# Tideman's Election Model

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.

### Incomplete beta function

For α≥1 and β≥1 let

Ix(α,β)   =   Γ(α+β)/[Γ(α)Γ(β)] · ∫0<u<xuα-1(1-u)β-1du

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. Ix(α,β) is the probability that a Beta(α,β)-distributed random variable u obeys 0≤u<x; note

I1(α,β)=1,       I0(α,β)=0,     and       Ix(α,β)+I1-x(β,α)=1.

Letting η=α+β, the expected value of u is r=α/η and the variance of u is (1-r)r/(η+1).

### The model

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:

x=(j-i)/(C+1),   y=1-(j+i)/(C+1),   r=(1/2)exp([a1 + a2x + a3y2]x),   η=[(1-2r)r/2]a4,   α=rη,   β=η-α

where Tideman's preferred parameter values (designed to best-fit data from 87 real-world ranked-ballot elections) are

a1 = -0.532±0.028 ,   a2 = -0.789±0.055,   a3 = -2.486±0.010,   a4 = -1.281±0.011.

(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.)

### Pairwise defeat-probability matrix D

From this it follows (in the V→∞ limit) that the probability Dij that candidate i will defeat candidate j pairwise by voter-majority, is the same as the probability that 0≤u<1/2, which is

Dij   =   I1/2ij, βij).

### Warning

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→∞.

### Numerical calculations

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

Prob(undefeated winner exists) = 2∑A=1..CB=1..C DAB.

I wrote a (rather slow, but apparently accurate to 8 significant digits) computer program to compute the Dij and then the sum-of-products above.

The table below gives my calculations of

1. The probability (based on Tideman's central estimates of a1,a2,a3,a4) that a Condorcet winner exists in C-candidate elections
2. The same probability, but now written as mean±stddev, based on Tideman's error-bar-equipped estimates of those same fitting parameters, and under the assumption these 4 errors are independent.
CProb(Cond.Winner exists)With error bar
111.0000000000±0.0000000000
21.0000000001.0000000000±0.0000000000
30.98829008220.9881513238±0.001994394165
40.97489590710.9746722419±0.003549113867
50.96485836180.9645768206±0.004620496654
60.95681017980.9564885975±0.005407542225
70.94962984940.9492788131±0.006047017500
80.94269843000.9423242788±0.006614385888
90.93571011890.9353168500±0.007147770799
100.92852377420.9281141056±0.007665731809
110.9210820434
120.9133700949
130.9053945858
140.8971727514
150.88872663520.8882590394±0.01019604600
160.8800799963
170.8712566469
180.8622795708
190.8531704841
200.84394963720.8434511869±0.01262449172
210.8346357628
220.8252460951
230.8157964365
240.8063012426
250.79677371720.7962656662±0.01484845837
260.7872259066
270.7776687941
280.7681123896
290.7585658089
300.74903735500.7485375062±0.01680625596
400.65611621560.6556738400±0.01985794091
500.57013472820.5697865061±0.02181859904
600.49278333490.4925483674±0.02285077960
700.42431608990.4242004911±0.02313931045
800.36433427430.3643351016±0.02285682763
900.31215371380.3122648049±0.02215455082
1000.26702044590.2680084049±0.02182221980
150 0.1262199699±0.01483359052
2000.055891047420.05635966735±0.008921947048
3000.01485059262

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%.