We test a simple quantum decision model of the Ellsberg paradox. We find that the theoretical predictions of the model are in conformity with our experimental results. The predictions of our quantum model are not statistically significantly different from those of the source dependent model. The source dependent model requires the specification of probability weighting functions in order to fit the evidence. On the other hand, our quantum model makes no recourse to probability weighting functions. This suggests that much of what is normally attributed to probability weighting may actually be due to quantum probability. When we replace quantum probability by Kolmogorov probability in our model, then the Ellsberg paradox reemerges. Hence, we make essential use of quantum probability theory. All our development uses no more than standard linear algebra and real numbers, which are very familiar to economists. This makes our paper accessible to a wider audience than the quantum community. JEL Classification: D01, D81, D91
We test a simple quantum decision model of the Ellsberg paradox. We find that the theoretical predictions of the model are in conformity with our experimental results. The predictions of our quantum model are not statistically significantly different from those of the source dependent model. The source dependent model requires the specification of probability weighting functions in order to fit the evidence. On the other hand, our quantum model makes no recourse to probability weighting functions. This suggests that much of what is normally attributed to probability weighting may actually be due to quantum probability. When we replace quantum probability by Kolmogorov probability in our model, then the Ellsberg paradox reemerges. Hence, we make essential use of quantum probability theory. All our development uses no more than standard linear algebra and real numbers, which are very familiar to economists. This makes our paper accessible to a wider audience than the quantum community. JEL Classification: D01, D81, D91
We report the results of an experiment we performed to test the matching probabilities for the Ellsberg paradox predicted by the quantum decision model of al-Nowaihi and Dhami (2016). We find that the theoretical predictions of that model are in conformity with our experimental results. This supports the thesis that violations of classical (Kolmogorov) probability theory may not be due to irrational behaviour but, rather, due to inadequacy of classical probability theory for the description of human behaviour.
The expected utility hypothesis and Savage's Sure-Thing Principle are violated in real life decisions, as shown by the Allais and Ellsberg paradoxes. The popular explanation in terms of ambiguity aversion is not completely accepted. As a consequence, uncertainty is still problematical in economics. To overcome these difficulties a distinction between risk and ambiguity has been introduced which depends on the existence of a Kolmogorovian probabilistic structure modeling these uncertainties. On the other hand, evidence of everyday life suggests that context plays a fundamental role in human decisions under uncertainty. Moreover, it is well known from physics that any probabilistic structure modeling contextual interactions between entities structurally needs a non-Kolmogorovian framework admitting a quantum-like representation. For this reason, we have recently introduced a notion of contextual risk to mathematically capture situations in which ambiguity occurs. We prove in this paper that the contextual risk approach can be applied to the Ellsberg paradox, and elaborate a sphere model within our hidden measurement formalism which reveals that it is the overall conceptual landscape that is responsible of the disagreement between actual human decisions and the predictions of expected utility theory, which generates the paradox. This result points to the presence of a quantum conceptual layer in human thought which is superposed to the usually assumed classical logical layer, and conceptually supports the thesis of several authors suggesting the presence of quantum structure in economics and decision theory.
Many paradoxical findings in decision-making that have resisted explanations by standard decision theories have accumulated over the past 50 years. Recent advances based on quantum probability theory have successfully accounted for many of these puzzling findings. Critics, however, claim that quantum probability theory is less constrained than standard probability theory, and hence quantum models only fit better because they are more complex than standard decision models. In this article, for the first time, a Bayesian method was used to quantitatively compare the 2 types of decision models, which is a method that evaluates models with respect to accuracy, parsimony, and robustness. A large experiment was used to compare the best-known models of each type, matching in their numbers of parameters, but possibly differing in the complexity of their functional forms. Surprisingly, the Bayesian model comparison overwhelmingly favored the quantum model, indicating that its success is due to its robust ability to make accurate predictions rather than accidental fits afforded by increased complexity.
Organizational decision making is often explored with theories from the heuristics and biases research program, which have demonstrated great value as descriptions of how people in organizations make decisions. Nevertheless, rational analysis and classical probability theory are still seen by many as the best accounts of how decisions should be made and classical probability theory is the preferred framework for cognitive modelling for many researchers. The focus of this work is quantum probability theory, an alternative probabilistic framework. Results in decision making, which appear paradoxical from a perspective of classical probability theory, may make perfect sense if one adopts quantum probability theory. We review some cognitive models of decision making based on quantum probability theory. Each of these models is based on a challenge to prescription from classical probability theory. The transition from labeling a particular behavior as irrational, by classical probability standards, to (potentially) rational (or, at any rate, not fallacious), raises interesting possibilities, including that of characterizing certain heuristics in formal, probabilistic terms.
How can we use quantum theory to model other phenomena of interest to researchers in cognition and decision making? Quantum theory is not easy for researchers in cognition and decision making to accept. In fact, quantum mechanics was not easy for physicists to accept either, but it was forced on them by several paradoxical findings that could not be explained using classical physics. We have a similar problem in cognition and decision making – there are numerous paradoxical findings that just seem irrational according to classic probability theory. For example, under some conditions, people judge the probability of event A and B to be greater than the probability of event B , which is called the conjunction fallacy (Tversky & Kahneman, 1983). Also, under the same conditions, they judge the probability of A or B to be less than the probability of event A (Carlson & Yates, 1989), which is called the disjunction fallacy. In this chapter we examine how quantum probability theory explains these and other puzzling results from human probability judgment research. This chapter has two main parts. In the first part, we use a quantum model to derive qualitative predictions for conjunction errors, disjunction errors, and other closely related findings. The first section provides a general set of predictions that do not depend on specific assumptions about the features used to represent events, and the predictions are parameter free. In the second part, we examine the quantitative predictions of the quantum model for a Bayesian inference task, which we use to explain order effects on inference.
We consider several puzzles of bounded rationality. These include the Allais- and Ellsberg paradox, the disjunction effect, and related puzzles. We argue that the present account of quantum cognition—taking quantum probabilities rather than classical probabilities—can give a more systematic description of these puzzles than the alternate treatments in the traditional frameworks of bounded rationality. Unfortunately, the quantum probabilistic treatment does not always provide a deeper understanding and a true explanation of these puzzles. One reason is that quantum approaches introduce additional parameters which possibly can be fitted to empirical data but which do not necessarily explain them. Hence, the phenomenological research has to be augmented by responding to deeper foundational issues. In this article, we make the general distinction between foundational and phenomenological research programs, explaining the foundational issue of quantum cognition from the perspective of operational realism. This framework is motivated by assuming partial Boolean algebras (describing particular perspectives). They are combined into a uniform system (i.e. orthomodular lattice) via a mechanism preventing the simultaneous realization of perspectives. Gleason’s theorem then automatically leads to a distinction between probabilities that are defined by pure states and probabilities arising from the statistical mixture of pure states. This formal distinction relates to the conceptual distinction between risk and ignorance. Another outcome identifies quantum aspects in dynamic macro-systems using the framework of symbolic dynamics. Finally, we discuss several ideas that are useful for justifying complementarity in cognitive systems.