Consider an election in which each of the n voters casts a vote consisting of a strict preference ranking of the three candidates A, B, and C. In the limit as n→∞, which scoring rule maximizes, under the assumption of Impartial Anonymous Culture (uniform probability distribution over profiles), the probability that the Condorcet candidate wins the election, given that a Condorcet candidate exists? We produce an analytic solution, which is not the Borda (...) Count. Our result agrees with recent numerical results from two independent studies, and contradicts a published result of Van Newenhizen (Economic Theory 2, 69–83. (1992)). (shrink)

Many voting rules and, in particular, the plurality rule and Condorcet-consistent voting rules satisfy the simple-majority decisiveness property. The problem implied by such decisiveness, namely, the universal disregard of the preferences of the minority, can be ameliorated by applying unbiased scoring rules such as the classical Borda rule, but such amelioration has a price; it implies erosion in the implementation of the widely accepted majority principle . Furthermore, the problems of majority decisiveness and of the erosion (...) in the majority principle are not necessarily severe when one takes into account the likelihood of their occurrence. This paper focuses on the evaluation of the severity of the two problems, comparing simple-majoritarian voting rules that allow the decisiveness of the smallest majority larger than 1/2 and the classical Borda method of counts. Our analysis culminates in the derivation of the conditions that determine, in terms of the number of alternatives k, the number of voters n, and the relative (subjective) weight assigned to the severity of the two problems, which of these rules is superior in light of the dual majoritarian approach. (shrink)

In a three-candidate election, a scoring rule s (s in [0,1]) assigns 1, s, and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule s (...) has the greatest Condorcet efficiency? After recalling the known answer to this question, we investigate the impact of social homogeneity on the optimal value of ?. One of the most salient results we obtain is that the optimality of the Borda rule (s=1/2) holds only if the voters act in an independent way. (shrink)

In a three-candidate election, a scoring rule λ, λ∈[0,1], assigns 1,λ and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule λ has the greatest (...) Condorcet efficiency? After recalling the known answer to this question, we investigate the impact of social homogeneity on the optimal value of λ. One of the most salient results we obtain is that the optimality of the Borda rule (λ=1/2) holds only if the voters act in an independent way. (shrink)

De Finetti introduced the concept of coherent previsions and conditional previsions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring (...) rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti’s two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti’s fundamental theorem of prevision to deal with conditional previsions. (shrink)

It is commonly assumed that when we assign different credences to a proposition, a perfect compromise between our opinions simply ‘splits the difference’ between our credences. I introduce and defend an alternative account, namely that a perfect compromise maximizes the average of the expected epistemic values that we each assign to alternative credences in the disputed proposition. I compare the compromise strategy I introduce with the traditional strategy of compromising by splitting the difference, and I argue that my strategy (...) is a reasonable characterization of epistemic compromise. (shrink)

In many voting systems, voters’ preferences on a set of candidates are represented by linear orderings. In this context, scoring rules are well-known procedures to aggregate the preferences of the voters. Under these rules, each candidate obtains a fixed number of points, sk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_k$$\end{document}, each time he/she is ranked k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}th by one voter and the candidates are ordered according to the total number (...) of points they receive. In order to identify the best scoring rule to use in each situation, we need to know which properties are met by each of these procedures. Although some properties have been analyzed extensively, there are other properties that have not been studied for all scoring rules. In this paper, we consider two desirable social choice properties, the Pareto-optimality and the immunity to the absolute loser paradox, and establish characterizations of the scoring rules that satisfy each of these specific axioms. Moreover, we also provide a proof of a result given by Saari and Barney, where the scoring rules meeting reversal symmetry are characterized. From the results of characterization, we establish some relationships among these properties. Finally, we give a characterization of the scoring rules satisfying the three properties. (shrink)

The aim of this paper is to apply the accuracy based approach to epistemology to the case of higher order evidence: evidence that bears on the rationality of one's beliefs. I proceed in two stages. First, I show that the accuracy based framework that is standardly used to motivate rational requirements supports steadfastness—a position according to which higher order evidence should have no impact on one's doxastic attitudes towards first order propositions. The argument for this will require a generalization (...) of an important result by Greaves and Wallace for the claim that conditionalization maximizes expected accuracy. The generalization I provide will, among other things, allow us to apply the result to cases of self-locating evidence. In the second stage, I develop an alternative framework. Very roughly, what distinguishes the traditional approach from the alternative one is that, on the traditional picture, we're interested in evaluating the expected accuracy of conforming to an update procedure. On the alternative picture that I develop, instead of considering how good an update procedure is as a plan to conform to, we consider how good it is as a plan to make. I show how, given the use of strictly proper scoring rules, the alternative picture vindicates calibrationism: a view according to which higher order evidence should have a significant impact on our beliefs. I conclude with some thoughts about why higher order evidence poses a serious challenge for standard ways of thinking about rationality. (shrink)

Epistemic scoring rules are the en vogue tool for justifications of the probability norm and further norms of rational belief formation. They are different in kind and application from statistical scoring rules from which they arose. In the first part of the paper I argue that statistical scoring rules, properly understood, are in principle better suited to justify the probability norm than their epistemic brethren. Furthermore, I give a justification of the probability norm applying statistical (...) class='Hi'>scoring rules. In the second part of the paper I give a variety of justifications of norms for rational belief formation employing statistical scoring rules. Furthermore, general properties of statistical scoring rules are investigated. Epistemic scoring rules feature as a useful technical tool for constructing statistical scoring rules. (shrink)

We focus on the dichotomous choice model, which goes back as far as Condorcet (1785; Essai sur l'application de l'analyse a la probabilité des décisions rendues a la pluralité des voix, Paris). A group of experts is required to select one of two alternatives, of which exactly one is regarded as correct. The alternatives may be related to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision. (...) A decision rule is optimal if it maximizes the probability of the group to make a correct choice. In this paper we assume the correctness probabilities of the experts to be independent random variables, selected from some given distribution. Moreover, the ranking of the members in the team is (at least partly) known. Thus, one can follow rules based on this ranking. The polar different rules are the expert and the majority rules. The probabilities of the two polar rules being optimal were compared in a series of papers. The main purpose of this paper is to outline the results, providing exact formulas or estimates for these probabilities. We consider a variety of distributions and show that for all of these distributions the asymptotic behaviour of the probabilities of the two polar rules follows the same patterns. (shrink)

ABSTRACTSuppose that beliefs come in degrees. How should we then measure the accuracy of these degrees of belief? Scoring rules are usually thought to be the mathematical tool appropriate for this job. But there are many scoring rules, which lead to different ordinal accuracy rankings. Recently, Fallis and Lewis [2016] have given an argument that, if sound, rules out many popular scoring rules, including the Brier score, as genuine measures of accuracy. I respond to this argument, (...) in part by noting that the argument fails to account for verisimilitude—that certain false hypotheses might be closer to the truth than other false hypotheses are. Oddie [forthcoming], however, has argued that no member of a very wide class of scoring rules can appropriately handle verisimilitude. I explain how to respond to Oddie's argument, and I recommend a class of weighted scoring rules that, I argue, genuinely measure accuracy while escaping the arguments of Fallis and Lewis as well as Oddie. (shrink)

Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms of the probability calculus, but Predd et al. [9] have shown (...) that given a finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal. (shrink)

Strictly proper scoring rules have been studied widely in statistical decision theory and recently in experimental economics because of their ability to encourage assessors to honestly provide their true subjective probabilities. In this article, we study the spherical scoring rule by analytically examining some of its properties and providing some new geometric interpretations for this rule. Moreover, we state a theorem which provides an axiomatic characterization for the spherical scoring rule. The objective of (...) this analysis is to provide a better understanding of one of the most commonly available scoring rules, which could aid decision makers in the selection of an appropriate tool for evaluating and assessing probabilistic forecasts. (shrink)

Can there be good reasons for judging one set of probabilistic assertions more reliable than a second? There are many candidates for measuring "goodness" of probabilistic forecasts. Here, I focus on one such aspirant: calibration. Calibration requires an alignment of announced probabilities and observed relative frequency, e.g., 50 percent of forecasts made with the announced probability of.5 occur, 70 percent of forecasts made with probability.7 occur, etc. To summarize the conclusions: (i) Surveys designed to display calibration curves, from which (...) a recalibration is to be calculated, are useless without due consideration for the interconnections between questions (forecasts) in the survey. (ii) Subject to feedback, calibration in the long run is otiose. It gives no ground for validating one coherent opinion over another as each coherent forecaster is (almost) sure of his own long-run calibration. (iii) Calibration in the short run is an inducement to hedge forecasts. A calibration score, in the short run, is improper. It gives the forecaster reason to feign violation of total evidence by enticing him to use the more predictable frequencies in a larger finite reference class than that directly relevant. (shrink)

Dutch Book Arguments. B is susceptibility to sure monetary loss (in a certain betting set-up), and F is the formal role played by non-Pr b’s in the DBT and the Converse DBT. Representation Theorem Arguments. B is having preferences that violate some of Savage’s axioms (and/or being unrepresentable as an expected utility maximizer), and F is the formal role played by non-Pr b’s in the RT.

When choosing a voting rule to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider (...) in this article the menu constituted by the three well-known scoring rules {Borda, Plurality, and Antiplurality}. Under the Impartial Culture assumption, which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules. (shrink)

Fitelson and McCarthy have proposed an accuracy measure for confidence orders which favors probability measures and Dempster-Shafer belief functions as accounts of degrees of belief and excludes ranking functions. Their accuracy measure only penalizes mistakes in confidence comparisons. We propose an alternative accuracy measure that also rewards correct confidence comparisons. Thus we conform to both of William James’ maxims: “Believe truth! Shun error!” We combine the two maxims, penalties and rewards, into one criterion that we call prioritized accuracy optimization. (...) That is, PAO punishes wrong comparisons and rewards right comparisons. And it requires to prioritize being right und avoiding to be wrong in a specific way. Thus PAO is both, a scoring rule and a decision rule. It turns out that precisely confidence orders representable by two-sided ranking functions satisfy PAO. The point is not to argue that PAO is the better accuracy goal. The point is only that ranking theory can also be supported by accuracy considerations. Thus, those considerations by themselves cannot decide about rational formats for degrees of belief, but are part and parcel of an overall normative assessment of those formats. (shrink)

Men are living in the midst of a world pervaded by ideas.* Ideas, which provide men with help in their will to act and to think according to rules, which offer them guidance in their lives, are mustered into systems which are called ideologies.Ideologies are issued by the inclusive society, by special, functional groups inside the society and, beginning with a given stage in the development of human society, by ideological movements which profess to provide men (...) not only with guidance for special tasks or functions, but for life as a whole. So, individual men are faced with various sets of ideologies, more or less in agreement between themselves, more 'or less authoritative for them. They must choose between the suggestions, the rules, the orders given to them. As men are always in search of some degree of consistency in their ways of thinking and acting, they fit together the ideas they choose, unconsciously, into systems, into personal, private, individual ideologies. (shrink)

Epistemic utility theory seeks to establish epistemic norms by combining principles from decision theory and social choice theory with ways of determining the epistemic utility of agents’ attitudes. Recently, Moss, 1053–69, 2011) has applied this strategy to the problem of finding epistemic compromises between disagreeing agents. She shows that the norm “form compromises by maximizing average expected epistemic utility”, when applied to agents who share the same proper epistemic utility function, yields the result that agents must form compromises by splitting (...) the difference between their credence functions. However, this “split the difference” norm is in conflict with conditionalization, since applications of the two norms don’t commute. A common response in the literature seems to be to abandon the procedure of splitting the difference in favor of compromise strategies that avoid non-commutativity. This would also entail abandoning Moss’ norm. I explore whether a different response is feasible. If agents can use epistemic utility-based considerations to agree on an order in which they will apply the two norms, they might be able to avoid diachronic incoherence. I show that this response can’t save Moss’ norm, because the agreements concerning the order of compromising and updating it generates are not stable over time, and hence cannot avoid diachronic incoherence. I also show that a variant of Moss’ norm, which requires that the weights given to each agent’s epistemic utility change in a way that ensures commutativity, cannot be justified on epistemological grounds. (shrink)

The Condorcet efficiency of a social choice procedure is usually defined as the probability that this procedure coincides with the majority winner (or majority ordering) in random samples, given a majority winner exists (or given the majority ordering is transitive). Consequently, it is in effect a conditional probability that two sample statistics coincide, given certain side conditions. We raise a different issue of Condorcet efficiencies: What is the probability that a social choice procedure applied to a sample matches (...) with the majority preferences of the population from which the sample was drawn? We investigate the canonical case where the sample statistic is itself also majority rule and the samples are drawn from real world distributions gathered from national election surveys in Germany, France, and the United States. We relate the results to the existing literature on majority cycles and social homogeneity. We find that these samples rarely display majority cycles, whereas the probability that a sample misrepresents the majority preferences of the underlying population varies dramatically and always exceeds the probability that the sample displays cyclic majority preferences. Social homogeneity plays a fundamental role in the type of Condorcet efficiency investigated here. (shrink)

We consider two no-show paradoxes, in which a voter obtains a preferable outcome by abstaining from a vote. One arises when the casting of a ballot that ranks a candidate in first place causes that candidate to lose the election, superseded by a lower-ranked candidate. The other arises when a ballot that ranks a candidate in last place causes that candidate to win, superseding a higher-ranked candidate. We show that when there are at least four candidates and when voters (...) may express indifference, every voting rule satisfying Condorcet’s principle must generate both of these paradoxes. (shrink)

In ‘Normative Uncertainty as a Voting Problem’, William MacAskill argues that positive credence in ordinal-structured or intertheoretically incomparable normative theories does not prevent an agent from rationally accounting for her normative uncertainties in practical deliberation. Rather, such an agent can aggregate the theories in which she has positive credence by methods borrowed from voting theory—specifically, MacAskill suggests, by a kind of weighted Borda count. The appeal to voting methods opens up a promising new avenue for theories of rational choice (...) under normative uncertainty. The Borda rule, however, is open to at least two serious objections. First, it seems implicitly to ‘cardinalize’ ordinal theories, and so does not fully face up to the problem of merely ordinal theories. Second, the Borda rule faces a problem of option individuation. MacAskill attempts to solve this problem by invoking a measure on the set of practical options. But it is unclear that there is any natural way of defining such a measure that will not make the output of the Borda rule implausibly sensitive to irrelevant empirical features of decision-situations. After developing these objections, I suggest an alternative: the McKelvey uncovered set, a Condorcet method that selects all and only the maximal options under a strong pairwise defeat relation. This decision rule has several advantages over Borda and mostly avoids the force of MacAskill’s objection to Condorcet methods in general. (shrink)

"Margaret L. King has put together a highly representative selection of readings from most of the more significant—but by no means the most obvious—texts by the authors who made up the movement we have come to call the 'Enlightenment.' They range across much of Europe and the Americas, and from the early seventeenth century until the end of the eighteenth. In the originality of the choice of texts, in its range and depth, this collection offers both wide coverage and striking (...) insights into the intellectual transformation which has done more than any other to shape the world in which we live today. It is _simply the best introduction to the subject now available_."_ —Anthony Pagden, UCLA, and author of _The Enlightenment and Why It Still Matters_ Contents:_ Chronology, Introduction _Chapter One - Casting Out Idols: 1620–1697_ _Idols, or false notions: _Francis Bacon, _The New Instrument_ _I think, therefore I am: _René Descartes, Discourse on Method _God, or Nature: _Baruch Spinoza, _Ethics_ _The system of the world: _Isaac Newton, _Mathematical Principles of Natural Philosophy_ _He searched for truth throughout his life: _Pierre Bayle, _Historical and Critical Dictionary_ _Chapter Two - _The Learned Maid: 1638–1740 _A face raised toward heaven:_ Anna Maria van Schurman, _Whether the Study of Letters Befits a Christian Woman_ _The worlds I have made:_ Margaret Cavendish, _The Blazing World_ _A finer sort of cattle:_ Bathsua Makin, _An Essay to Revive the Ancient Education of Gentlewomen_ _I warn you of the world:_ Madame de Maintenon, _Letter: On the Education of the Demoiselles of Saint-Cyr_, and _Instruction: On the World_ _The daybreak of your reason:_ Émilie Du Châtelet, _Fundamentals of Physics_ _Chapter Three - _A State of Perfect Freedom: 1689–1695 _The chief criterion of the True Church:_ John Locke, _Letter on Toleration_ _Freedom from any superior power on earth:_ John Locke, _Second Treatise on Civil Government_ _A white paper, with nothing written on it:_ John Locke, _Essay Concerning Human Understanding_ _Let your rules be as few as possible:_ John Locke, _Some Thoughts Concerning Education_ _From death, Jesus Christ restores all to life:_ John Locke, _The Reasonableness of Christianity, as Delivered in the Scriptures_ _Chapter Four - All Things Made New: 1725–1784_ _In the wilderness, they are reborn:_ Giambattista Vico, _The New Science_ _Without these Names, nothing can be known,_ Carl Linnaeus, _System of Nature_ _All the clouds at last are lifted:_ Anne Robert Jacques Turgot, _The Successive Advancement of the Human Mind_ _A genealogical or encyclopedic tree of knowledge:_ Jean le Rond d’Alembert, _Preliminary Discourse_ _Dare to know! :_ Immanuel Kant, _What Is Enlightenment?_ _Chapter Five - Mind, Soul, and God: 1740–1779_ _The narrow limits of human understanding:_ David Hume, _Anof a Book Lately Published_ _The soul is but an empty word:_ Julien Offray de La Mettrie, _Man a Machine_ _All is reduced to sensation:_ Claude Adrien Helvétius, _On the Mind_ _An endless web of fantasies and falsehoods:_ Paul-Henri Thiry, baron d’Holbach, _Common Sense_ _Let each believe that his own ring is real:_ Gotthold Ephraim Lessing, _Nathan the Wise_ _Chapter Six - Crush That Infamous Thing: 1733–1764_ _This is the country of sects:_ Voltaire, _Philosophical Letters_ _Disfigured by myth, until enlightenment comes:_ Voltaire, _The Culture and Spirit of Nations_ _The best of all possible worlds:_ Voltaire, _Candide_ _Are we not all children of the same God?:_ Voltaire, _Treatise on Tolerance_ _If a book displeases you, refute it! :_ Voltaire, _Philosophical Dictionary_ _Chapter Seven - Toward the Greater Good: 1748–1776_ _Things must be so ordered that power checks power,_ Charles de Secondat, baron de Montesquieu, _The Spirit of the Laws_ _Complete freedom of trade must be ensured:_ François Quesnay, _General Maxims for the Economic Management of an Agricultural Kingdom_ _The nation's war against the citizen: Cesare_ Beccaria, _On Crimes and Punishments_ _There is no peace in the absence of justice:_ Adam Ferguson, _An Essay on the History of Civil Society_ _Led by an invisible hand:_ Adam Smith, _An Inquiry into the Nature and Causes of the Wealth of Nations_ _Chapter Eight - Encountering Others: 1688–1785_ _Thus died this great man:_ Aphra Behn, _Oroonoko: or The Royal Slave_ _Not one sins the less for not being Christian: _Lady Mary Wortley Montagu, _Embassy Letters_ _Do you not restore to them their liberty?:_ Guillaume-Thomas Raynal, _Philosophical and Political History of European Colonies and Commerce in the Two Indies_ _Some things which are rather interesting:_ Captain James Cook, _Voyage towards the South Pole, and Round the World_ _The inner genius of my being:_ Johann Gottfried von Herder, _Ideas for a Philosophy of the History of Humankind_ _Chapter - Nine Citizen of Geneva: 1755–1782_ _The most cunning project ever to enter the human mind: _Jean-Jacques Rousseau, _Discourse on the Origin and Foundations of Human Inequality_ _The supreme direction of the General Will:_ Jean-Jacques Rousseau, The Social Contract _Two lovers from a small town at the foot of the Alps,_ Jean-Jacques Rousseau, _Julie, or the New Heloise_ _Build a fence around your child’s soul:_ Jean-Jacques Rousseau, _Emile, or On Education_ _This man will be myself:_ Jean-Jacques Rousseau, _Confessions_ _Chapter Ten - Vindications of Women: 1685–1792_ _No higher design than to get her a husband:_ Mary Astell, _Reflections on Marriage_ _The days of my bondage begin:_ Anna Stanisławska, _Orphan Girl_ _A dying victim dragged to the altar:_ Denis Diderot, _The Nun_ _Created to be the toy of man:_ Mary Wollstonecraft, _Vindication of the Rights of Woman_ _Man, are you capable of being just?:_ Olympe de Gouges, _Declaration of the Rights of Woman as Citizen_ _Chapter Eleven - American Reverberations: 1771–1792_ _I took upon me to assert my freedom:_ Benjamin Franklin, _Autobiography_ _Freedom has been hunted round the globe:_ Thomas Paine, _Common Sense_ _Endowed by their Creator with certain unalienable Rights:_ Thomas Jefferson and Others, _Declaration of Independence_ _A safeguard against faction and insurrection:_ James Madison, _Federalist No. 10_ _An end to government by force and fraud:_ Thomas Paine, _The Rights of Man_ _Chapter Twelve - Enlightenment's End: 1790–1794_ _A partnership of the living, the dead, and those unborn:_ Edmund Burke, _Reflections on the Revolution in France_ _The future destiny of the human species:_ Nicolas de Condorcet, _A Sketch of a Historical Portrait of the Progress of the Human Mind_ Texts and Studies, Index. 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There is an ongoing debate about which rule we ought to use for scoring probability estimates. Much of this debate has been premised on scoring-rule monism, according to which there is exactly one best scoring rule. In previous work, I have argued against this position. The argument given there was based on purely a priori considerations, notably the intuition that scoring rules should be sensitive to truthlikeness relations if, and only if, (...) such relations are present among whichever hypotheses are at issue. The present paper offers a new, quasi-empirical argument against scoring-rule monism. This argument uses computational simulations to show that different scoring rules can have different economical consequences, depending on the context of use. (shrink)

In Ruling Majorities and Reasoning Pluralities, Professor Saul Levmore explores the “division of labor” between the various thresholds of agreement required for collective action—supermajority, simple majority, or plurality rule. His particular emphasis is on the choice between the last two options. To improve our understanding of this choice in various settings, Professor Levmore considers the relationship between two well-known contributions to the study of group decisionmaking, namely, the Condorcet Jury Theorem and the Condorcet Criterion, which have (...) not generally been treated together. This essay explores the relationship between these two insights in the context of judicial decisionmaking. Counterintuitively, the essay demonstrates that while the Condorcet Criterion continues to hold great promise as a tool of decision in en banc appellate courts, the Condorcet Jury Theorem is most appealing in judicial contexts that appear less collegial and thus less like a jury. In contexts in which jurists engage in deliberation and in which they anticipate that their outputs will have precedential effect, the Jury Theorem has little if any role to play. In contrast, the Theorem might prove more useful in evaluating low-level judicial outputs or decisions that otherwise are not expected to be treated as precedent. Thus, common law trial court decisions and civil law court decisions are more likely to be useful benchmarks in a Condorcet Jury Theorem analysis. It is because such courts do not anticipate that their decisions will have the force of precedent that they are more likely to be derived from governing legal texts, and thus more prone to characterizations of accuracy, than are judicial outputs that are driven by policy concerns and are generated with the understanding that they will be treated as precedent. (shrink)

Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by (...) PMR, and the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections. (shrink)

This paper defends the communitarian account of meaning against Boghossian’s (Wittgensteinian) arguments. Boghossian argues that whilst such an account might be able to accommodate the infinitary characteristic of meaning, it cannot account for its normativity: he claims that, since the dispositions of a group must mirror those of its members, the former cannot be used to evaluate the latter. However, as this paper aims to make clear, this reasoning is fallacious. Modelling the issue with four (justifiable) assumptions, it shows that (...) Condorcet’s ‘Jury Theorem’ can be used to prove that the dispositions of the majority of the members of a group can differ from those of any individual member in a way that makes it possible to use communal dispositions as a standard with which individual dispositions can be assessed. Moreover, the argument of the paper is also shown to have general implications for the use of formal methods in the explanation of the nature of certain fallacious inferences. (shrink)

I argue that many of the priority rankings that have been proposed by effective altruists seem to be in tension with apparently reasonable assumptions about the rational pursuit of our aims in the face of uncertainty. The particular issue on which I focus arises from recognition of the overwhelming importance and inscrutability of the indirect effects of our actions, conjoined with the plausibility of a permissive decision principle governing cases of deep uncertainty, known as the maximality rule. I (...) conclude that we lack a compelling decision theory that is consistent with a long-termist perspective and does not downplay the depth of our uncertainty while supporting orthodox effective altruist conclusions about cause prioritization. GPI Working Paper No. 2-2019. (shrink)

The first goal of this study was to validate the Rational-Experiential Inventory (REI) and the Cognitive Reflection Test (CRT) through checking their relation to the transitivity axiom. The second goal was to test the relation between decision strategies and cognitive style as well as the relation between decision strategies and the transitivity of preferences. The following characteristics of strategies were investigated: requirements for trade-offs, maximization vs. satisficing and option-wise vs. attribute-wise information processing. Respondents were given choices between two multi-attribute options. (...) The options were designed so that the choice indicated which strategy was applied. Both the REI-R and the CRT were found to be good predictors of the transitivity of preferences. Respondents who applied compensatory strategies and the maximization criterion scored highly on the REI-R and in the CRT, whereas those who applied the satisficing rule scored highly on the REI-R but not in the CRT. Attribute-wise information processing was related to low scores in both measurements. Option-wise information processing led to a high transitivity of preferences. (shrink)

Abstract: In this article, I offer a proposal to clarify what I believe is the proper relation between value maximization and stakeholder theory, which I call enlightened value maximization. Enlightened value maximization utilizes much of the structure of stakeholder theory but accepts maximization of the long-run value of the firm as the criterion for making the requisite tradeoffs among its stakeholders, and specifies long-term value maximization or value seeking as the firm’s objective. This proposal therefore solves the problems that (...) arise from the multiple objectives that accompany traditional stakeholder theory. I also discuss the Balanced Scorecard, which is the managerial equivalent of stakeholder theory, explaining how this theory is flawed because it presents managers with a scorecard that gives no score—that is, no single-valued measure of how they have performed. Thus managers evaluated with such a system (which can easily have two dozen measures and provides no information on the tradeoffs between them) have no way to make principled or purposeful decisions. The solution is to define a true (single dimensional) score for measuring performance for the organization or division (and it must be consistent with the organization’s strategy), and as long as their score is defined properly, (and for lower levels in the organization it will generally not be value) this will enhance their contribution to the firm. (shrink)

We use a theorem from M. J. Schervish to explore the relationship between accuracy and practical success. If an agent is pragmatically rational, she will quantify the expected loss of her credence with a strictly proper scoring rule. Which scoring rule is right for her will depend on the sorts of decisions she expects to face. We relate this pragmatic conception of inaccuracy to the purely epistemic one popular among epistemic utility theorists.

Decision-making competence reflects individual differences in the susceptibility to decision-making errors, measured using tasks common from behavioral decision research (e.g., framing effects, under/overconfidence, following decision rules). Prior research demonstrates that those with higher decision-making competence report lower incidence of health-risking and antisocial behaviors, but there has been less focus on intermediate mechanisms that may impact real-world decisions, and, in particular, those implicated by normative models. Here we test the associations between measures of youth decision-making competence (Y-DMC) and one such mechanism, (...) the degree to which individuals make choices consistent with maximizing expected value (EV). Using a task involving hypothetical gambles, we find that greater EV sensitivity is associated with greater Y-DMC. Higher Y-DMC scores are associated with (a) choosing risky options when expected value favors those options and (b) avoiding risky options when expected value favors a certain option. This relationship is stronger for gambles that involved potential losses. The results suggest that Y-DMC captures decision processes consistent with standard normative evaluations of risky decisions. (shrink)

One might think that if the majority of virtue signallers judge that a proposition is true, then there is significant evidence for the truth of that proposition. Given the Condorcet Jury Theorem, individual virtue signallers need not be very reliable for the majority judgment to be very likely to be correct. Thus, even people who are skeptical of the judgments of individual virtue signallers should think that if a majority of them judge that a proposition is true, then that (...) provides significant evidence that the proposition is true. We argue that this is mistaken. Various empirical studies converge on the following point: humans are very conformist in the contexts in which virtue signalling occurs. And stereotypical virtue signallers are even more conformist in such contexts. So we should be skeptical of the claim that virtue signallers are sufficiently independent for the Condorcet Jury Theorem to apply. We do not seek to decisively rule out the relevant application of the Condorcet Jury Theorem. But we do show that careful consideration of the available evidence should make us very skeptical of that application. Consequently, a defense of virtue signalling would need to engage with these findings and show that despite our strong tendencies for conformism, our judgements are sufficiently independent for the Condorcet Jury Theorem to apply. This suggests new directions for the debate about the epistemology of virtue signalling. (shrink)

This paper reports completeness results for dyadic deontic logics in the tradition of Hansson’s systems. There are two ways to understand the core notion of best antecedent-worlds, which underpins such systems. One is in terms of maximality, and the other in terms of optimality. Depending on the choice being made, one gets different evaluation rules for the deontic modalities, but also different versions of the so-called limit assumption. Four of them are disentangled, and compared. The main observation of this (...) paper is that, even in the partial order case, the contrast between maximality and optimality is not as significant as one could expect, because the logic remains the same whatever notion of best is used. This is established by showing that, given analogous properties for the betterness relation, the same system is sound and complete with respect to its intended modelling. The chief result of this paper concerns Åqvist’s system F supplemented with the principle of cautious monotony. It is established that, under the maximality rule, F + is sound and complete with respect to the class of models in which the betterness relation is required be reflexive and smooth . From this, a number of spin-off results are obtained. First and foremost, it is shown that a similar determination result holds for optimality; that is, under the optimality rule, F + is also sound and complete with respect to the class of models in which the betterness relation is reflexive and smooth . Other spin-off results concern classes of models in which further constraints are placed on the betterness relation, like totalness and transitivity. (shrink)

I argue that many of the priority rankings that have been proposed by effective altruists seem to be in tension with apparently reasonable assumptions about the rational pursuit of our aims in the face of uncertainty. The particular issue on which I focus arises from recognition of the overwhelming importance and inscrutability of the indirect effects of our actions, conjoined with the plausibility of a permissive decision principle governing cases of deep uncertainty, known as the maximality rule. I (...) conclude that we lack a compelling decision theory that is consistent with a longtermist perspective and does not downplay the depth of our uncertainty, while also supporting orthodox effective altruist conclusions about cause prioritization. (shrink)

Maximizers in isolated Prisoner's Dilemmas are doomed to frustration. But in Braybrooke's view maximizers might do better in a series, securing Pareto-optimal arrangements if not from the very beginning, at least eventually. Given certain favourable special conditions, it can be shown according to Braybrooke and shown even without question-begging motivational or value assumptions, that in a series of Dilemmas maximizers could manage to communicate a readiness to reciprocate, generate thereby expectations of reciprocation, and so give rise to optimizing reciprocations (...) class='Hi'>which, in due course, would reinforce expectations, the net result of all this being an increasingly stable practice to mutual benefit. In this way neighbours could learn to be good neighbours: they could learn to respect each other's property, to engage in reciprocal assistance, and perhaps even to make and keep promises covering a range of activities. So maximizers are, Braybrooke holds, capable of a society of sorts. Out of the ashes they might build something. But even under favourable conditions they could not build much and most conditions would defeat them almost entirely, for many-person Dilemmas whether isolated or in series would be quite beyond them, question-begging motivational assumptions aside. In settings at all like those in which we live, utility maximization assuming only ‘traditional’ motivation is self-defeating without remedy, and most certainly not an adequate basis for social life.' The probable inference from as a conception of individual rationality: for surely truly rational agents would be if not perfectly well designed at least better designed for communal living than utility maximizers would be assuming only ‘traditional’, non-question-begging values or motivation. Or, if this inference is denied and maximization is not challenged as a conception of rationality, we can conclude instead, Braybrooke has more recently suggested, that the advice, ‘Be rational,’ is incomplete, and when addressed to groups should be supplemented by, ‘And be somewhat ‘untraditional’ in your values; respect at least some rules that would make coordination and cooperation possible even in many-person interaction problems.’. (shrink)

This paper provides a distance based analysis of the Borda rule with respect to Condorcet’s criterion. It shows that the minimal Condorcet consistency present in the Borda rule, whenever a Condorcet winner (the alternative that wins against every other alternative in a pairwise contest) exists, disappears in the case of voting cycles. First, it is shown that for certain preference profiles the Borda winner is furthest from being a Condorcet winner. Second, it is shown (...) that there exist preference profiles for which the Borda winner is closest from being a Condorcet loser (the alternative that loses against every other alternative in a pairwise contest). (shrink)

The dominant rule of economic evaluation within health care posits that resources are distributed in order to maximize health benefit. There are instances, however, where the public has demonstrated that they do not prefer such an allocation scheme, particularly in the context of life-saving interventions. Objectives Deviations from preferences of maximizing health benefit have important implications on both financial and distributive levels. This study sought to specify the circumstances in which respondent preferences are inconsistent with maximizing health benefit. (...) Methods Ninety respondents recruited from the London School of Economics and Political Science completed a questionnaire comprised of a series of paired profiles involving various combinations of life-saving or quality-of-life enhancing interventions. Results The results indicate that saving a life holds value beyond that captured by traditional health benefit measurement and that the value of saving a life is not consistent across ages. More specifically, the value of saving a life was age-dependent and markedly attenuated for older-age patients. Conclusions Many respondents were willing to overlook maximizing health benefit in order to rescue a life in immediate peril, and showed a diminished sense of moral imperative to rescue older-age patients. In light of difficulties related to the implementation of larger-scale policies incorporating Rule of Rescue concerns, the most realistic approaches will likely involve adopting smaller-scale policies that address issues such as do not resuscitate and living wills. Potential policy solutions such as age or monetary thresholds for life-saving interventions may be favoured in a research context; however, their overall social feasibility is questionable. (shrink)

Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of (...) set and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a ‘one step back from disaster’ rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee’s Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable. (shrink)

Maxims play a crucial role in Kant's ethical philosophy, but there is significant disagreement about what maxims are. In this two-part essay, I survey eight different views of Kantian maxims, presenting their strengths, and their weaknesses. Part I: Established Approaches, begins with Rüdiger Bubner's view that Kant took maxims to be what ordinary people of today take them to be, namely pithily expressed precepts of morality or prudence. Next comes the position, most associated with Rüdiger Bittner and Otfried Höffe, that (...) maxims are Lebensregeln, or 'life-rules'– quite general rules for how to conduct oneself based on equally general outlooks on how the world is. These first two interpretations make sense of Kant's claim, made in his anthropological and pedagogical writings, that we have to learn how to act on maxims, but they become less plausible in light of Kant's probable view that people always act on maxims – after all, how can people learn how to act on something they always act on anyway? The next two views, each advanced, at different times, by Onora O'Neill, make better sense of the fact that people always act on maxims, for they hold that maxims are intentions – either specific intentions, such as 'to open the door', or general intentions, such as 'to make guests feel welcome'– and it is perfectly sensible to claim that people always act on intentions. However, they face the same problem as the two previous views, which is that if people always act on maxims, what sense does it make to say they also have to learn how to act on them? Henry Allison, the main representative of the fifth view, claims, on the basis of Kant's doctrine of the 'highest maxim', that maxims are principles organized hierarchically, such that an agent endorses one maxim because she endorses a more general maxim. Unfortunately for Allison, there is little direct textual support for his claim that maxims are organized hierarchically. (shrink)

Most plausible moral theories must address problems of partial acceptance or partial compliance. The aim of this paper is to examine some proposed ways of dealing with partial acceptance problems as well as to introduce a new Rule Utilitarian suggestion. Here I survey three forms of Rule Utilitarianism, each of which represents a distinct approach to solving partial acceptance issues. I examine Fixed Rate, Variable Rate, and Optimum Rate Rule Utilitarianism, and argue that a new approach, (...) Maximizing Expectation Rate Rule Utilitarianism, better solves partial acceptance problems. (shrink)

The level of enjoyment in participating in sports activities is one component that causes young athletes to decide to stop or become more motivated to pursue sports activities. Practicing and participating in competitions are the main activities in sports coaching interactions towards optimal performance. This study aims to determine the effect of modifying the match rules implemented in youth soccer competitions on the level of enjoyment of players. Using an experimental method with 20 soccer schools participating in a competition with (...) modified game rules. Players registered by his team amounted to an average of 13.8 players aged 9-12 years. The number of players who are willing to fill out the pleasure level questionnaire is 238 players. Each player was asked to fill out a questionnaire before and after the competition, the questionnaire was analyzed using the mean and percentage test technique. The results of data analysis showed the average score before participating in the competition was 4.10 while after participating in the competition it was 4.47, which means an increase of 8.88%. The results showed that the level of enjoyment of the players increased after participating in the competition. Modification of the rule of the game in competition affects the level of enjoyment of players in the interaction of youth soccer coaching. Coaches, administrators, and match organizers are expected to have a positive paradigm of thinking and policy making towards the organization of competitions for young soccer players. (shrink)

Maxims play a crucial role in Kant's ethical philosophy, but there is significant disagreement about what maxims are. In this two-part essay, I survey eight different views of Kantian maxims, presenting their strengths and their weaknesses. In Part II: New Approaches, I look at three more recent views in somewhat greater detail than I do the five treatments canvassed in 'Recent Works on Kantian Maxims I: Established Approaches'. First, there is Richard McCarty's Interpretation, which holds that Kant's understanding of (...) maxims can be illuminated by placing them in the context of the Wollfian tradition, according to which maxims are the major premises of practical syllogisms. The next subject Maria Schwartz, holds that careful attention to Kant's distinction between rules and maxims, as well as Kant's concept of happiness, allows us to make sense of almost all of Kant's remarks on maxims. It may be, however, that on Schwartz's view agents turn out to perform actions as opposed to thoughtlessly habitual behaviors much less often than is plausible. This leads to the final approach, exemplified by Jens Timmermann, which is that Kant understands maxims equivocally. I claim that something like Timmermann's approach is the only way to make sense of all of what Kant has to say on maxims. (shrink)

The choice of a social decision rule for a federal assembly affects the welfare distribution within the federation. But which decision rules can be recommended on welfarist grounds? In this paper, we focus on two welfarist desiderata, viz. (i) maximizing the expected utility of the whole federation and (ii) equalizing the expected utilities of people from different states in the federation. We consider the European Union as an example, set up a probabilistic model of decision making and explore (...) how different decision rules fare with regard to the desiderata. We start with a default model, where the interests, and therefore the votes of the different states are not correlated. This default model is then abandoned in favor of models with correlations. We perform computer simulations and find that decision rules with a low acceptance threshold do generally better in terms of desideratum (i), whereas the rules presented in the Accession Treaty and in the (still unratified) Constitution of the European Union tend to do better in terms of desideratum (ii). The ranking obtained regarding desideratum (i) is fairly stable across different correlation patterns. (shrink)

The existence of the obligation to follow rules in sport is widely accepted, but there are only a few studies that provide accounts that justify it. Building upon Wolff's challenge to traditional political theories, this study proposes a theory that limits the level of normativity to which participants in sport contests are bound in an effort to maximize their autonomy. Instead of constructing a unitary theory of obligations to follow sport rules, a pluralistic account is offered, one that allows (...) for multiple sources of normativity, thus augmenting the freedom of communities to play games according to their values. (shrink)

David Gauthier and Edward McClennen have claimed that it could be rational to form an intention to A because it maximizes utility to intend to A, and that acting on such an intention could be rational even if it maximizes utility not to A. Michael Bratman has objected to this way of thinking, claiming that it is equivalent to the familiar rule-utilitarian mistake of rule-worship. The purpose of this paper is to argue that, so long as (...) one is aware at the time of forming an intention to A that it maximizes utility not to A, then acting on that intention need not be rule worship, but the result of a rational refusal to reconsider an issue which has already been adequately considered. (shrink)

In this paper, we use an experimental design to compare the performance of elicitation rules for subjective beliefs. Contrary to previous works in which elicited beliefs are compared to an objective benchmark, we consider a purely subjective belief framework. The performance of different elicitation rules is assessed according to the accuracy of stated beliefs in predicting success. We measure this accuracy using two main factors: calibration and discrimination. For each of them, we propose two statistical indexes and we compare (...) the rules’ performances for each measurement. The matching probability method provides more accurate beliefs in terms of discrimination, while the quadratic scoring rule reduces overconfidence and the free rule, a simple rule with no incentives, which succeeds in eliciting accurate beliefs. Nevertheless, the matching probability appears to be the best mechanism for eliciting beliefs due to its performances in terms of calibration and discrimination, but also its ability to elicit consistent beliefs across measures and across tasks, as well as its empirical and theoretical properties. (shrink)

Structured argumentation formalisms, such as ASPIC +, offer a formal model of defeasible reasoning. Usually such formalisms are highly parametrized and modular in order to provide a unifying framework in which different forms of reasoning can be expressed. This generality comes at the price that, in their most general form, formalisms such as ASPIC + do not satisfy important rationality postulates, such as non-interference. Similarly, links to other forms of knowledge representation, such as reasoning with maximal consistent sets of (...) rules, are insufficiently studied for ASPIC + although such links have been established for other, less complex forms of structured argumentation where defeasible rules are absent. Clearly, for a formal model of defeasible reasoning it is important to understand for which range of parameters the formalism displays a behavior that adheres to common standards of consistency, logical closure and logical relevance and can be adequately described in terms of other well-known forms of knowledge representation. In this paper we answer this question positively for a fragment of ASPIC + without the attack form undercut by showing that it satisfies all standard rationality postulates of structured argumentation under stable and preferred semantics and is adequate for reasoning with maximal consistent sets of defeasible rules. The study is general in that we do not impose any other requirements on the strict rules than to be contrapositable and propositional and in that we also consider priorities among defeasible rules, as long as they are ordered by a total preorder and lifted by weakest link. In this way we generalize previous similar results for other structured argumentation frameworks and so shed further light on the close relations between assumption-based argumentation and ASPIC +. (shrink)

The basic idea of rule-utilitarianism is that right action should be defined in terms of what would be required by rules which would maximize either actual or expected utility if those rules gained general acceptance, or perhaps general compliance. Rule-utilitarians face a dilemma. They must characterize 'general acceptance' either as 100% acceptance, or as something less. On the first horn of the dilemma, rule-utilitarianism in vulnerable to the charge of utopianism; on the second, it is open (...) to the charge of arbitrariness and lack of philosophical depth. I press this objection, and develop and defend an alternative version of rule-utilitarianism which can evade the dilemma. I call this new version 'variable-rate rule-utilitarianism'. (shrink)