Nicolaus Tideman used data from 87 real-world rank-order-ballot elections to fit a statistical model of elections. It is described in chapter 9 of his book Collective decisions and voting (Ashgate 2006) and we will describe it again here. Our description is more concise and we shall also redo some of Tideman's calculations.
For α≥1 and β≥1 let
denote the "normalized incomplete beta function." Many properties of this function are described in §26.5 (pages 944-945 of my copy) of M.Abramowitz & I.A.Stegun: Handbook of Mathematical Functions, NBS (10th corrected printing) 1972. For example, it gives series and continued fraction expansions. And here is an alleged calculator. I_{x}(α,β) is the probability that a Beta(α,β)-distributed random variable u obeys 0≤u<x; note
Letting η=α+β, the expected value of u is r=α/η and the variance of u is (1-r)r/(η+1).
There are C candidates numbered 1,2,...,C. There also are V voters, V→∞, who each cast rank-order ballots. We shall mostly be interested in the V→∞ limit. If 1≤i<j≤C then Tideman's model is that a random voter ranks candidate i below candidate j with probability u which is a "beta-distributed random variable" on the real interval (0,1) where the beta distribution is defined by parameters α and β calculable from i and j with 1≤i<j≤C by this procedure:
where Tideman's preferred parameter values (designed to best-fit data from 87 real-world ranked-ballot elections) are
(We have also given ±1σ error bars on Tideman's fit-values.) Note that if i<j then this causes r<1/2 and hence α<β and hence i is more likely to defeat j than the reverse. (If i>j then swap i and j, compute α and β as above, and finally swap them.)
From this it follows (in the V→∞ limit) that the probability D_{ij} that candidate i will defeat candidate j pairwise by voter-majority, is the same as the probability that 0≤u<1/2, which is
Tideman's model does not easily allow you to generate a single random rank-order ballot!
But it does allow you to easily generate a random C×C pairwise-margins matrix (sampled from his distribution) arising from V voters in the limit V→∞.
Tideman, in the bottom row of table 9.11 page 118 of his book Collective decisions and voting (Ashgate 2006) calculates his best estimate of the real-world probability of a Condorcet winner existing in a C-candidate, (V→∞)-voter election. Unfortunately, when I calculated these numbers myself, I got slightly different results than Tideman. (Also Tideman himself emailed me some numbers extending the tables in his book up to C=320, and these tables also disagree slightly both with those in his book and with my own computations.) The exact probability, in this model, that a "Condorcet winner" (who defeats every rival pairwise by voter majority) exists is
I wrote a (rather slow, but apparently accurate to 8 significant digits) computer program to compute the D_{ij} and then the sum-of-products above.
The table below gives my calculations of
C | Prob(Cond.Winner exists) | With error bar |
---|---|---|
1 | 1 | 1.0000000000±0.0000000000 |
2 | 1.000000000 | 1.0000000000±0.0000000000 |
3 | 0.9882900822 | 0.9881513238±0.001994394165 |
4 | 0.9748959071 | 0.9746722419±0.003549113867 |
5 | 0.9648583618 | 0.9645768206±0.004620496654 |
6 | 0.9568101798 | 0.9564885975±0.005407542225 |
7 | 0.9496298494 | 0.9492788131±0.006047017500 |
8 | 0.9426984300 | 0.9423242788±0.006614385888 |
9 | 0.9357101189 | 0.9353168500±0.007147770799 |
10 | 0.9285237742 | 0.9281141056±0.007665731809 |
11 | 0.9210820434 | |
12 | 0.9133700949 | |
13 | 0.9053945858 | |
14 | 0.8971727514 | |
15 | 0.8887266352 | 0.8882590394±0.01019604600 |
16 | 0.8800799963 | |
17 | 0.8712566469 | |
18 | 0.8622795708 | |
19 | 0.8531704841 | |
20 | 0.8439496372 | 0.8434511869±0.01262449172 |
21 | 0.8346357628 | |
22 | 0.8252460951 | |
23 | 0.8157964365 | |
24 | 0.8063012426 | |
25 | 0.7967737172 | 0.7962656662±0.01484845837 |
26 | 0.7872259066 | |
27 | 0.7776687941 | |
28 | 0.7681123896 | |
29 | 0.7585658089 | |
30 | 0.7490373550 | 0.7485375062±0.01680625596 |
40 | 0.6561162156 | 0.6556738400±0.01985794091 |
50 | 0.5701347282 | 0.5697865061±0.02181859904 |
60 | 0.4927833349 | 0.4925483674±0.02285077960 |
70 | 0.4243160899 | 0.4242004911±0.02313931045 |
80 | 0.3643342743 | 0.3643351016±0.02285682763 |
90 | 0.3121537138 | 0.3122648049±0.02215455082 |
100 | 0.2670204459 | 0.2680084049±0.02182221980 |
150 | 0.1262199699±0.01483359052 | |
200 | 0.05589104742 | 0.05635966735±0.008921947048 |
300 | 0.01485059262 |
I also wrote another computer program which generates random Tideman-statistics elections (in the form of a pairwise margins matrix sampled from the exactly correct distribution) and also does the same for the simpler (but presumably less realistic) random elections model. Using these matrices, the program can carry out Monte Carlo simulations to answer questions about election behavior for Tideman-statistics and random-election-model statistics elections, for any election method that depends only on the pairwise matrix.
Example eligible election methods: Borda, Nanson-Condorcet, Least-Reversal-Condorcet, Copeland-Condorcet. Example election methods that cannot be handled: Instant Runoff, Plurality, Approval, least-median-rank, and Range Voting.
One interesting result found by this simulator is that in Tideman elections with large numbers of candidates, the Smith set tends to be the entire set of candidates. The probability of that approaches 100%.