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.{{Citation needed|reason="Impossible"For isexample, for a pretty2-dimensional boldEuclidean claimspatial model with 6 candidates, andthere needsare <math>6! = 720</math> possible rankings, but it is geometrically impossible to beconstruct backeda upvoter withand evidence.candidate distribution Alsowhich produces more than 101 distinct ballots, themeaning claimat thatleast "violations86% of unrestrictedballots domainwill arenever commonemerge in low-dimensionalany simulations"voter alsoor needscandidate citationdistribution.}} CertainElections votinginvolving methodsmore withthan arbitrary101 ballotunique restrictionsballots mayare beimpossible fundamentallyin unablesuch toa capturescenario, and these correspond to the informationvast availablemajority inof anpossible electorateelections. See [[space of possible elections]] and [[limitations of spatial models of voting]] for more details.