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## Quantum models for decision making

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School of psychology, Shaanxi Normal University & Key Laboratory of behavioral and cognitive neuroscience of Shaanxi Province, Xi’an 710062, China

 基金资助: 国家社会科学基金教育学青年课题.  CCA110105陕西师范大学教师教育研究专项.  JSJY2017006

Abstract

During the recent decade, quantum decision-making models were established based on the mathematical structure and methodologies of quantum mechanics. Owing to its unique theoretical structure, the quantum decision-making models can be applied to explain problems that violate the classical decision models, especially for judgments under uncertainty and decisions under conflicts. Quantum decision-making models have been used to explain phenomena such as disjunction effect, conjunction fallacy and interference of categorization on decision making which are difficult to account for with classical decision models. A model called quantum question equality has been tested for its accurate prediction of order effects. Being a new research field contributing to analysis of decision making, quantum decision-making models worth further investigations both in theoretical and applicative levels.

Keywords： quantum ; decision making ; model ; probability

XIN Xiaoyang, XU Chenhong, CHEN Hongyu, LI Ying. Quantum models for decision making. Advances in Psychological Science[J], 2018, 26(8): 1365-1373 doi:10.3724/SP.J.1042.2018.01365

## 1 引言

### 4.2 分类-决策中的干涉效应

C-then-D D-alone
p (G) p (A|G) p (B) p (A|B) pT(A) p (A)
0.17 0.41 0.83 0.63 0.59 0.69

p (A) = pT(A) = p (G) p (A|G) + p (B) p (A|B) (1)

(注：彩图见电子版)

### 4.4 量子问题等式

Wang和Busemeyer (2013)提出的量子问题等式(quantum question equality), 是一种对于顺序效应的先验性预测模型。这一模型可以精确地量化预测顺序效应大小, 证明量子决策模型不仅仅是一种后验性的模型(Yearsley & Busemeyer, 2016)。在实验中, 被试会被问到两个问题(假设为A和B), 并以不同顺序出现(AB或是BA)。p (Ay, Bn)表示被试对问题A肯定回答后对问题B的回答为否定的概率, 同理p (Bn, Ay)表示被试对B问题作出否定回答后再对A作出肯定回答的概率, 同理p (An, By)和p (An, By)的定义也类似。运用量子理论构建的量子问题等式为：[p (Ay, Bn)-p (Bn, Ay)]= -[p (An, By)-p (By, An)]。

Busemeyer和Wang (2015)在70个不同地区中使用不同的问题进行了实验, 其实验结果如图5所示。横坐标表示p (Ay, Bn)与p (Bn, Ay)的差值, 也就是Ay与Bn的顺序效应, 即量子问题等式左边的部分, 称为第一顺序效应(The First Order Effect); 纵坐标表示p (An, By)与p (By, An)的差值, 即An与By的顺序效应, 为量子问题等式左边部分。图中每个点表示在一个地区进行的实验结果, 图中共有70个点。运用线性拟合的结果显示, 拟合出直线斜率为-1, 相关系数r值为-0.82, 验证了量子问题等式的预测准确性。

### 图5

(资料来源：Busemeyer & Wang, 2015)

## 5 总结与展望

The authors have declared that no competing interests exist.

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The disjunction effect violates Savage’s sure-thing principle: that is, if is preferred over regardless of whether relevant outcome occurs, then should always be preferred over [L.J. Savage, The Foundations of Statistics, New York, Wiley, 1954]. We tested “reason-based” and “reluctance-to-think” accounts of the disjunction effect. According to the former account, the disjunction effect occurs when different reasons underlie the preference for under versus the preference for under not . According to the latter account, the disjunction effect is due to the failure to consider preferences when is unknown. We tested these accounts by varying the number of reasons underlying choices in the and not conditions. Consistent with the reason-based account, when only one reason was available, the disjunction effect was reduced. In addition, we propose a new method of measuring the disjunction effect under different conditions based on the logic proposed by Lambdin and Burdsal (2007) [C. Lambdin, C. Burdsal, The disjunction effect reexamined: relevant methodological issues and the fallacy of unspecified percentage comparisons, Organizational Behavior and Human Decision Processes 103 (2007) 268–276].

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People can often outperform statistical methods and machine learning algorithms in situations that involve making inferences about the relationship between causes and effects. While people are remarkably good at causal reasoning in many situations, there are several instances where they deviate from expected responses. This paper examines three situations where judgments related to causal inference problems produce unexpected results and describes a quantum inference model based on the axiomatic principles of quantum probability theory that can explain these effects. Two of the three phenomena arise from the comparison of predictive judgments (i.e., the conditional probability of an effect given a cause) with diagnostic judgments (i.e., the conditional probability of a cause given an effect). The third phenomenon is a new finding examining order effects in predictive causal judgments. The quantum inference model uses the notion of incompatibility among different causes to account for all three phenomena. Psychologically, the model assumes that individuals adopt different points of view when thinking about different causes. The model provides good fits to the data and offers a coherent account for all three causal reasoning effects thus proving to be a viable new candidate for modeling human judgment.

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Abstract One of the basic axioms of the rational theory of decision under uncertainty IS Savage's (1954) sure-thing principle (STP) It states that if prospect x is preferred to y knowing that Event A occurred, and if x IS preferred to y knowing that A did not occur, then x should be preferred to y even when it is not known whether A occurred We present examples in which the decision maker has good reasons for accepting x if A occurs, and different reasons for accepting X if A does not occur Not knowing whether or not A occurs, however, the decision maker may lack a clear reason for accepting X and may opt for another option We suggest that, in the presence of uncertainty, people are often reluctant to think through the implications of each outcome and, as a result, may violate STP This interpretation is supported by the observation that STP is satisfied when people are made aware of their preferences given each outcome

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Many decision making tasks in life involve a categorization process, but the effects of categorization on subsequent decision making has rarely been studied. This issue was explored in three experiments ( N = 721), in which participants were shown a face stimulus on each trial and performed variations of categorization-decision tasks. On C-D trials, they categorized the stimulus and then made an action decision; on X-D trials, they were told the category and then made an action decision; on D-alone trials, they only made an action decision. An interference effect emerged in some of the conditions, such that the probability of an action on the D-alone trials (i.e., when there was no explicit categorization before the decision) differed from the total probability of the same action on the C-D or X-D trials (i.e., when there was explicit categorization before the decision). Interference effects are important because they indicate a violation of the classical law of total probability, which is assumed by many cognitive models. Across all three experiments, a complex pattern of interference effects systematically occurred for different types of stimuli and for different types of categorization-decision tasks. These interference effects present a challenge for traditional cognitive models, such as Markov and signal detection models, but a quantum cognition model, called the belief-action entanglement (BAE) model, predicted that these results could occur. The BAE model employs the quantum principles of superposition and entanglement to explain the psychological mechanisms underlying the puzzling interference effects. The model can be applied to many important and practical categorization-decision situations in life.

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The purpose of this tutorial is to give an introduction to quantum models, with a particular emphasis on how to build these models in practice. Examples are provided by the study of order effects on judgements, and we will show how order effects arise from the structure of the theory. In particular, we show how to derive the recent discovery of a particular constraint on order effects implied by quantum models, called the Quantum Question (QQ) Equality, which does not appear to be derivable from alternative accounts, and which has been experimentally verified to high precision. However the general theory and methods of model construction we will describe are applicable to any quantum cognitive model. Our hope is that this tutorial will give researchers the confidence to construct simple quantum models of their own, particularly with a view to testing these against existing cognitive theories.