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Decision Theory with Prospect Interference and Entanglement

V. I. Yukalov, D. Sornette

arXiv:1102.2738v1math-phcs.AIphysics.soc-phquant-ph

TL;DR

The paper addresses how classical decision theory can account for human paradoxes involving uncertainty, composite prospects, and subjective interference. It develops Quantum Decision Theory using separable Hilbert spaces and interference terms, showing that interference alternation combined with uncertainty aversion explains the disjunction effect and that the framework also explains the conjunction fallacy. Its quantitative estimates agree with experiments, while its predictions are explicitly aggregate and contextual.

  • Problem

    Existing decision theory does not provide a general quantitative account of decision interference and related disjunction and conjunction effects in human decision making.

  • Method

    QDT represents composite prospects and intentions in separable Hilbert spaces, using interference terms and aggregate probabilistic predictions.

  • Results

    Interference alternation together with uncertainty aversion explains the disjunction effect, while interference terms explain the conjunction fallacy; quantitative estimates agree with experiments.

  • Takeaways & Limitations

    The conjunction fallacy is a sufficient condition for the disjunction effect, and the paper proposes experiments examining their combined interplay.

  • Takeaways & Limitations

    QDT’s interference terms are contextual across decision makers and times, and its predictions concern aggregate population behavior rather than specific individuals.

Abstract

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We present a novel variant of decision making based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage's sure-thing principle, known as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analysed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect and novel experiments testing the combined interplay between the two effects are suggested.

1 Introduction

The paper introduces Quantum Decision Theory as a Hilbert-space-based mathematical framework for modeling departures from classical rational decision theory. It uses quantum-theoretical techniques as a formal language, not as a claim that brains or decision makers are quantum objects.

  • 1 Introduction: QDT generalizes objective probabilities into nonlinear subjective probabilities to explain disjunction and conjunction effects quantitatively.The disjunction effect violates Savage’s sure-thing principle, while the conjunction effect treats a specific conjunction as more probable than a general event.
  • 1 Introduction: The framework models decisions as discrete selections from large sets of entangled options and accounts for hidden variables involved in decision making.Its underlying structure is the mathematical theory of separable Hilbert spaces.
  • 1 Introduction: QDT focuses on mathematical modeling of functional decision processes rather than physiological mechanisms of the brain.The authors present it as a mathematical tool for complex systems, analogous to differential equations used beyond their original planetary application.
  • 1 Introduction: The paper places QDT among approaches that use quantum techniques as a convenient language for generalizing classical probability to classical problems.This differs from theories that attempt to model the brain itself as a quantum object.
  • 1 Introduction: QDT uses Hilbert-space mathematics without assuming that the human brain, consciousness, or decision maker has quantum physical properties.The authors distinguish mathematical analogy from quantum mechanics as a physical theory.

2 Foundations of Quantum Decision Theory

QDT represents decision makers, intentions, and composite prospects within Hilbert spaces, allowing entangled decision states and noncommutative subsequent decisions. Composite prospects are central because they incorporate multiple interconnected intended actions.

  • 2 Foundations of Quantum Decision Theory: QDT constructs action probabilities using complex separable Hilbert spaces and extends quantum-mechanical measurement theory from simple actions to composite prospects.The paper’s aim is a detailed theory applicable to real-life decision-making problems.
  • 2 Foundations of Quantum Decision Theory: An intention is a thought about doing something, while action modes are concrete implementations or representations of that intention.Examples distinguish an intention such as marrying from representations such as marrying a particular person.
  • 2 Foundations of Quantum Decision Theory: The mode space is modeled as a Hilbert space containing intention states represented as linear combinations of basis states.The expansion coefficients are decision-maker-defined weights for the corresponding representation states.
  • 2 Foundations of Quantum Decision Theory: A prospect is a conjunction of several intended actions or intention representations, reflecting the interconnected intentions involved in actual decisions.The formalism therefore treats prospects as composite objects rather than isolated actions.
  • 2 Foundations of Quantum Decision Theory: Entangled prospect and strategic states generally prevent prospect probabilities from factorizing into products of separate terms.The framework therefore characterizes decision making as naturally entangled.
  • 2 Foundations of Quantum Decision Theory: Noncommutativity of subsequent decisions models dynamic inconsistency, in which decision makers change plans after experiencing outcomes.The paper attributes this effect to entanglement between intention representations and intention interference.

3 Prospect Interference

QDT locates decision interference in composite, uncertain prospects represented in a multidimensional mind space. Its interference terms alternate in sign, depressing some action probabilities while enhancing others.

  • 3 Prospect Interference: A general theory was needed to explain why and when decision interference appears, predict it, and evaluate it quantitatively.QDT addresses this gap by treating interference within composite prospects.
  • 3 Prospect Interference: For two intentions, prospect probabilities decompose into partial probabilities plus interference terms, such as p(πA) = p(AW) + p(AG) + q(πA).The interference terms represent deliberation between the simultaneous intention representations.
  • 3 Prospect Interference: Interference appears when a decision concerns a composite entangled prospect containing several intention representations realized simultaneously.The two-intention example combines choosing a friend with choosing a type of work.
  • 3 Prospect Interference: Interference requires uncertainty and a mind dimensionality of at least two; in one-dimensional minds, intentions do not interfere.The paper identifies these as necessary conditions for intention interference.
  • 3 Prospect Interference: The interference-alternation theorem states that normalized prospect probabilities have interference terms whose total vanishes.If any term is nonzero, some terms must be positive and others negative.
  • 3 Prospect Interference: Negative interference depresses some probabilities, whereas positive interference enhances others, linking uncertainty-related effects to decision anomalies.The authors associate probability depression with uncertainty aversion.

4 Interference Quarter Law

The paper develops the interference quarter law to estimate subjective interference effects at the aggregate level. It combines probability-based assumptions with interference alternation to make QDT quantitatively predictive.

  • 4 Interference Quarter Law: Prospect probabilities combine a utility factor with an interference or attraction factor representing subjective prospect quality and emotions.The interference term reflects subconscious feelings, emotions, and biases.
  • 4 Interference Quarter Law: Interference terms arise for composite prospects, and interference alternation constrains their signs, but these properties alone do not determine their magnitudes.The quarter law supplies an additional estimation principle.
  • 4.1 Aggregate Nature of Quantum Decision Theory: QDT interprets probabilities in the frequentist sense, making its predictions statistical statements about aggregate population behavior rather than exact individual decisions.Expected interference values correspond to ensembles of decision makers under given conditions.
  • 4 Interference Quarter Law: Under uniform, noninformative priors, the binary-prospect calculation yields the interference-quarter law.The calculation assumes expected probabilities under square roots equal 1/2.
  • 4 Interference Quarter Law: For arbitrary prospects under general conditions, the expected modulus of the interference term is q = 1/4.This estimate is used to predict average behavior and quantify the influence of emotions.
  • 4 Interference Quarter Law: The derived interference properties enable analysis of interference’s influence on human decision making and explanation of paradoxical effects.The paper summarizes these properties as composite-prospect occurrence, sign alternation, and quarter-law magnitude estimation.

5 Disjunction Effect

The disjunction effect is Savage’s sure-thing principle violation: preferences for an action under each known event do not carry over when the event is unknown.

  • The sure-thing principle states that preferring A to B under either X1 or X2 should imply preferring A to B when which event occurred is unknown.
  • The disjunction effect names the observed failure of this implication under uncertainty.

5.1 Sure-Thing Principle

Classical probability derives the sure-thing principle from conditional preferences across events and uses it as a normative consistency condition for decisions under uncertainty.

  • Classical probability represents preference for A over B as P(A) > P(B), with conditional preferences written P(A|Xj) > P(B|Xj).
  • Theorem 2 states that if P(A|Xj) > P(B|Xj) for every j, then P(AX) > P(BX).
  • The conclusion follows immediately from Eqs. (75) and (76) under assumption (73).
  • Savage treated this classical proposition as a normative assumption about consistent human decisions under uncertainty, making violations empirically testable.

5.2 Examples Illustrating the Disjunction Effect

Experiments across gambling, purchasing, and stock-market decisions show that people often choose an action when outcomes are known but reject it when the outcome is unknown, violating the sure-thing principle.

  • Numerous empirical studies have reported violations of the sure-thing principle in human decision making.
  • To gamble or not to gamble?: In a two-step gamble, most participants accepted the second gamble after either winning or losing, but a minority accepted it when the first outcome was unknown.
  • To buy or not to buy?: In the vacation scenario, most students bought the vacation after either passing or failing the exam, satisfying the conditional preference pattern.
  • To sell or not to sell?: The stock-market example concerns selling or holding stocks when the presidential election outcome is unknown, extending the effect to deliberation about a future event.
  • Disjunction effects require multiple possible events and uncertainty about which event has occurred or will occur.
  • Reason-based explanations and their limits are reviewed before introducing the QDT account.

5.3 Reason-Based Analysis

Reason-based analysis explains disjunction effects through the availability of reasons for known outcomes, but the paper argues that this explanation is vague and ad hoc.

  • Reason-based analysis explains choices by balancing reasons for and against alternatives.
  • When outcomes are known, decision makers may find a reason to choose; under uncertainty, they may lack one and abstain.
  • The paper criticizes “reason” as too vague and subjective to quantify or define qualitatively.
  • Mental accounting can supply a rationale for accepting a second gamble after a known win by treating the initial gain differently from ordinary wealth.
  • Under ignorance about the first gamble, an equally plausible rationale is to disregard the unknown outcome and gamble again.
  • Because opposite actions can receive equally reasonable justifications, the paper characterizes the reason-based account as ad hoc and lacking real explanatory power.

5.4 Quantitative Analysis within Quantum Decision Theory

The section develops a quantitative QDT account of the disjunction effect, deriving interference terms whose signs and magnitudes explain suppressed action under uncertainty. Calculations and experiments support a priori estimates based on uncertainty aversion and interference laws.

  • Theoretical framework: QDT treats the disjunction effect as interference between intentions when decisions are made under uncertainty.The approach differs from earlier ad hoc interference assumptions by embedding the effect within a general theory.
  • Interference alternation: The interference-alternation condition requires the interference terms for acting and remaining inactive to have opposite signs and sum to zero.Without this condition, the probability equations are incomplete and the disjunction effect cannot be explained.
  • Interference alternation: When conditional action probabilities exceed conditional inaction probabilities, uncertainty factors must compensate through smaller magnitude for action than for inaction.The framework links this alternation to the structure of the uncertainty factors and human aversion to uncertainty.
  • Interpretation: Providing detailed explanations of possible outcomes may reduce the disjunction effect by encouraging people to act despite uncertainty.The paper contrasts this account with standard reason-based rationalization and interprets the effect as an emotional reaction associated with uncertainty aversion.
  • Quantitative mechanism: Uncertainty aversion predicts negative interference suppressing action and positive interference enhancing inaction under uncertainty.The resulting ordering is p(AX) < p(BX), which constitutes the disjunction effect.
  • Quantitative validation: The interference amplitudes observed in the examples are close to 0.25, matching the interference-quarter law within typical experimental accuracy.The paper therefore claims that the strength of the disjunction effect can be predicted without knowing the experiment’s results.

6 Conjunction Fallacy

The section explains the conjunction fallacy in QDT as a consequence of interference under uncertainty. It derives conditions for the fallacy, compares them with experiments, and limits the analysis to independently deciding individuals.

  • Scope: The analysis focuses on separate individuals without social interaction, whereas group communication substantially weakens the conjunction fallacy.The paper describes social interaction as causing decoherence that destroys interference, leaving group decisions outside its main scope.
  • Definition and evidence: The conjunction fallacy is the observed violation of the standard rule that a conjunction cannot be more probable than a separate event.People may judge p(AX) < p(AXj) for some j, contrary to classical probability theory.
  • QDT formulation: QDT explains the conjunction fallacy through two intentions: deciding about primary feature A and deciding about secondary feature X1 or X2.The prospect probability combines conditional probabilities with an interference term.
  • QDT formulation: Under uncertainty, uncertainty aversion makes q(AX) negative; the conjunction fallacy occurs when its amplitude is sufficiently large to produce a positive conjunction error.QDT therefore predicts the fallacy when uncertainty is present and the negative interference crosses the derived threshold.
  • Comparison with experiments: For compatible characteristics, average probabilities p(AX) = 0.537 and p(AX1) = 0.567 coincide within 20% statistical errors, so no conjunction fallacy arises.The paper attributes this case to a lack of uncertainty because the features are similar.
  • Relation to the disjunction effect: The conjunction fallacy is sufficient for the disjunction effect, but the reverse does not hold because conjunction requires sufficiently large interference amplitudes.Both effects are attributed to interference between probabilities under uncertainty.

7 Conclusion

Quantum Decision Theory applies Hilbert-space mathematics to decision making without requiring decision makers to be quantum objects. It models entangled decisions, non-commutativity, and intention interference, explaining uncertainty-related disjunction and conjunction effects with quantitative agreement with experiments.

  • Theory and scope: The theory characterizes each decision maker with an individual strategic state and models several unusual properties, including entangled decisions and interference between intentions.It is applicable to non-quantum decision makers and provides everyday interpretations of these formally quantum-described features.
  • Theory and scope: Quantum Decision Theory is a general mathematical approach applied throughout the paper to different decision-making effects rather than fitted to particular cases.Its formal tools derive from Hilbert spaces and quantum theory, but imply no quantum nature of decision makers.
  • Interference and uncertainty: Interference alternation produces positive and negative terms whose combined effects can generate human decision paradoxes and logical fallacies under uncertainty.The theorem links these alternating signs to uncertainty aversion, which suppresses action probabilities while enhancing passivity.
  • Explained effects: The conjunction fallacy is explained by intention interference together with uncertainty aversion, and it is shown to be sufficient for the disjunction effect.The paper also formulates an interference-quarter law that predicts the amplitude of interference terms and the quantitative violation of the sure-thing principle.
  • Empirical agreement: Detailed quantitative comparisons with experiments on the disjunction effect and conjunction fallacy confirm the validity of the derived laws.The interference estimates are reported as being in good agreement with disjunction-effect data and excellent agreement with analyzed conjunction-fallacy experiments.
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