| Budget Amount *help |
¥2,150,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥150,000)
Fiscal Year 2007: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2006: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
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| Research Abstract |
Principal achievements of this project are as follows.
(1) Applications of algebraic topology to social choice theory.
(i) In the case where there are two individuals and three alternatives (or under the assumption of free-triple property) the Arrow impossibility theorem for social welfare functions that there exists no social welfare function (transitive binary social choice rule) which satisfies Pareto principle, independence of irrelevant alternatives, and has no dictator is equivalent to the Brouwer fixed point theorem on a 2-dimensional ball (circle).
(ii) Arrow impossibility theorem for binary social choice rules that there exists no binary social choice rule which satisfies transitivity Pareto principle, independence of irrelevant alternatives (IIA), and has no dictator, and Amartya Sen's liberal paradox for binary social choice rules that there exists no binary social choice rule which satisfies acyclicity, Pareto principle and the minimal liberalism are topologically equivalent.
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2) HEX game theorem and theorems of social choice theory.
(i) Under some assumptions about marking rules of HEX games, the HEX game theorem is equivalent to the Arrow impossibility theorem of social choice theory that there exists no binary social choice rule which satisfies transitivity Pareto principle, independence of irrelevant alternatives and has no dictator.
(ii) Under some assumptions about marking rules of HEX games, the HEX game theorem is equivalent to the Duggan-Schwartz theorem for strategy-proof social choice correspondences that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator.
(3) Computability of social choice rules.
(i) If there exists a dictator for a social choice function, it is computable in the sense of Type two computability, but if there exists no dictator it is not computable.
(ii) The problem whether a transitive binary social choice rule satisfying Pareto principle and independence of irrelevant alternative (IIA) has a dictator or has no dictator in an infinite society is an unsolvable problem, that is, there exists no ideal computer program for a transitive binary social choice rule satisfying Pareto principle and IIA that decides whether the binary social choice rule has a dictator or has no dictator. And it is equivalent to unsolvability of the halting problem. Less
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