Pothos & Busemeyer (P&B) provide an intriguing overview of quantum probability as a framework for modeling decision making. In this commentary, we concentrate on two aspects of their article: the relationship between classical probability (CP) and quantum probability (QP) and its implications for ideas about rationality in humans and other animals.
In order to evaluate the contribution that QP makes to our understanding of cognitive processes, it is necessary to be clear about the relationship between CP and QP. In Section 4.3, P&B reject the view that QP provides a better account of human decision making “simply because it is more flexible.” We are not completely happy with some of the statements that P&B make about this issue. From a mathematical point of view, CP is embedded as a special case in the more general non-commutative (also referred to as “quantum”) probability theory. There are no cases of CP distributions and the evolution thereof that cannot be represented in a quantum probabilistic framework. Once the Heisenberg matrix mechanics is used (equivalent to the Schrödinger wave vector notation used by P&B) it can be shown that any commutative quantum theory can be written as a CP theory, and vice versa, CP theory can be written as a commutative quantum theory. Hence, CP is strictly a special case of non-commutative (quantum) probability (see Streater Reference Streater2000). To clarify the connection to P&B by example: what they call the “law” of double stochasticity is merely a special case in quantum theory, that of projective measurement, which is a subset of all (generalised) measurements.
The discussion about “QP theory violating the law of total probability” would be helped by clarifying the law of total probability. It states that the probabilities of all elements in a set sum up to the probability of the whole set. In quantum mechanics the whole can be less than the sum of its parts because of superposition of states. An equation such as the one in the target article's appendix
is more helpful if stripped down to the “superposition part” (the state is “separable”), that is, the only information that matters is
Our final topic is the relationship between rational decision making and natural selection. In this context, it is worth noting that the decisions of nonhuman animals violate the principles of rational decision making (Houston et al. Reference Houston, McNamara and Steer2007b). Given that natural selection is expected to favour optimal choices, there is a tension between natural selection and the observed behaviour of animals. One response is to argue that animals only need to make approximately optimal decisions in particular circumstances – the circumstances in which they have evolved. This idea is related to the suggestion that decision making is based on heuristics, as mentioned by P&B in Section 4.2. A more interesting possibility is that what appear to be suboptimal choices are part of an optimal strategy for an environment that is richer than was originally envisaged. For example, Houston et al. (Reference Houston, McNamara and Steer2007a) show that apparent violations of transitivity can be generated by the optimal state dependent strategy given that the options presented to the animal will persist into the future. In essence, the violations appear because the experimenter does not have the same view of the world as the animal. Another example is provided by the work of McNamara et al. (Reference McNamara, Trimmer and Houston2012) on state-dependent valuation.
With these examples, we are trying to illustrate that the macroscopic world of decisions is more complex than traditional models of decision theory assume. P&B (sect. 5) write: “If we cannot assume an objective reality and an omniscient cognitive agent, then perhaps the perspective-driven probabilistic evaluation in quantum theory is the best practical rational scheme. In other words, quantum inference is optimal, for when it is impossible to assign probabilities to all relevant possibilities and combinations concurrently.” We are not convinced that understanding the macroscopic world of decisions requires a step as dramatic as that of abandoning objective reality in the way that the quantum world does.
Pothos & Busemeyer (P&B) provide an intriguing overview of quantum probability as a framework for modeling decision making. In this commentary, we concentrate on two aspects of their article: the relationship between classical probability (CP) and quantum probability (QP) and its implications for ideas about rationality in humans and other animals.
In order to evaluate the contribution that QP makes to our understanding of cognitive processes, it is necessary to be clear about the relationship between CP and QP. In Section 4.3, P&B reject the view that QP provides a better account of human decision making “simply because it is more flexible.” We are not completely happy with some of the statements that P&B make about this issue. From a mathematical point of view, CP is embedded as a special case in the more general non-commutative (also referred to as “quantum”) probability theory. There are no cases of CP distributions and the evolution thereof that cannot be represented in a quantum probabilistic framework. Once the Heisenberg matrix mechanics is used (equivalent to the Schrödinger wave vector notation used by P&B) it can be shown that any commutative quantum theory can be written as a CP theory, and vice versa, CP theory can be written as a commutative quantum theory. Hence, CP is strictly a special case of non-commutative (quantum) probability (see Streater Reference Streater2000). To clarify the connection to P&B by example: what they call the “law” of double stochasticity is merely a special case in quantum theory, that of projective measurement, which is a subset of all (generalised) measurements.
The discussion about “QP theory violating the law of total probability” would be helped by clarifying the law of total probability. It states that the probabilities of all elements in a set sum up to the probability of the whole set. In quantum mechanics the whole can be less than the sum of its parts because of superposition of states. An equation such as the one in the target article's appendix
is more helpful if stripped down to the “superposition part” (the state is “separable”), that is, the only information that matters is
Our final topic is the relationship between rational decision making and natural selection. In this context, it is worth noting that the decisions of nonhuman animals violate the principles of rational decision making (Houston et al. Reference Houston, McNamara and Steer2007b). Given that natural selection is expected to favour optimal choices, there is a tension between natural selection and the observed behaviour of animals. One response is to argue that animals only need to make approximately optimal decisions in particular circumstances – the circumstances in which they have evolved. This idea is related to the suggestion that decision making is based on heuristics, as mentioned by P&B in Section 4.2. A more interesting possibility is that what appear to be suboptimal choices are part of an optimal strategy for an environment that is richer than was originally envisaged. For example, Houston et al. (Reference Houston, McNamara and Steer2007a) show that apparent violations of transitivity can be generated by the optimal state dependent strategy given that the options presented to the animal will persist into the future. In essence, the violations appear because the experimenter does not have the same view of the world as the animal. Another example is provided by the work of McNamara et al. (Reference McNamara, Trimmer and Houston2012) on state-dependent valuation.
With these examples, we are trying to illustrate that the macroscopic world of decisions is more complex than traditional models of decision theory assume. P&B (sect. 5) write: “If we cannot assume an objective reality and an omniscient cognitive agent, then perhaps the perspective-driven probabilistic evaluation in quantum theory is the best practical rational scheme. In other words, quantum inference is optimal, for when it is impossible to assign probabilities to all relevant possibilities and combinations concurrently.” We are not convinced that understanding the macroscopic world of decisions requires a step as dramatic as that of abandoning objective reality in the way that the quantum world does.