Axiomatic Study of Solutions

这个项目是数学家们雄心勃勃的努力的一部分，目的是确定和严格研究个人和集体选择的基本原则。当这些原则应用于具体问题时，它们以公理的形式出现，事实证明，它们在为经济问题的解决方案提供新的见解以及统一经济理论的不同领域方面具有极其强大的作用。本项目研究并刻画了满足增广原理及其“对偶”--一致性原理的解。一致性是这样一种性质，即无论任何子问题的结果是什么，其中子问题大致意味着通过分别考虑来自原始问题的代理及其资源的子集而从原始问题派生的问题。扩充的工作是相反的：任何问题的解决方案都应该与一个更大的问题的解决方案保持一致，而给定的问题是其子问题。这些原则适用于以下几类经济问题：讨价还价、破产和税收、成本分配、公有制经济中的资源分配。这条基础研究路线很重要，因为它帮助我们理解哪些社会目标组合是可行的，哪些是不可行的。例如，该项目确定林达尔通信--公共产品经济中分配的主要理论工具之一--何时满足社会选择的基本原则，并开发精确的工具来量化给定通信未能满足给定要求的程度。

This project is part of an ambitious effort by mathematical economists to identify and rigorously study fundamental principles of individual and collective choice. These principles, when applied to specific problems, take the form of axioms and have proved extremely powerful in providing new insights into the solutions to economic problems and in unifying the varied areas of economic theory. This project studies and characterizes solutions satisfying the augmentation principle an its "dual", the consistency principle. Consistency is the property that whatever outcome of any subproblem where subproblem roughly means a problem derived from the original by considering separately a subset of the agents, and their resources, from the original problem. Augmentation works in reverse: any solution to a problem should be consistent with the solution to a larger problem of which the given problem is a subproblem. The principles are applied to the following classes of economic problems: bargaining, bankruptcy and taxation, cost allocation, resource allocation in economies with public godds. This line of basic research is important because it helps us understand the boundary between which combinations of social objectives are feasible and which are not. For example, the project determines when the Lindahl correspondence, one of the main theoretical tools for allocation in public good economies, satisfies fundamental principles of social choice and develops precise tools for quantifying the extent to which a given correspondence fails to meet a given requirement.