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Strategic Choice in Hilbert Space

Charles E. Smith, Christopher Zorn

Published Jan 1, 2017• 0 citations

Abstract

A small-but-growing number of social scientists have in recent years begun to explore the purchase of formalisms and a probability theory originally developed to accommodate nonclassical experimental results in particle physics. Motivated by the desire to explain empirical outcomes that do not fit comfortably with the axioms of rational choice theory—and that are thus related tenets of classical probability theory—these scholars have begun to examine and embrace the theory of probability namesaked for Max Born as a general framework for understanding and modeling choice. Born’s (1926) account of probability differs from the familiar, classical context of Kolmogorov both in terms of its mathematical exposition/foundation and with respect to its governing axioms. The Born theory is expressly geometric as opposed to set-theoretic. Its axioms are formalized in the (usually) complex planes of Hilbert spaces, where distances are most generally conceptualized as metrics between (sometimes high-dimensional) spaces as opposed to points. In this framework, certain of Kolmogorov’s set-theoretic axioms can be relaxed, and empirical results that do not agree with them can be systematically accommodated. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1057/978-1-137-49276-0_7

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The situation in probability theory is more or less analogous to the situation in geometry in the 19th century. Many scientists begin to understand that the Kol-mogorov axiomatic approach [1], 1933, to the modern theory of probability cannot describe all probabilistic phenomena observed in nature. Hence, the theory of probability based on this axiomatic approach is not a unique and universal probabilistic formalism. It is only one model, namely, the Kolmogorov probabilistic model. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Aufféves and Grangier, which in turn are inspired by Gleason's theorem.

In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra

\mathcal{B}

X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on

\mathcal{B}

X constitute a convex set whose extreme points are degenerate probability measures.