吉伯德-萨特思韦特防策略不可能性定理的逻辑初探

自20世纪70年代吉伯德-萨特思韦特防策略不可能性定理确立后，引起了数学、经济学、计算机科学和哲学等诸多领域中学者的广泛关注。然而，在国内外文献中，对吉伯德-萨特思韦特防策略不可能性定理的逻辑研究还比较少。基于此，借助公理化的方法探讨吉伯德-萨特思韦特防策略不可能性定理，并形式化地给出定理的内容和证明，力图把精细的逻辑推理应用于复杂的过程分析中，使防策略投票这一过程形式化、清晰化，以期能指导人们进行合理、有效的决策活动。     

Expected Utility Consistent Extensions of Preferences

We consider the problem of extending a (complete) order over a set to its power set. The extension axioms we consider generate orderings over sets according to their expected utilities induced by some assignment of utilities over alternatives and probability distributions over sets. The model we propose gives a general and unified exposition of expected utility consistent extensions whilst it allows to emphasize various subtleties, the effects of which seem to be underestimated – particularly in the literature on strategy-proof social choice correspondences.      

On Asymptotic Strategy-Proofness of Classical Social Choice Rules

We show that, when the number of participating agents n tends to infinity, all classical social choice rules are asymptotically strategy-proof with the proportion of manipulable profiles being of order O (1/n).     

Oligarchy for Social Choice Correspondences and Strategy-Proofness

We study the existence of a group of individuals which has some decisive power for social choice correspondences that satisfy a monotonicity property which we call modified monotonicity. And we examine the relation between modified monotonicity and strategy-proofness of social choice correspondences according to the definition by Duggan and Schwartz (2000). We will show mainly the following two results. (1) Modified monotonicity implies the existence of an oligarchy. An oligarchy is a group of individuals such that it has some decisive power (semi-decisiveness), and at least one of the most preferred alternatives of every its member is always chosen by any social choice correspondence. (2) Strategy-proofness of social choice correspondences is equivalent to modified monotonicity.     

Strategy-proof and Symmetric Social Choice Functions for Public Good Economies

We study economies with one private good and one pure public good, and consider the following axioms of social choice functions. Strategy-proofness says that no agent can benefit by misrepresenting his preferences, regardless of whether the other agents misrepresent or not, and whatever his preferences are. Symmetry says that if two agents have the same preference, they must be treated equally. Anonymity says that when the preferences of two agents are switched, their consumption bundles are also switched. Individual rationality says that a social choice function never assigns an allocation which makes some agent worse off than he would be by consuming no public good and paying nothing. In Theorem 1, we characterize the class of strategy-proof, budget-balancing, and symmetric social choice functions, assuming convexity of the cost function of the public good. In Theorem 2, we characterize the class of strategy-proof, budget-balancing, and anonymous social choice functions. In Theorem 3, we characterize the class of strategy-proof, budget-balancing, symmetric, and individually rational social choice functions.     

Unanimity and Resource Monotonicity

In the context of indivisible public objects problems (e.g., candidate selection or qualification) with “separable” preferences, unanimity rule accepts each object if and only if the object is in everyone’s top set. We establish two axiomatizations of unanimity rule. The main axiom is resource monotonicity, saying that resource increase should affect all agents in the same direction. This axiom is considered in combination with simple Pareto (there is no Pareto improvement by addition or subtraction of a single object), independence of irrelevant alternatives, and either path independence or strategy-proofness.