Collaborative Research: Quantum Decision Theory

对人类判断和决策的研究揭示了一些矛盾的发现，这些发现难以在共同的理论框架下解释。这包括违反决策的确定性公理，推理和决策之间的相互作用，违反决策的约简公理，违反概率论的合取公理和析取公理，以及对判断的顺序影响。在过去，使用经典决策理论的变体提出了单独和不相关的解释。本研究基于一种新的量子决策理论，对所有这些矛盾的结果提出了统一的解释。经典决策理论是建立在经典概率论基础上的。概率被赋予定义为一个全称集合的子集的事件，它服从布尔代数的所有定律。量子决策理论是建立在量子概率论基础上的。定义为希尔伯特空间的子空间的事件，其概率服从布尔代数除分配公理外的所有定律。从分配公理出发，经典概率论坚持其最重要的定理之一，即总概率定律。因为量子逻辑不必服从分配定律，量子概率也不必服从总概率定律。相反，量子概率论必须遵守另一个被称为双重随机定律的定律，而经典的概率模型不遵守这个定律。因此，这两种概率论本质上是不同的，关键的问题是哪一套规则能更好地描述人类的行为。本研究的直接目标是严格比较基于经典概率论和量子概率论建立的决策模型。为了严格比较量子与经典决策概率模型，将进行一系列实验。实验集中在全概率定律和双随机定律的检验上，这两类模型做出了主要的、定性不同的预测。本研究将实现三个目标：(1)发展一种新的人类推理和决策的量子理论；(2)利用人类推理和决策行为对全概率和双随机的基本定律进行新的实证检验；(3)根据新的实证发现严格比较和对比经典模型和量子模型的决策。只有当量子模型或经典模型在准确性和简洁性两方面对现象提供了更好的科学解释时，才会被优先考虑。该研究项目的长远目标是通过从量子而不是经典概率原理为社会和行为科学建立概率和动态系统来开辟新的道路。以前，这些领域的理论家依赖于基于从经典物理学借鉴的基本假设的数学模型（例如随机微分方程）。这些基本假设是什么？它们是否过于严格？社会和行为科学家也面临着从经典概率观点来看仍然自相矛盾的发现。这些悖论表明，这些领域的测量可能并不总是遵循全概率定律，并需要不同的假设。这个研究项目也将有助于培养两所主要州立大学的本科生、研究生和博士后水平的学生。除了学生培训外，研究人员还将努力培养量子认知领域的科学家。他们在年度认知科学会议上进行了一整天的辅导，并计划在未来继续这些辅导。他们还计划在《数学心理学杂志》上组织一期关于量子认知的特刊。我们将在研究生阶段开设新的量子认知和决策的研究生课程，最后，我们将开发一个资源网站，提供社会和行为科学的量子理论的教程和参考信息。

Research on human judgment and decision making has revealed a number of paradoxical findings that have resisted explanation under a common theoretical framework. These include violations of the sure thing axiom of decision making, interactions between inferences and decisions, violations of the reduction axiom of decision making, violations of the conjunctive and disjunctive axioms of probability theory, and order effects on judgments. In the past, separate and disconnected explanations have been proposed using variants of classic decision theory. This research proposes a unifying explanation for all of these paradoxical results based on a new quantum decision theory. Classic decision theory is based on classic probability theory. Probabilities are assigned to events defined as subsets of a universal set, which obey all the laws of Boolean algebra. Quantum decision theory is based on quantum probability theory. Probabilities are assigned to events defined as subspaces of a Hilbert space, which obey all the laws of Boolean algebra except the distributive axiom. Following from the distributive axiom, classic probability theory adheres to one of its most important theorems, the law of total probability. Because quantum logic does not have to obey the distributive law, quantum probabilities do not have to obey the law of total probability. Instead, quantum probability theory must obey another law called the doubly stochastic law, which the classic probability model does not obey. Hence, the two probability theories are fundamentally different and the critical question is which set of rules provides a better description of human behavior. The immediate goal of this research is to rigorously compare decision models built upon classical probability theory with those built from quantum probability theory. To rigorously compare quantum versus classical probability models of decision making, a series of experiments will be conducted. The experiments focus on tests of the law of total probability and tests of the law of double stochasticity, where the two classes of models make major and qualitatively different predictions. The research will accomplish three objectives: (1) develop a new quantum theory of human inference and decision making, (2) conduct new empirical tests of the fundamental laws of total probability and double stochasticity using human inference and decision behavior, and (3) rigorously compare and contrast classic versus quantum models of decision making with respect to the new empirical findings. A quantum or classic model will be preferred only if it provides a superior scientific explanation of the phenomena with respect to both accuracy and parsimony.The broad and long-term goal of this research program is to break new ground and pioneer a new path by building probabilistic and dynamic systems for social and behavioral sciences from quantum rather than classical probability principles. Previously, theorists in these fields have relied on mathematical models (e.g. stochastic differential equations) based on fundamental assumptions borrowed from classical physics. What are these fundamental assumptions? Are they overly restrictive? Social and behavioral scientists also face findings that remain paradoxical from a classic probability point of view. These paradoxes suggest that measurements in these fields may not always obey the law of total probability and entail different assumptions. This program of research also will contribute to the training of students at the undergraduate, graduate, and post doctoral levels at two major state universities. In addition to student training, the investigators will make an effort to train scientists in the area quantum cognition. They have conducted a full day tutorial at the annual Cognitive Science meeting and they plan to continue these tutorials in the future. They also plan to organize a special issue on Quantum Cognition in the Journal of Mathematical Psychology. New graduate courses on quantum cognition and decision making will be prepared and presented at the graduate level, and finally, a resource web site will be developed with tutorial and reference information on quantum theory for social and behavioral sciences.