Ambiguity and uncertainty in Ellsberg and Shackle

This paper argues that Ellsberg’s and Shackle’s frameworks for discussing the limits of the (subjective) probabilistic approach to decision theory are not as different as they may appear. To stress the common elements in their theories Keynes’s Treatise on Probability provides an essential starting point. Keynes’s rejection of well-defined probability functions, and of maximisation as a guide to human conduct, is shown to imply a reconsideration of what probability theory can encompass, that is in the same vein of Ellsberg’s and Shackle’s concern in the years of the consolidation of Savage’s new probabilistic mainstream. The parallel between Keynes and the two decision theorists is drawn by means of a particular assessment of Shackle’s theory of decision, namely, it is interpreted in the light of Ellsberg’s doctoral dissertation. In this thesis, published only as late as 2001, Ellsberg developed the details and devised the philosophical background of his criticism of Savage as first put forward in the famed 1961 QJE article. The paper discusses the grounds on which the ambiguity surrounding the decision maker in Ellsberg’s urn experiment can be deemed analogous to the uncertainty faced by Shackle’s entrepreneur taking “unique decisions.” The paper argues also that the insights at the basis of the work of both Shackle and Ellsberg, as well as the criteria for decision under uncertainty they put forward, are relevant to understand the development of modern decision theory.uncertainty, weight of argument, non-additive probability

Contrary to T. Hirai, Ramsey’s Critiques of Keynes in 1922 and 1926 Were Completely Wrong

This paper covers Harai’s analysis, contained in his section, titled “Keynes as a philosopher”, of Keynes’s logical theory of probability. Harai spends too much of his time repeating Ramsey’s claims about having uncovered serious errors in the structure of Keynes’s relational propositional theory. .For 100 years, Keynes’s logical theory has been interpreted and misevaluated through the eyes of an ignorant 18 year old teenager. The result has been the proliferation and spread of what can be called the “Ramsey myth”.The “Ramsey myth” is that an 18 year old teenager appeared at Cambridge University in 1921.This 18 year old teenager was a genius who wrote a three page review in the Jan., 1922 issue of Cambridge Magazine, which supposedly destroyed, devastated and demolished the logical foundations of Keynes’s A Treatise on Probability, which was his relational, propositional logic founded on Boole’s relational, propositional logic. Russell countered this in his review, but was ignored (See Brady, 2016a). In 1931, it is further supposed that Keynes then capitulated to Ramsey and repudiated his own logical theory of probability, accepting some version of Ramsey’s subjectivist theory. This myth is what Hirai’s paper is based on. It was false in 1921 and it is false today in 2022

Distorted optimal transport

Classic optimal transport theory is built on minimizing the expected cost between two given distributions. We propose the framework of distorted optimal transport by minimizing a distorted expected cost. This new formulation is motivated by concrete problems in decision theory, robust optimization, and risk management, and it has many distinct features compared to the classic theory. We choose simple cost functions and study different distortion functions and their implications on the optimal transport plan. We show that on the real line, the comonotonic coupling is optimal for the distorted optimal transport problem when the distortion function is convex and the cost function is submodular and monotone. Some forms of duality and uniqueness results are provided. For inverse-S-shaped distortion functions and linear cost, we obtain the unique form of optimal coupling for all marginal distributions, which turns out to have an interesting ``first comonotonic, then counter-monotonic" dependence structure; for S-shaped distortion functions a similar structure is obtained. Our results highlight several challenges and features in distorted optimal transport, offering a new mathematical bridge between the fields of probability, decision theory, and risk management

Shackle versus Savage: non-probabilistic alternatives to subjective probability theory in the 1950s

G.L.S Shackle’s rejection of the probability tradition stemming from Knight's definition of uncertainty was a crucial episode in the development of modern decision theory. A set of methodological statements characterizing Shackle’s stance, abandoned for long, especially after Savage’s Foundations, have been re-discovered and are at the basis of current non-expected utility theories, in particular of the non-additive probability approach to decision making. This paper examines the discussion between Shackle and his critics in the 1950s. Drawing on Shackle’s papers housed at Cambridge University Library as well as on printed matter, we show that some critics correctly understood two aspects of Shackle’s theory which are of the utmost importance in our view: the non-additive character of the theory and the possibility of interpreting Shackle’s ascendancy functions as a specific distortion of the weighting function of the decision maker. It is argued that Shackle neither completely understood criticisms nor appropriately developed suggestions put forward by scholars like Kenneth Arrow, Ward Edwards, Nicholas Georgescu- Roegen. Had he succeeded in doing so, we contend, his theory might have been a more satisfactory alternative to Savage’s theory than it actually was.uncertainty, decision theory, non-additive measures

Risk exchange with distorted probabilities Topic 2: Risk finance and risk transfer

Abstract We study the equilibrium in a risk exchange, where agents' preferences are characterised by generalised (rank-dependent) expected utility, i.e. by a concave utility and a convex probability distortio

Ambiguity and Economic Models

This thesis is based on a collection of essays and studies the behavioral consequences of the concept of ambiguity for a variety of economic models. After introducing the reader to the fundamentals of decision-theory, I proceed by considering a Hotelling duopoly game under demand ambiguity. Firms' preferences are assumed to be of the Choquet-expected-utility-type. In this framework, I derive firms' subgame-perfect product differentiation. It turns out that confidence is a differentiation force when firms are sufficiently optimistic. This finding has important consequences for the interpretation of various applications of Hotelling models under uncertainty treated by Król (2012). Subsequently, ambiguity is implemented in the context of primary prevention. In particular, I contemplate a physician-counseling model where Choquet-expected-utility-maximizing patients face ambiguity with respect to the relationship between their level of adherence to a preventive regime and the resulting probability of disease. In this framework, I examine the effect of confidence and optimism on preventive activities. It turns out that the effect of optimism on prevention is determined by two concurrent effects, which are denoted as "perceived efficacy effect" and "expected marginal utility effect". The perceived efficacy effect captures the fact that optimists and pessimists might differ in their assessment of the preventive regime's capability to reduce the underlying probability of disease. The expected marginal utility effect takes into account that a shift in the perceived disease probability might increase or decrease marginal gains or losses from additional units of prevention. In the following step, I introduce information into the previous setting. Information is modeled by means of an imprecise signal provided by the physician. Patients update their beliefs in the light of new information by using one of the three major updating rules for neo-additive capacities introduced by Eichberger et al. (2010). It turns out that Knightian uncertainty can, depending on the underlying updating rule, provide an explanation for poor patient compliance as well as excessive preventive behavior. The ensuing chapter turns to the famous Blackwell's theorem and its extension to MEU-preferences proposed by Ҫelen (2012) showing that Ҫelen’s notion of a value of information under MEU is incompatible with dynamic consistency. Finally, I conclude with an application of ambiguity to monopoly pricing. Using the most prominent models of decision-making under ambiguity, I derive the implications of ambiguity for monopoly pricing. In the Choquet case with neo-additive capacities, I can demonstrate that pessimism reduces the resulting monopoly price whereas confidence decreases (increases) monopoly prices if the monopolist is sufficiently optimistic (pessimistic)

Modelling of and empirical studies on portfolio choice, option pricing, and credit risk

This thesis develops and applies a statistical spanning test for mean-coherent regular risk portfolios. Similarly in spirt to Huberman and Kandel (1987), this test can be implemented by means of a simple semi-parametric instrumental variable regression, where instruments have a direct link with a stochastic discount factor. Applications to different asset classes are studied. The results are compared to the conventional mean-variance approach. The second part of the thesis concerns option pricing under stochastic volatility and credit risk modelling. It is shown that modelling dynamics of the implied prices of volatility risk can improve out-of-sample option pricing performance. Finally, an equity-based structural model of credit risk with a constant elasticity of volatility assumption is discussed. This model might be particularly suitable for analysis of high yield fixed income instruments, where correlation between credit spreads and equity returns is substantial.

The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems

In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities