Modeling combination of question order effect, response replicability effect, and QQ-equality with quantum instruments

Abstract We continue to analyze basic constraints on the human decision making from the viewpoint of quantum measurement theory (QMT). As it has been found, the conventional QMT based on the projection postulate cannot account for the combination of the question order effect (QOE) and the response replicability effect (RRE). This was an alarming finding for quantum-like modeling of decision making. Recently, it was shown that this difficulty can be resolved by using of the general QMT based on quantum instruments. In the present paper we analyze the problem of the combination of QOE, RRE, and the well-known QQ-equality (QQE). This equality was derived by Busemeyer and Wang, and it was shown (in a joint paper with Solloway and Shiffrin) that statistical data from many social opinion polls satisfy it. Here we construct quantum instruments satisfying QOE, RRE and QQE. The general features of our approach are formalized with postulates that generalize (the Wang–Busemeyer) postulates for quantum-like modeling of decision making. Moreover, we show that our model closely reproduces the statistics of the well-known Clinton–Gore Poll data with a prior belief state independent of the question order. This model successfully corrects for the order effect in the data to determine the “genuine” distribution of the opinions in the Poll. The paper also provides an accessible introduction to the theory of quantum instruments — the most general mathematical framework for quantum measurements.