How often do condorcet cycles happen? Probably less than 1 election in 1000.
I used two voter models to simulate elections.
Unifom random preference model: Each voter gets a preference [0.0..1.0) for each each choice.
Spatial normal random model: Voters and choices exist in N-dimensional space and are distributed on the same normal distribution. Voters prefer choices based on closeness in space.
Both models produce some sort of close election where any choice has the same probability of wining. I think intuitively the uniform random model is less like reality. Real voters don't have independent variables for their preference for different choices but actually choices are likely correlated: Someone voting for a far-left choice is more likely to have a moderate-left choice as their second choice.
I manually ran a handful of trials this morning and got some first immediate patterns.
The uniform random model resulted in 20-30% of votes going to condorcet cycle resolution.
The spatial model resulted in 1 in 1000 or fewer votes going to condorcet cycle resolution.
Simulated election had 1000 or 10000 voters, 5 or 7 choices, and 2 to 7 dimensions. Dimensionality had no consistent effect on likelihood of cycle.
Next work could be to regularize the search on a few parameters and plot the results, but right now I feel pretty good based on the spatial model that I can say that condorcet cycles should be rare. These are artificially close elections. Normal elections will be even less likely to have a cycle.
Test program source:
https://github.com/brianolson/voteutil/blob/master/go/cyclething/cyclething.go
