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Theory and Decision

Dynamic consistency of expected utility under non-classical (quantum) uncertainty

V. I. Danilov, Ariane Lambert‐Mogiliansky, Vassili Vergopoulos

Published Mar 19, 2018• 14 citations
Publisher
DOI
10.1007/s11238-018-9659-7

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Quantum cognition in decision-making is a recent and rapidely growing field. In this paper we develop an expected utility theory in a context of non-classical (quantum) 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 Neuman-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.
Fluctuations can challenge the possibility of improving work extraction from quantum correlations. This uncertainty in the work extraction process can be addressed by resorting to the expected utility hypothesis, which can provide an optimal method for work extraction. We study a bipartite quantum system and examine the role of quantum correlations in a daemonic work extraction performed by certain local operations and classical communication. Specifically, we demonstrate and explain how, depending on the so-called absolute risk aversion, a non-neutral risk agent, influenced by fluctuations, views quantum correlations differently from a neutral risk agent that is affected solely by the average work.
We define three uncertainty regimes. The first regime, we call the uncertainty regime. This is the regime used in the Black-Scholes model. The non-classical uncertainty regime is defined either in terms of so called or in terms of Intuitively, both types of non-classical uncertainty are augmented versions of classical For instance, with Heisenberg uncertainty it can be shown that the 'Reduction of Compound Lotteries' axiom in the von Neumann-Morgenstern expected utility model fails. e -Uncertainty can be shown to be a mixture of Heisenberg Uncertainty and classical We consider two models under uncertainty. The first model introduces the relationship between the SchrA¶dinger equation and a particular Brownian motion. Using Black-Scholes methodology, we find that although the portfolio change does not involve a Wiener process we can not say that the portfolios return is the risk-free rate. In the second model we augment the Brownian motion of the first model and we use again Black-Scholes methodology to derive an option price. The portfolio return is again not risk free and we obtain stochastic call and put prices. Finally, we provide for an appreciation of an option pricing model under so called e-Uncertainty. d Finance 2004