Voting Questions that Might Interest

Computing Specialists

Nicolaus TidemanVirginia Tech

Things to Explore

• Frequency of cycles (a function of the number of candidates and the number of voters)

• Susceptibility to strategic voting• Frequency with which the choice of a voting

rule makes a difference in the outcome

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Such analyses require- either lots of voting data - or data simulated with a statistical model of the

process that generates voting outcomes.

A Spatial Model of Elections

• Candidates have known locations in “attribute space.”

• Every voter has spherical preferences in attribute space (or indifference contours are concentric hyper-ellipses of the same shape and orientation)

• Voters’ ideal points are drawn independently from a multivariate normal distribution in attribute space.

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How many rankings can be observed?

• If there are n candidates, then all rankings can be observed if and only if the dimensionality of the space of ideal points is greater than or equal to n – 1.

• For the number of rankings that can be observed in other cases, see Good and Tideman, “Sterling Numbers and a Geometric Structure from Voting Theory” Journal of Combinatorial Theory 23 (July 1977), 34 45.‑

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Test of the Spatial Model• Use ranking data from three-candidate

elections(1) 913 “elections” compiled from the

American National Election Survey (ANES)

(2) 883 three-candidate comparisons from elections

counted by the UK Electoral Reform Society (ERS)(3) 82,754 “elections” compiled from the

German Politbarometer (1977 – 2008)5

• Five degrees of freedom per election in the data, four degrees of freedom in the model.

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Prediction of the Spatial Model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

pCAB = 0.1, pCBA = 0.1

pBCA = 0.1

pBCA = 0.6

…

pABC

pACB

(scaled)

(scaled)

Along each curve:Vary pBAC and find the combination of pABC and pACB that satisfy the spatial model.

Results – Distance from model(.061 for random points)

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-0.04 0.01 0.06

-0.005

0.000

0.005

0.010

0.015

0.020

ANES avg 500 ANES adj 500 ERS avg 500 ERS adj 500

1/sqrt(N)

55%60%65%70%75%80%85%90%95%

100%

1 2 3 4 5 6

Shar

e of

PM

RW

log(even Number of voters)

Shares of Strict and Weak Pairwise Majority Rule Winners (PMRW)

IAC: Strict PMRW IAC: Weak PMRWSpatial Model: Strict PMRW Spatial Model: Weak PMRWERS data: Strict PMRW ANES data: Strict PMRWGehrlein (TD, 2002)

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Simulations: The Frequencies of Cycles

What might account for excess distance?

• Lack of agreement about locations?• Non-normality of ideal points?• Non-independence of probabilities?

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Þ All invite using the Pólya (Dirichlet-multinomial) distribution

Things to do

• Get more data—particularly on elections• Replicate our results with additional data• Estimate parameters for simulating elections

Challenge: Can you simulate elections so well that I can’t tell the difference between real data and simulated data?

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More Things to Do

• Improve the speed of spatial model estimation(help us develop a new program for Gaussian quadrature)

• Extend the model to four candidates in three dimensions (quadrature in three dimensions)(co-author and programmer for the effort: Florenz Plassmann, fplass@binghamton.edu)

Would you like to help?

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Quadrature in Three Dimensions

Slice the tri-variate normal distribution into• 24 central triangular cones• 36 central planar wedges• 36 (?) triangular columns• x tetrahedronsPut them together in the right combinationsWould you like to help?

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Conjecture• If one wants to know the integrals (of any

point-symmetric function) over each of the n! cones into which an (n-1) space is divided by the bisecting hyperplanes of the line segments connecting n points, then it is sufficient to compute the (n-1)! integrals in (n-1)-space associated with the cones where one of the points is closest or farthest, supplemented by various integrals in (n-2)-space or lower.

Would you like to prove it?13

Another Line of Inquiry: STV

• Counting votes by STV– Compute the quota: n/(k+1)– Allocate each vote to the candidate ranked first– Distribute surpluses– Delete the candidate with the fewest votes– Start again

• Problems of sequential elimination• Alternative of global comparisons: CPO-STV

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The Computational Problem of CPO-STV• Counting votes for a CPO-STV comparison

– Compute the quota: n/(k+1)– Allocate each vote to the candidate ranked first– Distribute surpluses – Stop.– Evaluate the matrix of paired comparisons of

outcomes by your favorite matrix evaluation rule • There is a potential problem of a

combinatorial explosion. How serious is it?• What approximations are interesting?Would you like to explore it?

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Are you interested?• Florenz and I will provide our data and

programs to anyone who is interested• If you would like to collaborate, get in touch.• If you would like to buy my book, Collective

Decisions and Voting (Ashgate, 2006) use promotion code S1DTS20 at the UK Ashgate Web site during the next three months, and you should get it for 20% off (£52 rather than £65).

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Thank you!

Florenz PlassmannBinghamton University

fplass@binghamton.edu

Nicolaus TidemanVirginia Tech

ntideman@vt.edu

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