Choquet utility depending on the state of nature

This article, which is part of the general framework of mathematics applied to economics, is a decision-making model in total ignorance. Such an environment is characterized by the absence of a law of distribution of the states of nature allowing having good forecasts or anticipations. Based primarily on the integral of Choquet, this model allows aggregating the different states of nature in order to make a better decision. This integral of Choquet imposes itself with respect to the complexity of the environment and also by its relevance of aggregation of the interactive or conflicting criteria. The present model is a combination of the Schmeidler model and the Brice Mayag algorithm for the determination of Choquet 2-additive capacity. It fits into the framework of subjective models and provides an appropriate response to the Ellsberg paradox. Cet article qui s'inscrit dans le cadre général des mathématiques appliquées à l'économie est un modèle de prise de décision dans l'ignorance totale. Un tel environnement est caractérisé par l'absence d'une loi de distribution des états de la nature permettant d'avoir des bonnes prévisions ou anticipations. Se basant principalement sur l'intégrale de Choquet, ce modèle permet d'agréger les différents états de la nature afin de prendre une meilleure décision. Cette intégrale de Choquet s'impose par rapport à la complexité de l'environnement et aussi par son caractère pertinent d'agrégation des critères interactifs ou conflictuels. Le présent modèle est une combinaison du modèle de Schmeidler et de l'algorithme de Brice Mayag pour la détermination de la capacité 2-additive de Choquet. Il s'inscrit dans le cadre des modèles subjectifs et apporte une réponse appropriée au paradoxe d'Ellsberg.

LEONARD SAVAGE, THE ELLSBERG PARADOX, AND THE DEBATE ON SUBJECTIVE PROBABILITIES: EVIDENCE FROM THE ARCHIVES

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner, and Cedric Smith, who were supporters of the newly developed subjective approach but could not understand Savage’s insistence on the strict version he shared with Bruno de Finetti. The episode is well known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defense of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

Representing Attitudes Towards Ambiguity in Hilbert Space: Foundations and Applications

We provide here a general mathematical framework to model attitudes towards ambiguity which uses the formalism of quantum theory as a “purely mathematical formalism, detached from any physical interpretation”. We show that the quantum-theoretic framework enables modelling of the Ellsberg paradox, but it also successfully applies to more concrete human decision-making tests involving financial, managerial and medical decisions. In particular, we elaborate a mathematical representation of various empirical studies which reveal that attitudes of managers towards uncertainty shift from ambiguity seeking to ambiguity aversion, and viceversa, thus exhibiting hope effects and fear effects. The present framework provides a promising direction towards the development of a unified theory of decisions in the presence of uncertainty.

LEONARD SAVAGE, THE ELLSBERG PARADOX AND THE DEBATE ON SUBJECTIVE PROBABILITIES: EVIDENCE FROM THE ARCHIVES

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner and Cedric Smith, who were supporters of the newly developed subjective approach, but could not understand Savage’s insistence on the strictversion he shared with Bruno de Finetti. The episode is well-known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defence of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

Quantum Decision Theory, Bounded Rationality and the Ellsberg Paradox

We test a simple quantum decision model of the Ellsberg paradox. We find that the theoretical predictions of the model are in conformity with our experimental results. The predictions of our quantum model are not statistically significantly different from those of the source dependent model. The source dependent model requires the specification of probability weighting functions in order to fit the evidence. On the other hand, our quantum model makes no recourse to probability weighting functions. This suggests that much of what is normally attributed to probability weighting may actually be due to quantum probability. When we replace quantum probability by Kolmogorov probability in our model, then the Ellsberg paradox reemerges. Hence, we make essential use of quantum probability theory. All our development uses no more than standard linear algebra and real numbers, which are very familiar to economists. This makes our paper accessible to a wider audience than the quantum community. JEL Classification: D01, D81, D91

Alternative Preferences

The Allais and Ellsberg paradoxes are presented. Various generalizations of expected utility motivated by these and other paradoxes are discussed, including betweenness preferences, rank‐dependent preferences, multiple prior max‐min preferences, and prospect theory. For betweenness preferences, which include weighted utility and disappointment aversion, an investor’s marginal utility is proportional to a stochastic discount factor. Disappointment averse utility and rank‐dependent utility have first‐order risk aversion. Multiple prior max‐min utility is one way to accomodate the Ellsberg paradox (ambiguity aversion or Knightian uncertainty). The dynamic consistency of updating multiple priors is discussed.