The three most common responses to Taurek’s ‘numbers problem’ are saving the greater number, equal chance lotteries and weighted lotteries. Weighted lotteries have perhaps received the least support, having been criticized by Scanlon What We Owe to Each Other ( 1998 ) and Hirose ‘Fairness in Life and Death Cases’ ( 2007 ). This article considers these objections in turn, and argues that they do not succeed in refuting the fairness of a weighted lottery, which remains a (...) potential solution to cases of conflict. Moreover, it shows how these responses actually lead to a new argument for weighted lotteries, appealing to fairness and Pareto-optimality. (shrink)
Adversaries of Moral Luck (AMLs) are at pains to explain why wrongdoers are liable to bear burdens (punishment, compensation etc.) which are related to the harm they cause, because the consequences of what we do are a matter of luck. One attempt to solve this problem suggests that wrongdoers who cause more harm are liable to bear a greater burden not because they are more blameworthy but rather because they get the short straw in a liability lottery (represented by the (...) apparently indeterminate causal process). In this paper I argue that this attempt fails on several grounds. Apart from the fact that it is hard to see how the implementation of liability lotteries can be motivated and the fact that such scheme presupposes a political order (whereas the notion of liability does not seem to presuppose one), detaching liability from the outcomes of a culpable action undermines whichever justifications there were for imposing liability in the first place. Moreover, relying on the determination of the causal process as a good indication of the wrongdoer’s degree of culpability is mistaken, because the luck brought about through the causal process is not necessarily the only element involved in cases of harmful conduct which lies beyond the wrongdoers’ control. (shrink)
In this paper we experimentally investigate the disparity between willingness-to-accept (WTA) and willingness-to-pay (WTP) for risky lotteries. The direction of the income effect is reversed by endowing subjects with the highest price of a lottery when asking the WTP question. Our results show that the income effect is too small to be the only source of the disparity. Since the disparity concentrates on a subsample of subjects, parametric and nonparametric tests of the WTA-WTP ratio may lead to contradictory results. (...) The disparity is significantly reduced when background risk is introduced. That is, putting subjects always into a risky position could improve the contingent valuation method, which is often concerned with the assessment of risky situations such as health risks, automobile safety, etc. (shrink)
In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They (...) illustrate this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (shrink)
This paper addresses an argument offered by John Hawthorne gainst the propriety of an agent’s using propositions she does not know as premises in practical reasoning. I will argue that there are a number of potential structural confounds in Hawthorne’s use of his main example, a case of practical reasoning about a lottery. By drawing these confounds out more explicitly, we can get a better sense of how to make appropriate use of such examples in theorizing about norms, knowledge, and (...) practical reasoning. I will conclude by suggesting a prescription for properly using lottery propositions to do the sort of work that Hawthorne wants from them. (shrink)
Sylvia Wenmackers (2012). Ultralarge and Infinite Lotteries. In B. Van Kerkhove, T. Libert, G. Vanpaemel & P. Marage (eds.), Logic, Philosophy and History of Science in Belgium II (Proceedings of the Young Researchers Days 2010). Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten.score: 14.0
By exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. We solve the 'adding problems' that occur in these two contexts using a similar strategy, based on non-standard analysis.
If payoffs are tickets for binary lotteries, which involve only two money prizes, then rationality requires expected value maximization in tickets. This payoff scheme was increasingly used to induce risk neutrality in experiments. The experiment presented here involved lottery choice and evaluation tasks. One subject group was paid in binary lottery tickets, another directly in money. Significantly greater deviations from risk neutral behavior are observed with binary lottery payoffs. This discrepancy increases when subjects have easy access to the alternatives' (...) expected values and mean absolute deviations. Behavioral regularities are observed at least as often as with direct money payoffs. (shrink)
This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem.
Knowledge and Lotteries is organized around an epistemological puzzle: in many cases, we seem consistently inclined to deny that we know a certain class of propositions, while crediting ourselves with knowledge of propositions that imply them. In its starkest form, the puzzle is this: we do not think we know that a given lottery ticket will be a loser, yet we normally count ourselves as knowing all sorts of ordinary things that entail that its holder will not suddenly acquire (...) a large fortune. After providing a number of specific and general characterizations of the puzzle, Hawthorne carefully examines the competing merits of candidate solutions. In so doing, he explores a number of central questions concerning the nature and importance of knowledge, including the relationship of knowledge to assertion and practical reasoning, the status of epistemic closure principles, the merits of various brands of scepticism, the prospects for a contextualist account of knowledge, and the potential for other sorts of salience-sensitive accounts. Along the way, he offers a careful treatment of pertinent issues at the foundations of semantics. His book will be of interest to anyone working in the field of epistemology, as well as to philosophers of language. (shrink)
Lotteries have long been used to resolve competing claims, yet their recent implementation to allocate school places in Brighton and Hove, England led to considerable public outcry. This article argues that, given appropriate selection is impossible when parties have equal claims, a lottery is preferable to an auction because it excludes unjust influences. Three forms of contractualism are discussed and the fairness of lotteries is traced to the fact that they give each person an equal chance, as a (...) surrogate for their equal claim to the good. It is argued that this can be a reason to favour an artificially-constructed lottery to a 'natural' lottery where there is suspicion that the latter may be biased. (shrink)
I. Introduction 1.1 George M avrodes seems to say that, as reasonable persons, on reading reports of winners of really big lotteries believe these reports, so must reasonable persons, on hearing or reading testimony for what in their view would be miracles, believe this testimony. That a randomly selected entrant should win a big lottery is immensely improbable, and yet a single report can reverse this improbability. He says it shows “that there is nothing incredible, or even unusual, in (...) the power of a single testimony to reverse an enormous initial improbability” (Mavrodes 2005, p. 317) Of miracles he says that “almost every aficionado” of them holds that they are improbable in the same way, and, he seems to say, to the same effect. (shrink)
We conducted five experiments that reveal some main contours of the folk epistemology of lotteries. The folk tend to think that you don't know that your lottery ticket lost, based on the long odds ("statistical cases"); by contrast, the folk tend to think that you do know that your lottery ticket lost, based on a news report ("testimonial cases"). We evaluate three previous explanations for why people deny knowledge in statistical cases: the justification account, the chance account, and the (...) statistical account. None of them seems to work. We then propose a new explanation of our own, the formulaic account, according to which some people deny knowledge in statistical cases due to formulaic expression. (shrink)
THIS PAPER ANSWERS RECENT CRITICISMS OF HUME’S SKEPTICISM WITH REGARD TO MIRACLES BY THOSE WHO ARGUE THAT THERE ARE COUNTEREXAMPLES, ILLUSTRATED BY LOTTERIES, TO HUME’S ACCOUNT OF HOW THE TRUTH OF REPORTS ABOUT IMPROBABLE EVENTS MUST BE EVALUATED. THE AUTHOR FIRST SHOWS THAT THESE ARGUMENTS ARE ANALOGOUS TO BUTLER’S CRITICISM OF HUME’S PREDECESSORS IN THE DEBATE ABOUT MIRACLES. IT IS THEN ARGUED THAT EACH OF THESE CRITICISMS COLLAPSES THE DISTINCTION BETWEEN PROBABILITIES PERTAINING TO EVENTS QUA UNIQUE OCCURRENCES AND PROBABILITIES (...) PERTAINING TO EVENTS QUA TOKENS INSTANTIATING EVENT-TYPES. (shrink)
In this article I criticize the non-consequentialist Weighted Lottery (WL) solution to the choice between saving a smaller or a larger group of people. WL aims to avoid what non-consequentialists see as consequentialism's unfair aggregation by giving equal consideration to each individual's claim to be rescued. In so doing, I argue, WL runs into another common objection to consequentialism: it is excessively demanding. WL links the right action with the outcome of a fairly weighted lottery, which means that an agent (...) can only act rightly if s/he has actually run the lottery. In many actual cases, this involves epistemic demands that can be almost impossible to meet. I argue that plausible moral principles cannot make such extreme epistemic demands. (shrink)
A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that is intrinsically (...) vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness. (shrink)
The lottery paradox plays an important role in arguments for various norms of assertion. Why is it that, prior to information on the results of a draw, assertions such as "My ticket lost" seem inappropriate? This paper is composed of two projects. First, I articulate a number of problems arising from Williamson's analysis of the lottery paradox. Second, I propose a relevant alternatives theory, which I call the Non-Destabilizing Alternatives Theory (NDAT), that better explains the pathology of asserting lottery propositions, (...) while permitting assertions of what I call fallible propositions such as "My car is in the driveway.". (shrink)
This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance (...) lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting. (shrink)
The sensitivity of expected utility choice to slight variations in the description of lotteries is considered. This sensitivity is allowed to influence actual choice in what is called the expected utility with perturbed lotteries model because the slight variations are used to represent vagueness regarding the dollar-prize, probability description of a lottery. Example illustrate how this sensitivity can affect actual choice for an otherwise expected utility decision-maker and provide an explanation for some of the anomolous evidence on risky (...) choice. (shrink)
In this imaginative and provocative book, Barbara Goodwin explores the question of how lottery systems can achieve egalitarian social justice in societies with seemingly ineradicable inequalities. She begins with the utopian fable of Aleatoria, a country not unlike our own in the not-too-distant-future, where most goods are distributed by lottery--even the right to have children. She then analyzes the philosophical arguments for and against lottery distribution and a comparison of "justice by lottery" with other contemporary theories of justice. Goodwin also (...) applies her theory to practical problems in the real world which could be--or have been--justly resolved by the use of lotteries, such as military drafts, jury duty, and immigration eligibility. She demonstrates that in many areas, including that of political power, a regular and random reallocation of goods would be a fairer and more democratic method than the distributive systems found in liberal democracies today. (shrink)
The literature concerning the possibility and desirability of using new pharmacological and possible future genetic techniques to enhance human characteristics is well-established and the debates follow some well-known argumentative patterns. However, one argument in particular stands out and demands attention. This is the attempt to tie the moral necessity of moral enhancement to the hypothesised risks that allowing cognitive enhancement will bring. According to Persson and Savulescu, cognitive enhancement should occur only if the risks they think it to poses are (...) mitigated by moral enhancement. By this they mean the compulsory and universal amplification of the disposition of altruism and the inflation of our sense of fairness, by chemical and/or genetic means. Their claim is important, intriguing and unsettling. This paper focuses on three central, but relatively neglected, features of their argument. First, there is a pernicious ambiguity in the language of ‘risk’ used by Persson and Savulescu where they tend to conflate ‘risk’ and ‘uncertainty’. Second, their use of the lottery analogy to render their position more plausible is unconvincing. It tends to distort rather than illuminate the relevant considerations. Third, Persson and Savulescu do not adequately take into account the social and individual benefits that enhancing cognition could have. If they did, it would be apparent that those benefits alone would outweigh the considerations used to justify accompanying CE with ME. (shrink)
: In some cases in which rational and moral agents experience moral uncertainty, they are unable to assign exact degrees of moral value—in a non-arbitrary way—to some of the different acts available to them, and so are unable to choose with certainty the best act. This article presents a new justification for the use of lotteries in this kind of situation. It is argued that sometimes the only rational thing for a morally motivated agent to do here is to (...) use a lottery. (shrink)
It is a platitude in epistemology to say that knowledge excludes luck. Indeed, if one can show that an epistemological theory allows ‘lucky’ knowledge, then that usually suffices to warrant one in straightforwardly rejecting the view. Even despite the prevalence of this intuition, however, very few commentators have explored what it means to say that knowledge is incompatible with luck. In particular, no commentator, so far as I am aware, has offered an account of what luck is and on this (...) basis identified what it means for a true belief to be non-lucky. It is just such a view that I propose, however, and I hope to give a flavour of what this strategy involves here. In particular, I have two goals in this paper. The first is to outline the general contours of the position and show how such a view can account for the attraction of adducing a safety condition on knowledge, with all the epistemic benefits that this principle holds. Relatedly, I will also explain how an anti-luck epistemology can assist us in determining the best formulation of this principle. The second goal of the paper is to show anti-luck epistemology in action by highlighting how such a view can deal with the various problems posed by lottery-style examples. (shrink)
The lottery paradox can be solved if epistemic justification is assumed to be a species of permissibility. Given this assumption, the starting point of the paradox can be formulated as the claim that, for each lottery ticket, I am permitted to believe that it will lose. This claim is ambiguous between two readings, depending on the scope of ‘permitted’. On one reading, the claim is false; on another, it is true, but, owing to the general failure of permissibility to agglomerate, (...) does not generate the paradox. The solution generalizes to formulations of the paradox in terms of rational acceptability and doxastic rationality. (shrink)
Abstract Several philosophers have claimed that S knows p only if S’ s belief is safe, where S's belief is safe iff (roughly) in nearby possible worlds in which S believes p, p is true. One widely held intuition many people have is that one cannot know that one's lottery ticket will lose a fair lottery prior to an announcement of the winner, regardless of how probable it is that it will lose. Duncan Pritchard has claimed that a chief advantage (...) of safety theory is that it can explain the lottery intuition without succumbing to skepticism. I argue that Pritchard is wrong. If a version of safety theory can explain the lottery intuition, it will also lead to skepticism. Content Type Journal Article Category Original Article Pages 1-26 DOI 10.1007/s10670-011-9305-z Authors Dylan Dodd, Department of Philosophy, Northern Institute of Philosophy, University of Aberdeen, Aberdeen, UK Journal Erkenntnis Online ISSN 1572-8420 Print ISSN 0165-0106. (shrink)
John Hawthorne’s recent monograph Knowledge and Lotteries1 is centred on the following puzzle: Suppose you claim to know that you will not be able to afford to summer in the Hamptons next year. Aware of your modest means, we believe you. But suppose you also claim to know that a ticket you recently purchased in a multi-million dollar lottery is a loser. Most of us have the intuition that you do not know that your ticket is a loser. However, your (...) alleged knowledge of not being able to afford to summer in the Hamptons puts you in a position to know that your ticket is a loser. For the proposition that you will not be able to afford to summer in the Hamptons entails the proposition that you will lose the lottery. And the following principle, what Hawthorne calls ‘Single Premise Closure’ ( p. 34), is very plausible: If you know that p, p entails q, and you competently deduce q from p thereby coming to believe that q (all the while retaining your knowledge of p), then you come to know q. (shrink)
Thomas Kroedel argues that we can solve a version of the lottery paradox if we identify justified beliefs with permissible beliefs. Since permissions do not agglomerate, we might grant that someone could justifiably believe any ticket in a large and fair lottery is a loser without being permitted to believe that all the tickets will lose. I shall argue that Kroedel’s solution fails. While permissions do not agglomerate, we would have too many permissions if we characterized justified belief as sufficiently (...) probable belief. If we reject the idea that justified beliefs can be characterized as sufficiently probably beliefs, Kroedel’s solution is otiose because the paradox can be dissolved at the outset. (shrink)
One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on (...) the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (shrink)
There are many ordinary propositions we think we know. Almost every ordinary proposition entails some lottery proposition which we think we do not know but to which we assign a high probability of being true (for instance:I will never be a multi-millionaire entails I will not win this lottery). How is this possible – given that some closure principle is true? This problem, also known as the Lottery puzzle, has recently provoked a lot of discussion. In this paper I discuss (...) one of the most promising answers to the problem: Stewart Cohens contextualist solution, which is based on ideas about the salience of chances of error. After presenting some objections to it I sketch an alternative solution which is still contextualist in spirit. (shrink)
I seem to know that I won't experience spaceflight but also that if I win the lottery, then I will take a flight into space. Suppose I competently deduce from these propositions that I won't win the lottery. Competent deduction from known premises seems to yield knowledge of the deduced conclusion. So it seems that I know that I won't win the lottery; but it also seems clear that I don't know this, despite the minuscule probability of my winning (if (...) I have a lottery ticket). So we have a puzzle. It seems to generalize, for analogues of the lottery-proposition threaten almost all ordinary knowledge attributions. For example, my apparent knowledge that my bike is parked outside seems threatened by the possibility that it's been stolen since I parked it, a proposition with a low but non-zero probability; and it seems that I don't know this proposition to be false. Familiar solutions to this family of puzzles incur unacceptable costs?either by rejecting deductive closure for knowledge, or by yielding untenable consequences for ordinary attributions of knowledge or of ignorance. After canvassing and criticizing these solutions, I offer a new solution free of these costs. Knowledge that p requires an explanatory link between the fact that p and the belief that p. This necessary but insufficient condition on knowledge distinguishes actual lottery cases from typical, apparently analogous ?quasi-lottery? cases. It does yield scepticism about my not winning the lottery and not experiencing spaceflight, but the scepticism doesn't generalize to quasi-lottery cases such as that involving my bike. (shrink)
Hume’s main argument against rational belief in miracles might seem to rule out rational belief in other antecedently improbable occurrences as well--for example, a certain person’s having won the lottery. Dorothy Coleman has recently defended Hume against the lottery counterexample, invoking Hume’s distinction between probability of chances and probability of causes. I argue that Coleman’s defence fails.
Fairness is a central, but under-theorized, notion in moral and political philosophy. This paper makes two contributions. Firstly, it criticizes Broome’s seminal account of fairness in ( 1990–1991 ) Proc Aristotelian Soc 91:87–101, showing that there are problems with restricting fairness to a matter of relative satisfaction and holding that it does not itself require the satisfaction of the claims in question. Secondly, it considers the justification of lotteries to resolve cases of ties between competing claims, which Broome claims (...) as support for his theory, and contrasts random procedures to contests of skill, which may also be considered lotteries in a broader sense. I offer no alternative account of fairness of my own, but hope to point the way for future research on the nature of fairness. (shrink)
John Hawthorne’s marvelous book contains a wealth of arguments and insights based on an impressive knowledge and understanding of contemporary discussion. We can address only a small aspect of the topic. In particular, we will offer our own answers to two questions about knowledge that he discusses.
This paper revisits a puzzle that arises for theories of knowledge according to which one can know on the basis of merely inductive grounds. No matter how strong such theories require inductive grounds to be if a belief based on them is to qualify as knowledge, there are certain beliefs (namely, about the outcome of fair lotteries) that are based on even stronger inductive grounds, while, intuitively, they do not qualify as knowledge. This paper discusses what is often regarded (...) as the most promising classical invariantist solution to the puzzle, namely, that beliefs about the outcomes of fair lotteries do not qualify as knowledge because they are too lucky to do so (or, relatedly, because they do not satisfy a safety condition on knowledge), while other beliefs based on potentially weaker inductive grounds are not too lucky (or, relatedly, because they are safe). A case is presented that shows that this solution to the puzzle is actually not viable. It is argued that there is no obvious alternative solution in sight and that therefore the puzzle still awaits a classical invariantist solution. (shrink)
This paper explores how the Bayesian program benefits from allowing for objective chance as well as subjective degree of belief. It applies David Lewis’s Principal Principle and David Christensen’s principle of informed preference to defend Howard Raiffa’s appeal to preferences between reference lotteries and scaling lotteries to represent degrees of belief. It goes on to outline the role of objective lotteries in an application of rationality axioms equivalent to the existence of a utility assignment to represent preferences (...) in Savage’s famous omelet example of a rational choice problem. An example motivating causal decision theory illustrates the need for representing subjunctive dependencies to do justice to intuitive examples where epistemic and causal independence come apart. We argue to extend Lewis’s account of chance as a guide to epistemic probability to include De Finetti’s convergence results. We explore Diachronic Dutch book arguments as illustrating commitments for treating transitions as learning experiences. Finally, we explore implications for Martingale convergence results for motivating commitment to objective chances. (shrink)
For several centuries, economists, sociologists, and philosophers have been concerned with the magnitude and e¤ects of inequality. Economists have concentrated on inequality in income and wealth, and have linked this inequality to social welfare, aggregate savings and investment, economic development, and other issues. They have explained the observed degree of inequality by the e¤ect of random shocks, inherited position, and inequality..