The April 2008 Notices of the American Mathematical Society has an article on voting by Don Saari. Mathematics and voting is the theme for this year’s Mathematics Awareness Month. It’s well worth a look as an introduction to some questions in the area. As usual, Saari’s strong (some might say bombastic) personality shines through.
Michael Trick has a pointer to a piece at TierneyLab about an apparent error committed throughout the psychological literature. An example: monkeys are given a choice between red and blue M&Ms. The ones who chose red over blue are then given a choice between blue and green, and about 2/3 of them chose green.
Under the assumption that the monkeys like each colour equally, the conclusion was drawn that their first choice influences their second in an irrational manner (“blue was bad, so still is”). However, under the assumption that each monkey has a preference order for the colours, the observed data would be exactly as expected.
This is not the time to get started on a discussion of my prejudices about mathematics abuse and woolly thinking in the “soft” sciences and humanities. The link above should give lots of interesting reading.
It is always good to remember the difference between mathematical models and reality. A coauthor and I recently had a paper on voting rejected by an economics journal because the assumptions we made were considered to be too unrealistic. Then a friend sent me a link to a fascinating New York Times magazine article that discusses the difficulties of automated vote counting. It seems that in practice (in the USA at least) the error rates of electronic voting machines are enormous, the source code is not available, the election workers are not properly trained, etc.
New Zealand still uses paper ballots, although I am not sure how they are counted. For large electorates such as in the USA, some form of automation is probably necessary. Given the inclusive nature of the voting process, and the multitude of errors that voters can make, it is obviously quite difficult to come up with a voting process that is simple enough for voters and yet amenable to quick and accurate tallying, and the possibility of recounts.
A salutary reminder of the need to balance theory and practice!
Geoffrey Pritchard and Mark C. Wilson, submitted to Journal of Economic Theory, February 2007 (20 pages).
Abstract: We consider the problem of manipulation of elections using positional voting rules under Impartial Culture voter behaviour. We consider both the logical possibility of coalitional manipulation, and the number of voters that must be recruited to form a manipulating coalition. It is shown that the manipulation problem may be well approximated by a very simple linear program in two variables. This permits a comparative analysis of the asymptotic (large-population) manipulability of the various rules. It is seen that the manipulation resistance of positional rules with 5 or 6 (or more) candidates is quite different from the more commonly analyzed 3- and 4-candidate cases.
Mark C. Wilson and Geoffrey Pritchard, Mathematical Social Sciences 54 (2007), 244-256.
Abstract: We show how powerful algorithms recently developed for counting lattice points and computing volumes of convex polyhedra can be used to compute probabilities of a wide variety of events of interest in social choice theory. Several illustrative examples are given.
Geoffrey Pritchard and Mark C. Wilson, Social Choice and Welfare 29 (2007), 487-513.
We consider 3-candidate elections under a general scoring rule and derive precise conditions for a given voting situation to be strategically manipulable by a given coalition of voters. We present an algorithm that makes use of these conditions to compute the minimum size M of a manipulating coalition for a given voting situation.
The algorithm works for any voter preference model — here we present numerical results for IC and for IAC, for a selection of scoring rules, and for numbers of voters up to 150. A full description of the distribution of M is obtained, generalizing all previous work on the topic.
The results obtained show interesting phenomena and suggest several conjectures. In particular we see that rules “between plurality and Borda” behave very differently from those “between Borda and antiplurality”.