, Volume 59, Issue 2, pp 101-120

The problem of indeterminacy in approval, multiple, and truncated voting systems

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It is well known that a plurality election need not reflect the true sentiments of the electorate. Some of the proposed reform procedures, such as approval and cumulative voting, share the characteristics that there are several ways to tally each voter's preferences. Voting systems that permit truncated ballots share this feature. It is shown that the election results for any such procedure can be highly indeterminate; all possible election results can occur with the same choice of sincere voters. This conclusion of indeterminacy holds even when measures of voters' sentiments, such as the existence of a Condorcet winner or even much stronger measures, indicate there is considerable agreement among the voters. Then, multiple systems are compared with all standard tallying procedures. For instance, a corollary asserts it is probable for the plurality voting method to elect the Condorcet winner while approval voting has an indeterminate outcome.

The work of both authors was supported, in part, by NSF Grant IST 8415348. This work was partially stimulated by comments concerning the assertions about approval voting and other systems that are discussed in (Saari, 1987). In particular, we are pleased to thank S. Brams, Carl Simon, and the other participants for their questions after one of us presented (Saari, 1987) at the July, 1984, ‘Workshop on Mathematical Models in Political Science’ held at the University of Minnesota in Minneapolis.