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Impossibility Theorems and the Universal Algebraic Toolkit

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  • Mario Szegedy
  • Yixin Xu

Abstract

We elucidate a close connection between the Theory of Judgment Aggregation (more generally, Evaluation Aggregation), and a relatively young but rapidly growing field of universal algebra, that was primarily developed to investigate constraint satisfaction problems. Our connection yields a full classification of non-binary evaluations into possibility and impossibility domains both under the idempotent and the supportive conditions. Prior to the current result E. Dokow and R. Holzman nearly classified non-binary evaluations in the supportive case, by combinatorial means. The algebraic approach gives us new insights to the easier binary case as well, which had been fully classified by the above authors. Our algebraic view lets us put forth a suggestion about a strengthening of the Non-dictatorship criterion, that helps us avoid "outliers" like the affine subspace. Finally, we give upper bounds on the complexity of computing if a domain is impossible or not (to our best knowledge no finite time bounds were given earlier).

Suggested Citation

  • Mario Szegedy & Yixin Xu, 2015. "Impossibility Theorems and the Universal Algebraic Toolkit," Papers 1506.01315, arXiv.org.
  • Handle: RePEc:arx:papers:1506.01315
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    References listed on IDEAS

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    1. List, Christian & Pettit, Philip, 2002. "Aggregating Sets of Judgments: An Impossibility Result," Economics and Philosophy, Cambridge University Press, vol. 18(1), pages 89-110, April.
    2. Sen, Amartya K, 1979. "Personal Utilities and Public Judgements: Or What's Wrong with Welfare Economics?," Economic Journal, Royal Economic Society, vol. 89(355), pages 537-558, September.
    3. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations with abstentions," Journal of Economic Theory, Elsevier, vol. 145(2), pages 544-561, March.
    4. Rubinstein, Ariel & Fishburn, Peter C., 1986. "Algebraic aggregation theory," Journal of Economic Theory, Elsevier, vol. 38(1), pages 63-77, February.
    5. Wilson, Robert, 1975. "On the theory of aggregation," Journal of Economic Theory, Elsevier, vol. 10(1), pages 89-99, February.
    6. Kenneth J. Arrow, 1950. "A Difficulty in the Concept of Social Welfare," Journal of Political Economy, University of Chicago Press, vol. 58(4), pages 328-328.
    7. Dokow, Elad & Holzman, Ron, 2010. "Aggregation of binary evaluations," Journal of Economic Theory, Elsevier, vol. 145(2), pages 495-511, March.
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