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 2
3 = 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.
Received: 31 July 1996 / Accepted: 1 October 1996