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Cycling of Simple Rules in the Spatial Model

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  • David Austen-Smith
  • Jeffrey S. Banks

Abstract

McKelvey [4] proved that for strong simple preference aggregation rules applied to multidimensional sets of alternatives, the typical situation is that either the core is nonempty or the top-cycle set includes all available alternatives. But the requirement that the rule be strong excludes, inter alia, all supermajority rules. In this note, we show that McKelvey's theorem further implies that the typical situation for any simple rule is that either the core is nonempty or weak top-cycle set (equivalently, the core of the transitive closure of the rule) includes all available alternatives. Moreover, it is often the case that both of these statements obtain.

Suggested Citation

  • David Austen-Smith & Jeffrey S. Banks, 1998. "Cycling of Simple Rules in the Spatial Model," Discussion Papers 1246, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    • Handle: RePEc:nwu:cmsems:1246
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    References listed on IDEAS

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    1. Norman Schofield, 1983. "Generic Instability of Majority Rule," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 50(4), pages 695-705.
    2. Cohen, Linda, 1979. "Cyclic sets in multidimensional voting models," Journal of Economic Theory, Elsevier, vol. 20(1), pages 1-12, February.
    3. Schofield, Norman, 1980. "Generic properties of simple Bergson-Samuelson welfare functions," Journal of Mathematical Economics, Elsevier, vol. 7(2), pages 175-192, July.
    4. Schofield, Norman, 1984. "Social equilibrium and cycles on compact sets," Journal of Economic Theory, Elsevier, vol. 33(1), pages 59-71, June.
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    Cited by:

    1. Daniel Cardona & Antoni Rubí-Barceló, 2014. "On the efficiency of equilibria in a legislative bargaining model with particularistic and collective goods," Public Choice, Springer, vol. 161(3), pages 345-366, December.
    2. Kevin Roberts, 2007. "Condorcet cycles? A model of intertemporal voting," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(3), pages 383-404, October.
    3. Banks, Jeffrey S. & Duggan, John, 2008. "A Dynamic Model of Democratic Elections in Multidimensional Policy Spaces," Quarterly Journal of Political Science, now publishers, vol. 3(3), pages 269-299, October.
    4. Lee, Barton E., 2022. "Gridlock, leverage, and policy bundling," Journal of Public Economics, Elsevier, vol. 212(C).
    5. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2002. "Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections," Journal of Economic Theory, Elsevier, vol. 103(1), pages 88-105, March.
    6. Gallego, Maria & Schofield, Norman, 2017. "Modeling the effect of campaign advertising on US presidential elections when differences across states matter," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 160-181.
    7. Akira Okada & Ryoji Sawa, 2016. "An evolutionary approach to social choice problems with q-quota rules," KIER Working Papers 936, Kyoto University, Institute of Economic Research.

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