Realistic opinion aggregation: Lehrer-Wagner with a finite set of opinion values

An allocation problem is a type of aggregation problem in which the values of individuals' opinions on some set of variables (canonically a set of mutually exclusive and exhaustive possibilities) sum to a constant. This paper shows that for realistic allocation problems, namely ones in which the set of possible opinion values is finite, the only universal aggregation methods that satisfy two commonly invoked conditions are the dictatorial ones. The two conditions are, first, that the aggregate opinion on any variable depends only on the individuals' opinions on that variable and not on their opinions on any other variable and, secondly, that the aggregate opinion minimally preserves unanimous individual opinion. Connections between this result and similar ones in the literature on judgment aggregation are explored and their significance assessed.