Abstract
Abstract A voting model is considered in which the philosophy of the traditional plurality method of selecting a winner is accepted but it is desired to compensate for the fact that voters may not have been able to evaluate all of the nominees, for example, voters voting for “best foreign film” from a list of nominees, consumers buying (that is, voting for) a product of a certain brand from a selection of available brands, and respondents selecting an answer to a multiple-choice question for which the possible alternative responses are not the same for all respondents. The model can be interpreted as a generalization to selections from possibly more than two nominees of the basic Bradley-Terry odds-ratio model of paired comparisons. Nominees (k) correspond to probabilities (pk ), and the probability that a voter votes for an evaluated nominee k is pk divided by the sum of the pi 's of the nominees evaluated by the voter. This article's major contribution is a convenient matrix representation of the log-likelihood function (2.3) and its gradient (2.5), from which maximum-likelihood estimates can be obtained easily using the method of steepest ascent. The resulting estimates can be used to determine a winner in several ways, to rank the nominees, and to estimate selection probabilities for voters considering any combination of nominees. An algorithm for identifying and eliminating degenerate special cases is given. A program that performs the calculations is available from the author. Interestingly, given the maximum likelihood estimator (MLE) vector, choosing the “winner” as the nominee with the highest estimated pk is not equivalent to choosing the “winner” as the nominee that would have received the highest expected number of votes among only the voters who already voted, if they had all evaluated all the nominees.
Published Version
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