Contribution
Order effects in surveys were known and catalogued (Moore 2002 named four patterns), and quantum-probability models had been fitted to them, but a fit says little. This paper derives from the quantum model a constraint with no free parameters that order effects must meet, the QQ equality, and tests it on six data sets. It also gives a similarity index h that sets the direction of the order effect, and argues that Bayesian and Markov models do not generally impose the constraint.
Key insight
If the second answer is computed from the state left by the first, as a projection, then the probability that the two answers agree is a quadratic form symmetric in the two projectors. Order can move probability between yes–yes and no–no, and between yes–no and no–yes, but it cannot move it between “agree” and “disagree”. The model’s surprise is in what order cannot change.
Assumptions
- Belief is a unit vector S in an N-dimensional Hilbert space, the same for both orders (in the 2014 paper, a common density matrix across a population).
- Each answer is a subspace with an orthogonal projector; the answers to one question are mutually orthogonal and their projectors sum to I.
- p(answer) = ‖P S‖², and the post-answer state is P S/‖P S‖ (Lüders’ rule).
- The critical one: “the only factor that changes the context or the state for answering a question is answering its preceding question” (§4). Information read out between questions breaks it.
- Binary answers. Respondents who gave neither yes nor no are dropped (Table 1 note).
Key results
- Order effect (Eq. 1a, Appendix): Γ_C = TP_C − p(Cy) = 2·Re⟨S|P_G P_C (I − P_G)|S⟩ = 2p(GyCy) − 2h·√p(Cy)·√p(Gy), with h = R cos φ, R = |⟨S|P_G P_C|S⟩|/(‖P_C S‖‖P_G S‖) ∈ [0, 1], so −1 ≤ h ≤ 1. Commuting projectors give Γ = 0: non-commutativity is necessary for an order effect in this model.
- Same h for both questions, because Re⟨S|P_C P_G|S⟩ = Re⟨S|P_G P_C|S⟩. Eliminating h between Eqs. 1a and 1b gives the QQ equality: q = [p(AyBn) + p(AnBy)] − [p(ByAn) + p(BnAy)] = 0.
- h is observable (Eq. 2): h = (0.5·Γ_C + p(GyCy))/√(p(Cy)p(Gy)), and likewise from Γ_G; for Clinton–Gore the two estimates are .8397 and .8421.
- Directions of effect (Table 2): consistency, contrast, additive and subtractive effects correspond to h relative to the ratios a = p(ByAy)/√(p(Ay)p(By)) and b = p(AyBy)/√(p(Ay)p(By)).
- Data (Table 1): q = −.0031 (z = .109), −.0031 (z = .090), −.0189 (z = .742), .1514 (z = 5.30; χ²(1) = 28.57) for Rose–Jackson, .0757 (z = 1.216) and .0430 (z = .831). I recomputed Clinton–Gore (.2214 − .2246 = −.0032) and Rose–Jackson (.3419 − .1905 = .1514) from the proportions.
- A three-question test is proposed, not carried out: in a two-dimensional model, h_AB and h_BC predict h_AC.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Under Lüders-rule projection from a common state, the probability of disagreeing answers is order-invariant for any projectors in any dimension | strong (proof) | Appendix; re-derived independently in LIT-264 §3 |
| C2 | In this model, order effects require non-commuting projectors | strong (proof) | Appendix |
| C3 | Five question-order data sets satisfy the QQ equality | moderate: six data sets, N about 100–500 per order | Table 1, §5 |
| C4 | The equality fails when information is inserted between questions | weak: one case, explained after the model was formulated but stated as a prediction | Rose–Jackson, §5 |
| C5 | Bayesian and Markov models do not generally satisfy the QQ equality | weak: an informal argument; the Markov proof is withheld | §6 |
| C6 | Quantum probability is “a powerful natural explanation” for survey order effects | not established by the evidence here | §8 |
Method
Not an algorithm. The test is a one-degree-of-freedom comparison: a z-test of q = p_AB − p_BA against zero with variance 2p(1 − p)/N, and a likelihood-ratio χ² comparing binomial models of the two 2 × 2 tables with and without the constraint p_AB = p_BA.
Concepts
- non-comparative / comparative context: a question asked first, or second, after the other question (Moore’s terms).
- order effect Γ: TP(answer asked second) − p(answer asked first).
- similarity index h: the normalised real part of ⟨S|P_G P_C|S⟩; the cosine between the two “yes” rays in the two-dimensional example.
- law of reciprocity: the paper’s name, after Peres, for |⟨S_B|S_A⟩|² = |⟨S_A|S_B⟩|².
- compatible / incompatible questions: commuting or non-commuting projectors; incompatible pairs are expected to be new or unusual combinations whose answers are constructed on the spot (§8.1).
Connections
It rests on Moore (2002) for the data and the taxonomy, Peres (1998) for reciprocity, and Niestegge (2008), who derived the same property for an axiomatic analysis of quantum theory. The 2014 PNAS paper, LIT-795, tests the equality on 72 studies. LIT-264 re-derives it in two lines from P Q P + (I − P)(I − Q)(I − P) = I − (P + Q) + (P Q + Q P) and shows it implies noncontextuality in the Contextuality-by-Default sense. The general framework is LIT-316, unread here.
Bearing on the record
- Sources THEORY-166, with LIT-795 (primary) and LIT-264.
- THEORY-013. This paper is the model LIT-264 uses to argue that a quantum model can predict the absence of contextuality. Nothing here speaks to contextuality; the paper does not use the word.
- On the derivation. The paper attributes the equality to the law of reciprocity for transitions between rays (§4, note 3). The Appendix uses less: only Re⟨S|P_G P_C|S⟩ = Re⟨S|P_C P_G|S⟩, which holds because projectors are self-adjoint. The equality is a property of self-adjoint projectors under Lüders updating from a common state, not of rays in particular. This is my reading of the Appendix, and it agrees with the re-derivation in LIT-264.
- On “order sensitivity establishes non-commutativity”. Within the model, an order effect needs non-commuting projectors. The paper itself (§6) says a Bayesian model with order events and a Markov model with memory both produce order effects, so the order effect alone does not select the quantum model. What the paper offers in its place is the QQ equality.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Six data sets, four of them chosen from a review of order effects.
- The classical alternatives are dismissed rather than tested: the Bayesian argument shows only that an order-conditioned model is unconstrained, and the Markov proof is not given. A classical model constrained to satisfy the equality is not considered here (the 2014 paper concedes one can be built).
- Rose–Jackson is “predicted” to fail by an explanation stated alongside the data; no independent case with inserted information is tested.
- Respondents without a yes/no answer are dropped, which may not be random with respect to order.
Open questions
- Does a classical model with memory and order-dependent updating that satisfies the QQ equality also reproduce the order effects observed? A worked counterexample would answer C5.
- The three-question prediction of §7 (h_AC from h_AB and h_BC) is untested here.
- Whether the Rose–Jackson failure is reproduced by the extended model with transformations between questions, fitted without access to the q value.