Projective Expected Utility

📝 Abstract

Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.

💡 Analysis

Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.

📄 Content

arXiv:0802.3300v1 [quant-ph] 22 Feb 2008 Projective Expected Utility Pierfrancesco La Mura Department of Microeconomics and Information Systems HHL - Leipzig Graduate School of Management Jahnallee 59, 04109 Leipzig (Germany) Abstract Motivated by several classic decision-theoretic para- doxes, and by analogies with the paradoxes which in physics motivated the development of quantum me- chanics, we introduce a projective generalization of ex- pected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting de- cision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium. Introduction John von Neumann (1903-1957) is widely regarded as a founding father of many fields, including game theory, deci- sion theory, quantum mechanics and computer science. Two of his contributions are of special importance in our context. In 1932, von Neumann proposed the first rigorous founda- tion for quantum mechanics, based on a calculus of projec- tions in Hilbert spaces. In 1944, together with Oskar Mor- genstern, he gave the first axiomatic foundation for the ex- pected utility hypothesis (Bernoulli 1738). In both cases, the frameworks he pioneered are still at the core of the re- spective fields. In particular, the expected utility hypothesis is still the de facto foundation of fields such as finance and game theory. The von Neumann - Morgenstern axiomatization of ex- pected utility, and later on the subjective formulations by Savage (1954) and Anscombe and Aumann (1963) were im- mediately greeted as simple and intuitively compelling. Yet, in the course of time, a number of empirical violations and paradoxes (Allais 1953, Ellsberg 1961, Rabin and Thaler 2001) came to cast doubt on the validity of the hypothesis as a foundation for the theory of rational decisions in condi- tions of risk and subjective uncertainty. In economics and in the social sciences, the shortcomings of the expected utility hypothesis are generally well known, but often tacitly ac- cepted in view of the great tractability and usefulness of the corresponding mathematical framework. In fact, the hypoth- esis postulates that preferences can be represented by way of a utility functional which is linear in probabilities, and linearity makes expected utility representations particularly tractable in models and applications. The experimental paradoxes, which in the context of physics motivated the introduction of quantum mechanics, bear interesting relations with some of the decision-theoretic paradoxes which came to challenge the status of expected utility in the social sciences: in both cases, the anomalies can be regarded as violations of an appropriately defined notion of independence. In physics, quantum mechanics was intro- duced as a tractable and empirically accurate mathematical framework in the presence of such violations. In economics and the social sciences, the importance of accounting for violations of the expected utility hypothesis has long been recognized, but so far none of its numerous alternatives has emerged as dominant, sometimes due to a lack of mathemat- ical tractability, or to the ad-hoc nature of some axiomatic proposals. Motivated by these considerations, we would like to introduce a decision-theoretic framework which accom- modates the dominant paradoxes while retaining significant simplicity and tractability. As we shall see, this is obtained by weakening the expected utility hypothesis to its projec- tive counterpart, in analogy with the quantum-mechanical generalization of classical probability theory. The structure of the paper is as follows. The next section briefly reviews the von Neumann-Morgenstern framework. Sections 3 and 4, respectively, present Allais’ and Ellsberg’s paradoxes. Section 5 introduces a mathematical framework for projective expected utility, and a representation result. Section 6 contains a subjective formulation for projective ex- pected utility, and a corresponding representation result. In section 7 there is a brief discussion, and in sections 8 and 9, respectively, we show how Allais’ and Ellsberg’s paradoxes can be accommodated within the new framework. Section 10 discusses the multi-agent case and the issue of existence of strategic equilibrium, while the last section concludes. von Neumann - Morgenstern Expected Utility Let S be a finite set of outcomes, and ∆be the set of probability functions defined on S, taken to represent risky prospects (or lotteries). Next, let ⪰be a complete and transitive binary relation defined on ∆× ∆, representing a decision-maker’s preference ordering over lotteries. Indif- ference of p, q ∈∆is defined as [p ⪰q and q ⪰p] and denoted as p ∼q, while strict preference of p over q is de- fined as [p ⪰q and not q ⪰p], and denoted by p ≻q. The preference ordering is assumed to satisfy the following two co

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