Editing Condorcet paradox (section)

=== Spatial model ===
A study of three-candidate elections analyzed 12 different models of voter behavior, and found the [[spatial model of voting]] to be the most accurate to real-world [[Ranked voting|ranked-ballot]] election data.  Analyzing this spatial model, they found the likelihood of a cycle to decrease to zero as the number of voters increases, with likelihoods of 5% for 100 voters, 0.5% for 1000 voters, and 0.06% for 10,000 voters.<ref name=":3">{{Citation|last1=Tideman|first1=T. Nicolaus|title=Modeling the Outcomes of Vote-Casting in Actual Elections|date=2012|url=http://link.springer.com/10.1007/978-3-642-20441-8_9|work=Electoral Systems|editor-last=Felsenthal|editor-first=Dan S.|at=Table 9.6 Shares of strict pairwise majority rule winners (SPMRWs) in observed and simulated elections|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-20441-8_9|isbn=978-3-642-20440-1|quote=Mean number of voters: 1000 … Spatial model: 99.47% [0.5% cycle likelihood] … 716.4 [ERS data] … Observed elections: 99.32% … 1,566.7 [ANES data] … 99.56%|access-date=2021-11-12|last2=Plassmann|first2=Florenz|editor2-last=Machover|editor2-first=Moshé}}</ref>

Another spatial model found likelihoods of 2% or less in all simulations of 201 voters and 5 candidates, whether two or four-dimensional, with or without correlation between dimensions, and with two different dispersions of candidates.<ref name=":4" />{{Rp|page=31|location=|quote=% Condorcet winners 99+ 99 99+ 99+ 98 98 98 99}}