Store:QLMfr11

6.2. Response replicability effect for sequential questioning

The approach based on identification of the order effect with noncommutative representation of questions (Wang and Busemeyer, 2013) was criticized in paper (Khrennikov et al., 2014). To discuss this paper, we recall the notion of response replicability. Suppose that a person, say John, is asked some question $A$ and suppose that he replies, e.g, “yes”. If immediately after this, he is asked the same question again, then he replies “yes” with probability one. We call this property $A-A$ response replicability. In quantum physics,  $A-A$ response replicability is expressed by the projection postulate.The Clinton–Gore opinion poll as well as typical decision making experiments satisfy $A-A$ response replicability. Decision making has also another feature -  $A-A$response replicability. Suppose that after answering the $A$-question with say the “yes”-answer, John is asked another question $B$. He replied to it with some answer. And then he is asked $A$ again. In the aforementioned social opinion pool, John repeats her original answer to $A$, “yes” (with probability one).

This behavioral phenomenon we call $A-B-A$ response replicability. Combination of  $A-A$ with  $A-B-A$ and $B-A-B$ response replicability is called the response replicability effect RRE.

6.3. “QOE+RRE”: described by quantum instruments of non-projective type

In paper (Khrennikov et al., 2014), it was shown that by using the von Neumann calculus it is impossible to combine RRE with QOE. To generate QOE, Hermitian operators  ${\widehat {A}},{\widehat {B}}$ should be noncommutative, but the latter destroys$A-B-A$ response replicability of $A$. This was a rather unexpected result. It made even impression that, although the basic cognitive effects can be quantum-likely modeled separately, their combinations cannot be described by the quantum formalism.

However, recently it was shown that theory of quantum instruments provides a simple solution of the combination of QOE and RRE effects, see Ozawa and Khrennikov (2020a) for construction of such instruments. These instruments are of non-projective type. Thus, the essence of QOE is not in the structure of observables, but in the structure of the state transformation generated by measurements’ feedback. QOE is not about the joint measurement and incompatibility (noncommutativity) of observables, but about sequential measurement of observables and sequential (mental-)state update. Quantum instruments which are used in Ozawa and Khrennikov (2020a) to combine QOE and RRE correspond to measurement of observables represented by commuting operators ${\widehat {A}},{\widehat {B}}$. Moreover, it is possible to prove that (under natural mathematical restriction) QOE and RRE can be jointly modeled only with the aid of quantum instruments for commuting observables.