[Paper Review] Belief revision in quantum decision theory: gambler's and hot hand fallacies

Riccardo Franco|ArXiv.org|Jan 29, 2008

Sports Analytics and Performance14 references3 citations

TL;DR

This paper proposes a quantum decision theory framework using entangled qubits and conditional phase shifts to model belief revision in human judgment, specifically explaining the gambler's fallacy (negative recency) and hot hand fallacy (positive recency). The model shows that psychological influence from prior outcomes induces quantum correlations, with interference terms capturing deviations from rational probability, and partial trace operations demonstrate how erasing prior information restores rationality.

ABSTRACT

In the present article we introduce a quantum mechanism which is able to describe the creation of correlations in the evaluation of random independent events: such correlations, known as positive and negative recency, correspond respectively to the hot hand's and to the gambler's fallacies. Thus we propose a description of these effects in terms of qubits, which may become entangled, forming a system which can not be described completely only in terms of its constituents. We show that such formalism is able to describe and interpret the experimental results, thus providing a general and unifying framework for the cognitive heuristics.

Motivation & Objective

• To develop a quantum formalism that explains how prior outcomes influence subjective probability judgments in random sequences.
• To model the gambler’s fallacy (negative recency) and hot hand fallacy (positive recency) as quantum correlations arising from belief revision.
• To provide a clear psychological interpretation of quantum parameters—specifically, the phase shift—as a mechanism for updating beliefs.
• To unify the description of cognitive heuristics like the disjunction effect and framing effect under a single quantum-like representation framework.
• To demonstrate that quantum entanglement and interference can model single-subject autocorrelation in judgment, not just inter-subject correlations.

Proposed method

• Modeling the opinion state of a subject about two independent random events using a two-qubit quantum system.
• Applying a conditional phase shift operator to induce correlations between qubits, representing psychological influence from prior outcomes.
• Using density matrices and partial trace operations to simulate the erasure of prior information and restore rational probability judgments.
• Incorporating interference terms in the probability formula: $ P(1) = P_b(1) + I( heta) $, where $ I( heta) = ext{Re}[e^{i heta} ho_{01}] $, to capture judgment deviations.
• Defining the phase factor $ heta $ as a psychological parameter that reflects emotional or cognitive influence from previous outcomes.
• Applying the formalism to experimental data from roulette and basketball, showing that interference terms can reproduce observed deviations from 50% probability.

Experimental results

Research questions

• RQ1 How can quantum entanglement and conditional phase shifts model the gambler’s fallacy and hot hand fallacy in human judgment?
• RQ2 What is the role of the phase parameter in representing psychological belief revision in response to prior outcomes?
• RQ3 How do interference effects in quantum probability reproduce empirical deviations from rational probability in sequential decision-making?
• RQ4 In what way does the partial trace operation model the loss of memory or psychological erasure of prior outcomes?
• RQ5 Can a single-subject quantum model of autocorrelation explain both positive and negative recency effects in judgment?

Key findings

• The conditional phase shift operator provides a clear, interpretable mechanism for modeling psychological influence on belief revision, with phase as a psychological parameter.
• Experimental data from roulette and basketball show that estimated probabilities deviate from 50% by ±5% (e.g., P(1) = 0.55 in competitive scenarios, P(1) = 0.45 in random ones), which the model reproduces via interference terms.
• The interference term $ I( heta) $ is maximized when $ ext{Re}[e^{i heta} ho_{01}] $ is large, and it is bounded by the magnitude of the coherence term, showing that maximal interference is smaller at 70% than at 50% rational probability.
• Partial trace over the first qubit results in a mixed state with $ P(0) = P(1) = 0.5 $, demonstrating that erasing prior information restores rational probability, consistent with the absence of recency effects.
• The model explains that both gambler’s and hot hand fallacies arise from quantum correlations between opinion states, even when events are stochastically independent.
• The framework unifies multiple cognitive heuristics, suggesting that quantum-like representations can model belief updating across different decision-making phenomena.