Diachronic Dutch Books and Sleeping Beauty

Hitchcock advances a diachronic Dutch Book argument (DDB) for a 1/3 answer to the Sleeping Beauty problem. Bradley and Leitgeb argue that Hitchcock’s DDB argument fails. We demonstrate the following: (a) Bradley and Leitgeb’s criticism of Hitchcock is unconvincing; (b) nonetheless, there are serious reasons to worry about the success of Hitchcock’s argument; (c) however, it is possible to construct a new DDB for 1/3 about which such worries cannot be raised.     

Reflecting on diachronic Dutch books

Conditionalization 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.     

Generalized Probabilism: Dutch Books and Accuracy Domination

This paper explores De Finetti’s generalized versions of Dutch Book and Accuracy Domination theorems. Following proposals due to Jeff Paris, we construe these as underpinning a generalized probabilism appropriate to belief states against either a classical or a non-classical background. Both results are straightforward corollaries of the separating hyperplane theorem; their geometrical relationship is examined. It is shown that each point of Accuracy Domination for b induces a Dutch Book on b; but Dutch Books may need to be ‘scaled’ in order to find a point of Accuracy-Domination. Finally, diachronic Dutch Book defences of conditionalization are examined in the general setting. The formulation and limitations of the generalized conditionalization this delivers are examined.     

Credence and self-location

All parties to the Sleeping Beauty debate agree that it shows that some cherished principle of rationality has to go. Thirders think that it is Conditionalization and Reflection that must be given up or modified; halfers think that it is the Principal Principle. I offer an analysis of the Sleeping Beauty puzzle that allows us to retain all three principles. In brief, I argue that Sleeping Beauty’s credence in the uncentered proposition that the coin came up heads should be 1/2, but her credence in the centered proposition that the coin came up heads and it is Monday should be 1/3. I trace the source of the earlier mistakes to an unquestioned assumption in the debate, namely that an uncentered proposition is just a special kind of centered proposition. I argue that the falsity of this assumption is the real lesson of the Sleeping Beauty case.     

Sleeping Beauty, evidential support and indexical knowledge: reply to Horgan

Terence Horgan defends the thirder position on the Sleeping Beauty problem, claiming that Beauty can, upon awakening during the experiment, engage in “synchronic Bayesian updating” on her knowledge that she is awake now in order to justify a 1/3 credence in heads. In a previous paper, I objected that epistemic probabilities are equivalent to rational degrees of belief given a possible epistemic situation and so the probability of Beauty’s indexical knowledge that she is awake now is necessarily 1, precluding such updating. In response, Horgan maintains that the probability claims in his argument are to be taken, not as claims about possible rational degrees of belief, but rather as claims about “quantitative degrees of evidential support.” This paper argues that the most plausible account of quantitative degree of support, when conjoined with any of the three major accounts of indexical thought in such a way as to plausibly constrain rational credence, contradicts essential elements of Horgan’s argument.     

Subjective probabilities and betting quotients

This paper addresses the problem of why the conditions under which standard proofs of the Dutch Book argument proceed should ever be met. In particular, the condition that there should be odds at which you would be willing to bet indifferently for or against are hardly plausible in practice, and relaxing it and applying Dutch book considerations gives only the theory of upper and lower probabilities. It is argued that there are nevertheless admittedly rather idealised circumstances in which the classic form of the Dutch Book argument is valid.Probability is the very guide of life' (Bishop Butler).     

Sleeping beauty should be imprecise

The traditional solutions to the Sleeping Beauty problem say that Beauty should have either a sharp 1/3 or sharp 1/2 credence that the coin flip was heads when she wakes. But Beauty’s evidence is incomplete so that it doesn’t warrant a precise credence, I claim. Instead, Beauty ought to have a properly imprecise credence when she wakes. In particular, her representor ought to assign \(R(H\!eads)=[0,1/2]\) . I show, perhaps surprisingly, that this solution can account for the many of the intuitions that motivate the traditional solutions. I also offer a new objection to Elga’s restricted version of the principle of indifference, which an opponent may try to use to collapse the imprecision.     

Sleeping Beauty and shifted Jeffrey conditionalization

In this paper, I argue for a view largely favorable to the Thirder view: when Sleeping Beauty wakes up on Monday, her credence in the coin’s landing heads is less than 1/2. Let’s call this “the Lesser view.” For my argument, I (i) criticize Strict Conditionalization as the rule for changing de se credences; (ii) develop a new rule; and (iii) defend it by Gaifman’s Expert Principle. Finally, I defend the Lesser view by making use of this new rule.     

Diachronic Dutch Books and Evidential Import

A handful of well‐known arguments (the ‘diachronic Dutch book arguments’) rely upon theorems establishing that, in certain circumstances, you are immune from sure monetary loss (you are not ‘diachronically Dutch book‐able’) if and only if you adopt the strategy of conditionalizing (or Jeffrey conditionalizing) on whatever evidence you happen to receive. These theorems require non‐trivial assumptions about which evidence you might acquire—in the case of conditionalization, the assumption is that, if you might learn that e, then it is not the case that you might learn something else that is consistent with e. These assumptions may not be relaxed. When they are, not only will non‐(Jeffrey) conditionalizers be immune from diachronic Dutch bookability, but (Jeffrey) conditionalizers will themselves be diachronically Dutch bookable. I argue: 1) that there are epistemic situations in which these assumptions are violated; 2) that this reveals a conflict between the premise that susceptibility to sure monetary loss is irrational, on the one hand, and the view that rational belief revision is a function of your prior beliefs and the acquired evidence alone, on the other; and 3) that this inconsistency demonstrates that diachronic Dutch book arguments for (Jeffrey) conditionalization are invalid.     

The puzzle of the hats

The Puzzle of the Hats is a betting arrangement which seems to show that a Dutch book can be made against a group of rational players with common priors who act in the common interest and have full trust in the other players’ rationality. But we show that appearances are misleading—no such Dutch book can be made. There are four morals. First, what can be learned from the puzzle is that there is a class of situations in which credences and betting rates diverge. Second, there is an analogy between ways of dealing with situations of this kind and different policies for sequential choice. Third, there is an analogy with strategic voting, showing that the common interest is not always served by expressing how things seem to you in social decision-making. And fourth, our analysis of the Puzzle of the Hats casts light on a recent controversy about the Dutch book argument for the Sleeping Beauty.     

On the Expected Utility Objection to the Dutch Book Argument for Probabilism*

The Dutch Book Argument for Probabilism assumes Ramsey's Thesis (RT), which determines the prices an agent is rationally required to pay for a bet. Hedden (2013) argues that Ramsey's Thesis is wrong. He claims that Maximise Subjective Expected Utility (MSEU) determines those prices, and it often disagrees with Ramsey's Thesis. I suggest two responses to Hedden's objection. First, we might be permissive: agents are permitted to pay any price that is required or permitted by RT, and they are permitted to pay any price that is required or permitted by MSEU. This allows us to give a revised version of the Dutch Book Argument for Probabilism, which I call the Permissive Dutch Book Argument. Second, I suggest that even Hedden should admit that RT gives the correct answer in certain very limited cases, and I show that, together with MSEU, this very restricted version of RT gives a new pragmatic argument for Probabilism, which I call the Bookless Pragmatic Argument.     

Inertia,Optimism and Beauty

The best arguments for the 1/3 answer to the Sleeping Beauty problem all require that when Beauty awakes on Monday she should be uncertain what day it is. I argue that this claim should be rejected, thereby clearing the way to accept the 1/2 solution.     

Horgan on Sleeping Beauty

With the notable exception of David Lewis, most of those writing on the Sleeping Beauty problem have argued that 1/3 is the correct answer. Terence Horgan has provided the clearest account of why, contrary to Lewis, Beauty has evidence against the proposition that the coin comes up heads when she awakens on Monday. In this paper, I argue that Horgan’s proposal fails because it neglects important facts about epistemic probability.     

Agnostic empiricism versus scientific realism: Belief in truth matters

This paper aims to defend scientific realism against two versions of agnostic empiricism: a naive agnostic position, which suggests that the only rational option is to remain agnostic as to the truth of theoretical assertions, and van Fraassen's more sophisticated agnostic empiricism - which may be called "Hypercritical Empiricism". It first argues that given semantic realism, naive agnostic empiricism cannot be maintained: there is no relevant epistemic difference between theoretical assertions and observational ones. It then focuses on van Fraassen's more sophisticated alternative to scientific realism and suggests that an attitude towards science which involves less than aiming at theoretical truth and believing in theories would be, in some concrete respect that empiricists should recognize, worse off than the recommended realist attitude. To this end, the paper develops the so-called conjunction argument into a diachronic argument for scientific realism.     

The Coherence Argument Against Conditionalization

I re-examine Coherence Arguments (Dutch Book Arguments, No Arbitrage Arguments) for diachronic constraints on Bayesian reasoning. I suggest to replace the usual game–theoretic coherence condition with a new decision–theoretic condition ('Diachronic Sure Thing Principle'). The new condition meets a large part of the standard objections against the Coherence Argument and frees it, in particular, from a commitment to additive utilities. It also facilitates the proof of the Converse Dutch Book Theorem. I first apply the improved Coherence Argument to van Fraassen's (1984) Reflection principle. I then point out the failure of a Coherence Argument that is intended to support Conditionalization as a naive, universal, update rule. I also point out that Reflection is incompatible with the universal use of Conditionalization thus interpreted. The Coherence Argument therefore defeats the naive view on Bayesian learning that it was originally designed to justify. This revised version was published online in August 2006 with corrections to the Cover Date.     

A forward looking decision rule for imprecise credences

Adam Elga (Philosophers’ Imprint, 10(5), 1–11, 2010) presents a diachronic puzzle to supporters of imprecise credences and argues that no acceptable decision rule for imprecise credences can deliver the intuitively correct result. Elga concludes that agents should not hold imprecise credences. In this paper, I argue for a two-part thesis. First, I show that Elga’s argument is incomplete: there is an acceptable decision rule that delivers the intuitive result. Next, I repair the argument by offering a more elaborate diachronic puzzle that is more difficult for imprecise Bayesians to avoid.     

Synchronic Bayesian updating and the Sleeping Beauty problem: reply to Pust

I maintain, in defending “thirdism,” that Sleeping Beauty should do Bayesian updating after assigning the “preliminary probability” 1/4 to the statement S: “Today is Tuesday and the coin flip is heads.” (This preliminary probability obtains relative to a specific proper subset I of her available information.) Pust objects that her preliminary probability for S is really zero, because she could not be in an epistemic situation in which S is true. I reply that the impossibility of being in such an epistemic situation is irrelevant, because relative to I, statement S nonetheless has degree of evidential support 1/4.     

For whom the Bell arguments toll

We will formulate two Bell arguments. Together they show that if the probabilities given by quantum mechanics are approximately correct, then the properties exhibited by certain physical systems must be nontrivially dependent on thetypes of measurements performedand eithernonlocally connected orholistically related to distant events. Although a number of related arguments have appeared since John Bell's original paper (1964), they tend to be either highly technical or to lack full generality. The following arguments depend on the weakest of premises, and the structure of the arguments is simpler than most (without any loss of rigor or generality). The technical simplicity is due in part to a novel version of the generalized Bell inequality. The arguments are self contained and presuppose no knowledge of quantum mechanics. We will also offer a Dutch Book argument for measurement type dependence.