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Modeling the Disjunction Effect within Classical Probability: A New Decision Process Model and Comparison with Quantum-like Models

Ryo Nasu, Yoshihiro Maruyama

Abstract

The disjunction effect in human decision making is often taken to show that the classical law of total probability is violated, motivating quantum-like models. We re-examine this claim for the Prisoner's Dilemma disjunction effect. Under the mental-event reading of the opponent-choice events, the conventional classical decision-process model implicitly builds in a certainty-only premise: its standard partition assumptions leave no room for ambiguity, forcing every participant to be certain that the opponent will defect or will cooperate. We relax this by introducing a new classical model in which each participant carries a continuous expectation parameter representing the anticipated likelihood of opponent defection, and the participant pool is partitioned by expectation level; the resulting ambiguity set is precisely the union of the interior expectation bins. In contrast, under the quantum-like event semantics, ambiguous pure states are generic (dense and of full unitarily invariant measure on the unit sphere), so "certainty states" are mathematically exceptional. We prove that an instance of our classical model can realize any empirically observed triple of defection rates across the three information conditions, including strong disjunction-effect patterns, while strictly obeying the classical law of total probability. We further prove that for any such triple produced by a standard quantum-like model of the same experiment, there exists a classical instance reproducing it exactly. In this sense, classical and quantum-like approaches have the same observable-rate expressiveness; their substantive difference lies in how ambiguity is represented and in their respective event semantics, not in a breakdown of classical probability.

Modeling the Disjunction Effect within Classical Probability: A New Decision Process Model and Comparison with Quantum-like Models

Abstract

The disjunction effect in human decision making is often taken to show that the classical law of total probability is violated, motivating quantum-like models. We re-examine this claim for the Prisoner's Dilemma disjunction effect. Under the mental-event reading of the opponent-choice events, the conventional classical decision-process model implicitly builds in a certainty-only premise: its standard partition assumptions leave no room for ambiguity, forcing every participant to be certain that the opponent will defect or will cooperate. We relax this by introducing a new classical model in which each participant carries a continuous expectation parameter representing the anticipated likelihood of opponent defection, and the participant pool is partitioned by expectation level; the resulting ambiguity set is precisely the union of the interior expectation bins. In contrast, under the quantum-like event semantics, ambiguous pure states are generic (dense and of full unitarily invariant measure on the unit sphere), so "certainty states" are mathematically exceptional. We prove that an instance of our classical model can realize any empirically observed triple of defection rates across the three information conditions, including strong disjunction-effect patterns, while strictly obeying the classical law of total probability. We further prove that for any such triple produced by a standard quantum-like model of the same experiment, there exists a classical instance reproducing it exactly. In this sense, classical and quantum-like approaches have the same observable-rate expressiveness; their substantive difference lies in how ambiguity is represented and in their respective event semantics, not in a breakdown of classical probability.
Paper Structure (20 sections, 3 theorems, 80 equations, 2 figures, 1 table)

This paper contains 20 sections, 3 theorems, 80 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Background and setup for the proposed decision-process model
  5. Review of arguments for violations of the classical law of total probability
  6. Two readings of the opponent-choice events
  7. The conventional decision-process model
  8. Related work: classical accounts of the disjunction effect
  9. Proposal of a new classical decision-process model
  10. Quantum-like decision-process model and relation to the proposed classical model
  11. Mathematical preliminaries and quantum analogue of the law of total probability
  12. Interpreting the quantum-like decision-process model
  13. Comparing the quantum-like and proposed classical models
  14. Mathematical Results
  15. Event semantics and how ambiguity enters the three models
  16. ...and 5 more sections

Key Result

Theorem 5.1

Let interpreted as the observed defection rates in the Prisoner's Dilemma experiment under the "unknown", "told defect", and "told cooperate" conditions, respectively. Then there exists a three-bin instance of the classical decision-process model of Section section:classic, with expectation parameter $\ and bin-wise defection probabilities such that: In particular, such a model exists even when

Figures (2)

  • Figure 1: Classical decision-process models. (a) Conventional model. Participants are split into two certainty states, "opponent defects" ($\Pr(\mathrm{D_{op}})$) or "opponent cooperates" ($\Pr(\mathrm{C_{op}})$), as implied by \ref{['eq:empty_DC']}, \ref{['eq:sure_DC']}; choices then satisfy equation \ref{['eq:LTP_DC']}. (b) Proposed model. Participants are grouped by an expectation parameter $\mathbb P\in[0,1]$ into bins $I_k$ with weights $\Pr(\mathbb P\in I_k)$; actions follow $\Pr(\mathrm{D_{pa}}\mid\mathbb P\in I_k)$, yielding \ref{['eq:LTP_P']} under \ref{['eq:empty_P']}, \ref{['eq:sure_P']}. See Sections \ref{['subsec:decision-process_model_review']} and \ref{['section:classic']} for details.
  • Figure 2: Event semantics across the three decision-process models. (a) Conventional classical (Boolean) reading: the certainty events exhaust the space, $\llbracket \mathrm{D_{op}}\rrbracket\cup\llbracket \mathrm{C_{op}}\rrbracket=\llbracket\top\rrbracket=\Omega$, so ambiguity has no semantic place. (b) Proposed classical reading: we keep set semantics but introduce an expectation parameter $\mathbb P\in[0,1]$; the certainty events are the extremes, $\llbracket \mathrm{C_{op}}\rrbracket=\llbracket \mathbb P\in I_1\rrbracket$ and $\llbracket \mathrm{D_{op}}\rrbracket=\llbracket \mathbb P\in I_K\rrbracket$, while interior levels account for $\llbracket\top\rrbracket\setminus(\llbracket \mathrm{D_{op}}\rrbracket\cup\llbracket \mathrm{C_{op}}\rrbracket)$. (c) Quantum-like reading: events are closed subspaces $\mathscr H_{\mathrm D},\mathscr H_{\mathrm C}\subset\mathscr H$ with $\mathscr H_{\mathrm D}\oplus\mathscr H_{\mathrm C}=\llbracket\top\rrbracket=\mathscr H$; superposition states lie in $\mathscr H$ but in neither subspace, so $\llbracket \mathrm{D_{op}}\rrbracket\cup\llbracket \mathrm{C_{op}}\rrbracket\subsetneq\llbracket\top\rrbracket$.

Theorems & Definitions (6)

  • Theorem 5.1: Classical realization of a strong disjunction triple
  • proof
  • Theorem 5.2: Classical realization of quantum disjunction probabilities
  • proof
  • Theorem 6.1: Ambiguity sets in the three decision-process models
  • proof