The allocation of vaccines and therapeutics for Covid‐19 obviously raises ethical questions, and physicians and ethicists have begun to address them. Writers have identified various criteria that should guide allocation decisions, but the criteria often conflict and need to be balanced against one another. This article proposes a model for thinking about how different considerations that are relevant to the distribution of vaccines and scarce treatments for Covid‐19 could be integrated into an allocation procedure. The model employs the construct of (...) a weighted lottery, which is a construct that has been employed in other contexts that involve the distribution of scarce resources. The article highlights the advantages of applying a weighted lottery to the Covid‐19 context and offers an illustration for how it might work in practice. The primary aim of the article is to articulate the structural features of a weighted lottery for this context and to bring out its advantages over other methods for allocating Covid‐19 medications. (shrink)

After providing a novel taxonomy of lottery procedures for fairly distributing scarce goods, I defend a new weighted lottery theory. This taxonomy is necessary because the debate between unweighted and weighted lottery theorists overlooks a range of cases, overlap cases, in which conducting an unweighted lottery is impossible or implausible. Therefore, to account for all such cases, lottery theorists must adopt a weighted lottery. However, while no extant weighted lottery (...) is adequate in overlap cases, my new weighted lottery theory is. I conclude by discussing my theory’s practical implications. (shrink)

Faced with a choice between saving one stranger and saving a group of strangers, some people endorse weighted lotteries, which give a strictly greater chance of being saved to the group of strangers than the single stranger. In this paper I attempt to criticize this view. I first consider a particular version of the weighted lotteries, Frances Kamm's procedure of proportional chances, and point out two implausible implications of her proposal. Then, I consider weighted lotteries in general, (...) and claim (1) that the correct thing to distribute is not the chance of being saved but the good of being saved, (2) that assigning some chance to the single stranger is not the only way to give a positive (and equal) respect to the people concerned, and (3) that the weighted lottery appears to be deceptive since it would show the respect to the single stranger in a negligible way. (shrink)

Shortages of new therapeutics to treat coronavirus disease (COVID-19) have forced clinicians, public health officials, and health systems to grapple with difficult questions about how to fairly allocate potentially life-saving treatments when there are not enough for all patients in need (1). Shortages have occurred with remdesivir, tocilizumab, monoclonal antibodies, and the oral antiviral Paxlovid (2) -/- Ensuring equitable allocation is especially important in light of the disproportionate burden experienced during the COVID-19 pandemic by disadvantaged groups, including Black, Hispanic/Latino and (...) Indigenous communities, individuals with certain disabilities, and low-income persons. However, many health systems have resorted to first-come, first-served approaches to allocation, which tend to disadvantage individuals with barriers in access to care (3). There is mounting evidence of racial, ethnic, and socioeconomic disparities in access to medications for COVID-19 (4, 5). -/- One potential method to promote equitable allocation is to use a weighted lottery, which is an allocation strategy that gives all eligible patients a chance to receive the scarce treatment while also allowing the assignment of higher or lower chances according to other ethical considerations (6). We sought to assess the feasibility of implementing a weighted lottery to allocate scarce COVID-19 medications in a large U.S. health system and to determine whether the weighted lottery promotes equitable allocation. (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)

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)

If we can save the lives of only one of multiple groups of people, we might be inclined simply to save whichever group is largest. We may worry, though, that automatically saving the largest group fails to take each saveable individual sufficiently into account, offering some of these individuals no chance at all of being rescued. Still wanting to give larger groups higher chances of survival, we may then say that we ought to employ a proportionally weighted lottery (...) to determine which group to save. In this paper, I argue that this would be a mistake. Given the most plausible way of specifying it, the weighted-lottery view itself fails to treat each saveable individual with equal moral respect. (shrink)

I argue that Schmidtet al, while correctly diagnosing the serious racial inequity in current ventilator rationing procedures, misidentify a corresponding racial inequity issue in alternative ‘unweighted lottery’ procedures. Unweighted lottery procedures do not ‘compound’ (in the relevant sense) prior structural injustices. However, Schmidtet aldo gesture towards a real problem with unweighted lotteries that previous advocates of lottery-based allocation procedures, myself included, have previously overlooked. On the basis that there are independent reasons to prefer lottery-based allocation of (...) scarce lifesaving healthcare resources, I develop this idea, arguing that unweighted lottery procedures fail to satisfy healthcare providers’ duty to prevent unjust population-level health outcomes, and thus that lotteries weighted in favour of Black individuals (and others who experience serious health injustice) are to be preferred. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:328 HUME, MIRACLES AND LOTTERIES This paper addresses recent criticisms of Hume's skepticism with regard to miracles, by 1 2 Sorensen and Hambourger who argue that there are counterexamples, illustrated by lotteries, to Hume's account of how the truth of reports of improbable events (either first or second hand) must be evaluated. They believe these counterexamples are sufficient to prove that Hume's argument against the believability of miracles, defined (...) as violations of laws of nature caused by God, is unsound. Their arguments merit consideration not only in their own right but also on the basis of historical precedent, since they have in common an assumption that is found in Butler's criticism of Hume's predecessors in this debate. The bulk of my paper deals with Hambourger, who presents the most detailed version of the 'Butler' criticism. Sorensen's version can be answered in light of the evaluation of Hambourger 's argument. Hume's Argument Hume's argument is not against the possibility of miracles but against the possibility that it could ever be reasonable to believe a miracle 4 had occurred. His argument can be summed up as follows: in judging the credibility of a report of an improbable event, the probability of the event must be weighed against the probability that the report could be mistaken. The probability of an event is determined by the degree to which it conforms to known laws of nature. Laws of nature are formulated on the basis of uniform experience. Assuming that a miracle is a violation of laws of nature, any judgment that an event is miraculous presupposes a 329 judgment that there exists a uniform pattern of causation to which it is opposed, which pattern would constitute empirical proof against the miracle's occurrence. Testimony, on the other hand, is shown by experience not to be uniformly reliable. Consequently, no report of a miracle will have credibility since there will always be a uniform experience that is full proof against it, which therefore outweighs whatever merits testimony may have. In Hume's words, [N]o testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavors to establish.... When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened.... If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till- then, can he pretend to command my belief or opinion (E 115-116). Hambourger 's Criticism Hambourger maintains that Hume's argument depends on a principle which he calls the "principle of relative likelihood." He restates this principle as follows: Suppose that someone or, perhaps, a group of people testify to the truth of a proposition P that, considered by itself, is improbable. Then to evaluate the testimony, one must weight (sic) the probability that P is true against the probability that the informants are lying or mistaken. If it is more likely that P is true than that the informants are lying or mistaken, then, on balance, the 330 testimony renders P more likely than not, and it may be reasonable for one to believe that P. However, if it is as likely, or even more likely, that the informants are lying or mistaken than it is that P is true, then, on balance, the testimony does not render P more likely true than false, and it would not be,reasonable for one to believe that P. As a counterexample to this principle, Hambourger asks us to suppose that a lottery is held in which there are one million entrants, each of whom has a one in a million chance of winning, and that a reliable newspaper, the New York Times, reports that the winner of the lottery is Smith. He further asks us to suppose that the Times errs only once out of every 10,000 times. Thus, the probability that the Times ' report is wrong is... (shrink)

Numerous instruments have been developed to elicit numerical values that represent the strength of preference for different health states. However, relatively few studies have attempted to analyse the reasoning processes that people employ when they are asked to answer questions based on these elicitation methods. The lottery equivalents method is a preference elicitation instrument that has recently received some attention in the literature. This study attempts a qualitative analysis of the use of this instrument on a group of 25 (...) relatively highly educated respondents. For each of three health states considered in the study, a substantial number of respondents refused to trade the chance of survival for a possible improvement in the health state. Therefore, many respondents violated an assumption that is necessary for the lottery equivalents instrument to generate cardinal health state values. These findings place a question mark against the usefulness of the lottery equivalents method, and add weight to the suspicion that ‘preferences’ are constructed according to how questions are framed. (Published Online July 31 2007). (shrink)

We develop a model of Disappointment in which disappointment and elation arise from comparing the outcome received, not with an expected value as in previous models, but rather with the other individual outcomes of the lottery. This approach may better reflect the way individuals are liable to experience disappointment. The model obtained accounts for classic behavioral deviations from the normative theory, offers a richer structure than previous disappointment models, and leads to a Rank-Dependent Utility formulation in a transparent way. (...) Thus, our disappointment model may provide a clear psychological rationale for the subjective transformation of probabilities. (shrink)

John Broome has proposed a theory of fairness according to which fairness requires that agents’ claims to goods be satisfied in proportion to the relative strength of those claims. In the case of competing claims for a single indivisible good, Broome argues that what fairness requires is the use of a weighted lottery as a surrogate to satisfying the competing claims: the relative chance of each claimant's winning the lottery should be set to the relative strength of (...) each claimant's claim. In this journal, James Kirkpatrick and Nick Eastwood have objected that the use of weighted lotteries in the case of indivisible goods is unacceptable. In this article, I explain why Kirkpatrick and Eastwood's objection misses its mark. (shrink)

In “Weighing Up Weighted Lotteries: Scarcity, Overlap Cases, and Fair Inequalities of Chance”, Gerard Vong addresses intriguing problems in which it is impossible to give an equal chance of receiving a good to a set of equal claimants, because goods can be distributed only via groups which have overlapping membership. Vong proposes a rule for distributing chances that he argues is sensitive to both comparative and absolute fairness. This comment discusses some formal difficulties with Vong’s proposal and argues that (...) it is better to separate the two concerns and to allow the tradeoffs between them to depend on characteristics of the goods to be distributed. (shrink)

John Broome argues that fairness requires that claims are satisfied in proportion to their strength. Broome holds that, when distributing indivisible goods, fairness requires the use of weighted lotteries as a surrogate to satisfy proportionally each candidate's claims. In this article, we present two arguments against Broome's account of fairness. First, we argue that it is almost impossible to calculate the weights of the lotteries in accordance with the requirements of fairness. Second, we argue that Broome rules out those (...) methods whose use might provide some resolution to this problem. From these arguments, we conclude that, contra Broome, fairness does not require the proportional satisfaction of claims. (shrink)

You must either save a group of m people or a group of n people. If there are no morally relevant diff erences among the people, which group should you save? is problem is known as the number problem. e recent discussion has focussed on three proposals: (i) Save the greatest number of people, (ii) Toss a fair coin, or (iii) Set up a weighted lottery, in which the probability of saving m people is m / m + (...) n , and the probability of saving n people is n / m + n . is contribution examines a fourth alternative, the mixed solution, according to which both fairness and the total number of people saved count. It is shown that the mixed solution can be defended without assuming the possibility of interpersonal comparisons of value. (shrink)

When faced with multiple claims to a particular good, what does distributive justice require? To answer this question, we need a substantive moral theory that will enable us assign relative moral weights to the parties' claims. But this is not all we need. Once we have assessed the moral weight of each party's claim, we still need to decide what method of distribution to employ, for there are two methods open to us. We could take the winner-take-all approach, and award (...) the good to the party with the strongest claim. On the other hand, we could divide the good proportionally, according to the relative strength of each party's claim. Because the choice between these two methods of distribution can have a dramatic impact on the resulting pattern of distribution, the choice presents a question of justice. But this is a question of justice that is often overlooked. As a result, we currently employ the principle of proportionality far less often than justice actually requires. If we focus on the question of distributive method, however, we are not only better able to understand how certain reasons enter into our all-things-considered moral judgments, we are also able to explain some perplexing but common aspects of our moral beliefs: how rights can be said to have peremptory force, yet still be balanced against other important interests; how justice can sometimes require compromise, yet sometimes require victory; and how a moral theory can avoid being too demanding while still being demanding enough. Key Words: autonomy • equality • indivisibility • inviolability • weighted lotteries • rights. (shrink)

Philosophical theories of fairness propose to divide a good that several individuals have a claim to in proportion to the strength of their respective claims. We suggest that currently, these theories face a dilemma when dealing with a good that is indivisible. On the one hand, theories of fairness that use weighted lotteries are either of limited applicability or fall prey to an objection by Brad Hooker. On the other hand, accounts that do without weighted lotteries fall prey (...) to three fairness paradoxes. We demonstrate that division methods from apportionment theory, which has hitherto been ignored by philosophical theories of fairness, can be used to provide fair division for indivisible goods without weighted lotteries and without fairness paradoxes. (shrink)

There is ongoing debate on how to fairly allocate scarce critical care resources to patients with COVID-19. The debate revolves around two views: those who believe that priority for scarce resources should primarily aim at saving the most lives (SML) or at saving the most life-years, and those who believe that public health should focus on health equity to address health disparities and social determinants of health. I argue that maximizing medical outcomes by saving the greatest number of patients is (...) not a plausible strategy for combating COVID-19. There are reasons of fairness to give each patient who can meet general eligibility requirements a chance of being saved. Rather than focusing on outcome maximization, a better solution would be the individualist lottery that takes account of probability of survival and duration of treatment. Although the individualist lottery allocates scarce resources in a fair way that is responsive to health equity concerns, it still gives considerable weight to the concern of SML. Thus, this procedure can be reasonably accepted by all key stakeholders. (shrink)

What is required for something to be evidence for a hypothesis? In this fascinating, elegantly written work, distinguished philosopher of science Peter Achinstein explores this question, rejecting typical philosophical and statistical theories of evidence. He claims these theories are much too weak to give scientists what they want--a good reason to believe--and, in some cases, they furnish concepts that mistakenly make all evidential claims a priori. Achinstein introduces four concepts of evidence, defines three of them by reference to "potential" evidence, (...) and characterizes the latter using a novel epistemic interpretation of probability. The resulting theory is then applied to philosophical and historical issues. Solutions are provided to the "grue," "ravens," "lottery," and "old-evidence" paradoxes, and to a series of questions. These include whether explanations or predictions furnish more evidential weight, whether individual hypotheses or entire theoretical systems can receive evidential support, what counts as a scientific discovery, and what sort of evidence is required for it. The historical questions include whether Jean Perrin had non-circular evidence for the existence of molecules, what type of evidence J. J. Thomson offered for the existence of the electron, and whether, as is usually supposed, he really discovered the electron. Achinstein proposes answers in terms of the concepts of evidence introduced. As the premier book in the fabulous new series Oxford Studies in Philosophy of Science, this volume is essential for philosophers of science and historians of science, as well as for statisticians, scientists with philosophical interests, and anyone curious about scientific reasoning. (shrink)

A surprisingly large number of lottery prizes go unclaimed every year. This leads us to suspect that what people bet on is not only money, but also good mood. We conducted three studies to explain, from an emotional perspective, why people play lottery games. We first conducted two survey studies to assess mood state reported by online (Study 1a) and offline lottery buyers (Study 1b) at different stages of lottery play. The results revealed that participants’ highest (...) mood appeared before knowing whether they had won. In Study 2, we manipulated the means of reward (lottery tickets vs. cash) and compared participants’ mood changes at different stages of a rewards game in the laboratory. We found the following: first, lottery group participants were generally in a better mood; second, 42% of lottery group participants did not come to the laboratory to collect scratch cards; and third, lottery group participants took more time to return to the laboratory to check their tickets than participants in the cash group. In Study 3, we examined whether priming good or bad mood could influence participants’ preferences for cash versus lottery tickets. The results revealed that participants who were primed for poor mood had a higher preference for lottery tickets compared with their good mood counterparts. These findings suggest that what our participants sought in lottery play was not only money, but improved mood. (shrink)

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to (...) include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions. (shrink)

Suppose we can save either a larger group of persons or a distinct, smaller group from some harm. Many people think that, all else equal, we ought to save the greater number. This article defends this view (with qualifications). But unlike earlier theories, it does not rely on the idea that several people's interests or claims receive greater aggregate weight. The argument starts from the idea that due to their stakes, the affected people have claims to have a say in (...) the rescue decision. As rescuers, our primary duty is to respect these procedural claims, which we must do by doing what these people would decide, in a process where each is given an equal vote on the matter. So in cases where each votes in their own self-interest, respect for their equal right to decide, or their autonomy, will lead us to save the greater number. The argument is explained in detail, with special attention to the questions of how, exactly, it avoids aggregation, and of why majority rule is superior to lottery procedures. The view has further advantages. Especially, it explains the “partial” relevance of numbers in cases involving unequal harms, and it does so in a way that dissolves the appearance of paradox that besets theories of “partial aggregation.”. (shrink)

Crafted within a knowledge-first epistemological framework, Mona Simion’s engaging and wide-ranging work ensures that both the Knowledge Norm of Assertion (KNA) and Classical Invariantism (CI) can be part of a viable and productive research program.Dissatisfied with current strategies on offer in the literature, she successfully counters objections to the pair sourced in “shiftiness intuitions”—intuitions that seem to indicate that mere changes in practical context can impact the propriety of assertions and knowledge attributions. For example, in Keith DeRose’s famous pair of (...) low stakes versus high stakes bank cases, the consequences of Keith’s acting on The bank is open on Saturday if it were false change from trivial in low stakes to catastrophic in high stakes. We are to suppose that the proposition is true, and that Keith has access to the same quantity and quality of evidence for it in both cases. In low stakes, but not in high stakes, we are inclined to think that Keith can appropriately assert to his inquiring wife “The bank is open on Saturday.” In low stakes, Keith’s knowledge ascription “I know that the bank is open on Saturday” strikes us as aptly asserted and true. In high stakes only a knowledge denial on Keith’s part strikes us as aptly asserted and true.If we want to hang on to KNA, it looks like we will have to abandon CI and concede that ‘knows’ or knowledge is sensitive to changes in practical considerations. If we want to retain CI, we can try to say that, while Keith does still know that the bank is open on Saturday in high stakes, he does not have sufficient warrant to properly assert that it is. But such a move seems to run counter to KNA. We appear to be stuck in the Shiftiness Dilemma.Keen to get us out of the dilemma and to preserve the idea that epistemically good thinking and asserting are independent of practical concerns, Simion suggests a strategy that can also be used to protect other epistemic speech-act norms and notions from similar threats of practical shiftiness. She notes that having an impact on the degree of epistemic warrant required is not enough to make a norm an epistemic one. Fair enough. If I am given strong practical reason (a million dollars, or a gun to the head) not to adopt full belief until I have gathered more evidence, there is potentially some nonepistemic norm at work in the demand for further evidence. So it is a live possibility (indeed, a plausible one, as Simion would say) that our intuitions in the Shiftiness Dilemma are being misdescribed: they are tracking all-things-considered propriety of assertion, not epistemic propriety. In high stakes, practical norms override the epistemic norm KNA. Keith’s asserting “The bank is open on Saturday” would be epistemically proper but all-things-considered improper.Simion backs up her account with an etiological-function origin story and typology for the norms of assertion. Here is her characterization of etiological function for traits, artifacts, and actions. E-function.A token of type T has the e-function of type B of producing effect E in a system S iff (1) tokens of T produced E in the past, (2) producing E resulted in benefit of type B in S/S’s ancestors, and (3) producing E’s having B-benefited S’s ancestors contributes to the explanation of why T exists in S.In this schema, norms governing an action are typed by the functions or purposes of the action, which in turn are typed by the benefits the action delivers. The epistemic norm of assertion will be the one that characteristically and reliably enables participants in the practice of assertion to secure its primary epistemic good, which Simion identifies as knowledge (testimonial knowledge in hearers generated under normal conditions by the assertions of knowledgeable speakers). Moves in a practice either directly or indirectly aim at fulfilling the goal of the practice. Assertion is a move in the practice of inquiry, which has the generation of knowledge as its aim. Hence, Simion concludes, assertion aims at generating knowledge.This account helps to explain why, in cases like high stakes, Keith all-things-considered ought not to assert “The bank is open on Saturday,” even though he knows it is true. Prudential norms of assertion override the epistemic KNA, for when multiple functions come into conflict, the ones that are more important to the survival of the organism will generate weightier normative constraints.Simion then applies her account to the general category of constatives—for example, speech acts that express a speaker’s belief and the intention that the hearer form or continue to hold the like belief. Employing the taxonomy of Bach and Harnish, she argues that, as expressions of belief and hence varieties of assertion, constatives are subject to knowledge norms as well. Bach and Harnish identify fifteen different subcategories of constatives, each characterized in terms of the kind of belief and intention they express. The norms governing a species have to be at least as strong as the norms governing the corresponding type. As species of the type assertion, each subcategory, then, is subject to at least the necessary direction of KNA. Ultimately, we arrive at a knowledge norm for every speech act on the list, such as The Knowledge Norm of Conjecture (KNC):One’s conjecture that p is epistemically permissible only if one knows that there is reason, but not sufficient reason, to believe that p. Most of the proposed interpretations of various constatives are on the whole reasonable enough, though J. L. Austin would surely be unhappy to see so many different speech acts fashioned into varieties of assertion. Simion concludes with reflections on conjecturing, telling, and moral assertion.Construing the challenge of shiftiness intuitions in terms of the Shiftiness Dilemma allows Simion to declare success when the compatibility of KNA and CI with the relevant data of intuition is secured. Ensuring empirical adequacy for an explanatory framework is an important step, but pronouncements of victory by way of inference to the best explanation would be premature.Other candidates have been shown to match KNA’s familiar explanatory successes (involving lottery propositions, Moorean statements, etc.). Unlike KNA, alternatives that require only epistemically justified or rational belief or credence can allow that a subject is epistemically proper to assert what they believe when their belief is epistemically rational, even under conditions where there are defeating reasons that they are not in a position to be aware of. Furthermore, such norms seem better suited for philosophical or heated moral discussions where there is little consensus and individuals routinely fail to know that what they assert is true. Simion sounds the alarm that the Shiftiness Dilemma threatens to generalize “to epistemic normativity as a whole” (xi–xii), but norms concerning rational credence seem particularly immune.As Simion notes, the etiological-function model of norms that she proposes could be rebuilt with a different central epistemic aim at its core. Indeed, Millikan-inspired models along these lines are appealed to in other areas in philosophy where the concept of information takes center stage—a concept that may be more amenable to rigorous definition and use in dialogue with the sciences than knowledge. We can wonder, also, what special weight the etiological-function model accords to a knowledge-first perspective on epistemic normativity when the model is malleable in this way.The effectiveness of Simion’s strategy for shielding epistemically good thinking and assertion from practical shiftiness becomes less apparent when we try to apply it to Keith’s belief or judgment that the knowledge ascription/denial he asserts is true. It is unclear whether we are to suppose that, in high stakes, Keith epistemically ought to believe/judge that his asserted knowledge denial is false, but all-things-considered he ought to believe/judge that it is true. We do not have to accept that the knowledge-first, etiological function account of normativity for assertion carries over to belief, of course, but then what progress have we made in accounting for the shiftiness in Keith’s judgment that he knows the bank is open on Saturday in low stakes but not in high stakes?Furthermore, there are other ways to generate problems for fallibilist Classical Invariantists in low versus high stakes pairs of cases. Here is a simplified version of a puzzle from the debate over pragmatic encroachment. Consider low cases where you meet the invariant standards for knowing that p, but your justified confidence falls short of certainty. Among these cases, there must be some where you also meet the invariant standards for knowing that if p is true, then among your choices of action ϕ-ing results in the best overall outcome. Say, for instance, that you have a machine with a ϕ button, and your machine, which you know to be extremely reliable, tells you what the cost or benefit will be if you push the button and p is true versus if p is false. If you ϕ (push the ϕ button) and p is true, you win one hundred dollars; if you ϕ and p is false, you lose one hundred dollars. You face the choice: to ϕ or not to ϕ. Since you know that p, and you know that if p is true, then among your choices of action ϕ-ing results in the best overall outcome, you rationally conclude that ϕ-ing results in the best overall outcome, and hence that you ought to ϕ. If you know which choice of action has the best overall outcome, that is what you all-things-considered should do. Now consider high, where the quantity and quality of your evidence for p is the same, and you still know that your machine is extremely reliable. This time, however, the machine tells you that if you push ϕ and p is true, you win one hundred dollars, but if you push ϕ and p is false, you lose $1 million. Your rational confidence in p is not high enough to justify pushing the button, so you rationally conclude that you ought not to ϕ. Since in high nothing evidentially relevant has changed (all that has changed is the machine informing you of a new, more serious cost of acting on p if p is false), the classical invariantist will have to say that you still know that ϕ, and you still know that if p is true, then among your choices of action ϕ-ing results in the best overall outcome. So, according to CI, you should still be able to reason properly as follows: you know that p, and you know that if p is true, then among your choices of action ϕ-ing results in the best overall outcome, so you know that ϕ-ing results in the best overall outcome, hence you ought to ϕ. Classical invariantists owe us an explanation concerning how we can avoid the paradox here without conceding that knowledge is practically shifty. (shrink)

We present an experiment in which we add a common delay in a choice between two risky prospects. The results show that delay produces the same change in preferences as in the well-documented common ratio effect in risky lotteries. The added common delay acts as if the probabilities were divided by some common ratio. Moreover, we show that there is a strong magnitude effect, in the sense that the effect of delay depends on the magnitude of the outcome. The results (...) are consistent with the recently introduced probability time trade-off (PTT) model by Baucells and Heukamp. We present a parameterization of the model based on the experimental results, showing that the value function exhibits increasing relative risk aversion, the weighting function is s-shaped, and the intrinsic discount rate is decreasing. (shrink)

The majority consensus in the empirical literature is that probability weighting functions are typically inverse-S shaped, that is, people tend to overweight small and underweight large probabilities. A separate stream of literature has reported event-splitting effects and shown that they can explain violations of expected utility. This leads to the questions whether the observed shape of weighting functions is a mere consequence of the coalesced presentation and, more generally, whether preference elicitation should rely on presenting lotteries in a canonical split (...) form instead of the commonly used coalesced form. We analyze data from a binary choice experiment where all lottery pairs are presented in both split and coalesced forms. Our results show that the presentation in a split form leads to a better fit of expected utility theory and to probability weighting functions that are closer to linear. We thus provide some evidence that the extent of probability weighting is not an ingrained feature, but rather a result of processing difficulties. (shrink)

Prospect theory predicts a four-fold risk attitude, which means that people are risk seeking for low-probability gain and high-probability loss and risk averse for low-probability loss and high-probability gain because they overweight probability when it is low. The four-fold pattern of risk attitude has been supported by several former studies with mainstream industrialized populations but has never previously been tested in a non-industrialized society. In this work, we examined the robustness of the four-fold risk attitude in the DuLong minority ethnicity (...) in China, which is a small society with only 4,000 members that is isolated from modern civilization. We used simple lotteries for gain and loss with different probabilities to elicit the risk attitude of 37 DuLong villagers. Our results support prospect theory predictions in that DuLong people are risk seeking for low-probability gain and risk averse for low-probability loss. However, although they showed a tendency to decrease their degree of risk seeking for gain, their risk attitude did not reverse when the probability of the prospect increased to 50%. In summary, our results suggest a right-shifted weighting function in this non-industrialized small society. The deviation might be caused by the particular living situation of the DuLong people, their sensitivity to monetary payoffs, and the elicitation procedure. (shrink)

The article presents an Internet experiment where subjects sequentially bid for basic gifts and binary-lotteries on these gifts in incentive compatible Vickrey auctions. Subjects exhibit uniformly pessimistic prize-weighting in spite of precautions to reduce suspicion and prohibit collusion. The bids for lotteries are close to the minimal payable value, even when the probability of obtaining a better prize is larger than 50%. Prize-weighting becomes even more conservative as the distance in value of payable prizes increases. The twofold aversive affect appears (...) for three distinct groups of students; we demonstrate, however, that the same subjects overweight small win-probabilities in standard binary-choice. (shrink)

Discounted utility theory and its generalizations use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity–concavity of discount function. This test can be used with any utility function and any preferences over risky lotteries. The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function.

According to the ‘Description–Experience gap’, when people are provided with the descriptions of risky prospects they make choices as if they overweight the probability of rare events; but when making decisions from experience after exploring the prospects’ properties, they behave as if they underweight such probability. This study revisits this discrepancy while focusing on information-search in decisions from experience. We report findings from a lab-experiment with three treatments: a standard version of decisions from description and two versions of decisions from (...) experience: with and without a ‘history table’ recording previously sampled events. We find that people sample more from lotteries with rarer events. The history table proved influential; in its absence search is more responsive to cues such as a lottery’s variance while in its presence the cue that stands out is the table’s maximum capacity. Our analysis of risky choices captures a significant DE gap which is mitigated by the presence of the history table. We elicit probability weighting functions at the individual level and report that subjects overweight rare events in experience but less so than in description. Finally, we report a measure that allows us to compare the type of DE gap found in studies using choice patterns with that inferred through valuation and find that the phenomenon is similar but not identical across the two methods. (shrink)

In Kőszegi and Rabin’s reference-dependent model of preferences, the chance of obtaining a better outcome can reduce an agent’s expected utility through an increase in the stochastic reference point. This means that individuals may prefer stochastically dominated lotteries. In this sense, hope, understood as a small probability of a better outcome, can be a curse. While Kőszegi and Rabin focus on a linear specification of the utility function, we show that this effect occurs more broadly. Using fairly plausible assumptions and (...) parameter values, we specify the conditions under which it occurs, as well as the type of lotteries in which this should be expected. We then show that while a simple subjective transformation of probability into weights of the reference point may in some cases mitigate the issue, in others, it can intensify it or even generate new ones. Finally, we extend the model by adding the individual’s current reference point to the stochastic reference point. We show that this modification can reconcile Kőszegi and Rabin’s model with the apparent empirical infrequency of stochastically dominated choices while maintaining its main qualitative results. (shrink)

To understand how small business entrepreneurs respond to government policy one has to know their risk and time preferences. Are they risk averse, or have high discount rates, such that they are hard to motivate? We have conducted a set of field experiments in Denmark that will allow a direct characterization of small business entrepreneurs in terms of these traits. We build on experimental tasks that are well established in the literature. The results do not suggest that small business entrepreneurs (...) are more or less risk averse than the general population under the assumption of Expected Utility Theory. However, we generally find an S-shaped probability weighting function for both small business entrepreneurs and non-entrepreneurs, with entrepreneurs being more optimistic about the chance of occurrence for the best outcome in lotteries with real monetary outcomes. The results also point to a significant difference in individual discount rates between entrepreneurs and non-entrepreneurs: entrepreneurs are willing to wait longer for certain rewards than the general population. (shrink)

When individuals cannot make up their mind, they sometimes use a random decision-making aid such as a coin to make a decision. This aid may also elicit affective reactions: A person flipping a coin may (dis)like the outcome, and thus decide according to this feeling. We refer to this process as catalysing decisions and to the aid as catalyst. We investigate whether using a catalyst may not only elicit affect but also result in more affect-based decision making. We used different (...) online studies that examine affect-driven decisions by investigating scope insensitivity (indirect behavioural measure) and self-reported weight given to feelings versus reasons in hypothetical donation decisions. Study 1a showed that a catalyst (a lottery wheel) lead to more scope insensitive (i.e. affect-driven) donations. Study 1b included several changes and did not replicate these results. Study 2 (preregistered) examined scope insensitivity but did not replicate previous results; Study 3 (preregistered) looked at the weight given to feelings versus reason. Although catalyst (compared to control) participants descriptively reported relying more on feelings, this difference did not reach significance. In contrast to lay beliefs, results do not indicate support for the hypothesis that using a catalyst results in more affect-based hypothetical donation decisions. (shrink)

Risk attitude is known to be a key determinant of various economic and financial choices. Behavioral studies that aim to evaluate the role of risk attitudes in contexts of this type, therefore, require tools for measuring individual risk tolerance. Recent developments in decision theory provide such tools. However, the methods available can be time consuming. As a result, some practitioners might have an incentive to prefer “fast and frugal” methods to clean but more costly methods. In this article, we focus (...) on a tractable procedure initially proposed by Holt and Laury (2002) to elicit risk attitude. We generalize this method to measure utility and risk aversion as follows. First, we allow measurement of probabilistic risk attitude through violations of expected utility due to probability weighting. Second, we use the outcome scale rather than the probability scale in the menu of choices. Third, we compare sure payoffs with lotteries instead of comparing non-degenerate lotteries. A within-subject experimental study illustrates the gains in tractability and bias minimization that can result from such an extension. (shrink)

Au xviiie siècle, la figure insistante de la «femme philosophe» s'articule à un imaginaire ambivalent de la différence des sexes, entre hantise d'une confusion délétère et quête d'un modèle d'harmonie. La femme travestit-elle la philosophie? Les Lumières ont-elles un genre?

The lottery paradox shows seemingly plausible principles of rational acceptance to be incompatible. It has been argued that we shouldn’t be concerned by this clash, since the concept of (categorical) belief is otiose, to be supplanted by a quantitative notion of partial belief, in terms of which the paradox cannot even be formulated. I reject this eliminativist view of belief, arguing that the ordinary concept of (categorical) belief has a useful function which the quantitative notion does not serve. I (...) then propose a solution to the paradox it engenders. (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)

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 Timothy Williamson’s analysis of the lottery paradox. Second, I propose a relevant alternatives theory, which I call the Non-Destabilizing Alternatives Theory , 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.” Le paradoxe de la loterie joue un rôle important dans l’argumentation visant à défendre diverses normes de l’assertion. Comment se fait-il que, avant que les résultats d’un tirage soient connus, des assertions comme «Mon billet a perdu» semblent inappropriées? Cet article se compose de deux projets. Premièrement, je relève certains problèmes issus de l’analyse du paradoxe de la loterie par Timothy Williamson. Deuxièmement, je propose une théorie des alternatives pertinentes que j’appelle la «théorie des alternatives non-déstabilisantes» , et qui explique d’une meilleure façon la pathologie de l’assertion de propositions concernant la loterie, tout en permettant des assertions faillibles, telles que «Ma voiture est dans l’entrée». (shrink)

We talk and think about our beliefs both in a categorical and in a graded way. How do the two kinds of belief hang together? The most straightforward answer is that we believe something categorically if we believe it to a high enough degree. But this seemingly obvious, near-platitudinous claim is known to give rise to a paradox commonly known as the 'lottery paradox' – at least when it is coupled with some further seeming near-platitudes about belief. How to (...) resolve that paradox has been a matter of intense philosophical debate for over fifty years. This volume offers a collection of newly commissioned essays on the subject, all of which provide compelling reasons for rethinking many of the fundamentals of the debate. (shrink)

In a recent article, Douven and Williamson offer both (i) a rebuttal of various recent suggested sufficient conditions for rational acceptability and (ii) an alleged ‘generalization’ of this rebuttal, which, they claim, tells against a much broader class of potential suggestions. However, not only is the result mentioned in (ii) not a generalization of the findings referred to in (i), but in contrast to the latter, it fails to have the probative force advertised. Their paper does however, if unwittingly, bring (...) us a step closer to a precise characterization of an important class of rationally unacceptable propositions—the class of lottery propositions for equiprobable lotteries. This helps pave the way to the construction of a genuinely lottery-paradox-proof alternative to the suggestions criticized in (i). (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 shows that the following three individually highly plausible theses are jointly incompatible: highly probable propositions are justifiably believable, justified believability is closed under conjunction introduction, known contradictions are not justifiably believable. This paper argues that a satisfactory solution to the lottery paradox must reject as versions of the paradox can be generated without appeal to either or and proposes a new solution to the paradox in terms of a novel account of justified believability.

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)