Unrestricted domain: Difference between revisions

Mapping voter opinion to multi-dimensional space is tricky. When theorists choose four or more dimensions for their models, the result is difficult to visualize in a three-dimensional space. Moreover, the space of voter opinions imposes fundamental limitations on the set of allowed elections, depending on the number of candidates, as there may be insufficient room in the space for all ranked ballots to occur. This geometric result implies that violations of unrestricted domain are common in low-dimensional simulations involving many candidates, with the vast majority of election scenarios being impossible to arise. For example, for a 2-dimensional Euclidean spatial model with 6 candidates, there are $<math>6! = 720$</math> possible rankings, but it is geometrically impossible to construct a voter and candidate distribution which produces more than 101 distinct ballots, meaning at least 86% of ballots will never emerge in any voter or candidate distribution. Elections involving more than 101 unique ballots are impossible in such a scenario, and these correspond to the vast majority of possible elections. See [[space of possible elections]] and [[limitations of spatial models of voting]] for more details.