen and Howson and Urbach 1993 have offered so-called depragmatized versions of the Dutch Book argument, which aim to avoid what they take as the problematic relationship between credence and preference in standard presentations. However, these depragmatized Dutch Book arguments face severe difficulties, which are discussed in detail in Maher 1997. These, along with some additional problems, are noted in Vineberg 2011.

Christensen, David. “Dutch-Book Arguments Depragmatized: Epistemic Consistency For Partial Believers.” Journal of Philosophy 93 (1996): 450–479.
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Argues against the claim that the inconsistency involved in violating the probability axioms can be understood, as Ramsey did, as a kind of inconsistency of preference. Attempts to retain the basic idea of the Dutch Book argument by taking credences, not as requiring the acceptance of bets but as merely sanctioning certain bets.
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Christensen, David. Putting Logic in its Place. New York: Oxford University Press, 2004.
DOI: 10.1093/0199263256.001.0001 Save Citation »Export Citation »E-mail Citation »
Endorses a revised version of his depragmatized Dutch Strategy argument offered in (1996), which jettisons its problematic claim that bets that are individually sanctioned are jointly sanctioned. Instead it maintains this claim only for simple agents with idealized preferences.
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Howson, Colin, and Peter Urbach. Scientific Reasoning: The Bayesian Approach. 2d ed. Chicago: Open Court, 1993.
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Aims to avoid the problematic connection in some versions of the Dutch Book argument between degrees of belief and the acceptance of bets by associating an agent’s credence in a claim merely with regarding certain bets as fair. The Dutch Book theorem is employed to argue that failure to satisfy the probability axioms involves regarding as fair a set of bets that cannot be fair, which is taken as parallel to having inconsistent full beliefs.
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Maher, Patrick. “Depragmatized Dutch Book Arguments.” Philosophy of Science 64.2 (1997): 291–305.
DOI: 10.1086/392552 Save Citation »Export Citation »E-mail Citation »
Provides a detailed analysis and strong criticism of the attempts to depragmatize the Dutch Book argument by Christensen 1996 and Howson and Urbach 1993.
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Vineberg, Susan. “Dutch Book Arguments.” In Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.
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Points out several serious problems with the attempts of Christensen and Howson and Urbach to depragmatize Dutch Book arguments, noting that Christensen 2004 and its attempt to circumvent the problems of his earlier (1996) paper fails.
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Dutch Books and Non-Classical Logic

Skyrms interprets Ramsey as taking the possibility of making a Dutch Book against someone who violates the probability axioms as suffering from a kind of inconsistency: an inconsistency that arises from the underlying logic. The usual axioms of probability are built upon standard classical logic, but the fact that they need not be is discussed in Weatherson 2003.

Weatherson, Brian. “From Classical to Intuitionistic Probability.” Notre Dame Journal of Formal Logic 44.2 (2003): 111–123.
DOI: 10.1305/ndjfl/1082637807 Save Citation »Export Citation »E-mail Citation »
Discusses how probability might be founded on intuitionistic logic and the motivations for moving to such a logic, with special focus on how adopting intuitionistic probability functions permits a response to certain objections to classical Bayesianism. A treatment of intuitionistic probability functions is given and of betting within the intuitionistic framework. A Dutch Book argument is then put forth for satisfying the intuitionistic probability axioms.
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Using Dutch Book Arguments to Measure Coherence

The Dutch Book argument can be read as showing that the probability axioms constrain rationally ideal credences. Actual agents fall short of this standard, and this gives us reason to discriminate between the various ways (or degrees to which) they fail to meet the ideal. One way of approaching this problem uses the Dutch Book argument itself as a way to measure the degree to which an agent is coherent. Different ways of measuring coherence are explored in papers Schervish, et al. 2002a and Schervish, et al. 2002b; and discussed in Staffel 2015.

Schervish, Mark, Teddy Seidenfeld, and Joseph Kadane. “Measuring Incoherence.” Sankhya: Indian Journal of Statistics 64 (2002a): 561–587.
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A highly technical paper that considers different ways of measuring degrees of coherence using several ways of normalizing.
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Schervish, Mark, Teddy Seidenfeld, and Joseph Kadane. “Two Measures of Incoherence: How Not to Gamble If You Must.” In Bayesian Statistics 7. Edited by José M. Bernardo, M. J. Bayarri, A. Philip Dawid, James O. Berger, D. Heckerman, A. F. M. Smith, and Mike West, 385–401. Oxford: Oxford University Press, 2002b.
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A technical paper related to Schervish, et al.’s paper “Measuring Incoherence.”
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Staffel, Julia. “Measuring the Overall Incoherence of Credence Functions.” Synthese 192 (2015): 1467–1493.
DOI: 10.1007/s11229-014-0640-x Save Citation »Export Citation »E-mail Citation »
Discusses the question of how to measure the degree to which an agent’s credences are incoherent and evaluates several Dutch Book measures of incoherence proposed in the literature. An argument for the superiority of one of the measures considered by Schervish, Seidenfeld, and Kadane is included. Considerably less technical than their papers.
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Diachronic Dutch Book Arguments

In addition to defending the axioms of probability as a constraint on rational credences, Dutch Book arguments have also been used to defend the principles of conditionalization and reflection, which involve the regulation of an agent’s credences over time. Unlike in the arguments for the axioms of probability, the Dutch Books purporting to establish these principles involve bets that are placed over time and are often referred to as “Dutch Strategies.”

Dutch Strategy Arguments for Conditionalization

According to the Principle of Conditionalization (also called the “Rule of Conditionalization” or the “Simple Conditionalization Rule”), a rational agent should change his or her probability for the hypothesis h when he or she learns e, and nothing stronger, by conditionalizing on e—that is: pr₂(h) = pr₁(h/e), where pr₂(h) is his or her posterior probability for h and pr₁(h/e) is his or her prior probability for h conditional on e. A Dutch Strategy argument for this principle was constructed by Lewis 1999 but first published and discussed in Teller 1973, although Hacking 1967 had previously argued that no such argument was possible. A Generalized Rule of Conditionalization (or “Jeffrey kinematics” or “Jeffrey’s Rule”) is discussed in Jeffrey 1983, and a Dutch Strategy argument for the rule is provided in Armendt 1980. There have been numerous discussions of the Dutch Book Strategy arguments for these principles. Some works, including Bacchus, et al. 1990; Howson and Franklin 1994; and Levi 2002 are dismissive of these Dutch Strategy arguments and maintain that the conditionalization rules do not constrain rational changes of belief. A different approach guides Skyrms 1987 and van Fraassen 1989. Those two works make explicit the assumptions under which these Dutch Books can be made, arguing for the normative status of the rules given those assumptions.

Armendt, Brad. “Is There a Dutch Book Argument for Probability Kinematics?” Philosophical Topics 47 (1980): 583–588.
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Establishes the existence of a Dutch Strategy against those who violate Jeffrey’s Generalized Rule of Conditionalization.
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Bacchus, F., H. E. Kyburg, and Mariam Thalos. “Against Conditionalization.” Synthese 85 (1990): 475–506.
DOI: 10.1007/BF00484837 Save Citation »Export Citation »E-mail Citation »
Argues that rational changes of belief are not constrained by the conditionalization rules. Criticizes the assumptions of the basic Dutch Book arguments for probabilism, which are taken also as applying to the Dutch Strategy arguments for conditionalization, with additional criticisms leveled specifically at these arguments.
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Hacking, Ian. “Slightly More Realistic Personal Probability.” Philosophy of Science 34 (1967): 311–325.
DOI: 10.1086/288169 Save Citation »Export Citation »E-mail Citation »
Argues that there is no Dutch Book argument for conditionalization.
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Howson, Colin, and Allan Franklin. “Bayesian Conditionalization and Probability Kinematics.” British Journal for the Philosophy of Science 45 (1994): 451–466.
DOI: 10.1093/bjps/45.2.451 Save Citation »Export Citation »E-mail Citation »
Adheres to the line (also in Howson and Urbach 1993, cited under Textbooks) that a proper Dutch Book indicates a kind of logical consistency but that Dutch Strategy arguments against those who do not update by conditionalization do not establish that consistency requires such updating.
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Jeffrey, Richard. The Logic of Decision. 2d ed. Chicago: University of Chicago Press, 1983.
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A chapter of this textbook on decision theory is devoted to probability kinematics and the philosophical motivation for extending the Simple Rule of Conditionalization to Jeffrey’s Generalized Rule of Conditionalization, which is presented and discussed (pp. 164–183).
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Levi, Isaac. “Money Pumps and Diachronic Dutch Books.” Philosophy of Science 69.3 (2002): S235–S247.
DOI: 10.1086/341849 Save Citation »Export Citation »E-mail Citation »
Argues that the Dutch Strategy argument that has been proposed in defense of the Rule of Conditionalization fails to establish it as a principle that rational changes of belief must conform to.
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Lewis, David. “Why Conditionalize?” In Papers in Metaphysics and Epistemology. Cambridge, UK: Cambridge University Press, 1999.
DOI: 10.1017/CBO9780511625343 Save Citation »Export Citation »E-mail Citation »
Presents his Dutch Book argument that was reported in Teller 1973, together with some useful commentary that makes clear the idealized circumstances under which he takes the Principle of Conditionalization to apply. The paper concludes with Lewis’s view that full rationality requires knowing in advance how one will change beliefs in response to information.
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Skyrms, Brian. “Dynamic Coherence and Probability Kinematics.” Philosophy of Science 54 (1987): 1–20.
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Observes that Hacking’s claim that there can be no Dutch Book argument for conditionalization is correct insofar as what is given is one set of initial probabilities for an agent and at a later time the agent has different probabilities after learning about a piece of evidence. However, Skyrms notes that a Dutch Strategy such as Lewis’s is possible once the problem is given additional structure that demands a rule for updating probabilities. Jeffrey’s rule and higher-order probabilities are discussed as well.
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Teller, Paul. “Conditionalization and Observation.” Synthese 26 (1973): 218–258.
DOI: 10.1007/BF00873264 Save Citation »Export Citation »E-mail Citation »
Presents Lewis’s Dutch Book argument for conditionalization, along with an extended discussion of the Conditionalization Rule and its application.
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van Fraassen, Bas C. Laws and Symmetry. New York: Oxford University Press, 1989.
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Points out that these Dutch Strategy arguments do not support the claim that conditionalization must be followed dogmatically, but only the weaker claim that insofar as one has an updating plan in cases where conditionalization applies it is the correct rule.
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Dutch Strategy Argument for Reflection

The Principle of Reflection is a proposed further constraint on credences requiring that an agent’s current credences be constrained by the possible future credences that she envisions. In one form the principle demands that for any proposition A, and any future time t, an agent’s current probability for A conditional on assigning it probability r at t is itself r, that is pr(A/pr_t(A) = r) = r. There is again a Dutch Strategy argument for Reflection, given by van Fraassen 1984, but the principle is controversial as there are a number of convincing counterexamples to the claim that it is a requirement of rationality. Talbott 1991 gives a counterexample of one important type that involves forgetting, with others given in Christensen 1991 and elsewhere. Van Fraassen 1995 defends the Principle of Reflection, but others reject it and instead attempt to blunt the force of the Dutch Strategy argument for it—(e.g., Briggs 2009, Christensen 1991, Mahtani 2012, and Mahtani 2015). Whereas Christensen claims that only the synchronic Dutch Book argument reveals a problematic inconsistency, Vineberg 1997 takes this argument to be unconvincing against the claim that reflection is a genuine norm, albeit one that a rational person can sometimes violate. Huttegger 2013 adds additional commentary in defense of Dutch Book arguments with emphasis on reflection.

Briggs, Rachael. “Distorted Reflection.” Philosophical Review 118.1 (2009): 59–85.
DOI: 10.1215/00318108-2008-029 Save Citation »Export Citation »E-mail Citation »
Argues that not all cases of Dutch Book vulnerability involve incoherence (in the sense of inconsistency), and employs a “suppositional test” to distinguish between mere Dutch Book vulnerability and genuine incoherence.
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Christensen, David. “Clever Bookies and Coherent Beliefs.” Philosophical Review 100.2 (1991): 229–247.
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Argues that only some Dutch Book arguments establish a problematic type of inconsistency.
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Huttegger, Simon M. “In Defense of Reflection.” Philosophy of Science 80.3 (2013): 413–433.
DOI: 10.1086/671427 Save Citation »Export Citation »E-mail Citation »
Defends the Dutch Strategy arguments for reflection, as well as making a new argument for reflection, and argues that the force of the counterexamples can be deflated.
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Mahtani, Anna. “Diachronic Dutch Book Arguments.” Philosophical Review 121.3 (2012): 443–450.
DOI: 10.1215/00318108-1574445 Save Citation »Export Citation »E-mail Citation »
Argues that Briggs’s suppositional test for coherence draws the line between Dutch Books that reveal genuine incoherence and those that do not in the wrong place, because it counts certain cases of self-doubt where intuitively the agent is coherent as cases of incoherence.
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Mahtani, Anna. “Dutch Books, Coherence, and Logical Consistency.” Nous 49 (2015): 522–537.
DOI: 10.1111/nous.12070 Save Citation »Export Citation »E-mail Citation »
Adds to the discussion in Mahtani 2012 by presenting a new definition of incoherence as involving the acceptance of a set of bets as fair that would result in a loss under any interpretation. The Dutch Book argument for reflection does not indicate incoherence in this sense, nor does the Dutch Book argument against the self-doubting agent; however, the new definition of coherence leaves open whether the Dutch Strategy argument for conditionalization works.
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Talbott, W. “Two Principles of Bayesian Epistemology.” Philosophical Studies 62 (1991) 135–150.
DOI: 10.1007/BF00419049 Save Citation »Export Citation »E-mail Citation »
Discusses the Principles of Conditionalization and Reflection, and gives a frequently cited counterexample to the Principle of Reflection that arises from everyday memory loss, in which one would also expect to violate conditionalization. It is suggested that the Dutch Strategies that one might face in such circumstances do not show that such violations are irrational.
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van Fraassen, Bas C. “Belief and the Will.” Journal of Philosophy 81 (1984): 235–256.
DOI: 10.2307/2026388 Save Citation »Export Citation »E-mail Citation »
Presents a Dutch Strategy argument for Reflection as a requirement of rationality, along with considerable discussion of graded judgments in support of a voluntarist epistemology.
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van Fraassen, Bas C. “Belief and the Problem of Ulysses and the Sirens.” Philosophical Studies 77 (1995): 7–37.
DOI: 10.1007/BF00996309 Save Citation »Export Citation »E-mail Citation »
Generalizes the Principle of Reflection to apply to vague opinions. Here the defense of the principle is not taken to rest on a Dutch Strategy argument, although one can be given for it, but on the requirements of epistemic integrity.
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Vineberg, Susan. “Dutch Books, Dutch Strategies and What They Show About Rationality.” Philosophical Studies 86 (1997): 185–201.
DOI: 10.1023/A:1017932926382 Save Citation »Export Citation »E-mail Citation »
Discusses Christensen’s influential claim that diachronic Dutch Book arguments, along with some others, are lacking in force, and maintains that his argument for this is inadequate.
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Additional Dutch Book and Betting Arguments

Dutch Book arguments have also been used in defense of a wide variety of claims beyond their use in supporting the axioms of probability, the Conditionalization Rules and Reflection. These involve Dutch Books that can be made against agents whose credences take on particular values in the Sleeping Beauty problem and agents who apply inference to the best explanation in a specific way, to certain groups of agents, and to agents with unbounded utilities whose credences are not concentrated at finitely many points. They have also been applied in arguments against the idea that rationality permits imprecise credences.

Self-Locating Belief and the Sleeping Beauty Problem

Dutch Book arguments have been applied in support of assigning particular credences in the case of the Sleeping Beauty problem. The standard version of the problem concerns an experiment in which a subject is put to sleep on Sunday and awakened briefly either once or twice depending on the flip of a fair coin that is tossed after she goes to sleep. After each awakening the subject is put back to sleep with an amnesiac that causes her to forget the awakening. The subject is awakened again on Wednesday and the experiment ends. Assuming that the subject knows the protocol of the experiment, the problem concerns the probability that she should have that the coin landed heads upon awakening within the experiment. A popular answer is one-third, which was first defended in Elga 2000, but others have defended the answer as one-half. Hitchcock 2004 gives a Dutch Book argument in defense of the answer of one-third. His Dutch Book argument is discussed and criticized in Bradley and Leitgeb 2006, Draper and Pust 2008, and Vineberg 2011.

Bradley, Darren, and Hannes Leitgeb. “When Betting Odds and Credences Come Apart: More Worries for Dutch Book Arguments.” Analysis 66.2 (2006): 119–127.
DOI: 10.1093/analys/66.2.119 Save Citation »Export Citation »E-mail Citation »
Argues that the Sleeping Beauty problem presents a case in which betting quotients and credences come apart, thus undermining Hitchcock’s Dutch Book argument for the answer of one-third.
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Draper, Kai, and Joel Pust. “Diachronic Dutch Books and Sleeping Beauty.” Synthese 164.2 (2008): 281–287.
DOI: 10.1007/s11229-007-9226-1 Save Citation »Export Citation »E-mail Citation »
This paper disputes Bradley and Leitgeb’s criticism of Hitchcock’s Dutch Book argument for the answer of one-third in the Sleeping Beauty problem but argues that Hitchcock’s Dutch Book argument is vulnerable to an objection from evidential decision theory. A different Dutch Book argument for the answer of one-third is offered to avoid the problem.
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Elga, Adam. “Self-Locating Belief and the Sleeping Beauty Problem.” Analysis 60.2 (2000): 143–147.
DOI: 10.1093/analys/60.2.143 Save Citation »Export Citation »E-mail Citation »
Introduced the “Sleeping Beauty” problem into the philosophical literature and argues for the answer of one-third.
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Hitchcock, Christopher. “Beauty and the Bets.” Synthese 139 (2004): 405–420.
DOI: 10.1023/B:SYNT.0000024889.29125.c0 Save Citation »Export Citation »E-mail Citation »
Constructs a Dutch Book for both the answer of one-half and the answer of one-third in the Sleeping Beauty problem, and then argues that only the latter yields a correct Dutch Book argument, defending the answer of one-third.
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Vineberg, Susan. “Dutch Book Arguments.” In Stanford Encyclopedia of Philosophy. Edited by Edward N. Zalta. 2011.
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Notes a problem with Hitchcock’s claim that his Dutch Book argument for the answer of one-third in the Sleeping Beauty problem involves a proper Dutch Book, whereas the argument for one-half does not.
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Miscellaneous Uses of Dutch Books

Dutch Books have been invoked to argue for a variety of additional claims. Both Elga 2010 and McGee 1999 are concerned with rational credences in conjunction with the principle of expected utility maximization. Elga uses a Dutch Book in arguing that rational choice theory requires that credences be sharp rather than vague, whereas McGee uses a Dutch Book to show the limitations of the principle of expected utility maximization. Dutch Books have also been used to argue against the compatibilist view that determinism is compatible with intermediate chances (see Schaffer 2007), and against the Principle of Inference to the Best Explanation (see van Fraassen 1989), although this use of the Dutch Book has been criticized, as in Ganson 2007. Finally, Gillies 1991 extends the Dutch Book argument to cover social groups.

Elga, Adam. “Subjective Probabilities Should Be Sharp.” Philosopher’s Imprint 10.5 (2010): 1–11.
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Uses Dutch Books to argue that imprecise credences are connected with irrational choices and thus that rational credences should be sharp.
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Ganson, Dorit. “Van Fraassen’s Dutch Book Argument Against Explanation.” In Causation and Explanation. By Dorit Ganson, 171–189. Cambridge, MA: MIT, 2007.
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Discusses van Fraassen’s Dutch Book argument against the Principle of Inference to the Best Explanation and gives some criticisms of it.
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Gillies, Donald. “Intersubjective Probability and Confirmation Theory.” British Society for the Philosophy of Science 42.4 (1991): 513–533.
DOI: 10.1093/bjps/42.4.513 Save Citation »Export Citation »E-mail Citation »
Gives a Dutch Book argument for social groups in defense of the idea of intersubjective probability. It is proposed as applying in cases where the individuals of a group have a common interest, and in particular where probabilities are used in confirmation theory.
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McGee, Vann. “An Airtight Dutch Book.” Analysis 59.264 (1999): 257–265.
DOI: 10.1093/analys/59.4.257 Save Citation »Export Citation »E-mail Citation »
It is shown that there is a Dutch Book against an agent, with an unbounded utility scale, whose probabilities are not concentrated at finitely many points. The result is used to argue against the idea that the rule that one ought to act so as to maximize expected utility can be taken as governing all choices.
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Schaffer, Jonathan. “Deterministic Chance?” British Journal for the Philosophy of Science 58 (2007): 113–140.
DOI: 10.1093/bjps/axm002 Save Citation »Export Citation »E-mail Citation »
A Dutch Book argument is given against the compatibilist view that determinism is compatible with chances different from 0 and 1.
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van Fraassen, Bas C. Laws and Symmetry. New York: Oxford University Press, 1989.
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Constructs a Dutch Book that is alleged to apply to those who employ the Principle of Inference to the Best Explanation.
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Dutch Book Arguments
by
Susan Vineberg
LAST REVIEWED: 08 October 2015
LAST MODIFIED: 31 March 2016
DOI: 10.1093/obo/9780195396577-0301

Introduction

General Overviews

Textbooks

The Basic Dutch Book Argument for Coherence

Technical Results

Interpretations and Criticisms of Dutch Book Arguments

Incoherence as a Pragmatic Defect

Incoherence as a Form of Inconsistency

Dutch Book Arguments and Decision Theory

Depragmatizing Dutch Book Arguments

Dutch Books and Non-Classical Logic

Using Dutch Book Arguments to Measure Coherence

Diachronic Dutch Book Arguments

Dutch Strategy Arguments for Conditionalization

Dutch Strategy Argument for Reflection

Additional Dutch Book and Betting Arguments

Self-Locating Belief and the Sleeping Beauty Problem

Miscellaneous Uses of Dutch Books

Philosophy

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In This Article Dutch Book Arguments

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Other Subject Areas

Forthcoming Articles

Dutch Book Arguments bySusan VinebergLAST REVIEWED: 08 October 2015LAST MODIFIED: 31 March 2016DOI: 10.1093/obo/9780195396577-0301

Introduction

General Overviews

Textbooks

The Basic Dutch Book Argument for Coherence

Technical Results

Interpretations and Criticisms of Dutch Book Arguments

Incoherence as a Pragmatic Defect

Incoherence as a Form of Inconsistency

Dutch Book Arguments and Decision Theory

Depragmatizing Dutch Book Arguments

Dutch Books and Non-Classical Logic

Using Dutch Book Arguments to Measure Coherence

Diachronic Dutch Book Arguments

Dutch Strategy Arguments for Conditionalization

Dutch Strategy Argument for Reflection

Additional Dutch Book and Betting Arguments

Self-Locating Belief and the Sleeping Beauty Problem

Miscellaneous Uses of Dutch Books

Philosophy

Article

Dutch Book Arguments
by
Susan Vineberg
LAST REVIEWED: 08 October 2015
LAST MODIFIED: 31 March 2016
DOI: 10.1093/obo/9780195396577-0301