Characterization of Time-Consistent Sets of Measures in Finite Trees

In this paper we give an alternative characterization for time-consistent sets of measures in a discrete setting. For each measure p in a time-consistent set P we get a distinct set of predictable processes which in return describe the p uniquely. This implies we get a one-to-one correspondence between time-consistent sets of measures and sets of predictable processes with specifc features.Multiple Priors, Time-Consistency, Ambiguity, Uncertainty Aversion

Merging of Opinions under Uncertainty

We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non-time-consistent environment.Dynamic Convex Risk Measures, Multiple Priors, Uncertainty, Robust Representation, Time-Consistency, Blackwell-Dubins

Merging of Opinions under Uncertainty

We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non -time-consistent environment.Dynamic Convex Risk Measures, Multiple Priors, Uncertainty, Robust Representation, Time-Consistency, Blackwell-Dubins.

Characterization of time-consistent sets of measures in finite trees

Bier M. Characterization of time-consistent sets of measures in finite trees. Working Papers. Institute of Mathematical Economics. Vol 434. Bielefeld: Universität Bielefeld; 2010.In this paper we give an alternative characterization for time-consistent sets of measures in a discrete setting. For each measure \mathbb{P} in a time-consistent set \mathcal{P} we get a distinct set of predictable processes which in return decribe the \mathbb{P} uniquely. This implies we get a one-to-one correspondence between time-consistent sets of measures and sets of predictable processes with specific features

Über dynamische Knightsche Unsicherheitsmodelle : Zeitkonsistenz und optimales Verhalten

Bier M. On dynamic Knightian uncertainty models : time-consistency and optimal behavior. Bielefeld (Germany): Bielefeld University; 2009.Diese Arbeit beschäftigt sich mit drei verschiedenen Fragestellungen bezüglich Zeitkonsistenz unter der Annahme von Knightscher Unsicherheit im Rahmen des von Epstein und Schneider eingeführten Entscheidungsmodells. Der erste Teil liefert eine alternative Beschreibung zeitkonsistenter Mengen von Maßen im Rahmen von endlichen Entscheidungsbäumen. Es zeigt, wie zeitkonsistente Mengen von Maßen mit der Hilfe von dichte-erzeugenden Funktionen beschrieben werden können und andersrum. Der zweite Teil diskutiert einen Dualitätsansatz bei optimalen Entscheidungen unter Unsicherheit. Es bearbeitet die Frage, wann die Minimierung nach Maßen und Maximierung nach Handlungen vertauscht werden können. Der dritte und letzte Teil beschäftigt sich mit einem Aspekt des Lernens unter Unsicherheit. Mit dem Ziel, Verschiebungen von Risikoeinschätzungen im Laufe der Zeit zu erklären, entspricht der erste Teil einem Versuch, ein entsprechendes Risikomaß explizit zu konstruieren. Da dies nicht möglich ist, nimmt der zweite und wesentliche Ansatz ein entsprechendes zeitkonsistentes Risikomaß als gegeben an und zeigt, was damit im Laufe der Zeit bei wachsender Information passiert. Dieser zweite Ansatz ist eine Verallgemeinerung des Theorems von Blackwell und Dubins auf kohärente und konvexe Risikomaße.In the framework of Knightian uncertainty more precisely in the model introduced by Epstein and Schneider 3 different questions concerning the aspect of time-consistency, in the sense of m-stability or rectangularity, are studied. The first part describes an alternative description of time-consistent sets in the special framework of finite decision trees. It shows how the characteristics of time-consistent sets of measures can be expressed with the help of density functions, more precisely density generators, and vice versa. The second part discusses a duality approach for optimal decisions under uncertainty. It gives an answer to the question: when is it possible to interchange the order of minimizing with respect to distributions with the maximization over actions. The third and final part deals with a concept of learning under uncertainty. With the aim of explaining shifts in risk assessments over time the first approach is to explicitly construct an appropriate risk measure. Showing that this cannot be done in a straightforward way, the second and more important approach takes the existence of such a risk measure for granted and shows what happens in the course of time as information increases. This second approach is a generalization of the Blackwell and Dubins Theorem to coherent as well as convex risk measures

Merging of opinions under uncertainty

Bier M, Engelage D. Merging of opinions under uncertainty. Working Papers. Institute of Mathematical Economics. Vol 433. Bielefeld: Universität Bielefeld; 2010.We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non-time-consistent environment