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On the relationship between Keynes's conception of evidential weight and the Ellsberg paradox
... But he argued that, in the first case, a greater weight supports the argument in favour of the conclusion that the probability is ½. It is regrettable that Ellsberg did not comment on Keynes's dealing of the urn example in his doctoral thesis recognising the ample similarities of his approach with Keynes's (on this point see Feduzi 2007). given different readings. ...
... 16 Keynes introduced the following coefficient: c = 2pw/(1+q)(1+w), where p is the probability of an event, q = 1-p the probability of its complement, and w is the weight. On the role of Keynes's coefficient c of probability and weight, see also Feduzi 2007. ...
This note argues that a representation of the epistemic state of the individual through a non-additive measure provides a novel account of Keynes’s view of probability theory proposed in his Treatise on Probability. The paper shows, first, that Keynes’s “non-numerical probabilities” can be interpreted in terms of decisional weights and distorsions of the probability priors. Second, that the degree of non-additivity of the probability measure can account for the confidence in the assessment without any reference to a second order probability. And, third, that the criterion for decision making under uncertainty derived in the non-additive literature incorporates a measure of the degree of confidence in the probability assessment. The paper emphasises the Keynesian derivation of Ellsberg’s analysis: the parallel between Keynes and Ellsberg is deemed to be significant since Ellsberg’s insights represent the main starting point of the modern developments of decision theory under uncertainty and ambiguity.
... Dados los 5 Puesto que la lógica estudia relaciones normativas, quizá sea mejor referirnos a seres ideales cuando hablemos de sentido común y consistencia. Esta distinción deja claro que cosas como la paradoja de Ellsberg, que se suele utilizar como argumento contra la posibilidad de asignar valores numéricos a las probabilidades (ver, p. ej.,Feduzi, 2007), confunden el enfoque normativo con el descriptivo. ...
Este articulo analiza el concepto de probabilidad en la obra de Keynes y propone un metodo para formalizar la nocion de la incertidumbre en la Teoría general utilizando el teorema de Bayes y el principio de maxima entropia. Una de las principales conclusiones es que, a pesar de compartir su rechazo del enfoque frecuentista de la estadistica, es poco razonable pensar que no se pueden determinar probabilidades numericas, aunque se cumpla algun criterio de objetividad.
... Since so much has been written on the subject, we will do no more here than point to Bewley (1986), Einhorn and Hogarth (1986) and Fox and Tversky (1995) as three interesting 11 See Friedman (2020) for an account of how historical debates about social insurance have been coloured by changing ideas about the foundations of probability. 12 Ellsberg does not mention Keynes in his 1961 paper, which was completed before he had encountered A Treatise on Probability (Feduzi, 2007). But he does go on to discuss Keynes in his subsequent 1962 PhD Dissertation (Ellsberg, 2001). ...
1. Introduction
This Special Issue marks the 100th anniversary of the first publication of two major books by economists on and around the themes of probability, risk and uncertainty: Frank Hyneman Knight’s Risk, Uncertainty, and Profit and John Maynard Keynes’ A Treatise on Probability. Knight’s book was written for economists, went on to become a classic within the discipline, and continues to be widely cited to this day on topics ranging from entrepreneurship to insurance design. Keynes’ book, in contrast, was written for a philosophical audience, initially ignored by economists save for a few early reviews,¹ and subsequently overshadowed by his own magnum opus, The General Theory of Employment, Interest and Money, published in 1936. Yet the General Theory, alongside Keynes’ 1937 reply to its critics, marked something of a return to themes first explored in A Treatise on Probability and this, spurred by rising interest in expectations with the advent of the Rational Expectations Hypothesis and the New Classical Economics in the late 1970s and early 1980s, inspired a small but continuing stream of research on his earlier work by economists.² Partly as a consequence of this research, Keynes’ A Treatise on Probability is now also routinely cited on the subject of uncertainty, sometimes in tandem with references to Knight, predominantly in various branches of heterodox economics, but also in more mainstream contributions to economics, decision theory and management.
... Upon this decision rule, it is possible to give formal preference of choosing from urn 1 instead of urn 2, as suggested by Keynes. On the significance of the weight of evidence for decision-making, see also Feduzi (2007) and (Basili and Zappia 2009). 15 Ellsberg did not refer to Keynes in his 1961 article, and acknowledged the importance of Keynes's critique of (frequency) probability only in his 1962 doctoral thesis, which he claimed to have written after the publication of the article (Ellsberg 1962, p. xlix Popper's paradox of ideal evidence, then, is part of a substantial body of research, concerning both previous well-known, though mostly disregarded, analyses such as Peirce's (1878Peirce's ( [1956) and Keynes's (1921Keynes's ( [1973), and later ones such as Ellsberg's (1961). ...
... Bu görüş nihai olarak belirsizlik altında tercihler teorisine Savage'ın yaklaşımına, yani belirsizliğin tamamen öznel olduğu ve olasılık değerlendirmesini kişinin tercihlerinin belirlediğine yol açar. Örneğin, iki kumar oyununun sonucu aynı ise, ancak biri diğerine tercih ediliyorsa, karar alıcı favori alternatif için daha yüksek kazanma olasılığı belirler (Feduzi, 2007). Öznelciler, herhangi bir olay için olasılık belirlenebileceğini varsayarlar. ...
... That is exactly the point of behavioral economics: humans behave differently from neoclassical assumptions; their motivation is more complex and decisions systematically deviate from the 12 Ellsberg used "ambiguity" but it is similar to uncertainty or what Keynes earlier labeled in the "Treatise on Probability" (1921) as "non-comparable probabilities." Ellsberg was obviously not aware of Keynes's "Treatise on Probability" (1921) when his paper was published in 1961 (see Feduzi 2007). ...
Behavioral economics, the analysis of economic decisions, has made enormous progress over the last decades and become accepted as a major field in economics. How is behavioral economics to be compared to the neoclassical model? As a revision of the neoclassical model enhancing the set of variables for motivation such as fairness in the utility function which is then to be maximized? Or is behavioral economics a revolution, a departure from the neoclassical axioms, a new model? This paper argues that many of the findings in behavioral economics are incompatible with the neoclassical model and have paved the way for a revolution in economics.
... Ellsberg is concerned with decisions under uncertainty, following the distinction drawn by Knight (1921) between measurable uncertainty (which Ellsberg calls risk) and unmeasurable uncertainty (which Ellsberg calls ambiguity). Similarly, John Maynard Keynes (1921) distinguishes between probabilities that can be assigned definite numerical values versus noncomparable probabilities for which the weight of evidence does not support numeric estimates (Feduzi 2007). Like Keynes, Ellsberg believes that in situations characterized by high ambiguity, rational actors may make decisions that cannot be explained by any assignment of numerical probabilities reflecting degrees of belief associated with events. ...
The precautionary principle is often proposed as a guide to action in environmental management or risk assessment, and has been incorporated in various legal and regulatory contexts. For many, it reflects the common sense notion of being safe rather than sorry, but it has attracted numerous critics. At times, proponents and critics talk at cross purposes, due to the multiplicity of ways the precautionary principle has been formulated. The approach taken here is to examine four general varieties of precaution, relating each to arguments made in various contexts by others. First, I examine the parallel between the precautionary principle and an argument referred to as Pascal's wager. Critics are right to dismiss versions of the precautionary principle that follow the logic of Pascal's wager, because that argument requires assumption of an infinite catastrophe, which is seldom the case in environmental decisions. Second, I explore precaution viewed as an instance of the phenomenon of ambiguity aversion as described by Daniel Ellsberg. Third, I evaluate precautionary perspectives on our duties to future generations, drawing inspiration from the views of Gifford Pinchot. Fourth, I consider the precautionary principle as an instance of Aldo Leopold's notion of intelligent tinkering. Although controversy persists, I find that a legitimate theoretical foundation exists to implement Ellsbergian, Pinchotian and Leopoldean varieties of precaution in environmental decision making. Additionally, I remark on the role of adaptive management and maintaining resilience in ecological and social systems as an approach to implementing the precautionary principle.
... It is interesting that versions of the urn example Keynes uses to illustrate the weight idea also appear independently inKnight (1921, p 223) and the seminal paper in which Ellsberg (1961) presented his eponymous paradox (seeFeduzi, 2007).at University of Cambridge on January 9, 2014 http://cje.oxfordjournals.org/ ...
The well-known Knightian distinction between quantifiable risk and unquantifiable uncertainty is at odds with the dominant
subjectivist conception of probability associated with de Finetti, Ramsey and Savage. Risk and uncertainty are rendered indistinguishable
on the subjectivist approach insofar as an individual’s subjective estimate of the probability of any event can be elicited
from the odds at which she would be prepared to bet for or against that event. The risk/uncertainty distinction has however
never quite gone away and is currently under renewed theoretical scrutiny. The purpose of this article is to show that de
Finetti’s understanding of the distinction is more nuanced than is usually admitted. Relying on usually overlooked excerpts
of de Finetti’s works commenting on Keynes, Knight and interval valued probabilities, we argue that de Finetti suggested a
relevant theoretical case for uncertainty to hold even when individuals are endowed with subjective probabilities. Indeed,
de Finetti admitted that the distinction between risk and uncertainty is relevant when different individuals sensibly disagree
about the probability of the occurrence of an event. We conclude that the received interpretation of de Finetti’s understanding
of subjective probability needs to be qualified on this front.
... As is well-know, in his study of a few decisional urn problems Ellsberg showed that, when contemplating urns with an unknown proportion of balls of different colours, rational agents were induced to violate deliberately Savage's axioms, a point we have just seen ranges from 0 to 1, as suggested by Keynes in Chapter 26 of the Treatise.17 It is regrettable that Ellsberg, who mainly reproduced Keynes's point in his examination of the twourn example, did not comment on Keynes's scrutiny of it (on this point seeFeduzi 2007).Treatise is the main reference for the authors who share, although on different grounds, the same distrust about the meaningfulness of assuming that subjective beliefs could be always represented by a single and fully reliable additive probability function. 19 ...
A by-product of the recent financial crisis has been the renewed interest in Keynes's works. Both in the press and in scientific journals, a crowd of commentators has emphasised the need to scrutinise the General Theory in order to gain a better understanding of the actual macro-dynamics of the economy and of the policy measures apt to help the economy recover from the downturn. But Keynes's thought has been given central prominence also with respect to the understanding of what went wrong at the microeconomic level, with specific reference to the role played by “irrational” agents animated by animal spirits. This paper supplements the influential analysis of George Akerlof and Robert Shiller’s Animal Spirits by arguing that a Keynesian explanation of the actual behaviour of individual agents is to be based more on the Treatise on Probability than on the General Theory itself. Indeed, while it is well-know that the rationale of Keynes's rejection of “Benthamite calculus” is best provided in the Treatise, less attention is usually given to the constructive analysis emerging from his criticism of contemporary probability theory. Through an assessment of Keynes's examination of “the application of probability to conduct” in the Treatise, the paper shows that most of the developments of what is usually referred to as behavioural finance have a Keynesian origin. In particular Keynes hinted at a decision rule different from mathematical expectation, a rule intended to mimic the behaviour of actual agents making decisions under uncertainty. The understanding of the current financial crisis, the paper concludes, would gain from a Keynesian assessment of the rationale for actual decisions as much as from the usual one concerning macroeconomic policy.
... It is interesting that versions of the urn example Keynes uses to illustrate the weight idea also appear independently inKnight (1921, p 223) and the seminal paper in which Ellsberg (1961) presented his eponymous paradox (seeFeduzi, 2007).at University of Cambridge on January 9, 2014 http://cje.oxfordjournals.org/ ...
In the insurance literature, it is often argued that private markets can provide insurance against ‘risks’ but not against ‘uncertainties’ in the sense of Knight ([1921]) or Keynes ([1921]). This claim is at odds with the standard economic model of risk exchange which, in assuming that decision-makers are always guided by point-valued subjective probabilities, predicts that all uncertainties can, in theory, be insured. Supporters of the standard model argue that the insuring of highly idiosyncratic risks by Lloyd's of London proves that this is so even in practice. The purpose of this article is to show that Bruno de Finetti, famous as one of the three founding fathers of the subjective approach to probability assumed by the standard model, actually made a theoretical case for uncertainty within the subjectivist approach. We draw on empirical evidence from the practice of underwriters to show how this case may help explain the reluctance of insurers to cover highly uncertain contingencies.
• 1 Introduction
• 2 Knight and Keynes on the Philosophy of Unknown Probabilities and Lloyd's of London
• 2.1 Knight
• 2.2 Keynes
• 3 Insuring Unique Events: The Subjectivist Viewpoint as Represented by de Finetti
• 4 The ‘Philosophy’ of Practitioners
• 5 De Finetti on Uncertainty in Knight and Keynes and on Insurability
• 5.1 De Finetti on Knight
• 5.2 De Finetti on Keynes
• 6 Empirical Evidence on Insurance Under Ambiguity
• 7 Conclusion
In this paper, we use the algebra to characterise decision‐makers’ representations of risk and uncertainty. We show that risk can be represented by objective probabilities on one part of the algebra, and that uncertainty can be represented by subjective probabilities on the other part. Decision‐makers are shown to maximise a generalised form of rank‐dependent expected utility. Their occasionally anomalous behaviour is discussed.
Various modern decision theories seek to capture the intuition behind Keynes's conception of evidential weight. Keynes was nevertheless hesitant about the practical relevance of weight in the process of rational decision making because of the 'stopping problem' of finding a rational principle to decide where to stop the process of acquiring information in forming a probability judgment before making a decision. This paper discusses the relevance of the stopping problem by way of an inquiry into the nature, properties and implications for rational decision making of Keynes's conception of evidential weight. It is argued that in practical choice situations the decision maker often decides where to stop the process of acquiring information by following Keynes's advice to consider the degree of completeness of the available information before making a decision. This method implies that the decision maker is able to arrive at an assessment of the dimension of what may be called her 'relevant ignorance'. By considering some examples of how the acquisition of new evidence may affect the decision maker's behaviour, it is argued that it is in fact possible to talk reasonably about relevant ignorance, or what are sometimes called 'unknown unknowns', and that this concept might explain a range of human behaviours. While this concept does not provide a rational principle to solve the stopping problem, it does provide a method of inquiry for dealing with a number of paradoxes not solvable within the Bayesian approach.
Speculation and Reasonableness: a non-Bayesian Theory of Rationality. In the recent discussion on financial markets two main currents have crossed. One revives Keynes's discussion on speculation and liquidity preference and the connection these concepts have with uncertainty and probability in A Treatise on Probability (TP, CW VIII). 1 The other moves from the Bayesian theory and focuses on situations which generate paradoxes and anomalies of rational behaviour in financial markets considered "efficient", namely situations which give rise to situations of uncertainty à la Keynes, Knight and Shackle. The first current analises the rationality or irrationality of speculative behaviour and liquidity preference and the role of conventions in financial markets. In various ways this opposes the explanation of financial markets as efficient markets and reconnects with Victoria Chick's idea that Keynes's liquidity and speculation find no place in Keynesian theory nor in Tobin who, in Chick's view, reduces uncertainty to calculable risk (Chick 1983:214-6). These studies are also linked to a postkeynesian view of uncertainty based upon non measurable probabilities. Many authors have recently often in a critical manner analysed Keynes's contribution to the analysis of financial markets. These include Cottrell, Davis, Lawlor, Mini, Pratten, Runde, Winslow. As said above, these contibutions are all connected with the discussion of the role and relevance of A Treatise on Probability (TP) in the interpretation of Keynes's method in economics. They raise two main questions pertinent to this paper: in the General Theory (GT, CW VII), is Keynes adopting a subjectivist Bayesian probability or a logical probability in line with the TP?; is he defending a notion of rationality-and, if so, of what type-or does he hold a hypothesis of irrationality of behaviour in financial markets? This reference to the different approaches to probability as one of the key to understanding the contrast between the different theories of the working of the financial market leads us to the second current of the recent discussion on financial markets mentioned above. The second current picks up and develops some of the criticisms to the Bayesian theory, applying them to financial markets. The conclusions reached are similar to those of Keynes's followers although the starting point, the Bayesian theory, is different. These contributions move from the efficient markets hypothesis but stress the anomalies in financial markets: the excessive reaction or volatility of prices, the anomaly of the price-yields relation, the apparent 1 The abbreviation CW stands for The Collected Writings of J.M. Keynes followed by volume and page number, TP for A Treatise on Probability (1921),TM for A Treatise on Money (1930), GT for The General Theory (1936). The second part of the article deals with Keynes's early 1910 Notes on Stock Exchange and speculation.
This paper is an attempt to go beyond dichotomic taxonomies of the concept of uncertainty. It argues for the need to distinguish between at least two types of situation that have been opposed to what neoclassical economics deals with under the rubric of uncertainty or risk. These situations are associated with the concepts of, respectively, ambiguity and fundamental uncertainty. It is also argued that Keynes refers both to situations of ambiguity and of fundamental uncertainty in his several writings, without explicitly distinguishing between them. Finally, the paper also discusses whether ambiguity and fundamental uncertainty come in (ordinal) degrees.
Proposes a descriptive model of how people make judgments under ambiguity that assumes an anchoring-and-adjustment process in which an initial estimate provides the anchor, and adjustments are made for what might be. The latter is modeled as the result of a mental simulation process that reflects not only the amount of ambiguity, which affects the size of the simulation but also one's attitude toward ambiguity, which affects the differential weighting of imagined probabilities. A 2-parameter model of this process is shown to be consistent with D. Ellsberg's (1961) original paradox, the nonadditivity of complementary probabilities, current psychological theories of risk, and J. M. Keynes's (1921) idea of the "weight of evidence." Four experiments were conducted to test the model. In Exps I and II, each with 32 volunteers, the model accurately predicted inferential judgments; in Exp III, with 20 Ss, the model predicted choices between gambles; and in Exp IV, with 112 masters of business students, it showed how buying and selling prices for insurance are influenced by one's attitude toward ambiguity. Results are discussed with respect to the importance of ambiguity in assessing uncertainty, the use of cognitive strategies in judgments under ambiguity, the role of ambiguity in risky choice, and extensions of the model. Further explanations of specific facets of the model are appended. (62 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
In Chapter 12 of the General Theory , on “The State of Long-Term Expectation,” Keynes writes: “It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain” (CW VII, p. 148). In a footnote to this sentence, Keynes points out that by “very uncertain” he does not mean the same as “very improbable” and refers to the chapter on “The Weight of Arguments” in his earlier Treatise on Probability (CW VIII, pp. 77–85). The purpose of this article, in the first place, is to provide an account of, and to sort out the relations between, Keynes's views on probability, uncertainty, and the weight of arguments.
In The Enterprise of Knowledge (Levi, 1980a), I proposed a general theory of rational choice which I intended as a characterization of a prescriptive theory of ideal rationality. A cardinal tenet of this theory is that assessments of expected value or expected utility in the Bayesian sense may not be representable by a numerical indicator or indeed induce an ordering of feasible options in a context of deliberation. My reasons for taking this position are related to my commitment to the inquiry-oriented approach to human knowledge and valuation favored by the American pragmatists, Charles Peirce and John Dewey. A feature of any acceptable view of inquiry ought to be that during an inquiry points under dispute ought to be kept in suspense pending resolution through inquiry.
A central part of Bayesianism is the doctrine that the decision maker's knowledge in a given situation can be represented by a subjective probability measure defined over the possible states of the world. This measure can be used to determine the expected utility for the agent of the various alternatives open to him. The basic decision rule is then that the alternative which has the maximal expected utility should be chosen. A fundamental assumption for this strict form of Bayesianism is that the decision maker's knowledge can be represented by a unique probability measure. The adherents of this assumption have produced a variety of arguments in favor of it, the most famous being the so-called Dutch book arguments. A consequence of the assumption, in connection with the rule of maximizing expected utility, is that in two decision situations which are identical with respect to the probabilities assigned to the relevant states and the utilities of the various outcomes the decisions should be the same. It seems to us, however, that it is possible to find decision situations which are identical in all the respects relevant to the strict Bayesian, but which nevertheless motivate different decisions. As an example to illustrate this point, consider Miss Julie who is invited to bet on the outcome of three different tennis matches. 1 As regards match A, she is very well-informed about the two players - she knows everything about the results of their earlier matches, she has watched them play several times, she is familiar with their present physical condition and the setting of the match, etc. Given all this information, Miss lulie predicts that it will be a very even match and that a mere chance will determine the winner. In match B, she knows nothing whatsoever about the relative strength of the contestants (she has not even heard their names before) and she has no other information that
The basic proposition of this paper is that Keynes's analysis of uncertainty has continuing relevance as a contribution to economic theory and decision theory. Keynes's original analysis and its subsequent development by Post Keynesians can provide the foundations for a more general theory of decision-making under uncertainty. Keynes's analysis can be extended to encompass both orthodox risk-based theories such as subjective expected utility theory as well as recent non-risk-based research on ambiguity and the heuristics and biases approach of Tversky and Kahneman.
Support theory represents probability judgment in terms of the support, or strength of evidence, of the focal relative to the alternative hypothesis. It assumes that the judged probability of an event generally increases when its description is unpacked into disjoint components (implicit subadditivity). This article presents a significant extension of the theory in which the judged probability of an explicit disjunction is less than or equal to the sum of the judged probabilities of its disjoint components (explicit subadditivity). Several studies of probability and frequency judgment demonstrate both implicit and explicit subadditivity. The former is attributed to enhanced availability, whereas the latter is attributed to repacking and anchoring.
An act maps states of nature to outcomes: deterministic outcomes, as well as random outcomes, are included. Two acts f and g are comonotonic, by definition, if it never happens that f(s) > f(t) and g(t) > g(s) for some states of nature s and t. An axiom of comonotonic independence is introduced here. It weakens the von Neumann-Morgenstern axiom of independence as follows: If f > g and if f, g and h are comonotonic then $f + (1 - $)h > $g + (1 - $)h. If a nondegenerate, continuous, and monotonic (state independent) weak order over acts satisfies comonotonic independence, then it induces a unique non-(necessarily-) additive probability and a von Neumann-Morgenstern utility. Furthermore, one can compute the expected utility of an act with respect to the nonadditive probability, using the Choquet integral. This extension of the expected utility theory covers situations, such as the Ellsberg paradox, which are inconsistent with additive expected utility. The concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included. Copyright 1989 by The Econometric Society.
In subjective expected utility (SEU), the decision weights people attach to events are their beliefs about the likelihood of events. Much empirical evidence, inspired by Ellsberg (1961) and others, shows that people prefer to bet on events they know more about, even when their beliefs are held constant. (They are averse to ambiguity, or uncertainty about probability.) We review evidence, recent theoretical explanations, and applications of research on ambiguity and SEU.
John Maynard Keynes is undoubtedly the most influential Western economist of the twentieth century. His emphasis on the nature and role of uncertainty in economic thought is a dominant theme in his writings. This book brings together a wide array of experts on Keynes' thought such as Gay Tulip Meeks, Sheila Dow and John Davis who discuss, analyse and criticise such themes as Keynesian probability and uncertainty, the foundations of Keynes' economics and the relationship between Keynes' earlier and later thought. The Philosophy of Keynes' Economics is a readable and comprehensive book that will interest students and academics interested in the man and his thought. © 2003 Jochen Runde and Sohei Mizuhara, selection and editorial matter; individual chapters, the contributors. All rights reserved.
Decision theory under uncertainty (from now on DT) has played a remarkable role in many scientific disciplines such as economics, management, sociology, political science, law, philosophy, psychology, biology, even physics1. Any scientific discipline involves a relationship between an observer and an object of study. DT may often intervene in order to clarify the influence of the observer on her object, or at least on its description, explanation, and forecast. More in general, inductive (or abductive) inferences based on empirical evidence are often guided by procedures and criteria which are decision-theoretic, or may be rephrased in these terms. In the social sciences, even the object of the analysis is often taken to be, or to be reducible to, a set of past, present, or prospective, individual or collective decisions. Though the impact of DT has been deep in many scientific disciplines, it is safe to say that it has been, and is more and more, remarkable in economics (Anand (1993, p.l)). Not by chance, many of its main contributors in this century, have been either economists (Knight, Keynes, Morgenstern, Allais, Shackle, Ellsberg, Quiggin, Yaari, Kreps, etc.) or scholars with a strong interest in economics (Ramsay, de Finetti, {ptvon} Neumann, Savage, Aumann, Machina, Schmeidler, Epstein, Gilboa, etc.).
: Ambiguity results from having limited knowledge of the process that generates outcomes. It is argued that many real-world processes are perceived to be ambiguous; moreover, as Ellsberg demonstrated, this poses problems for theories of probability operationalized via choices amongst gambles. A descriptive model of how people make judgments under ambiguity in tasks where data come from a source of limited, but not exactly known reliability, is proposed. The model assumes an anchoring-and-adjustment process in which data provides the anchor, and adjustments are made for what might have been. The latter is modeled as the result of a mental simulation process that incorporates the unreliability of the source and one's attitude toward ambiguity in the circumstances. A two-parameter model of this process is shown to be consistent with: Keynes' idea of the weight of evidence, the non-additivity of complementary probabilities, current psychological theories of risk, and Ellsberg's original paradox. The model is tested in four experiments at both the individual and group levels. In experiments 1-3, the model is shown to predict judgments quite well; in experiment 4, the inference model is shown to predict choices between gambles. The results and model are then discussed with respect to the importance of ambiguity in assessing perceived uncertainty; the use of cognitive strategies in judgments under ambiguity; the role of ambiguity in risky choice; and extensions of the model. (Author)
In chapter VI of his [1921] Keynes treats the probability of H on E as a property of the argument from E to H. The probability depends for its value on the balance between the favourableness and unfavourableness of the evidence that E states in relation to H. But he considers that there may be another respect in which some kind of quantitative comparison between arguments is possible. ‘This comparison turns,’ he says, ‘upon a balance, not between the favourable and the unfavourable evidence, but between the absolute amounts of relevant knowledge and relevant ignorance. As the relevant evidence at our disposal increases, the magnitude of the probability of the argument may either decrease or increase, according as the new knowledge strengthens the unfavourable or the favourable evidence; but something seems to have increased in either case,—we have a more substantial basis upon which to rest our conclusion.’ Keynes expresses this by saying that an accession of relevant evidence increases what he calls the ‘weight’ of an argument. Thus the weight of an argument is independent of the correctness or incorrectness with which such-or-such a probability is assigned to the argument and is not necessarily determined by the probable error of the argument’s conclusion (where that conclusion assigns a value to a magnitude). Keynes says that, metaphorically, the weight of the argument from E to H measures the sum of the favourable and unfavourable evidence that E states for H, and the probability measures the difference. But he does not suggest any method by which weights might be measured and in fact admits that often one cannot even compare the weights of different arguments. He thinks that, ‘in deciding on a course of action, it seems plausible to suppose that we ought to take account of the weight as well as the probability of different expectations’. But he finds it difficult to think of any clear example of this, and he does not feel sure that the theory of evidential weight has much practical significance.
Diagrams and Tables. Measures of Location. Measures of Dispersion and Skewness. Basic Ideas of Probability. Random Variables and Their Probability Distribution. Some Standard Discrete and Continuous Probability Distributions. Approximations to the Binomial and Poisson Distributions. Linear Functions of Random Variables and Joining Distributions. Sample Populations and Point Estimation. Interval Estimation. Hypothesis Tests for the Mean and Variance of Normal Distributions. Hypothesis Tests for the Binomial Parameter p,p. Hypothesis Tests for Independence and Goodness-of-Fit. Non-Parametric Hypothesis Tests. Correlation. Regression. Elements of Experimental Design and Analysis. Quality Control Charts and Acceptance Sampling.
The word “equivalent” of the original has been translated throughout as “exchangeable.” The original term (used also by Khinchin) and even the term “symmetric” (used by Savage and Hewitt) appear to admit ambiguity. The word “exchangeable,” proposed by Fréchet, seems expressive and unambiguous and has been adopted and recommended by most authors, including de Finetti.
Ramsey's critique of Keynes's ‘logical’ approach to probability is widely regarded as decisive, and his own ‘subjective’ approach and SEU framework are now familiar tools in economics. This paper challenges the standard view of Ramsey's critique and assesses the SEU model from a Keynesian viewpoint on probability. It consists of a summary of the two theories and an evaluation of Ramsey's criticisms and alternative. The two main conclusions are (i) that although Keynes yields to Ramsey on the question of the existence of logical probability relations, the formal structure of his theory is compatible with non-objectivist interpretations of probability; and (ii) that Ramsey makes far heavier demands on the capacities of the ‘rational agent’ than does Keynes.
Ambiguity is uncertainty about an option's outcome-generating process, and is characterized as uncertainty about an option's outcome probabilities. Subjects, in choice tasks, typically have avoided ambiguous options. Descriptive models are identified and tested in two studies which had subjects rank monetary lotteries according to preference. In Study 1, lotteries involved receiving a positive amount or nothing, where P denotes the probability of receiving the nonzero amount. Subjects were willing to forego expected winnings to avoid ambiguity near P = .50 and P = .75. Near P = .25, a significant percentage of subjects exhibited ambiguity seeking, with subjects, on average, willing to forego expected winnings to have the more ambiguous option. The observed behavior contradicts the viability of a proposed lexicographic model. Study 2 tested four polynomial models using diagnostic properties in the context of conjoint measurement theory. The results supported a sign dependence of ambiguity with respect to the probability level P, such that subjects' preference orderings over ambiguity reversed with changes in P. This behavior was inconsistent with all the three-factor polynomial models investigated. Further analyses failed to support a variant of portfolio theory, as well. The implications of these results for the descriptive modeling of choice under ambiguity are discussed.
Acts are functions from the set of states of the world into the set of consequences. Savage proposed axioms regarding a binary relation on the set of acts which are necessary and sufficient for it to be representable by the functional ʃu(·)dP for some real-valued (utility) function u on the set of consequences and a (probability) measure P on the set of states of the world. The Ellsberg paradox leads us to reject one of Savage's main axioms - the Sure Thing Principle - and develop a more general theory, in which the probability measure need not be additive.
Instead of minimizing the maximum risk it is proposed to restrict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount. Subject to this restriction one may wish to minimize the average risk with respect to some guessed a priori distribution suggested by previous experience. It is shown how Wald's minimax theory can be modified to yield analogous results concerning such restricted Bayes solutions. A number of examples are discussed, and some extensions of the above criterion are briefly considered.
History of Political Economy 36.4 (2004) 768-769
It may still be early for an overall assessment of the long-term achievements of the recent literature on Keynes's philosophical work—literature that not only analyzes Keynes's work but sometimes credits him with being a leading figure of twentieth-century philosophy. Yet time seems ripe for a survey of the different interpretations, whose authors are now given "the opportunity to present reasonably brief and accessible statements of their positions" (15). After the editors' introduction, contributions are grouped under six headings: "probability, uncertainty, and choice" (five essays); "continuity" (five); "social ontology" (three); "convention" (two); "methodology" (two); and "looking ahead" (two). These subjects are so closely interconnected, however, that it makes sense to organize this discussion of their contents around the key issue of the relationship between the Treatise on Probability and the General Theory. The fundamental question is whether the later Keynes's notion of face-saving rationality to ward off uncertainty derives from—or at least can be grafted onto—his early theory of probability. To be sure, the debate "shows no sign of abating" (1), but the excitement of the late 1980s, when the continuity thesis stirred Keynesian scholarship, has now vanished. While Anna Carabelli continues to maintain the persistence of a unique approach to probable knowledge throughout Keynes's writings, Bradley W. Bateman and John B. Davis convincingly revive the old-fashioned view that in the Treatise Keynes upheld a rationalistic theory of human belief and action, which he later abandoned. The other early champion of continuity, Rod O'Donnell, has always argued for a weaker version of the thesis, and most contributors take the middle road, insisting on the coexistence of elements of both continuity and discontinuity in Keynes's work.
Attention in The Philosophy of Keynes's Economics focuses on some concepts of the Treatise that, although noticed by Keynes's contemporaries, were given prominence only in the 1980s: nonmeasurable and noncomparable probabilities and evidential weight (all faint copies of von Kries's originals, according to Guido Fioretti). Do they represent the—albeit limited—"framework" in which Keynes's later remarks on uncertainty and convention would find place (Runde, 53), or rather do they anticipate the "more psychological and practical framework" of the later book, which thus stands in "dynamic continuity" with the Treatise (Gerrard, 242–43)? Charles R. McCann Jr. (45) concludes his analysis of the Treatise leaving both options open, although the latter, leaning towards greater discontinuity, seems more persuasive. All discussion of convention, in particular of the third behavioral rule—"do as the others do"—set out by Keynes in his famous 1937 article "The General Theory of Employment," shows how far he moved away from his juvenile attempts to prove the rationale of probability judgments.
Doubts still surround the nature of conventions, their origin, the kind of equilibrium they lead to, and its stability. Jörg Bibow, Paul Lewes, and Runde discuss the French school's bootstrap theory, and Mizuhara compares Davis's analysis, limited to the origin of conventions, with Runde's approach, delving into their intrinsic features. Related themes are Donald Gillies's intersubjective probability and Carabelli's insistence on Keynes's dislike of pure convention (222). The latter deserves consideration, although it curiously shows that the continuity thesis can stand on an antisubjectivist head as well as on antirationalist feet.
By contrast, no open verdict appears to be possible on whether Keynes maintained over time the principle of organic unity in social affairs: evidence for some discontinuity seems overwhelming, suggesting a shift from an early leaning toward atomism to an organicist view of society—as cited by Ted Winslow, although with hazardous hints at Whitehead's influence. Tony Lawson himself, another early supporter of continuity, admits that Keynes's views were not fixed over time (164).
Athol Fitzgibbons finds continuity in Keynes's elitism, derived from the intuitionist theory of knowledge that first warranted access to probability judgments and then vindicated the authoritarian role of "the wise" in state...
The General Theory of Employment, Interest, and Money / John Maynard Keynes Note: The University of Adelaide Library eBooks @ Adelaide.
Decisions under uncertainty depend not only on the degree of uncertainty but also on its source, as illustrated by Daniel Ellsberg's (1961) observation of ambiguity aversion. In this article, the authors propose the comparative ignorance hypothesis, according to which ambiguity aversion is produced by a comparison with less ambiguous events or with more knowledgeable individuals. This hypothesis is supported in a series of studies showing that ambiguity aversion, present in a comparative context in which a person evaluates both clear and vague prospects, seems to disappear in a noncomparative context in which a person evaluates only one of these prospects in isolation. Copyright 1995, the President and Fellows of Harvard College and the Massachusetts Institute of Technology.
In 1989, I. Gilboa and D. Schmeidler proposed an extension of subjective expected utility theory called maxmin expected utility. The author presents a new exposition of this theory and suggests an extension of maxmin expected utility that which does not allow strictly dominated alternatives to be chosen. Maxmin expected utility is related to the earlier notion of weight of evidence and an application to welfare economics is discussed. The author examines normative properties of maxmin expected utility and argues that the theory is not vulnerable to the Dutch book argument. Copyright 1994 by Royal Economic Society.
This paper addresses the claim that L. J. Savage's account of subjective utility theory models beliefs for all rational agents. Proposals for a two-dimensional model of belief are discussed and sources of criticism of subjective utility theory as a theory of rational choice are categorized. A theory of rational choice is proposed that gives conditions under which choices (including those made by "uncertainty" averters) can be judged to be rational. The paper corroborates A. K. Sen's findings, which show that rationality is not a behavioral entity. Copyright 1991 by Royal Economic Society.
Frank Knight's (1921) famous distinction between risk and uncertainty has entered the jargon of economics and decision theory. Yet it appears to have gone unnoticed that his formulation suffers from an elementary confusion. This note locates the source of the problem and proposes a reformulation that avoids it. Copyright 1998 by Oxford University Press.
In his economic writings, Keynes hints that his liquidity preference analysis is informed by the notion of evidential weight that first appears in his earlier Treatise on Probability. This paper reviews Keynes's distinction between probability and evidential weight, and offers an account of its influence in his later work on the incentive to hold liquid assets. Alternative representations of the Keynesian uncertainty/liquidity preference relation are assessed in the light of this discussion: James Tobin's mean-variance approach, the Keynesian fundamentalist view of Paul Davidson, and the flexibility approach represented by Louis Makowski. Copyright 1994 by Oxford University Press.
Contains two other essays as well: Further Considerations & Last Papers: Probability and Partial Belief.
The R. E. Lucas (1978) general equilibrium model of asset prices is extended to admit beliefs that are represented by a (nonsingleton) set of probability measures. A primary motivation is evidence, such as the Ellsberg paradox, that people are averse to vague or imprecise probabilities. Intertemporal utility functions embodying such aversion are formulated and then, in the context of a Lucas-style economy, the existence and characterization of equilibria are addressed. A noteworthy feature is that (under specified conditions) equilibria are indeterminate. Therefore, 'animal spirits' may play a role and sizable price volatility may result. Copyright 1994 by The Econometric Society.
In this paper, Knight's distinction between risk and uncertainty, and its significance for economic analysis are examined. The paper consists of a survey of some recent developments on the theory of choice under uncertainty and some applications of these theories to problems for which Bayesian Decision Theory has not proved entirely satisfactory. Two problems are examined in detail. The first is that of finance and insurance and the second is that of risk-taking behavior with special emphasis on lotteries. Copyright 1992 by Blackwell Publishers Ltd
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