In Richard Bradley’s book, Decision Theory with a Human Face, we have selected two themes for discussion. The first is the Bolker-Jeffrey theory of decision, which the book uses throughout as a tool to reorganize the whole field of decision theory, and in particular to evaluate the extent to which expectedutility theories may be normatively too demanding. The second theme is the redefinition strategy that can be used to defend EU theories against the Allais and Ellsberg paradoxes, (...) a strategy that the book by and large endorses, and even develops in an original way concerning the Ellsberg paradox. We argue that the BJ theory is too specific to fulfil Bradley’s foundational project and that the redefinition strategy fails in both the Allais and Ellsberg cases. Although we share Bradley’s conclusion that EU theories do not state universal rationality requirements, we reach it not by a comparison with BJ theory, but by a comparison with the non-EU theories that the paradoxes have heuristically suggested. (shrink)
We use the multiple price list method and a recursive expectedutility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over (...) gains and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent. (shrink)
Consider a subjective expectedutility preference relation. It is usually held that the representations which this relation admits differ only in one respect, namely, the possible scales for the measurement of utility. In this paper, I discuss the fact that there are, metaphorically speaking, two additional dimensions along which infinitely many more admissible representations can be found. The first additional dimension is that of state-dependence. The second—and, in this context, much lesser-known—additional dimension is that of act-dependence. The (...) simplest implication of their usually neglected existence is that the standard axiomatizations of subjective expectedutility fail to provide the measurement of subjective probability with satisfactory behavioral foundations. (shrink)
Behaviour norms are considered for decision trees which allow both objective probabilities and uncertain states of the world with unknown probabilities. Terminal nodes have consequences in a given domain. Behaviour is required to be consistent in subtrees. Consequentialist behaviour, by definition, reveals a consequence choice function independent of the structure of the decision tree. It implies that behaviour reveals a revealed preference ordering satisfying both the independence axiom and a novel form of sure-thing principle. Continuous consequentialist behaviour must be (...) class='Hi'>expectedutility maximizing. Other plausible assumptions then imply additive utilities, subjective probabilities, and Bayes' rule. (shrink)
The rule to maximize expectedutility is intended for decisions where options involve risk. In those decisions the decision maker's attitude toward risk is important, and the rule ought to take it into account. Allais's and Ellsberg's paradoxes, however, suggest that the rule ignores attitudes toward risk. This suggestion is supported by recent psychological studies of decisions. These studies present a great variety of cases where apparently rational people violate the rule because of aversion or attraction to risk. (...) Here I attempt to resolve the issue concerning expectedutility and risk. I distinguish two versions of the rule to maximize expectedutility. One adopts a broad interpretation of the consequences of an option and has great intuitive appeal. The other adopts a narrow interpretation of the consequences of an option and seems to have certain technical and practical advantages. I contend that the version of the rule that interprets consequences narrowly does indeed neglect attitudes toward risk. That version of the rule excludes the risk involved in an option from the consequences of the option and, contrary to what is usually claimed, cannot make up for this exclusion through adjustments in probability and utility assignments. I construct a new, general argument that establishes this in a rigorous way. On the other hand, I contend that the version of the rule that interprets consequences broadly takes account of attitudes toward risk by counting the risk involved in an option among the consequences of the option. I rebut some objections to this version of the rules, in particular, the objection that the rule lacks practical interest. Drawing upon the literature on 'mean-risk' decision rules, I show that this version of the rule can be used to solve some realistic decision problems. (shrink)
The Dutch Book Argument for Probabilism assumes Ramsey's Thesis (RT), which purports to determine the prices an agent is rationally required to pay for a bet. Recently, a new objection to Ramsey's Thesis has emerged (Hedden 2013, Wronski & Godziszewski 2017, Wronski 2018)--I call this the ExpectedUtility Objection. According to this objection, it is Maximise Subjective ExpectedUtility (MSEU) that determines the prices an agent is required to pay for a bet, and this often disagrees (...) with Ramsey's Thesis. I suggest two responses to Hedden's objection. First, we might be permissive: agents are permitted to pay any price that is required or permitted by RT, and they are permitted to pay any price that is required or permitted by MSEU. This allows us to give a revised version of the Dutch Book Argument for Probabilism, which I call the Permissive Dutch Book Argument. Second, I suggest that even the proponent of the ExpectedUtility Objection should admit that RT gives the correct answer in certain very limited cases, and I show that, together with MSEU, this very restricted version of RT gives a new pragmatic argument for Probabilism, which I call the Bookless Pragmatic Argument. (shrink)
This note is a generalization and improved interpretation of the main result of Karni and Schmeidler. A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expectedutility over acts (...) with state-dependent utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable. (shrink)
There are decision problems where the preferences that seem rational to many people cannot be accommodated within orthodox decision theory in the natural way. In response, a number of alternatives to the orthodoxy have been proposed. In this paper, I offer an argument against those alternatives and in favour of the orthodoxy. I focus on preferences that seem to encode sensitivity to risk. And I focus on the alternative to the orthodoxy proposed by Lara Buchak’s risk-weighted expectedutility (...) theory. I will show that the orthodoxy can be made to accommodate all of the preferences that Buchak’s theory can accommodate. (shrink)
ABSTRACTA common objection to the precautionary principle is that it is irrational. I argue that this objection goes beyond the often-discussed claim that the principle is incoherent. Instead, I argue, expectedutility theory is the source of several more sophisticated irrationality charges against the precautionary principle. I then defend the principle from these objections by arguing that the relevant features of the precautionary principle are part of plausible normative theories, and that the precautionary principle does not diverge more (...) from ideal expectedutility maximization than non-ideal expectedutility maximizing procedures, and may do better in real-world choices. (shrink)
We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are (...) also consistent with the rejection of all of the expectedutility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expectedutility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expectedutility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expectedutility’ condition popular in non-expectedutility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expectedutility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity. (shrink)
The lesson to be learned from the paradoxical St. Petersburg game and Pascal’s Mugging is that there are situations where expectedutility maximizers will needlessly end up poor and on death’s door, and hence we should not be expectedutility maximizers. Instead, when it comes to decision-making, for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero, regardless of the utilities associated with those possibilities.
Standard theories of expectedutility require that preferences are complete, and/or Archimedean. We present in this paper a theory of decision under uncertainty for both incomplete and non-Archimedean preferences. Without continuity assumptions, incomplete preferences on a lottery space reduce to an order-extension problem. It is well known that incomplete preferences can be extended to complete preferences in the full generality, but this result does not necessarily hold for incomplete preferences which satisfy the independence axiom, since it may obviously (...) happen that the extension does not satisfy the independence axiom. We show, for incomplete preferences on a mixture space, that an extension which satisfies the independence axiom exists. We find necessary and sufficient conditions for a preorder on a finite lottery space to be representable by a family of lexicographic von Neumann–Morgenstern ExpectedUtility functions. (shrink)
In this article, Savage’s theory of decision-making under uncertainty is extended from a classical environment into a non-classical one. The Boolean lattice of events is replaced by an arbitrary ortho-complemented poset. We formulate the corresponding axioms and provide representation theorems for qualitative measures and expectedutility. Then, we discuss the issue of beliefs updating and investigate a transition probability model. An application to a simple game context is proposed.
We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expectedutility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature (...) on strategy-proof social choice correspondences. (shrink)
Some early phase clinical studies of candidate HIV cure and remission interventions appear to have adverse medical risk–benefit ratios for participants. Why, then, do people participate? And is it ethically permissible to allow them to participate? Recent work in decision theory sheds light on both of these questions, by casting doubt on the idea that rational individuals prefer choices that maximise expectedutility, and therefore by casting doubt on the idea that researchers have an ethical obligation not to (...) enrol participants in studies with high risk–benefit ratios. This work supports the view that researchers should instead defer to the considered preferences of the participants themselves. This essay briefly explains this recent work, and then explores its application to these two questions in more detail. (shrink)
A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expectedutility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the (...) mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result. (shrink)
In this article we explore an argumentative pattern that provides a normative justification for expectedutility functions grounded on empirical evidence, showing how it worked in three different episodes of their development. The argument claims that we should prudentially maximize our expectedutility since this is the criterion effectively applied by those who are considered wisest in making risky choices (be it gamblers or businessmen). Yet, to justify the adoption of this rule, it should be proven (...) that this is empirically true: i.e. that a given function allows us to predict the choices of that particular class of agents. We show how expectedutility functions were introduced and contested in accordance with this pattern in the 18th century and how it recurred in the 1950s when Allais made his case against the neo-Bernoullians. (shrink)
Expected-utility theory has been a popular and influential theory in philosophy, law, and the social sciences. While its original developers, von Neumann and Morgenstern, presented it as a purely predictive theory useful to the practitioners of economic science, many subsequent theorists, particularly those outside of economics, have come to endorse EU theory as providing us with a representation of reason. But precisely in what sense does EU theory portray reason? And does it do so successfully? There are two (...) strikingly different answers to these questions in the literature. On the one hand, there is the view of people such as David Gauthier that EU theory is an implementation of the idea that reason's only role is instrumental. On the other hand, there is the view suggested by Leonard Savage that the theory is a “formal” and noninstrumental characterization of our reasoning process. (shrink)
In the present paper we study the framework of additive utility theory, obtaining new results derived from a concurrence of algebraic and topological techniques. Such techniques lean on the concept of a connected topological totally ordered semigroup. We achieve a general result concerning the existence of continuous and additive utility functions on completely preordered sets endowed with a binary operation ``+'', not necessarily being commutative or associative. In the final part of the paper we get some applications to (...)expectedutility theory, and a representation theorem for a class of complete preorders on a quite general family of real mixture spaces. (shrink)
This monographic chapter explains how expectedutility (EU) theory arose in von Neumann and Morgenstern, how it was called into question by Allais and others, and how it gave way to non-EU theories, at least among the specialized quarters of decion theory. I organize the narrative around the idea that the successive theoretical moves amounted to resolving Duhem-Quine underdetermination problems, so they can be assessed in terms of the philosophical recommendations made to overcome these problems. I actually follow (...) Duhem's recommendation, which was essentially to rely on the passing of time to make many experiments and arguments available, and evebntually strike a balance between competing theories on the basis of this improved knowledge. Although Duhem's solution seems disappointingly vague, relying as it does on "bon sens" to bring an end to the temporal process, I do not think there is any better one in the philosophical literature, and I apply it here for what it is worth. In this perspective, EU theorists were justified in resisting the first attempts at refuting their theory, including Allais's in the 50s, but they would have lacked "bon sens" in not acknowledging their defeat in the 80s, after the long process of pros and cons had sufficiently matured. This primary Duhemian theme is actually combined with a secondary theme - normativity. I suggest that EU theory was normative at its very beginning and has remained so all along, and I express dissatisfaction with the orthodox view that it could be treated as a straightforward descriptive theory for purposes of prediction and scientific test. This view is usually accompanied with a faulty historical reconstruction, according to which EU theorists initially formulated the VNM axioms descriptively and retreated to a normative construal once they fell threatened by empirical refutation. From my historical study, things did not evolve in this way, and the theory was both proposed and rebutted on the basis of normative arguments already in the 1950s. The ensuing, major problem was to make choice experiments compatible with this inherently normative feature of theory. Compability was obtained in some experiments, but implicitly and somewhat confusingly, for instance by excluding overtly incoherent subjects or by creating strong incentives for the subjects to reflect on the questions and provide answers they would be able to defend. I also claim that Allais had an intuition of how to combine testability and normativity, unlike most later experimenters, and that it would have been more fruitful to work from his intuition than to make choice experiments of the naively empirical style that flourished after him. In sum, it can be said that the underdetermination process accompanying EUT was resolved in a Duhemian way, but this was not without major inefficiencies. To embody explicit rationality considerations into experimental schemes right from the beginning would have limited the scope of empirical research, avoided wasting resources to get only minor findings, and speeded up the Duhemian process of groping towards a choice among competing theories. (shrink)
An expectedutility model of individual choice is formulated which allows the decision maker to specify his available actions in the form of controls (partial contingency plans) and to simultaneously choose goals and controls in end-mean pairs. It is shown that the Savage expectedutility model, the Marschak- Radner team model, the Bayesian statistical decision model, and the standard optimal control model can be viewed as special cases of this goal-control expectedutility model.
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 expectedutility of the valued act and its maximal and minimal utilities.
In recent attempts at deriving morality from rationality expectedutility theory has played a major role. In the most prominent such attempt, Gauthier'sMorals by Agreement, a mode of maximizing utility calledconstrained maximization is defended. I want to show that constrained maximization or any similar proposal cannot be coherently supported by expectedutility theory. First, I point to an important implication of that theory. Second, I discuss the question of what the place of constrained maximization in (...)utility theory might be. Third, I argue that no matter how we answer this question, expectedutility theory cannot provide the reason why a moral disposition like constrained maximization is to be preferred to its rivals. (shrink)
This paper proposes a view uniformly extending expectedutility calculations to both individual and group choice contexts. Three related cases illustrate the problems inherent in applying expectedutility to group choices. However, these problems do not essentially depend upon the tact that more than one agent is involved. I devise a modified strategy allowing the application of expectedutility calculations to these otherwise problematic cases. One case, however, apparently leads to contradiction. But recognizing the (...) falsity of proposition (1) below allows the resolution of the contradiction, and also allows my modified strategy to resolve otherwise paradoxical cases of group choice such as the Prisoners' Dilemma: -/- (1) lf an agent x knows options A and B are both available, and x knows that were he to do A he would be better off (in every respect) than were he to do B, then doing A is more rational for x than doing B. (shrink)
Its difficult to process the number of fish killed annually by the fishing industry. Nevertheless, governments are encouraging people to eat even more fishsee, e.g., the USDA dietary guidelinesand although animal advocates certainly dont concur with this advice, they generally havent prioritized fish in their lobbying efforts. Given the influence of utilitarianism on animal advocacy, the odds are good that this is motivated by an expectedutility calculation. For those concerned about fish, is there any way to defend (...) them against this calculation? I argue for an affirmative answer: once you factor in an asymmetry between fishing and terrestrial animal agriculture, the expectedutility calculation comes out in favor of devoting resources to reducing fishing. (shrink)
We contrast three decision rules that extend ExpectedUtility to contexts where a convex set of probabilities is used to depict uncertainty: Γ-Maximin, Maximality, and E-admissibility. The rules extend ExpectedUtility theory as they require that an option is inadmissible if there is another that carries greater expectedutility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expectedutility. We (...) show that, even when the option set is convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not E-admissibility, which is not a pairwise decision rule. E-admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes. (shrink)
The context-free weak ordering principle is viewed by many as a cornerstone of rational choice theory. McClennen, for example, claims that this principle is one of a pair on which '[t]he theory of rational choice and preference, as it has been developed in the past few decades by economists and decision theorists, rests', and Sen characterizes a version of context freedom as ‘a very basic requirement of rational choice’. But this principle is certainly not uncontroversial: there are examples of principle (...) is certainly not apper irrational. (shrink)
Independence is the condition that, if X is preferred to Y, then a lottery between X and Z is preferred to a lottery between Y and Z given the same probability of Z. Is it rationally required that one’s preferences conform to Independence? The main objection to this requirement is that it would rule out the alleged rationality of Allais and Ellsberg Preferences. In this paper, I put forward a sequential dominance argument with fairly weak assumptions for a variant of (...) Independence (called Independence for Constant Prospects), which shows that Allais and Ellsberg Preferences are irrational. Hence this influential objection (that is, the alleged rationality of Allais and Ellsberg Preferences) can be rebutted. I also put forward a number of sequential dominance arguments that various versions of Independence are requirements of rationality. One of these arguments is based on very minimal assumptions, but the arguments for the versions of Independence which are strong enough to serve in the standard axiomatization of ExpectedUtility Theory need notably stronger assumptions. (shrink)
Quantum cognition in decision making is a recent and rapidly growing field. In this paper, we develop an expectedutility theory in a context of non-classical uncertainty. We replace the classical state space with a Hilbert space which allows introducing the concept of quantum lottery. Within that framework, we formulate axioms on preferences over quantum lotteries to establish a representation theorem. We show that demanding the consistency of choice behavior conditional on new information is equivalent to the von (...) Neumann–Lüders postulate applied to beliefs. A dynamically consistent quantum-like agent may violate dynamic recursive consistency, however. This feature suggests interesting applications in behavioral economics as we illustrate in an example of persuasion. (shrink)