Topics in Decision Theory

Formal representations drawn from rational choice theory have been used in a variety of ways to fruitfully model the way in which actual agents are approximately rational. This analysis requires bridging between ideal normative theory, in which the mechanisms, representations, and other such internal parts are in an important sense interchangeable, and descriptive psychological theory, in which understanding the internal workings of the agent is often the main goal of the entire inquiry. In this paper, I raise a problem brought (...) on by this gap: for almost every theory of approximate rationality, there will be an empirically indistinguishable alternative that individuates the parts of the agent in a significantly different way. (shrink)

Uniformity in human actions and attitudes incumbent with the ceteris paribus clause of folk psychology lucidly transits moral thoughts into the domain of subject versus object-centric explorations. In Zettel, Wittgenstein argues, “Concepts with fixed limits would demand uniformity of behaviour, but where I am certain, someone else is uncertain. And that is the fact of nature.” (Wittgenstein 2007, 68). Reflecting on the moral principle of “ethical giving” revives a novel stance in modern moral philosophy. An “ethical giving” is a moral (...) position that looks at giving from the context of harmonizing the changing demands of situations with normative ethical principles. Despite giving more prominence to the query of intuition, the chapter brings up the justification of normative moral principles pertaining to their applications and purports. Human activities aimed at decision-making goals conducted by the moral compass are nothing but the sanction of certain ethical norms and rules. In this context, two principles, the normative aspect and the perspective aspect need to be expounded in parallel. The perspective-based principle seeks practical guides to action, whereas the normative aspect principle looks for the concrete moral justification behind it. Normatively based moral decisions hinge on the probabilities and interests of agents that can sometimes lead towards a better choice. Thomas Nagel’s thought sounds more appealing, since his idea of practical conflict bends towards “...conflict between values which are incomparable for reasons apart from uncertainty about the facts.” (Nagel 2013, 128). Positions on moral personality and equality of justice collide with the conception of verities of the agents’ reason that cogitate a minimal aspect on the morally significant properties (range properties), which can hardly be possessed by all human beings equally. This argument sounds reasonable in light of Singer’s thought of “the principle of equal consideration of interests.” (Singer 2003, 21–22) The concern of this section is deliberative about how an agent could balance the demand of moral situations and the spirit of the moral principle devoid of being close to the relative truth. The conception of the moral decision remains capricious in line with the agent’s mood, time, and situation. The germane queries are: (a) How could one get the simpliciter of moral balance in “ethical giving” or offering help to the poor, orphans, and so on, who need helps from the agents and society? (b) Can we find any universalized principle in defense of ethical giving? (shrink)

State-dependent utility is a problem for the behavioural branch of decision theory under uncertainty. It questions the very possibility that beliefs be revealed by choice data. According to the current literature, all models of beliefs are equally exposed to the problem. Moreover, the problem is solvable only when the decision-maker can influence the resolution of uncertainty. This article gives grounds to reject these two views. The various models of beliefs can be shown to be unequally exposed to the problem of (...) state-dependent utility. The problem can be argued to be solved even when the decision-maker has no influence over the resolution of uncertainty. The implications of such reappraisal for a philosophical appreciation of the revealed preference methodology are discussed. (shrink)

When subject to the choice-blindness effect, an agent gives reasons for making choice B, moments after making the alternative choice A. Choice blindness has been studied in a variety of contexts, from consumer choice and aesthetic judgement to moral and political attitudes. The pervasiveness and robustness of the effect is regarded as powerful evidence of self-ignorance. Here we compare two interpretations of choice blindness. On the choice error interpretation, when the agent gives reasons she is in fact wrong about what (...) her choice is. On the choice change interpretation, when the agent gives reasons she is right about what her choice is, but she does not realise that her choice has changed. In this paper, we spell out the implications of the two interpretations of the choice-blindness effect for self-ignorance claims and offer some reasons to prefer choice change to choice error. (shrink)

A philosophical account of risk, such as this book provides, states what risk is, which attitudes to it are rational, and which acts affecting risks are rational. Attention to the nature of risk reveals two types of risk, first, a chance of a bad event, and, second, an act’s risk in the sense of the volatility of its possible outcomes. The distinction is normatively significant because different general principles of rationality govern attitudes to these two types of risk. Rationality strictly (...) regulates attitudes to the chance of a bad event and is more permissive about attitudes to an act’s risk. Principles of rationality governing attitudes to risk also justify evaluating an act according to its expected utility given that the act’s risk, if any, belongs to every possible outcome of the act. For a rational ideal agent, the expected utilities of the acts available in a decision problem explain the agent’s preferences among the acts. Maximizing expected utility is just following preferences among the acts. This view takes an act’s expected utility, not just as a feature of a representation of preferences among acts, but also as a factor in the explanation of preferences among acts. The view extends to evaluations of combination of acts, either simultaneous or in a sequence. It takes account of an agent’s attitudes to an act’s risk without weakening the standard of expected-utility maximization. The book’s theory of risk also grounds a justification of the risk-return evaluation of investments that finance advances, and a generalized version of this type of evaluation that a professional may use to advise clients about risks and that a government regulatory agency may use to make decisions about risks on behalf of the public. (shrink)

Cooperation has been analyzed primarily in the context of theories of collective intentionality. These discussions have primarily focused on interactions between pairs or small groups of agents who know one another personally. Cooperative game theory has also been used to argue for a form of cooperation in large unorganized groups. Here I consider a form of minimal cooperation that can arise among members of potentially large organized groups (e.g., corporate teams, committees, governmental bodies). I argue that members of organized groups (...) can be minimally cooperative in virtue of playing roles in an organizational structure and having a common goal. The minimal form of cooperation I argue for is not grounded in collective intentions involving symmetric mental states, special collective intentional modes, or joint commitments. More generally, I show how considering minimal cooperation in the context of organized groups provides an opportunity to reevaluate the extent to which the social world and social phenomena depend on internalist mental factors (e.g., intentions, beliefs) and externalist non-mental factors (e.g., documents, laws, job descriptions). The view of minimal cooperation among members of organized groups I offer provides support for an externalist rather than internalist theory of at least one social phenomenon. (shrink)

En este artículo se analiza la teoría para la toma de decisiones y se contrasta la valía de los dilemas morales para explicar los mecanismos deliberativos. Se sostiene que una estrategia de decisión exitosa es aquella que es capaz de realizar los movimientos racionales, adaptativos y necesarios para llegar a un fin programado. Una aspiración de las éticas normativas y descriptivas es elaborar una teoría de la decisión práctica, sin importar los análisis racionales del modo en que se llegue al (...) fin. El artículo plantea los procesos relacionados al cambio y a la retroalimentación sin encasillar la teoría de las decisiones en posturas deontológicas, utilitaristas o emocionales. Se examina la valía de los dilemas morales y algunas críticas recurrentes y se concluye que, personales o impersonales, los dilemas parecen no explicar la complejidad de los procesos deliberativos. This article analyzes the theory for decision making and contrasts the value of moral dilemmas to explain the deliberative mechanisms. It is argued that a successful decision strategy is one that is capable of performing the rational, adaptive and necessary movements to reach a programmed end. It suggests that the processes related to change and feedback can be useful but insufficient explanations if the theory of decisions is framed to deontological, utilitarian or emotional postures. The value of moral dilemmas and some recurrent criticisms are examined and it is concluded that, personal or impersonal, the dilemmas do not seem to explain the complexity of the deliberative processes. (shrink)

The basic idea underlying a precautionary principle is often summarized as “better safe than sorry.” Even if it is uncertain whether an activity will lead to harm, for example, to the environment or to human health, measures should be taken to prevent harm. This demand is partly motivated by the consequences of regulatory practices of the past. Often, chances of harm were disregarded because there was no scientific proof of a causal connection between an activity or substance and chances of (...) harm, for example, between asbestos and lung diseases. When this connection was finally established, it was often too late to prevent severe damage. However, it is highly controversial how the vague intuition behind “better safe than sorry” should be understood as a principle. As a consequence, we find a multitude of interpretations ranging from decision rules over epistemic principles to procedural frameworks. To acknowledge this diversity, it makes sense to speak of precautionary principles (PPs) in the plural. PPs are not without critices. For example, it has been argued that they are paralyzing, unscientific, or promote a culture of irrational fear. This article systematizes the different interpretations of PPs according to their functions, gives an overview about the main lines of argument in favor of PPs, and outlines the most frequent and important objections made to them. (shrink)

In the dynamic choice literature, temptations are usually understood as temporary shifts in an agent’s preferences. What has been puzzling about these cases is that, on the one hand, an agent seems to do better by her own lights if she does not give into the temptation, and does so without engaging in costly commitment strategies. This seems to indicate that it is instrumentally irrational for her to give into temptation. On the other hand, resisting temptation also requires her to (...) act contrary to the preferences she has at the time of temptation. But that seems to be instrumentally irrational as well. I here consider the two most prominent types of argument why resisting temptation could nevertheless be instrumentally rational, namely two-tier and intra-personal cooperation arguments. I establish that the arguments either fail or are redundant. In particular, the arguments fail under the pervasive assumption in both decision theory and the wider literature on practical rationality that the agent’s preferences over the objects of choice are themselves the standard of instrumental rationality. And they either still fail or they become redundant when we give up that assumption. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

This article argues that Lara Buchak’s risk-weighted expected utility theory fails to offer a true alternative to expected utility theory. Under commonly held assumptions about dynamic choice and the framing of decision problems, rational agents are guided by their attitudes to temporally extended courses of action. If so, REU theory makes approximately the same recommendations as expected utility theory. Being more permissive about dynamic choice or framing, however, undermines the theory’s claim to capturing a steady choice disposition in the face (...) of risk. I argue that this poses a challenge to alternatives to expected utility theory more generally. (shrink)

This paper discusses random utility representations of the Luce model. Earlier works, such as McFadden, Yellott, and Strauss have discussed random utility representations under the assumption that utilities are additively separable in a deterministic and a random part. Under various conditions, they have established that a separable and independent random utility representation exists if and only if the random terms are type III extreme value distributed. This paper analyzes independent random utility representations without the separability condition and with an infinite (...) universal set of alternatives. Under these assumptions, it turns out that the most general random utility representation of the Luce model is a utility function that is an arbitrary strictly increasing transformation of a separable utility function with extreme value distributed random terms. (shrink)

The existence of a Richter–Peleg multi-utility representation of a preorder by means of upper semicontinuous or continuous functions is discussed in connection with the existence of a Richter–Peleg utility representation. We give several applications that include the analysis of countable Richter–Peleg multi-utility representations.

This paper studies decisions under ambiguity when attention is paid to extreme outcomes. In a purely subjective framework, we propose an axiomatic characterization of affine capacities, which are Choquet capacities consisting in an affine transformation of a subjective probability. Our main axiom restricts the well-known Savage’s Sure-Thing Principle to a change in a common intermediate outcome. The representation result is then an affine combination of the expected utility of the valued act and its maximal and minimal utilities.

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P be the probability of B on the subjunctive supposition that A. Is P equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. (...) On the other hand, I argue that even if we can equate P with P, we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped. (shrink)

This paper examines the preference-based approach to the identification of beliefs. It focuses on the main problem to which this approach is exposed, namely that of state-dependent utility. First, the problem is illustrated in full detail. Four types of state-dependent utility issues are distinguished. Second, a comprehensive strategy for identifying beliefs under state-dependent utility is presented and discussed. For the problem to be solved following this strategy, however, preferences need to extend beyond choices. We claim that this a necessary feature (...) of any complete solution to the problem of state-dependent utility. We also argue that this is the main conceptual lesson to draw from it. We show that this lesson is of interest to both economists and philosophers. (shrink)

This paper reports the results of an experiment designed to uncover the stochastic structure of individual preferences over lotteries. Unlike previous experiments, which have presented subjects with pair-wise choices between lotteries, our design allowed subjects to choose between two lotteries or (virtually) any convex combination of the two lotteries. We interpret the mixtures of lotteries chosen by subjects as a measure of the stochastic structure of choice. We test between two alternative interpretations of stochastic choice: the random utility interpretation and (...) the deterministic preferences interpretation. The main findings of the experiment are that the typical subject prefers mixtures of lotteries rather than the extremes of a linear lottery choice set. The distribution of choices does not change between a first and second asking of the same question. We argue that this provides support for the deterministic preferences interpretation over the random utility interpretation of stochastic choice. As a subsidiary result, we find a small proportion of subjects make choices that violate transitivity, but the level of intransitive choice falls significantly over time. (shrink)

When decision makers have more to gain than to lose by changing their minds, and that is the only relevant fact, they thereby have a reason to change their minds. While this is sage advice, it is silent on when one stands more to gain than to lose. The two envelope paradox provides a case where the appearance of advantage in changing your mind is resilient despite being a chimera. Setups that are unproblematically modeled by decision tables that are used (...) in the formulation of the two envelope paradox are described, and variations on them are stipulated. The problems posed by the paradoxical modeling are then contrasted with the variations. The paper concludes with a brief explanation of why the paradoxical modeling does not gain support from the fact that one envelope has twice the amount that is in the other. (shrink)

Ambivalence (as in practical conflicts, moral dilemmas, conflicting beliefs, and mixed feelings) is a central phenomenon of human life. Yet ambivalence is incompatible with entrenched philosophical conceptions of personhood, judgement, and action, and is denied or marginalised by thinkers of diverse concerns. This book takes a radical new stance, bringing the study of core philosophical issues together with that of ambivalence. The book proposes new accounts in several areas – including subjectivity, consciousness, rationality, and value – while elucidating a wide (...) range of phenomena expressive of ambivalence, from emotional ambivalence to self-deception. The book rejects the view that ambivalence makes a person divided, showing that our tension-fraught attitudes are profoundly unitary. Ambivalence is not tantamount to confusion or to paralysis: it is always basically rational, and often creative, active, and perceptive as well. The book develops themes from Wittgenstein, Davidson, Sartre, and Freud. It engages with contemporary debates in Analytic Philosophy in addition to work ranging from Aristotle to Cultural Studies and Empirical Psychology, and considers a rich set of examples from daily life and literature. (shrink)

In everyday life people frequently recognize that a person at a time may be more or less strongly motivated to carry out an intentional action and that “trying harder” frequently affects the successful completion of an intentional action. In “Rational Choice and Action Omnipotence,” John Pollock provides an original account of rational choice in which “trying to do an action” is a basic factor. This paper argues that Pollock’s “expected-utility optimality prescription” is deficient because it lacks a parameter for intensity (...) of trying. The paper also indicates specific ways in which this deficiency could be corrected. (shrink)

Although choice and preference are distinct categories, it may in some contexts be a useful idealization to treat choices as fully determined by preferences. In order to construct a general model of such preference-based choice, a method to derive choices from preferences is needed that yields reasonable outcomes for all preference relations, even those that are incomplete and contain cycles. A generalized choice function is introduced for this purpose. It is axiomatically characterized and is shown to compare favourably with alternative (...) constructions. (shrink)

We analyze the expressivity, succinctness, and complexity of a family of languages based on weighted propositional formulas for the representation of utility functions. The central idea underlying this form of preference modeling is to associate numerical weights with goals specified in terms of propositional formulas, and to compute the utility value of an alternative as the sum of the weights of the goals it satisfies. We define a large number of representation languages based on this idea, each characterized by a (...) set of restrictions on the syntax of formulas and the range of weights. Our aims are threefold. First, for each language we try to identify the class of utility functions it can express. Second, when different languages can express the same class of utility functions, one may allow for a more succinct representation than another. Therefore, we analyze the relative succinctness of languages. Third, for each language we study the computational complexity of the problem of finding the most preferred alternative given a utility function expressed in that language. (shrink)

This paper proposes a class of independence axioms for simple acts. By introducing the E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}$$\end{document}-cominimum independence axiom that is stronger than the comonotonic independence axiom but weaker than the independence axiom, we provide a new axiomatization theorem of simple acts within the framework of Choquet expected utility. Furthermore, in order to provide the axiomatization of simple acts, we generalize Kajii et al. into an infinite state space. Our axiomatization theorem relates (...) Choquet expected utility to multi-prior expected utility through the core of a capacity that is explicitly derived within our framework. Our result in this paper also derives Gilboa, Eichberger and Kelsey, and Rohde as a corollary. (shrink)

Nakata (Theory Decis 71:559–574, 2011) presents a model of acquisition of information where the agent does not know what pieces of information she is missing. In this note, we point out some technical problems in a few of Nakata’s results and show how to correct them.

Behind the utility function, which is the basis for economic and finance theory, is a philosophical and ethical approach based essentially on the Utilitarian and Hedonistic schools. Once qualitative, the utility function’s approach shifted to a quantitative one based on the work of the mathematician, D. Bernoulli. This quantitative approach is normative and based on a maximizing agent. In this paper, the “emotional well-being” function is developed which mixes the ethics of a rational economic individual with those of a more (...) complete individual who is simultaneously conditioned by economic, instinctive, social, religious, ethical, and esthetic values. This new utility function of emotional well-being is shown to remain within the envelope function of the Bernoullitype utility function. (shrink)

In a static decision problem the decision-maker faces just one up-front choice, while in a dynamic one there are some choices to be made after other choices or after the resolution of some uncertain events. The standard axioms for static decision-making require that choices under certainty or uncertainty be based on a weak order preference relation, and conform to the variously formulated independence condition for uncertainty; these axioms permit a derivation of the principle of maximizing expected utility. ;In a dynamic-choice (...) "exploitation" argument, a decision-maker who violates at least one of the standard axioms is placed in a dynamic choice situation, one in which she is vulnerable to exploitation by another agent seeking to profit from the violation. E. McClennen's Rationality and Dynamic Choice provides an evaluation of this approach to justifying the standard axioms. He identifies and questions two premises implicit in these arguments, the principles of Plan Reduction and Separability. ;However, it remains an open question in the literature whether the project would really be successful, even if these assumptions were justifiable. The dynamic exploitation arguments that have been considered address particular violations of the standard axioms, but do not show that every violation of the axioms makes one exploitable. McClennen conjectures, on the basis of a partial argument, that the generalization is true. ;This dissertation shows that, even granting the assumptions identified by McClennen, the standard axioms cannot be justified by means of the dynamic exploitation arguments, since there are violations of the axioms to which they are not applicable. I first identify two different kinds of dynamic exploitability and their necessary conditions, and then show that some decision-making procedures violating the standard axioms fail to conform to these necessary conditions, and thus are not dynamically exploitable. (shrink)

A choice function is rational if it describes the maximization of a preference relation. There are a variety of conditions that a preference relation can satisfy. We focus on orders, a complete and transitive relation. While alpha and beta characterize choice functions normalized by an order, they do not characterize all choice functions that describe the maximization of an order. We show that lambda and mu characterize choice functions that are representable by an order. The difference between a normal and (...) a representable choice function is that the former assumes that if two alternatives are in the image of a choice set, then the two alternatives are indifferent. Representability does not make this assumption. A distinction between the ordering properties of a preference relation and the uniformity of a family of preference relations is drawn. A family of preference relations $\{$R$\sb{\rm s}\}$ is uniform if and only if R$\sb{\rm s}$ restricted to S$\sp\prime$ equals R$\sb{\rm s}\sp\prime$. We argue that the failures of transitivity of a preference relation are better expressed as failures of uniformity, and suggest that not all failures of uniformity are failures of rationality. (shrink)

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to be borne out (...) by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment. (shrink)

Naive versions of decision theory take probabilities and utilities as primitive and use expected value to give norms on rational decision. However, standard decision theory takes rational preference as primitive and uses it to construct probability and utility. This paper shows how to justify a version of the naive theory, by taking dominance as the most basic normatively required preference relation, and then extending it by various conditions under which agents should be indifferent between acts. The resulting theory can make (...) all the decisions of classical expected utility theory, plus more in cases where expected utilities are infinite or undefined. Although the theory requires similarly strong assumptions to classical expected utility theory, versions of the theory can be developed with slightly weaker assumptions, without having to prove a new representation theorem for the weaker theory. This alternate foundation is particularly useful if probability is prior to preference, as suggested by the recent program to base probabilism on accuracy and alethic considerations rather than pragmatic ones. (shrink)

This paper presents a critical reflection on dynamic consistency as commonly used in economics and decision theory, and on the difficulty to test it experimentally. It distinguishes between the uses of the term dynamic consistency in order to characterize two different properties: the first accounts for the neutrality of individual preferences towards the timing of resolution of uncertainty whereas the second guarantees that a strategy chosen at the beginning of a sequential decision problem is immune to any reevaluation and will (...) effectively be implemented from then on in the decision problem. Although these two properties are equivalent under expected utility, this is not the case under non-EU. Building on the possible characteristics of individual dynamic preferences under risk, this paper proposes a conceptual categorization, that is experimentally testable, of possible sequential decision making behaviors of non-EU maximizers. (shrink)

Let X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }$$\end{document} be a set of outcomes, and let I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{I }$$\end{document} be an infinite indexing set. This paper shows that any separable, permutation-invariant preference order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} on XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X }^\mathcal{I }$$\end{document} admits an additive representation. That is: there exists a linearly ordered abelian group R\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document} and a ‘utility function’ u:X⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u:\mathcal{X }{{\longrightarrow }}\mathcal{R }$$\end{document} such that, for any x,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document} which differ in only finitely many coordinates, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∑i∈Iu-u≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i\in \mathcal{I }} \left[u-u\right]\ge 0$$\end{document}. Importantly, and unlike almost all previous work on additive representations, this result does not require any Archimedean or continuity condition. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} also satisfies a weak continuity condition, then the paper shows that, for anyx,y∈XI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x},\mathbf{y}\in \mathcal{X }^\mathcal{I }$$\end{document}, we have x≽y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{x}\succcurlyeq \mathbf{y}$$\end{document} if and only if ∗∑i∈Iu≥∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u\ge {}^*\!\sum _{i\in \mathcal{I }}u$$\end{document}. Here, ∗∑i∈Iu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\sum _{i\in \mathcal{I }} u$$\end{document} represents a nonstandard sum, taking values in a linearly ordered abelian group ∗R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^*\!\mathcal{R }$$\end{document}, which is an ultrapower extension of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R }$$\end{document}. The paper also discusses several applications of these results, including infinite-horizon intertemporal choice, choice under uncertainty, variable-population social choice and games with infinite strategy spaces. (shrink)

Several axiom systems for preference among acts lead to a unique probability and a state-independent utility such that acts are ranked according to their expected utilities. These axioms have been used as a foundation for Bayesian decision theory and subjective probability calculus. In this article we note that the uniqueness of the probability is relative to the choice of whatcounts as a constant outcome. Although it is sometimes clear what should be considered constant, in many cases there are several possible (...) choices. Each choice can lead to a different "unique" probability and utility. By focusing attention on statedependent utilities, we determine conditions under which a truly unique probability and utility can be determined from an agent's expressed preferences among acts. Suppose that an agent's preference can be represented in terms of a probability P and a utility U.That is, the agent prefers one act to another iff the expected utility of that act is higher than that of the other. There are many other equivalent representations in terms of probabilities Q, which are mutually absolutely continuous with P, and state-dependent utilities V, which differ from U by possibly different positive affine transformations in each state of nature. We describe an example in which there are two different but equivalent state-independent utility representations for the same preference structure. They differ in which acts count as constants. The acts involve receiving different amounts of one or the other of two currencies, and the states are different exchange rates between the currencies. It is easy to see how it would not be possible for constant amounts of both currencies to have simultaneously constant values across the differentstates. Savage (1954, sec. 5.5) discovered a situation in which two seemingly equivalent preference structures are represented by different pairs of probability and utility. He attributed the phenomenon to the construction of a "small world." We show that the small world problem is just another example of two different, but equivalent, representations treating different actsas constants. Finally, we prove a theorem (similar to one of Karni 1985) that shows how to elicit a unique state-dependent utility and does not assume that there are prizes with constant value. To do this, we define a new hypothetical kind of act in which both the prize to be awarded and the state of nature are determined by an auxiliary experiment. (shrink)

We examine a distinctive kind of problem for decision theory, involving what we call discontinuity at infinity. Roughly, it arises when an infinite sequence of choices, each apparently sanctioned by plausible principles, converges to a ‘limit choice’ whose utility is much lower than the limit approached by the utilities of the choices in the sequence. We give examples of this phenomenon, focusing on Arntzenius et al.’s Satan’s apple, and give a general characterization of it. In these examples, repeated dominance reasoning (...) (a paradigm of rationality) apparently gives rise to a situation closely analogous to having intransitive preferences (a paradigm of irrationality). Indeed, the agents in these examples are vulnerable to a money pump set-up despite having preferences that exhibit no obvious defect of rationality. We explore several putative solutions to such problems, particularly those that appeal to binding and to deliberative dynamics. We consider the prospects for these solutions, concluding that if they fail, the examples show that money pump arguments are invalid. (shrink)

The theory of direct reference suggests revising probability theory so that a probability attaches to a proposition given a way of understanding the proposition. The revisions make probabilities relative but do not change their structure.