Nebel argues for the Repugnant Conclusion via the “Intrapersonal Repugnant Conclusion,” on which certainty of a mediocre life is better for individuals than a sufficiently small chance of an excellent life. In this article, I deny that acceptance of the Intrapersonal Repugnant Conclusion leads us to the Repugnant Conclusion. I point out that on many views which avoid the Repugnant Conclusion we should discount very small probabilities down to zero. If we do, then Nebel’s crucial premise of Ex (...) Ante Pareto fails, because discounting at the individual level can fail to match up with discounting at the population level. (shrink)

When probability discounting (or probability weighting), one multiplies the value of an outcome by one's subjective probability that the outcome will obtain in decision-making. The broader import of defending probability discounting is to help justify cost-benefit analyses in contexts such as climate change. This chapter defends probability discounting under risk both negatively, from arguments by Simon Caney (2008, 2009), and with a new positive argument. First, in responding to Caney, I argue that (...) small costs and benefits need to be evaluated, and that viewing practices at the social level is too coarse-grained. Second, I argue for probability discounting using a distinction between causal responsibility and moral responsibility. Moral responsibility can be cashed out in terms of blameworthiness and praiseworthiness, while causal responsibility obtains in full for any effect which is part of a causal chain linked to one's act. With this distinction in hand, unlike causal responsibility, moral responsibility can be seen as coming in degrees. My argument is, given that we can limit our deliberation and consideration to that which we are morally responsible for and that our moral responsibility for outcomes is limited by our subjective probabilities, our subjective probabilities can ground probability discounting. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. (...) Much of the discussion on Independence generalizes to other views that also violate Independence. (shrink)

The topic of this thesis is how we should treat tiny probabilities of vast value. This thesis consists of six independent papers. Chapter 1 discusses the idea that utilities are bounded. It shows that bounded decision theories prescribe prospects that are better for no one and worse for some if combined with an additive axiology. Chapter 2, in turn, points out that standard axiomatizations of Expected Utility Theory violate dominance in cases that involve possible states of zero probability. Chapters (...) 3–6 discuss the idea that we should ignore tiny probabilities in practical decision-making. Chapter 3 argues that discounting small probabilities solves the ‘Intrapersonal Addition Paradox’ and thus helps avoid the Repugnant Conclusion. Chapter 4 explores what the most plausible version of this view might look like and what problems the different versions have. Chapter 5 focuses on one type of problem, namely, money pumps. The Independence Money Pump, in particular, presents a difficult challenge for those who wish to discount small probabilities. Finally, Chapter 6 discusses the implications of discounting small probabilities for the value of the far future. (shrink)

Orthodox decision theory is fanatical in the way it treats small probabilities of enormous value, if unbounded utility functions are allowed. Some have suggested a fix, Nicolausian discounting, according to which outcomes with small enough probabilities should be ignored when making decisions. However, there are lotteries involving only small-probability outcomes, none of which should intuitively be ignored. So the Nicolausian discounter needs a procedure for distinguishing the problematic cases of small-probability outcomes from the (...) unproblematic ones. In this paper, I present a dilemma for Nicolausian discounting: the view must be coupled either with a procedure that delivers fanatical verdicts of its own, as bad as those of orthodox decision theory, or with one that entails intransitive cyclic preferences. (shrink)

Many population axiologies avoid the Repugnant Conclusion by endorsing Superiority: some number of great lives is better than any number of mediocre lives. But as Nebel shows, RC follows from the Intrapersonal Repugnant Conclusion: a guaranteed mediocre life is better than a sufficiently small probability of a great life. This result is concerning because IRC is plausible. Recently, Kosonen has argued that IRC can be true while RC is false if small probabilities are discounted to zero. This (...) article details the unique problems created by combining Superiority with discounting. The resultant view, Superiority Discounting, avoids the Repugnant Conclusion only at the cost of the Preposterous Conclusion: near-certain hell for arbitrarily many people is better than near-certain heaven for arbitrarily many people. (shrink)

We conduct an experiment to investigate the degree to which deviations from exponential discounting can be accounted for by the hypothesis of hyperbolic discounting. Subjects are asked to choose between an earlier or later payoff in a series of 40 choice questions. Each question consists of a pair of monetary amounts determined by compounding a given base amount at a constant rate per period. Two bases (8 and 20 dollars), three compounding rates (low, medium and high) and three (...) delays (2, 4, and 6 weeks) are each used. There are also 2 initial periods (Today and 2 weeks) and there are two separate questionnaires, one with lower “realistic” compounding rates and the other with higher compounding rates, typical of those used in previous studies. We analyze the detailed patterns of choice in 6 groups of 6 related questions each (in which the base and rate is fixed but the initial period and delay varies), documenting the frequency of patterns consistent with exponential discounting and with hyperbolic discounting. We find that exponential discounting is the clear modal choice pattern in virtually all cases. Hyperbolic discounting is never the modal pattern (except in the sense that constant discounting is a special case of hyperbolic discounting). We also estimate a linear probability model that takes account of individual heterogeneity. The estimates show substantial increases in the probability of choosing the later option when the compounding rate increases, as one would expect. There are small, sometimes significant, increases in this probability when the delay is increased or the initial period is in the future. Such behavior is consistent with hyperbolic discounting, but can account for only a small proportion of choices. Overall, deviations from exponential discounting appear to be due to error, or to other effects not accounted for by hyperbolic discounting. Principal among these is an increase in later choices when the base is larger. (shrink)

Should you be willing to forego any sure good for a tiny probability of a vastly greater good? Fanatics say you should, anti-fanatics say you should not. Anti-fanaticism has great intuitive appeal. But, I argue, these intuitions are untenable, because satisfying them in their full generality is incompatible with three very plausible principles: acyclicity, a minimal dominance principle, and the principle that any outcome can be made better or worse. This argument against anti-fanaticism can be turned into a positive (...) argument for a weak version of fanaticism, but only from significantly more contentious premises. In combination, these facts suggest that those who find fanaticism counterintuitive should favor not anti-fanaticism, but an intermediate position that permits agents to have incomplete preferences that are neither fanatical nor anti-fanatical. (shrink)

We present a theory of decision by sampling (DbS) in which, in contrast with traditional models, there are no underlying psychoeconomic scales. Instead, we assume that an attribute’s subjective value is constructed from a series of binary, ordinal comparisons to a sample of attribute values drawn from memory and is its rank within the sample. We assume that the sample reflects both the immediate distribution of attribute values from the current decision’s context and also the background, real-world distribution of attribute (...) values. DbS accounts for concave utility functions; losses looming larger than gains; hyperbolic temporal discounting; and the overestimation of small probabilities and the underestimation of large probabilities. Ó 2005 Elsevier Inc. All rights reserved. (shrink)

The lesson to be learned from the paradoxical St. Petersburg game and Pascal’s Mugging is that there are situations where expected utility maximizers will needlessly end up (with high probability) poor and on death’s door, and hence we should not be expected utility maximizers. Instead, when it comes to decision-making, for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero, regardless of the utilities associated with those possibilities.

The lesson to be learned from the paradoxical St. Petersburg game and Pascal’s Mugging is that there are situations where expected utility maximizers will needlessly end up poor and on death’s door, and hence we should not be expected utility maximizers. Instead, when it comes to decision-making, for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero, regardless of the utilities associated with those possibilities.

This paper explores the consequences of applying two natural ideas from epistemology to decision theory: (1) that knowledge should guide our actions, and (2) that we know a lot of non-trivial things. In particular, we explore the consequences of these ideas as they are applied to standard decision theoretic puzzles such as the St. Petersburg Paradox. In doing so, we develop a “knowledge-first” decision theory and we will see how it can help us avoid fanaticism with regard to the St. (...) Petersburg puzzle and related puzzles. The result will be a decision theory that gives a novel, but well-motivated, reason for discounting small probabilities when making decisions. We examine the merits and demerits of such a decision theory. (shrink)

A small probability space representation of quantum mechanical probabilities is defined as a collection of Kolmogorovian probability spaces, each of which is associated with a context of a maximal set of compatible measurements, that portrays quantum probabilities as Kolmogorovian probabilities of classical events. Bell's theorem is stated and analyzed in terms of the small probability space formalism.

Shoot an arrow at a wall, and then paint a target around it so that the arrow sticks squarely in the bull's eye. Alternatively, paint a fixed target on a wall, and then shoot an arrow so that it sticks squarely in the bull's eye. How do these situations differ? In both instances the precise place where the arrow lands is highly improbable. Yet in the one, one can do no better than attribute the arrow's landing to chance, whereas in (...) the other one rightly attributes the arrow's landing to the archer's skill. ;Highly improbable events occur all the time, and by themselves may legitimately be attributed to chance. Yet when an event also conforms to a pattern given independently of that event , one rightly refuses to attribute the event to chance. Patterns given independently of events are called specifications. This dissertation explicates the relation of independence between specification and event, and sets precise bounds on what may count as a probability small enough to eliminate chance. Although this dissertation connects with how statistics eliminates chance, it considerably extends and clarifies the range of application for chance-elimination arguments based on small probabilities. ;The dual notions of specification and small probability together help justify a regulative principle of probability called the Law of Small Probability, which asserts that specified events of small probability do not occur by chance. This principle engenders a form of inference called the design inference, and a mode of explanation called design. To infer design in explaining an event is to eliminate decisively explanations of that event which appeal to law-like regularities or chance. (shrink)

Many scientists, businessmen, and government regulators believe that the criteria for acceptable societal risk are too stringent. Those who subscribe to this belief often accept the view which I call the probability-threshold position. Proponents of this stance maintain that society ought to ignore very small risks, i.e., those causing an average annual probability of fatality of less than 10–6.After examining the three major views in the risk-evaluation debate, viz., the probability-threshold position, the zero-risk position, and the (...) weighted-risk position, I focus on the arguments for the first of these views, since it is the position which currently undergirds most public policy (especially in the U.S.) regarding acceptable risk. After analyzing Arrow's argument from decision theory, Comar's and Gibson's argument from ontology, and Starr's and Whipple's argument from epistemology, I conclude that these defenses of the probability-threshold position err in a variety of ways. Most commonly, they fail because they tacitly accept the assumption that magnitude of probability, alone, provides a sufficient condition for judging the acceptability of a given risk. In the light of these errors, I suggest that it might be more desirable for risk assessors, decision theorists, and policymakers to weight various risk-cost-benefit parameters according to alternative ethical criteria, rather than to evaluate risks solely in terms of mathematical considerations. (shrink)

The design inference uncovers intelligent causes by isolating their key trademark: specified events of small probability. Just about anything that happens is highly improbable, but when a highly improbable event is also specified undirected natural causes lose their explanatory power. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence. This challenging and provocative 1998 book shows how incomplete undirected causes are (...) for science and breathes new life into classical design arguments. It will be read with particular interest by philosophers of science and religion, other philosophers concerned with epistemology and logic, probability and complexity theorists, and statisticians. (shrink)

The design inference uncovers intelligent causes by isolating the key trademark of intelligent causes: specified events of small probability. Just about anything that happens is highly improbable, but when a highly improbable event is also specified (i.e., conforms to an independently given pattern) undirected natural causes lose their explanatory power. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence.

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 (...) class='Hi'>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)

Analogies are offered to guide our explanatory responses to the fine-tuning of the universe. Situations that prompt us to reject an explanation involving a single chance event are presented as analogous to the fine-tuning. Thus, by analogy, we are prompted to reject an explanation of the fine-tuning involving a single universe fine-tuned by chance. But if the alleged analogues are not analogous they misguide us. I argue that the alleged analogues are not analogous and hence they do misguide our explanatory (...) responses to the fine-tuning. I use William Dembski’s work on eliminating chance explanations for “specified” events of small probability to illustrate the misguiding nature of the analogies. (shrink)

Organic evolution, or change, among the diatoms has proceeded with considerable regularity. The origins, the extinctions, and the increases in numbers of species and genera, on the whole, have submitted to law and order, as rules-within-limits . The changes of evolution have followed from two sorts of phenomena — 1) the origin and extinction of shortlived or unstable species, in a proportion which has been more or less constant, but different in the two groups of diatoms, Centricae and Pennatae; these (...) short-lived species died out in the subperiod of their origin even under favourable enough environment or surrounding conditions, because of the innate instability of these particular species: and 2) the origin of new species which demonstrated their innate capacity for permanence by remaining in the living condition from their origins mostly up to the present time. The continued or continual additions of new and long-living species, in proportion to the number available as possible and parental species, has given the geometrical progressions.Consequently evolution in the diatoms has been monopodial, or one-stemmed, and not sympodial or not with many substitutions of one species after another in successions. The two kinds of species-durations are well-known in other classes of plants and animals. Exactly the same processes of additions, subtractions, and multiplications, particularly as found in the centric diatoms, are very probable in these classes of plants and animals which have given the same answer or arithmetical result for the whole of their previous histories, in the form of the frequency distribution of the present sizes of their genera.The rules-within-limits of this basic evolution are described, defined, and discussed in their relations to various theories of evolution which perhaps only concern the working of factors of superimposed modifications that reduce the results of the pattern or plan of basic evolution.Die phylogenetische Entwicklung wurde im Bereich der Kieselalgen von bestimmten Gesetzmässigkeiten beherrscht. Die Entstehung neuer Arten und Gattungen, ihr Wiederverschwinden und die Zunahme des Formenreichtums, resultierend aus der Zahl der neuentstandenen, vermindert um die der wieder verschwindenen Arten und Gattungen und vollzog sich nach Regeln , die durch folgende Phänomene gekennzeichnet sind: Zwischen der Zahl der neuentstehenden und der Zahl der in der gleichen geologischen Subperiode aus inneren Gründen wieder aussterbenden Arten besteht eine mehr oder wenig feste Beziehung. Diese ist für die Centricae und Pennatae verschieden. Ein bestimmter Anteil der neuentstandenen Arten und Gattungen erwies sich aus inneren Gründen als „ausdauernd” und blieb bis auf den heutigen Tag erhalten. Die Zunahme der Arten- Zahl durch das Hinzukommen langlebiger Formen war stetig und vollzog sich nach Massgabe geometrischer Progressionen.Diese Tatsachen führen zu dem Schluss, dass die phylogenetische Entwicklung der Kieselalgen nach monopodialem, und nicht nach sympodialem Muster vorsichging. Auch in andern Pflanzengruppen und im Tierreich durften die gleichen Gesetzmässigkeiten herrschend sein. Am klarsten gelangen sie bei den Centricae zum, Ausdruck und werden vom Autor als das Fazit von Additions-, Subtraktions-, und Multiplikationsprozessen aufgefasst. Die heutige Situation bezüglich Häufigkeit der Grössen der Gattungen, in Zahl der Arten, ist das Endergebnis der summativen Wirkung dieser Einzelprozesse.Diese als ‚rules-within-limits’ bezeichneten Gesetzmässigkeiten werden im einzelnen beschrieben, erklärt und zusammenfassend dargestellt. Zu andern Evolutionstheorien in Beziehung gebracht, werden sie schliesslich für Erörterung des Problems benutzt, inwieweit noch andere evolutionsgeschichtliche Prozesse als übergeordnete Faktoren in den vom Verfasser dargestellten Grundplan des Evolutionsgeschehens eingreifen können.L'évolution organique ou le changement parmi les Diatomées s'est effectué avec une régularité considérable. Les origines et les extinctions, et les accroissements des nombres des espèces et des genres, se sont soumis en tout aux lois et aux ordres, commes des régularités à l'intérieur de légères déviations. Le changement de l'évolution fut la conséquence de deux sortes de phénomène: l'origine et l'extinction des espèces éphémères dans une proportion plus ou moins déterminée mais différente pour les deux classes des Diatomées, Centricae et Pennatae; ces espèces éphémères moururent dans la subpériode de leur origines malgré le caractère assez favorable des circonstances environnantes, à cause de l'évanescence inhérente aux espèces particulières, et l'origine des espèces nouvelles qui démontrèrent leur qualité innée de permanence tout en continuant de vivre depuis leurs origines, pour la plupart, jusqu'à présent. Les additions continuelles d'espèces nouvelles et permanentes, en nombres proportionnés aux nombres disponibles comme espèces parentes et possibles, donnèrent des accroissements par progressions géométriques.Par conséquent, l'évolution parmi les Diatomées fut monopodiale et non pas sympodiale. Les deux sortes d'espèces-durées sont bien connues parmi beaucoup d'autres classes de plantes et d'animaux. Les mêmes procédés d'additions, de soustractions et de multiplications, particulièrement tels qu'on les trouve dans les Centricae, sont très probables pour ces classes de plantes et d'animaux qui ont donné un résultat le plus semblable pour la somme totale de leur histoire précédente, sous la forme de la répartition des fréquences de leurs grandeurs générique actuelles.Les régularités à l'intérieur de légères déviations de cette évolution fondamentale sont détaillées, définies et discutées dans leurs relations aux diverses théories touchant les autres méthodes de l'évolution qui jouent le rôle de facteurs modifiants superposés au dessin général ou au dessein de l'évolution fondamentale. (shrink)

Statements that share an explanation tend to lend inductive support to one another. For example, being told that Many furniture movers have a hard time financing a house increases the judged probability that Secretaries have a hard time financing a house. In contrast, statements with different explanations reduce one another s judged probability. Being told that Many furniture movers have bad backs decreases the judged probability that Secretaries have bad backs. I pose two questions concerning such (...) class='Hi'>discounting effects. First, does the reduction depend on explanations being mutually incompatible or does it occur when explanations are deemed irrelevant to one another? I found that a small discounting effect occurred with statements that were blatantly unrelated. However, the discounting effect also depended on a factor external to the argument being judged; the composition of the argument set. Second, are explanation effects attributable to changes in the belief afforded statements or to response-specific changes resulting from misunderstanding of the probability rating task or response bias? The results implicate changes in belief. Prior belief influenced conditional probability more than argument strength judgements, as it would if participants understood the tasks in the same way as the experimenter. Also, conditional probability true and false judgements were complementary, suggesting no response bias. (shrink)

In a controlled laboratory experiment we use one sample of college students and one of mature executives to investigate how positive skew influences risky choices. In reduced-form regressions we find that both students and executives make riskier choices when lotteries display positive skew. We estimate decision models to explore three explanations for skew seeking choices: risk-loving, optimism and likelihood insensitivity. We find no role for love for risk as neither students nor executives have convex utility. Both optimism and likelihood insensitivity (...) play a part in skew seeking choices. Likelihood insensitivity is larger for students than for executives. Executives have more concave utility and are more optimistic than students, but this is found to be largely due to them being older. (shrink)

Tversky and Koehler's support theory attempts to explain why probability judgments are affected by the manner in which formally similar events are described. Support theory suggests that as the explicitness of a description increases, an event will be judged to be more likely. In the present experiment, experienced decision-makers from large, international accounting firms were given case-specific information about an audit client and asked to provide a series of judgments regarding the perceived likelihood of events. Unpacking a hypothesis into (...) four formally equivalent hypotheses more than doubled its perceived likelihood. These results are largely consistent with support theory, extending its generalizability to applied contexts involving business professionals. Further, the paper finds that task-specific factors, a previously untested area, can also affect support theory interpretations. (shrink)

Intergenerational impartiality requires putting the welfare of future generations at par with that of our own. However, rational choice requires weighting all welfare values by the respective probabilities of realization. As the risk of non-survival of mankind is strictly positive for all time periods and as the probability of non-survival is cumulative, the probability weights operate like discount factors, though justified on a morally justifiable and completely different ground. Impartial intertemporal welfare maximization is acceptable, though the welfare of (...) people in the very far future has lower effects as the probabilities of their existence are also lower. However, the effective discount rate on future welfare values (distinct from monetary values) justified on this ground is likely to be less than 0.1 per annum. Such discounting does not compromise environmental protection and sustainability unduly. The finiteness of our universe implies that the sum of our expected welfare to infinity remains finite, solving the paradox of having to compare different infinite values in optimal growth/conservation theories. (shrink)

For a finite universe of discourse, if Φ → and ~(Ψ → Φ) , then P(Ψ) > P(Φ), i.e., there is always a loss of information, there is an increase in probability, in a non reversible implication. But consider the two propositions, "All ravens are black", (i.e., "(x)(Rx ⊃ Bx)"), and "Some ravens are black" (i.e., "(∃x)(Rx & Bx)"). In a world of one individual, called "a", these two propositions are equivalent to "~Ra ∨ Ba" and "Ra & Ba" (...) respectively. However, (Ra & Ba) → (~Ra ∨ Ba) and ~[(~Ra ∨ Ba) → (Ra & Ba)]. Consequently, in a world of one individual it is more probable that all ravens are black than that some ravens are black! (shrink)

Time discounting is the phenomenon that a desired result in the future is perceived as less valuable than the same result now. Economic theories can take this psychological fact into account in several ways. In the economic literature the most widely used type of additive time discounting is exponential discounting. In exponential discounting, the fall of valuation depends by a constant factor on the length of the delay period. It is well known, however, that exponential time (...) discounting often does not describe well how people actually behave. Most people are averse to short delays in gratification now, while their future selves may not mind a bit of extra waiting. This behaviour can be described well by non-exponential discounting functions such as hyperbolic discounting. In hyperbolic discounting, valuations fall rapidly for small delay periods, but the fall gets slower for longer delay periods. Hyperbolic discounting captures phenomena such as procrastination, addiction and in general inconsistency over time. This chapter investigates whether forms of non-exponential discounting, in particular close to the so called Quasi-Hyperbolic model, could also be characterized in terms of dynamically consistent choices when individuals discount the welfare of future selves as well as their payoffs. (shrink)

Utility discounting in intertemporal economic modelling has been viewed as problematic, both for descriptive and normative reasons. However, positive utility discount rates can be defended normatively; in particular, it is rational for future utility to be discounted to take into account model-independent outcomes when decision-making under risk. The resultant values will tend to be smaller than descriptive rates under most probability assignments. This also allows us to address some objections that intertemporal considerations will be overdemanding. A principle for (...) utility discount rates is suggested which is rooted in probability discounting. Utility discounting is defended against objections from Parfit (1984) and Broome (2005); Broome (2012). A sample utility discount rate is estimated. [Open access]. (shrink)

To obtain effective fusion results of multi source evidences with different importance, an evidence fusion method with importance discounting factors based on neutrosopic probability analysis in DSmT framework is proposed. First, the reasonable evidence sources are selected out based on the statistical analysis of the pignistic probability functions of single focal elements. Secondly, the neutrosophic probability analysis is conducted based on the similarities of the pignistic probability functions from the prior evidence knowledge of the reasonable (...) evidence sources. Thirdly, the importance discounting factors of the reasonable evidence sources are obtained based on the neutrosophic probability analysis and the reliability discounting factors of the real-time evidences are calculated based on probabilistic-based distances. Fourthly, the real-time evidences are discounted by the importance discounting factors and then the evidences with the mass assignments of neutrosophic empty sets are discounted by the reliability discounting factors. Finally, DSmT+PCR5 of importance discounted evidences is applied. Experimental examples show that the decision results based on the proposed fusion method are different from the results based on the existed fusion methods. Simulation experiments of recognition fusion are performed and the superiority of proposed method is testified well by the simulation results. (shrink)

People differ in the extent to which they discount the values of future rewards. Behavioural economists measure these differences in terms of functions that describe rates of reduced valuation in the future – temporal discounting – as these vary with time. They measure differences in preference for risk – differing rates of probability discounting – in terms of similar functions that describe reduced valuation of rewards as the probability of their delivery falls. So-called ‘impulsive’ people, including (...) people disposed to addiction, tend to choose rewards in patterns that are described by unusually steeply sloped discount functions. The empirical literature is ambivalent as to whether this applies to pathological gamblers (PGs) who manifest most other behavioural patterns associated with addiction, with different, equally careful studies suggesting opposite conclusions (Petry & Casarella 1999; Holt, Green & Myerson 2003). This puzzle may arise from the fact that most previous experiments on discounting behaviour were not designed so as to allow effects of high temporal discounting to be distinguished from effects of low probability discounting. (Addiction has sometimes been associated with the latter because it seems to involve, at least in the starting addict, under appreciation of risk.) Our research project investigates both forms of discounting behaviour and their relationship to severity of risk of PG in a community sample of South African gamblers. (shrink)

The desirable gambles framework offers the most comprehensive foundations for the theory of lower pre- visions, which in turn affords the most general ac- count of imprecise probabilities. Nevertheless, for all its generality, the theory of lower previsions rests on the notion of linear utility. This commitment to linearity is clearest in the coherence axioms for sets of desirable gambles. This paper considers two routes to relaxing this commitment. The first preserves the additive structure of the desirable gambles framework and (...) the machinery for coherent inference but detaches the interpretation of desirability from the multiplicative scale invariance axiom. The second strays from the additive combination axiom to accommodate repeated gambles that return rewards by a non-stationary processes that is not necessarily additive. Unlike the first approach, which is a conservative amendment to the desirable gambles framework, the second is a rad- ical departure. Yet, common to both is a method for describing rewards called discounted utility. (shrink)

This paper formulates and axiomatizes utility models for denumerable time streams that make no commitment in regard to discounting future outcomes. The models address decision under certainty and decision under risk. Independence assumptions in both contexts lead to additive or multiplicative utilities over time periods that allow unambiguous comparisons of the relative importance of different periods. The models accommodate all patterns of future valuation. This discount-neutral feature is attained by restricting preference comparisons to outcome streams or probability distributions (...) on outcome streams that differ in at most a finite number of periods. (shrink)

Standard axioms of additively separable utility for choice over time and classic axioms of expected utility theory for choice under risk yield a generalized expected additively separable utility representation of risk-time preferences over probability distributions over sure streams of intertemporal outcomes. A dual approach is to use the analogues of the same axioms in a reversed order to obtain a generalized additively separable expected utility representation of time–risk preferences over intertemporal streams of probability distributions over sure outcomes. The (...) paper proposes an additional axiom, which is called risk-time reversal, for obtaining a special case of the two representations—expected discounted utility. The axiom of risk-time reversal postulates that if a risky lottery over streams of sure intertemporal outcomes and an intertemporal stream of risky lotteries yield the same probability distribution of possible outcomes in every point in time then a decision-maker is indifferent between the two. This axiom is similar to assumption 2 “reversal of order in compound lotteries” in Anscombe and Aumann :199–205, 1963, p. 201). (shrink)

Ainslie's insightful treatment of dynamically inconsistent choice stands in contrast to traditional views in psychology, economics, and philosophy. We comment on the form of the discounting function and on new findings regarding choice between delayed rewards. Finally, we argue that the positive correlation between temporal and probability discounting is inconsistent with the view that impulsivity represents a unitary trait.

We present a model for studying communities of epistemically interacting agents who update their belief states by averaging the belief states of other agents in the community. The agents in our model have a rich belief state, involving multiple independent issues which are interrelated in such a way that they form a theory of the world. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that (...) end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. It is shown that, under the assumptions of our model, an agent always has a probability of less than 2% of ending up in an inconsistent belief state. Moreover, this probability can be made arbitrarily small by increasing the number of independent issues the agents have to judge or by increasing the group size. A real-world situation to which this model applies is a group of experts participating in a Delphi-study. (shrink)

We describe a dual-process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non-numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single (...) events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non-numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P, P, P, disjunctions: P, P, P, and conditional probabilities P, P, P. They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning. (shrink)

A proposal for an objective interpretation of probability is introduced and discussed: probabilities as deriving from ranges in suitably structured initial-state spaces. Roughly, the probability of an event on a chance trial is the proportion of initial states that lead to the event in question within the space of all possible initial states associated with this type of experiment, provided that the proportion is approximately the same in any not too small subregion of the space. This I (...) would like to call the “natural-range conception” of probability. Providing a substantial alternative to frequency or propensity accounts of probability in a deterministic setting, it is closely related to the so-called “method of arbitrary functions”. It is explicated, confronted with certain problems, and some ideas how these might be overcome are sketched and discussed. (shrink)

We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery.

We present a new axiomatization of the classical discounted expected utility model, which is primarily used as a decision model for consumption streams under risk. This new axiomatization characterizes discounted expected utility as a model that satisfies natural extensions of standard axioms as in the one-period case and two additional axioms. The first axiom is a weak form of time separability. It only requires that the choice between certain constant consumption streams and lotteries should be made by just taking into (...) account the time periods where the consumption is different. The second axiom, the time–probability equivalence, requires that risk and time preferences basically work in the same way. Moreover, we prove that preferences satisfying the natural extensions of the standard axioms as well as the first axiom can be represented in a simple form relying on three functions linked to the risk or time preferences in simple situations. Finally, we illustrate that several examples that are not fully time separable satisfy all our axioms except for the time–probability equivalence. (shrink)