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Multiattribute Decision Modeling in Defense Applications
Published in Natalie M. Scala, James P. Howard, Handbook of Military and Defense Operations Research, 2020
One obvious approach to the problem of multiple stakeholders with different value structures is to simply ask them to vote on the alternative models. This approach should be viewed with extreme caution. The problem is that there is no way to aggregate votes on three or more alternatives that is without serious flaws. For instance, one might try “instant runoff,” in which each voter ranks the alternatives from most preferred to least. Then the alternative with the fewest first-place votes is eliminated and the votes distributed to the voters’ second choices. This is repeated until one alternative has a majority. But suppose 30% of the voters prefer A to B to C, 36% prefer B to A to C, and 34% prefer C to A to B. On the first ballot A is eliminated, and on the second B wins. This violates majority rule, since 64% of the voters preferred A to B. Another approach using voter ranks would be to weight the votes in a Borda count: first place among n alternatives receives n points, second place receives n − 1, and so forth. This method can also violate majority rule. If there are five voters and two prefer A to B to C, two prefer B to C to A, and one prefers C to A to B, then B wins, though 60% of the voters prefer A to B. In fact, Arrow (1950) proved that there is no way to aggregate votes for three or more alternatives that does not have a flaw like this or one just as serious. This is known as “Arrow’s Impossibility Theorem.” The conclusion is that voting is a poor way to make a group decision if there are more than two alternatives.
Representing voting rules in Łukasiewicz’s three-valued logic
Published in Journal of Applied Non-Classical Logics, 2022
Following the proof of Arrow’s impossibility theorem (1951), social choice theory developed as the study of the ways in which individual preferences can be aggregated to reach a collective or social preference. For example, given a distribution of votes expressed by the members of a group of people, a voting rule selects the collection of winning candidates or the number of seats won by political parties in representative bodies, like the Parliament or local councils. In the past decades, social choice techniques were applied to many other aggregation problems characterised by common structural features. The simplest case is that of a single person who rates m objects against each of n criteria in order to choose one of them. A more recent example was initiated by List and Pettit (2002): they explored the way in which the judgments held on some interconnected questions by the members of a group can be aggregated to form a collective set of judgments on those questions (List & Polak [2010] and Mongin [2018] are two surveys of the literature on this topic).