Formalizing Heuristics in Decision-Making: A Quantum Probability Perspective

One of the most influential research programs in psychology is that of Tversky and Kahneman’s (1973, 1983) on heuristics and biases in decision-making. Two characteristics of this program are, first, compelling empirical demonstrations that in some decision-making situations naïve observers violate the rules of classic probability (CP) theory and, second, that corresponding behavior can be explained with simple heuristics. Tversky and Kahneman’s work has led to a vast literature on what is the basis for psychological process in decision-making. Note that their work, however impactful, has not settled the debate of whether CP theory is suitable for modeling cognition or not. CP models have attracted enormous interest and they often do provide excellent coverage of cognitive processes (e.g., Oaksford and Chater, 2007; Griffiths et al., 2010; Tenenbaum et al., 2011). The idea of heuristics is appealing. First, they are simple. The assumption that human cognition is based on heuristics partly avoids the computational intractability problems which plague some formal approaches (cf. Sanborn et al., 2010). Second, they often allow an understanding of one process in terms of theory developed for other cognitive processes. Consider the representativeness and availability heuristics. According to the representativeness heuristic, judgments of frequency are driven by similarity and according to the availability heuristic by the ease of identifying related instances in memory. Thus, with these heuristics, an explanation for decision-making becomes one of similarity or memory. Third, heuristics often have strong empirical support. Tversky and Kahneman’s approach has been to motivate explanations based on heuristics by providing compelling demonstrations for violations of the standard approaches (in decision-making, CP theory). Other proponents of heuristic approaches have argued that heuristic schemes lead to better results (e.g., Gigerenzer and Todd, 1999). There is nothing wrong with heuristic approaches. But, there is a sense in which theoreticians have a bias for cognitive models based on formal frameworks, whether it is Bayesian probability, formal logic, or the quantum probability (QP) theory, which we discuss (cf. Elqayam and Evans, 2011). The properties of formal frameworks are interconnected. For example, all expressions in classical probability theory are based on a handful of axioms. Thus, one cannot accept the psychological relevance of one expression, but reject another: they are all related to each other. By contrast, heuristics, however successful, are somewhat interchangeable. Postulating the relevance of the representativeness heuristic does not necessitate the relevance of the availability heuristic (Pothos and Busemeyer, 2009a). The QP research program in psychology partly originated as an attempt to reconcile people’s violations of CP theory in decision-making situations with formal theory and examine whether it is possible to express formally some of the key heuristics in decision-making. QP theory is a theory for assigning probabilities to observables (Isham, 1989). Physicists are happy to employ CP theory in most cases but they believe that, ultimately, QP theory is the more appropriate choice. CP theory works by defining a sample space and expressing probabilities in terms of subsets of this space. A key property of this approach is the commutative nature of events and subsequent order independence for probabilities assigned to the joint events. QP is a geometric approach to probability. Events correspond to different subspaces and probabilities are computed by projections to these subspaces (note that projections have been discussed before in psychology; Sloman, 1993). Crucially, this makes probability assessment potentially order and context dependent and, e.g. (a suitable definition of), conjunction can fail commutativity. This and related interference effects lead to interesting predictions from QP theory. In the famous Linda experiment (Tversky and Kahneman, 1983), participants are told about Linda, who sounds like a feminist and are then asked to judge the probability of statements about her. The important comparison concerns the statements “Linda is a bank teller” and “Linda is a feminist and a bank teller.” The first statement is extremely unlikely. The second statement is a conjunction of the first statement and another one. Thus, according to CP theory, P(bank teller) ≥ (bank teller ∧ feminist). But, results violate CP theory, as most participants consider the statement “Linda is a bank teller and a feminist” as the more probable one (this is called the conjunction fallacy). Tversky and Kahneman’s explanation was that cognitive process is not based on CP theory, rather, participants employ a representativeness heuristic. They consider Linda as a very typical feminist, so that the characterization “bank teller and feminist” is probable, regardless of the bank teller part. One could also invoke an availability heuristic (as Tversky and Koehler, 1994 later did), whereby the statement “bank teller and feminist” activates memory instances similar to Linda. Figure 1 illustrates the QP theory explanation of the conjunction fallacy. The state vector is labeled as Psi and corresponds to what participants learn about Linda from the story. One 1D subspace corresponds to Linda being a feminist and another to a bank teller. We compute the probability for each possibility by projecting the state vector onto the corresponding subspace and squaring the length of the projection. If participants are asked to evaluate the probability that Linda is a just bank teller or just a feminist this is very unlikely and likely respectively. In QP theory, conjunction has to be typically defined as a sequential operation, i.e., Prob(A ∧ B) ≡ Prob(A ∧ then B).