Order effects#

Phenomenon. The answer to a question depends on what was asked before it. In a 1997 Gallup poll, 50% called Clinton honest when asked first and 57% when asked after Gore; Gore fell from 68% to 60% (Moore, 2002).

Use it for: surveys and questionnaires, user research, A/B tests of question wording, evaluation of human raters or language models, clinical interviews, any sequence of yes/no judgements.

Models#

Class

Parameters

Order effect

QQ equality

Notes

QuantumOrderModel4D

4 (state angles t1–t3, incompatibility phi)

yes

holds

Recommended. phi = 0 is exactly the Bayesian model, so it nests it

QuantumOrderModel

3 (angles a, b, g) + ranks (1/2 per question)

yes

holds

Fits the Clinton–Gore rates with ranks (1, 2); cannot represent order-free data with intermediate correlation

BayesOrderModel

3 (pA, pB, rho)

no

holds (trivially)

Classical baseline

AnchoringOrderModel

4 (pA, pB, wA, wB); option tied=True → 3

yes

violated unless wA = wB

Classical order-effect baseline

SaturatedOrderModel

6

any

any

Reference

ProjectiveQuestionModel

–

–

holds

Any state and projectors, any dimension (not fitted)

Mathematics#

A belief is a unit vector ψ; a “yes” to question q is a projector P_q. Answering applies the Lüders rule: p(yes) = ‖P_q ψ‖², then ψ ← P_q ψ / ‖P_q ψ‖. For two questions in either order every projective model satisfies the parameter-free QQ equality

p_AB(y,n) + p_AB(n,y) = p_BA(y,n) + p_BA(n,y)

(Wang and Busemeyer, 2013; supported in 70 national field experiments, Wang et al., 2014).

Data format#

design = ('AB', 'BA')                    # or None (default)
counts = {'AB': [yy, yn, ny, nn],        # A asked first; first letter = answer to the first question
          'BA': [yy, yn, ny, nn]}        # B asked first

Quick start#

from quantum_mind.core import compare
from quantum_mind.families.order_effects import (QuantumOrderModel4D, QuantumOrderModel, BayesOrderModel,
                                          AnchoringOrderModel, RANK_STRUCTURES, qq_test)
q, z, p = qq_test(counts)
results = compare([QuantumOrderModel4D, (QuantumOrderModel, {'structures': RANK_STRUCTURES}),
                   BayesOrderModel, AnchoringOrderModel], counts)
for r in results: print(type(r.model).__name__, r.bic)

Circuit#

quantum_mind.circuits.order_effects_circuit(model, 'AB', form='dynamic' | 'deferred'): the state is loaded with StatePreparation; each question is a binary projective measurement (subspace rotated to the computational basis, ancilla flagged, ancilla measured, rotation undone), so the post-measurement state follows the Lüders rule. 3D models use 2 system qubits; the 4D model also uses 2.

Limitations#

  • Aggregate fitting assumes a homogeneous population; individual differences can hide or mimic the structure (see the companion paper “Order-Aware Human Models for Robot Questioning and Trust”).

  • Standard projective models do not reproduce response replicability together with order effects (Khrennikov et al., 2014).

  • The QQ equality alone does not prove quantum-like processing. Boyer-Kassem, Duchêne and Guerci (2016) derived further predictions (the Grand Reciprocity equations) that non-degenerate versions of these models fail on most existing data sets; check them before claiming support.

References#

Moore (2002) Public Opinion Quarterly 66(1). Wang and Busemeyer (2013) Topics in Cognitive Science 5(4). Wang, Solloway, Shiffrin and Busemeyer (2014) PNAS 111(26). Khrennikov, Basieva, Dzhafarov and Busemeyer (2014) PLoS ONE 9. Boyer-Kassem, Duchêne and Guerci (2016) Mathematical Social Sciences 80, 33–46.