Win probabilities and simple majorities in probabilistic voting situations

This paper considers an election between candidatesA andB in which (1) voters may be uncertain about which candidate they will vote for, and (2) the winner is to be determined by a lottery betweenA andB that is based on their vote totals. This lottery is required to treat voters equally, to treat candidates equally, and to respond nonnegatively to increased support for a candidate. The setΛ _n of all such lottery rules based on a total ofn voters is the convex hull of aboutn/2 ‘basic’ lottery rules which include the simple majority rule. For odd values ofn ≥ 3 let\(\mu (n) = \tfrac{1}{2} + (n - 1)/[2n(n + 3)]\), and for even values ofn ≥ 4 let\(\mu (n) = \tfrac{1}{2} + [(n + 2)(2n(n + 2))^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} - n(n + 6)]/[n(n - 2)(n + 4)]\). With\(\bar p\) the average of then voters probabilities of voting forA, it is shown that withinΛ _n the simple majority rule maximizes candidateA's overall win probability whenever\(\bar p \geqslant \mu (n)\), and thatμ(n) is the smallest number for which this is true. Similarly, the simple majority rule maximizesB's overall win probability whenever\(1 - \bar p\) (the average of the voters probabilities of voting forB) is as large asμ(n).

Authors and Affiliations

1. The Pennsylvania State University, University Park, Penna., USA

   Peter C. Fishburn & William V. Gehrlein

This research was supported by the National Science Foundation, Grant SOC 75-00941.

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