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Cambridge University Press eBooks

What is the quantum advantage? Applications to decision making

Jerome R. Busemeyer, Peter Bruza

Published Jul 26, 2012• 8 citations
Publisher

Abstract

Can quantum theory provide new answers to puzzling results from decision research that have resisted formal explanations so far? This chapter presents several new contributions that quantum theory provides to decision research. However, in order to locate where quantum theory can make an important new contribution, we first need to provide some necessary background on decision theory and what has already been accomplished. Decision science is a highly developed field that has produced a variety of sophisticated mathematical theories (Gilboa, 2009; Wakker, 2010). Some of these theories are rational theories, which prescribe optimal ways to make decisions. Others are psychological theories that describe the way people actually make decisions. One of the most important rational decision theories is von Neumannn and Morgenstern's expected utility (EU) theory (von Neumann & Morgenstern, 1944). Expected utility theory is designed for decisions under risk ; that is, a choice among actions defined as an objective probability distribution over possible payoffs. An important extension of (EU) theory is Savage's (1954) subjective expected utility (SEU) theory . This extends utility theory to decisions under uncertainty ; that is, a choice among actions defined as functions that map un-certain events into payoffs, and the events may not have objective probabilities. A more general theory designed for decisions under both uncertainty and risk was presented by Anscombe and Aumann (1963). All of these theories are rational, in the sense that they are based on a small set of compelling axioms and the prescribed action logically follows from the axioms.

DOI
10.1017/cbo9780511997716.010

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Why should you be interested in quantum theory applied to Cognition and decision? Perhaps you are a physicist who is curious whether or not quantum principles can be applied outside of physics. In fact, that is one purpose of this book. Perhaps you are a cognitive scientist who is interested in representing concepts by vectors in a multidimensional feature space. This is essentially the way quantum theory works too. Perhaps you are a decision scientist who is trying to understand how people make decisions under uncertainty. Quantum theory could provide some interesting new answers. Generally speaking, quantum theory is a new theory for constructing probabilistic and dynamic systems , and in this book we apply this new theory to topics in cognition and decision. Later in this chapter we will give some specific examples, but let us step back at this point and try to understand the more general principles that support a quantum approach to cognition and decision. Six reasons for a quantum approach to cognition and decision Quantum physics is arguably the most successful scientific theoretical achievement that humans have ever created. It was created to explain puzzling findings that were impossible to understand using the older classical physical theory, and it achieved this by introducing an entirely new set of revolutionary principles. The older classical physical theory is now seen as a special case of the more general quantum theory. In the process of creating quantum mechanics, physicists also created a new theory of probabilistic and dynamic systems that is more general than the previous classic theory (Pitowski, 1989).

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We investigate how the choice of decision makers can be varied under the presence of risk and uncertainty. Our analysis is based on the approach we have previously applied to individual decision makers, which we now generalize to the case of decision makers that are members of a society. The approach employs the mathematical techniques that are common in quantum theory, justifying our naming as Quantum Decision Theory. However, we do not assume that decision makers are quantum objects. The techniques of quantum theory are needed only for defining the prospect probabilities taking into account such hidden variables as behavioral biases and other subconscious feelings. The approach describes an agent's choice as a probabilistic event occurring with a probability that is the sum of a utility factor and of an attraction factor. The attraction factor embodies subjective and unconscious dimensions in the mind of the decision maker. We show that the typical aggregate amplitude of the attraction factor is $1/4$, and it can be either positive or negative depending on the relative attraction of the competing choices. The most efficient way of varying the decision makers choice is realized by influencing the attraction factor. This can be done in two ways. One method is to arrange in a special manner the payoff weights, which induces the required changes of the values of attraction factors. We show that a slight variation of the payoff weights can invert the sign of the attraction factors and reverse the decision preferences, even when the prospect utilities remain unchanged. The second method of influencing the decision makers choice is by providing information to decision makers. The methods of influencing decision making are illustrated by several experiments, whose outcomes are compared quantitatively with the predictions of our approach.

arXiv:1409.0636•Sep 2, 2014•1 citation
Can quantum theory provide new answers to puzzling results from decision research that have resisted formal explanations so far? This chapter presents several new contributions that quantum theory provides to decision research. However, in order to locate where quantum theory can make an important new contribution, we first need to provide some necessary background on decision theory and what has already been accomplished. Decision science is a highly developed field that has produced a variety of sophisticated mathematical theories (Gilboa, 2009; Wakker, 2010). Some of these theories are rational theories, which prescribe optimal ways to make decisions. Others are psychological theories that describe the way people actually make decisions. One of the most important rational decision theories is von Neumannn and Morgenstern's expected utility (EU) theory (von Neumann & Morgenstern, 1944). Expected utility theory is designed for decisions under risk ; that is, a choice among actions defined as an objective probability distribution over possible payoffs. An important extension of (EU) theory is Savage's (1954) subjective expected utility (SEU) theory . This extends utility theory to decisions under uncertainty ; that is, a choice among actions defined as functions that map un-certain events into payoffs, and the events may not have objective probabilities. A more general theory designed for decisions under both uncertainty and risk was presented by Anscombe and Aumann (1963). All of these theories are rational, in the sense that they are based on a small set of compelling axioms and the prescribed action logically follows from the axioms.
Quantum theory has inspired many new ideas for understanding Cognition and decision making. Given the nascent state of this field, many new challenges still remain. What are the strengths and unique contributions of these ideas? What are likely next steps in its development? How can it be applied to practical problems in cognitive science? Is this a rational or an irrational system of reasoning? What, if any, are its connections to neuroscience? Should the hard problem of consciousness be considered? These are questions that we try to address in this chapter, albeit at times in a limited way. We end with a debate with a fictitious skeptic (actually previous reviewers) of these ideas. Let us start by reviewing what we have covered so far. Contributions to cognition and decision We begin by summarizing the conclusions reached in previous chapters regarding the usefulness of quantum theory for modelling phenomena in cognition and decision. Chapter 1 provided some motivation and Chapter 2 provided a theoretical introduction to quantum theory. Chapter 3 presented a quantum model to account for question order effects on survey research. An exact and parameter-free test was derived from the model, which produced surprisingly accurate predictions for results obtained from three different large Gallup poll survey studies. Chapter 4 described a quantum model of probability judgments to account for findings that produce violations of classic probability theory, including conjunction and disjunction fallacies and order effects on inference.
The core assumptions of modern economic utility theories such as rationality have been placed under scrutiny by recent studies in decision theory and behavioural economics and fi?nance. The main questions that were explored in these studies could be expressed as follows: do people always make rational choices? Do they make use of the normative rules of traditional logic and probability when dealing with risk and uncertainty? Are there any alternative methods to capture choices and reasoning when individuals are faced with uncertainty and risk? In this thesis I will elaborate on an interdisciplinary application of so-called QL (quantum-like) models to decision-making problems in ?finance and other socio-economic contexts. The core inquiry of this work is related to experimental tests of decision making with the help of investment related questions. Calculus and logic borrowed from quantum physics will be used to develop probabilistic models for the formation and evolution of decisions in ?finance context with the potential to generalize to other social decision- making settings. The main fi?ndings of this thesis can be summarized as follows: individuals exhibit preference reversals and context-dependent behaviour in different informational contexts. In particular, in condition of risk individuals exhibit uncertainty aversion and thus aversion to participating in other investment task, violating the postulates of classical probability theory. The core principle of rational (consequential) behaviour conceived by Savage (1954) [207], that is the "Savage Sure Thing" principle, was shown to be violated. In mathematical language, this violation is equivalent to the violation of the "Law of total probability". Implying contextuality of human judgement, different events cannot be jointly evaluated in a single probability space. Moreover, additional experimental fi?ndings showed that participants exhibited reference dependence in their risk attitudes depending on the preceding gain respective loss. These fi?ndings were statistically signi?cant and a quantum model for accommodating the above effects of contextual behaviour is introduced. A geometric depiction of the experimental fi?ndings was proposed. A so-called `state reconstruction' procedure from statistical data was performed via the inverse Born's rule (1926) [47], allowing the construction of a complete representation of agents' resolution from indeterminacy.
Of the three faculties of subjectivity, the empirical case for quantum cognition is the strongest. In the past decade a rapidly growing body of scholarship has emerged at the intersection of mathematical psychology and physics that not only models cognition in explicitly quantum terms, but has rigorously tested these models as well. While it remains to be seen how mainstream psychologists will respond, “quantum decision theory” has clearly come of age. Moreover, at the same time an almost completely independent literature has emerged on quantum game theory as well. This work is more in a purely formal vein and thus less well tested against real data, although its predictions about how people should behave in quantum games appear to conform to what experimental game theorists have found. I briefly take up quantum game theory at the end of this chapter, but my main focus will be on its decision-theoretic cousin. Quantum consciousness theory suggests that human beings are literally walking wave functions. Most quantum decision theorists would not go that far, and indeed – perhaps wary of controversy – they generally barely mention quantum consciousness, and then only to emphasize that they are making no claims about what is going on deep inside the brain (much less about consciousness), but are only interested in behavior. Instead, they have motivated their work in two other ways. First, they have highlighted the intuitive fit between quantum theory's indeterminism and the probabilistic character of human behavior. If we were founding the social sciences today and started – without metaphysical prejudice – by looking for useful models in physics, then quantum models would be a much more obvious import than classical ones. Second, quantum decision theorists have argued that their model can explain long-standing anomalies in human behavior under uncertainty. Beyond that they have remained agnostic about quantum consciousness, which makes quantum decision theory a form of what is called “weak” or “generalized” quantum theory, in which the formalism is detached from its physical basis and used simply as a tool to describe phenomena.
Following the ideas of Bohr, Von Neumann, and Benioff, we formulate quantum decision theory (QDT) as the quantum-mechanical theory of measurement for probability operators. QDT captures the effect of superposition of composite prospects, including many incorporated intentions. It is based on the hypothesis that the thought processes of real human beings involved in the definition and analysis of alternative prospects and scenarios do not necessarily separate them according to the recipes of standard probability theory and of classical utility theory. Our QDT formalizes systematically a broader class of decision making processes in which prospects can interact, interfere and remain entangled. The mathematical QDT is developed so as to be applicable to real-life decision making processes. We demonstrate that all known anomalies and paradoxes documented in the context of classical utility theory are reducible to just a few mathematical archetypes, all of which finding straightforward explanations in the framework of QDT. Interference terms, which are essential for resolving the paradoxes, quantify the aversion of human beings to uncertainty and/or to perceived potential loss resulting from their decisions.
Why should you be interested in quantum theory applied to Cognition and decision? Perhaps you are a physicist who is curious whether or not quantum principles can be applied outside of physics. In fact, that is one purpose of this book. Perhaps you are a cognitive scientist who is interested in representing concepts by vectors in a multidimensional feature space. This is essentially the way quantum theory works too. Perhaps you are a decision scientist who is trying to understand how people make decisions under uncertainty. Quantum theory could provide some interesting new answers. Generally speaking, quantum theory is a new theory for constructing probabilistic and dynamic systems , and in this book we apply this new theory to topics in cognition and decision. Later in this chapter we will give some specific examples, but let us step back at this point and try to understand the more general principles that support a quantum approach to cognition and decision. Six reasons for a quantum approach to cognition and decision Quantum physics is arguably the most successful scientific theoretical achievement that humans have ever created. It was created to explain puzzling findings that were impossible to understand using the older classical physical theory, and it achieved this by introducing an entirely new set of revolutionary principles. The older classical physical theory is now seen as a special case of the more general quantum theory. In the process of creating quantum mechanics, physicists also created a new theory of probabilistic and dynamic systems that is more general than the previous classic theory (Pitowski, 1989).
We investigate how the choice of decision makers can be varied under the presence of risk and uncertainty. Our analysis is based on the approach we have previously applied to individual decision makers, which we now generalize to the case of decision makers that are members of a society. The approach employs the mathematical techniques that are common in quantum theory, justifying our naming as Quantum Decision Theory. However, we do not assume that decision makers are quantum objects. The techniques of quantum theory are needed only for defining the prospect probabilities taking into account such hidden variables as behavioral biases and other subconscious feelings. The approach describes an agent's choice as a probabilistic event occurring with a probability that is the sum of a utility factor and of an attraction factor. The attraction factor embodies subjective and unconscious dimensions in the mind of the decision maker. We show that the typical aggregate amplitude of the attraction factor is 1/41/41/4, and it can be either positive or negative depending on the relative attraction of the competing choices. The most efficient way of varying the decision makers choice is realized by influencing the attraction factor. This can be done in two ways. One method is to arrange in a special manner the payoff weights, which induces the required changes of the values of attraction factors. We show that a slight variation of the payoff weights can invert the sign of the attraction factors and reverse the decision preferences, even when the prospect utilities remain unchanged. The second method of influencing the decision makers choice is by providing information to decision makers. The methods of influencing decision making are illustrated by several experiments, whose outcomes are compared quantitatively with the predictions of our approach.