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The Hex Game Theorem And The Arrow Impossibility Theorem: The Case Of Weak Orders

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  • Yasuhito Tanaka

Abstract

The Arrow impossibility theorem when individual preferences are weak orders is equivalent to the HEX game theorem. Because Gale showed that the Brouwer fixed point theorem is equivalent to the HEX game theorem, this paper indirectly shows the equivalence of the Brouwer fixed point theorem and the Arrow impossibility theorem. Chichilnisky showed the equivalence of her impossibility theorem and the Brouwer fixed point theorem, and Baryshnikov showed that the impossibility theorem by Chichilnisky and the Arrow impossibility theorem are very similar. Thus, Chichilnisky and Baryshnikov are precedents for the result—linking the Arrow impossibility theorem to a fixed point theorem.

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  • Yasuhito Tanaka, 2009. "The Hex Game Theorem And The Arrow Impossibility Theorem: The Case Of Weak Orders," Metroeconomica, Wiley Blackwell, vol. 60(1), pages 77-90, February.
    • Handle: RePEc:bla:metroe:v:60:y:2009:i:1:p:77-90
    DOI: 10.1111/j.1467-999X.2008.00332.x
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    1. Chichilnisky, Graciela, 1979. "On fixed point theorems and social choice paradoxes," Economics Letters, Elsevier, vol. 3(4), pages 347-351.
    2. Kotaro Suzumura, 2000. "Presidential Address: Welfare Economics Beyond Welfarist-Consequentialism," The Japanese Economic Review, Japanese Economic Association, vol. 51(1), pages 1-32, March.
    3. Yasuhito Tanaka, 2005. "On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem (forthcoming in ``Applied Mathematics and Computation''(Elsevier))," Public Economics 0506012, University Library of Munich, Germany, revised 17 Jun 2005.
    4. Chichilnisky, Graciela, 1980. "Social choice and the topology of spaces of preferences," MPRA Paper 8006, University Library of Munich, Germany.
    5. Chichilnisky, G., 1993. "Intersecting Families of Sets and the Topology of Cones in Economics," Papers 93-17, Columbia - Graduate School of Business.
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