We develop a theory of quantum rational decision making in the tradition of Anscombe and Aumann's axiomatisation of preferences on horse lotteries. It is essentially the Bayesian decision theory generalised to the space of Hermitian matrices. Among other things, this leads us to give a representation theorem showing that quantum complete rational preferences are obtained by means of expected utility considerations.
We develop a theory of quantum rational decision making in the tradition of Anscombe and Aumann's axiomatisation of preferences on horse lotteries. It is essentially the Bayesian decision theory generalised to the space of Hermitian matrices. Among other things, this leads us to give a representation theorem showing that quantum complete rational preferences are obtained by means of expected utility considerations.
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.
We consider the problem of gambling on a quantum experiment and enforce rational behaviour by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalised to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalised Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalisation (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.
We consider the problem of gambling on a quantum experiment and enforce rational behavior by a few rules. These rules yield, in the classical case, the Bayesian theory of probability via duality theorems. In our quantum setting, they yield the Bayesian theory generalized to the space of Hermitian matrices. This very theory is quantum mechanics: in fact, we derive all its four postulates from the generalized Bayesian theory. This implies that quantum mechanics is self-consistent. It also leads us to reinterpret the main operations in quantum mechanics as probability rules: Bayes' rule (measurement), marginalization (partial tracing), independence (tensor product). To say it with a slogan, we obtain that quantum mechanics is the Bayesian theory in the complex numbers.
Quantum decision theory, advanced earlier by the authors, and illustrated for lotteries with gains, is generalized to the games containing lotteries with gains as well as losses. The mathematical structure of the approach is based on the theory of quantum measurements, which makes this approach relevant both for the description of decision making of humans and the creation of artificial quantum intelligence. General rules are formulated allowing for the explicit calculation of quantum probabilities representing the fraction of decision makers preferring the considered prospects. This provides a method to quantitatively predict decision-maker choices, including the cases of games with high uncertainty for which the classical expected utility theory fails. The approach is applied to experimental results obtained on a set of lottery gambles with gains and losses. Our predictions, involving no fitting parameters, are in very good agreement with experimental data. The use of quantum decision making in game theory is described. A principal scheme of creating quantum artificial intelligence is suggested.
In this work, we study different parametrizations of Quantum Decision Theory (QDT) using experimental data made of choices between pairs of lotteries. We then compare QDT with two of most famous classical decision theories, Rank-Dependent Utility(RDU) and Cumulative Prospect Theory(CPT). At the aggregate level, assuming homogeneous preferences across subjects, we find that CPT-based QDT wins by far. At the individual level, we classify decision makers as RDU, CPT or QDT. Our major findings are: quantum factor plays a key role in describing subjects’ behaviour; there is a considerable heterogeneity across subjects so that the classic representative agent approach would be completely wrong for this sample.
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.