A SHORTCUT METHOD OF CALCULATING THE DISTRIBUTION OF ELECTION OUTCOMETYPES UNDER APPROVAL VOTING

The paper applies to approval voting, under which the voter casts a ba llot by casting one vote for each of k candidates, where k = 1, 2,..., m-1 and there are m candidates. I assume (following Brams and Fishbur n) that each of the voter's 2(m) -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,..., n votes; (m -2)-way ties, an d so on. The frequency distribution of outcome types varies only with m and n and is necessary to the calulation of the expected utilities o f successive ballots cast, in the same election, by a voter under a va riant of approval voting. This variant allows the voter to cast severa l complete ballots provided that he pays the respective prices, which could reasonably be based on the expected utilities. The paper describ es a shortcut method of calculating the distribution of outcome types when m = 4 and n 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 1 4 strategies available to the incremental voter. In going from n - 1 t o n, for n greater than or equal to 3, the number of outcome types inc reases by a factor of (n + 3)/n; whereas, the number of combinations o f strategies increases by a factor of 14.