Social Choice and Welfare

, Volume 15, Issue 2, pp 211-236

First online:

The paradox of multiple elections

  • Steven J. BramsAffiliated with Department of Politics, New York University, New York, NY 10003, USA
  • , D. Marc KilgourAffiliated with Department of Mathematics, Wilfrid Laurier University Waterloo, Ontario, Canada N2L 3C5
  • , William S. ZwickerAffiliated with Department of Mathematics, Union College, Schenectady, NY 12308, USA

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Assume that voters must choose between voting yes (Y) and voting no (N) on three propositions on a referendum. If the winning combination is NYY on the first, second, and third propositions, respectively, the paradox of multiple elections is that NYY can receive the fewest votes of the 23 = 8 combinations. Several variants of this paradox are illustrated, and necessary and sufficient conditions for its occurrence, related to the “incoherence” of support, are given.

The paradox is shown, via an isomorphism, to be a generalization of the well-known paradox of voting. One real-life example of the paradox involving voting on propositions in California, in which not a single voter voted on the winning side of all the propositions, is given. Several empirical examples of variants of the paradox that manifested themselves in federal elections – one of which led to divided government – and legislative votes in the US House of Representatives, are also analyzed. Possible normative implications of the paradox, such as allowing voters to vote directly for combinations using approval voting or the Borda count, are discussed.