A Shortcut Method of Calculating the Distribution of Election Outcome Types Under Approval Voting

Abstract

The paper applies to approval voting, under which the voter casts a ballot by casting one vote for each of k candidates, wherek=;1,2, ⋯ , m−1 and there are m candidates. I assume (following Brams and Fishburn) that each of the voter's 2=;−2 strategies is equally likely to be chosen. Election-outcome types include: the m-way tie;(m-1) -way ties with the runner-up trailing by 1,2,⋯,m votes; (m-2)-way ties, and so on. The frequency distribution of outcome types varies only with m and n and is necessary to the calculation of the expected utilities of successive ballots cast, in the same election, by a voter under a variant of approval voting. This variant allows the voter to cast several complete ballots provided that he pays the respective prices, which could reasonably be based on the expected utilities. The paper describes a shortcut method of calculating the distribution of outcome types when m=;4 andn rises to levels that make straightforward calculation computationally infeasible. The shortcut involves the combining of an outcome type, instead of each member of that type, with each of the 14 strategies available to the incremental voter. In going fromn-1 to n, for n≥3, the number of outcome types increases by a factor of (n+3)/n whereas, the number of combinations of strategies increases by a factor of 14.