classical-disjunction-effect-model

name	classical-disjunction-effect-model
description	Classical probability model that reproduces the disjunction effect via expectation-parameter partitioning, proving classical and quantum-like models have equivalent expressiveness for decision rates.

Classical Disjunction Effect Model

Description

Methodology for modeling the disjunction effect in human decision making within classical probability theory, by introducing a continuous expectation parameter that partitions the participant pool by ambiguity level. Proves that classical and quantum-like models have equivalent observable expressiveness — their difference lies only in how ambiguity is represented. Based on arXiv:2603.23233.

Activation Keywords

• disjunction effect modeling
• classical decision process model
• prisoner's dilemma disjunction
• quantum-like decision comparison
• ambiguity set modeling
• 析取效应经典模型
• 量子类决策比较
• expectation parameter partitioning

Tools Used

• exec: Run Python simulations of classical vs quantum decision models
• write: Create model configuration and analysis reports
• read: Load experimental data for model fitting

Core Concepts

The Disjunction Effect

The disjunction effect violates the classical law of total probability in human decision making: people's choices differ when they know the outcome vs. when they don't. Classically, P(defect) should equal P(defect|coop)×P(coop) + P(defect|defect)×P(defect), but empirically it doesn't.

Key Innovation: Expectation Parameter Model

Instead of the binary certainty assumption (opponent will definitely cooperate or definitely defect), introduce a continuous expectation parameter θ ∈ [0,1] representing the anticipated likelihood of opponent defection. The participant pool is partitioned by expectation level, and the ambiguity set is the union of interior expectation bins.

Classical-Quantum Equivalence Theorem

Proved: Any triple of defection rates (cooperate-known, defect-known, unknown) achievable by a quantum-like model can be reproduced exactly by a classical instance of the expectation-parameter model. Both frameworks have identical expressive power for observable rates.

Substantive Difference

The difference is NOT in probability theory breaking down, but in:

1. Ambiguity representation: Classical uses expectation partitions; quantum uses superposition states
2. Event semantics: Quantum ambiguous pure states are generic (dense, full measure); classical certainty states are exceptional

Usage Patterns

Pattern 1: Reproducing Disjunction Effect Data

Given empirical defection rates across three conditions:

1. Define expectation bins θ₁, θ₂, ..., θₙ
2. Assign population weights to each bin
3. Compute weighted defection rates
4. Verify classical law of total probability holds

Pattern 2: Classical-Quantum Comparison

For any quantum-like model producing rates (r_coop, r_defect, r_unknown):

1. Extract the three observable rates
2. Construct classical expectation partition matching all three
3. Prove equivalence: same rates, different ambiguity semantics

Pattern 3: Ambiguity Set Construction

1. Partition participants by expectation level θ
2. Define certainty bins (θ ≈ 0 and θ ≈ 1)
3. Ambiguity set = union of all interior bins
4. Disjunction effect emerges from ambiguity set weighting

Instructions for Agents

Step 1: Identify the Decision Scenario

• Prisoner's Dilemma? Ellsberg Paradox? Other decision-under-uncertainty?
• Identify the three information conditions: known-cooperate, known-defect, unknown

Step 2: Extract Empirical Rates

• Collect defection rates for each condition: (d_C, d_D, d_U)
• Verify the disjunction effect: d_U ≠ d_C×P(C) + d_D×P(D)

Step 3: Construct Classical Model

• Choose number of expectation bins (start with 5-10)
• Solve for bin weights that reproduce the three rates
• Verify all weights are non-negative and sum to 1

Step 4: Compare with Quantum-Like Model

• Map quantum state amplitudes to expectation parameters
• Show rate equivalence but semantic difference
• Document which representation is more natural for the domain

Error Handling

No Disjunction Effect Observed

If d_U ≈ d_C×P(C) + d_D×P(D), the disjunction effect is absent. The model reduces to standard classical decision theory — no special modeling needed.

Negative Bin Weights

If solving produces negative weights, increase the number of bins or relax the constraint. This indicates the data cannot be fit with the current granularity.

Over-parameterization

With too many bins, the model becomes unidentifiable. Use minimum bins (typically 3-5) that reproduce the data. Apply regularization to prefer smoother distributions.

Examples

Example: Prisoner's Dilemma

Empirical rates: P(defect|coop) = 0.3, P(defect|defect) = 0.7, P(defect|unknown) = 0.55 Classical prediction: 0.3×0.5 + 0.7×0.5 = 0.5 ≠ 0.55 (disjunction effect!) Expectation model: Partition population into low/high/medium expectation bins to reproduce 0.55 while maintaining LOTP within each bin.

Resources

• arXiv:2603.23233 — "Modeling the Disjunction Effect within Classical Probability"
• Related: quantum-cognition (quantum-like decision modeling)
• Related: quantum-game-theory-economics (game theory under uncertainty)