Zheng Wang, Jerome R. Busemeyer (2013), Topics in Cognitive Science 5(4):689–710 — DOI-10.1111/tops.12040
Key takeaways
- The model. A belief is a unit vector S. Each answer to a question is a subspace with projector P, and the projectors for one question’s answers are orthogonal and sum to I. An answer has probability ‖P S‖², and afterwards the state becomes P S/‖P S‖ (Lüders’ rule). So p(A_x then B_y) = ‖P_By P_Ax S‖².
- Order effects need non-commuting projectors. The order effect for question C is Γ_C = 2·Re⟨S|P_G P_C (I − P_G)|S⟩, which vanishes when the projectors commute (Appendix). Its size and sign are set by a similarity index h, the normalised real part of ⟨S|P_G P_C|S⟩, with −1 ≤ h ≤ 1; Table 2 maps ranges of h onto Moore’s four patterns (consistency, contrast, additive, subtractive).
- The QQ equality. Because Re⟨S|P_G P_C|S⟩ = Re⟨S|P_C P_G|S⟩, one h serves both order effects, and eliminating it gives p(AyBn) + p(AnBy) = p(ByAn) + p(BnAy): the chance of answering the two questions differently does not depend on the order. It holds in any dimension and for any projectors, with no free parameters, given that only the first answer changes the state before the second question.
- The test. On six data sets (Table 1): four Gallup polls from Moore (2002) and two of the authors’ laboratory replications. The q values are −.0031 (Clinton–Gore), −.0031 (Gingrich–Dole), −.0189 (white–black), .0757 and .0430 (the two laboratory studies), none significant by z or χ² test. The Rose–Jackson poll gives q = .1514 (z = 5.30, χ²(1) = 28.57). The authors predicted that failure, because each question was preceded by background information about the player, so more than the first answer changed the state.
- Against classical models. A Bayesian model has commutative events. It produces order effects only by conditioning on an order event, and then nothing constrains the sequences. A Markov model with memory in its state can produce order effects, but its transition probabilities need not be reciprocal. So “Bayesian and Markov models do not generally satisfy the QQ equality”. The Markov proof is “available upon request”, and not in the paper.
Standing in the record
Filed on 2026-10-09 at the owner’s request, from the bibliography of the
owner’s manuscript (work what-survives-translation) as it stood on
2026-10-08: a work the bibliography considered and the final reference
list dropped. See the curation entry of that day.
Read on 2026-10-09 (NOTE-636). It is the original statement of the QQ equality that LIT-795 tests on 72 surveys and that LIT-264 reads through Contextuality-by-Default: the equality implies that a question-order system is noncontextual. With them it is a source of THEORY-166. The authors’ book-length treatment is LIT-316, unread here.