不含完备性公理的混合空间上的期望效用理论
Expected utility theory on mixture spaces without the completeness axiom
摘要：
混合预序是混合空间（如凸集）上与混合操作相容的预序。在决策理论中，它满足强独立性的中心期望效用公理。我们考虑当一个混合前序有一个多表示，由实值，混合保持功能。如果是，它必须满足Herstein和Milnor（1953）的混合连续性公理。当混合空间的维数是可数的时，混合连续性对于混合保持多重表示是足够的，但当它是不可数的时，混合连续性就不足够了。我们最强的积极的结果是，混合物连续性是足够的结合一个新的公理，我们称之为可数支配，它限制了混合物的顺序复杂性的阿基米德结构。我们还考虑了当混合空间被赋予其自然弱拓扑时会发生什么。连续性（具有封闭的上集和下集）和封闭性（具有封闭的图）比混合连续性更强。我们表明，连续性是必要的，但不足以使一个混合前序有一个混合保持多重表示。封闭性也是必要的;我们把它作为一个开放的问题是否足够。最后，我们的结果有关的存在混合保持多表示，完全由严格增加的功能，和唯一性的结果。
英文摘要：
A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real-valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.
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