We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call *Countable (...) Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

Countenancing unbounded utility in ethics gives rise to deep puzzles in formal decision theory. Here, these puzzles are taken as an invitation to assess the most fundamental principles relating probability and value, with the aim of demonstrating that unbounded utility in ethics is compatible with a desirable decision theory. The resulting theory frames further discussion of Expected Utility Theory and of principles concerning symmetries of utility.

Pascal’s Wager is simply too good to be true—or better, too good to be sound. There must be something wrong with Pascal’s argument that decision-theoretic reasoning shows that one must (resolve to) believe in God, if one is rational. No surprise, then, that critics of the argument are easily found, or that they have attacked it on many fronts. For Pascal has given them no dearth of targets.

The Cable Guy is coming. You have to be home in order for him to install your new cable service, but to your chagrin he cannot tell you exactly when he will come. He will definitely come between 8.a.m. and 4 p.m. tomorrow, but you have no more information than that. I offer to keep you company while you wait. To make things more interesting, we decide now to bet on the Cable Guy’s arrival time. We subdivide the relevant part (...) of the day into two 4-hour long intervals, ‘morning’: (8, 12], and ‘afternoon’: (12, 4). You nominate an interval on which you will bet. If he arrives during your interval, you win and I will pay you $10; otherwise, I win and you will pay me $10. Notice that we stipulate that if he arrives exactly on the stroke of noon, then (8, 12] is the winning interval, since it is closed on the right; but we agree that this event has probability 0 (we have a very precise clock!). At first you think: obviously there is no reason to favour one interval over the other. Your probability distribution of his arrival time is uniform over the 8 a.m. – 4 p.m. period, and thus assigns probability 1/2 to each of the two 4-hour periods at issue. Whichever period you nominate, then, your expected utility is the same. The two choices are equally rational. But then you reason as follows. Suppose that you choose the morning interval. Then there will certainly be a period during which you will regard the other interval as.. (shrink)

There has been much discussion on the two-envelope paradox. Clark and Shackel (2000) have proposed a solution to the paradox, which has been refuted by Meacham and Weisberg (2003). Surprisingly, however, the literature still contains no axiomatic justification for the claim that one should be indifferent between the two envelopes before opening one of them. According to Meacham and Weisberg, "decision theory does not rank swapping against sticking [before opening any envelope]" (p. 686). To fill this gap in the literature, (...) we present a simple axiomatic justification for indifference, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using different decision principles such as maximin and maximax, and that our justification for indifference before opening applies here too. (shrink)

This paper deals with the two-envelope paradox. Two main formulations of the paradoxical reasoning are distinguished, which differ according to the partition of possibilities employed. We argue that in the first formulation the conditionals required for the utility assignment are problematic; the error is identified as a fallacy of conditional reasoning. We go on to consider the second formulation, where the epistemic status of certain singular propositions becomes relevant; our diagnosis is that the states considered do not exhaust the possibilities. (...) Thus, on our approach to the paradox, the fallacy, in each formulation, is found in the reasoning underlying the relevant utility matrix; in both cases, the paradoxical argument goes astray before one gets to questions of probability or calculations of expected utility. (shrink)

In the asymmetrical variant of the two-envelope paradox, the amount in envelope A is determined first, and then the amount in envelope B is determined to be either twice or half the amount in A by flipping a fair coin. Contra the common belief that B is preferable to A in this case, I show that the proposed arguments for this common belief all fail, and argue that B is not preferable to A if the expected values of the amounts (...) in the envelopes are infinite. Using the examples I deploy in my arguments against the common belief, I also refute certain proposed solutions to the two-envelope paradox and draw some general lessons. (shrink)

This paper proposes a new framework to solve the surprise examination paradox. I survey preliminary the main contributions to the literature related to the paradox. I introduce then a distinction between a monist and a dichotomic analysis of the paradox. With the help of a matrix notation, I also present a dichotomy that leads to distinguish two basically and structurally different notions of surprise, which are respectively based on a conjoint and a disjoint structure. I describe then how Quine's solution (...) and Hall's reduction apply to the version of the paradox corresponding to the conjoint structure. Lastly, I expose a solution to the version of the paradox based on the disjoint structure. (shrink)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

The Two Envelopes Problem is a beautiful and quite confusing problem in decision theory which is ca. 35 years old and has provoked at least 150 papers directly addressing the problem and displaying a surprising variety of different responses. This paper finds decisive progress in an approach of Priest and Restall in 2003, contends that the recent papers having appeared since did not really go beyond that paper, argues further that Priest’s and Restall’s solution is still not complete, and proposes (...) a completion of their solution. If the analysis is correct, this work has the potential of laying the Two Envelopes Problem at rest. (shrink)

Sutton ( 2010 ) claims that on our analysis (2007), the problem in the two-envelope paradox is an error in counterfactual reasoning. In fact, we distinguish two formulations of the paradox, only one of which, on our account, involves an error in conditional reasoning. According to Sutton, it is conditional probabilities rather than subjunctive conditionals that are essential to the problem. We argue, however, that his strategy for assigning utilities on the basis of conditional probabilities leads to absurdity. In addition, (...) we show that a crucial presupposition of Sutton’s argument — namely, that one can know that envelope A contains n simply on the basis of a stipulation — is mistaken. (shrink)

In the two-envelope problem, one is offered a choice between two envelopes, one containing twice as much money as the other. After seeing the contents of the chosen envelope, the chooser is offered the opportunity to make an exchange for the other envelope. However, it appears to be advantageous to switch, regardless of what is observed in the chosen envelope. This problem has an extensive literature with connections to probability and decision theory. The literature is roughly divided between those that (...) attempt to explain what is flawed in arguments for the advantage of switching and those that attempt to explain when such arguments can be correct if counterintuitive. We observe that arguments in the literature of the two-envelope problem that the problem is paradoxical are not supported by the probability distributions meant to illustrate the paradoxical nature. To correct this, we present a distribution that does support the usual arguments. Aside from questions about the interpretation of variables, algebraic ambiguity, modal confusions and the like, most of the interesting aspects of the two-envelope problem are assumed to require probability distributions on an infinite space. Our next main contribution is to show that the same counterintuitive arguments can be reflected in finite versions of the problem; thus they do not inherently require reasoning about infinite values. A topological representation of the problem is presented that captures both finite and infinite cases, explicating intuitions underlying the arguments both that there is an advantage to switching and that there is not. (shrink)

Recently we proposed “quantum language" (or,“the linguistic Copenhagen interpretation of quantum mechanics"), which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of Descartes=Kant epistemology. Namely, quantum language is the scientific final goal of dualistic idealism. It has a great power to describe classical systems as well as quantum systems. Thus, we believe that quantum language is the language in which science is written. The purpose of this preprint is to examine (...) and assert our belief (i.e.,“proposition in quantum language" ⇔“scientific proposition). We believe that it's one of main themes of scientific philosophy to make such language. (shrink)

Recently I proposed “quantum language” (or,“the linguistic Copenhagen interpretation of quantum mechanics”), which was not only characterized as the metaphysical and linguistic turn of quantum mechanics but also the linguistic turn of Descartes=Kant epistemology. Namely, quantum language is the scientific final goal of dualistic idealism. It has a great power to describe classical systems as well as quantum systems. In this research report, quantum language is seen as a fundamental theory of statistics and reveals the true nature of statistics.