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Public Choice

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01.03.2014

The Alternative Vote and Coombs Rule versus First-Past-the-Post: a social choice analysis of simulated data based on English elections, 1992–2010

verfasst von: Nicholas R. Miller

Erschienen in: Public Choice | Ausgabe 3-4/2014

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Abstract

This paper presents a social choice analysis, using simulated data based on English general elections from 1992 through 2010, of the properties of three voting rules: First-Past-the-Post, the Alternative Vote, and the Coombs Rule. More specifically, the paper examines (1) the plurality, anti-plurality, and Condorcet status of candidates in each election and the interrelationships among these statuses, (2) the effects of strict and partial single-peakedness of voter preferences, and (3) the identity of winners, Condorcet efficiency, and the relationship between votes and seats under the three voting rules. The analysis considers only the case of three candidates and, in the manner of basic social choice theory, the set of candidates and voter preferences over them are taken to be fixed.

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This system was proposed by Coombs (1964: 397–399) as “an alternative to the Hare [AV] system” in a side discussion of “majority decision from the point of view of unfolding theory” in what was primarily a treatise on the analysis of psychological data.

Thus Coombs, even more than AV, is vulnerable to what Australians call “donkey voting”—that is, casting ballots that rank candidates (beyond the first or several highest preferences) in an arbitrary order not reflecting preference, typically the order in which they are listed on the ballot.

Moreover, so far as I know, there is no standard way to break ties under any of the three systems. Allowing for ties would modify (and complicate) several of the definitions and propositions in the next section.

Aldrich et al. examine four alternatives to FPTP (Condorcet, AV, Coombs, and Borda) and conclude that the Liberals would have won such a hypothetical national vote in 2010 under all four systems except the one they actually advocated.

Under Anti-Plurality Voting, each voter casts a “negative” vote against one candidate, and the candidate with the fewest “negative” votes wins.

Given more than three candidates, preferences are single-peaked if this condition holds for every triple of candidates (Sen 1970: 167–168).

The numbers at the head of each column indicates the number of voters with that ranking. I am indebted to Dan Felsenthal for this example.

Point (b) generalizes to any number of candidates (Coombs 1964: 399; Grofman and Feld 2004: 650).

This data comes from Pippa Norris’s Shared Datasets website (http://www.hks.harvard.edu/fs/pnorris/Data/Data.htm). I am extremely grateful to Professor Norris for making this valuable data readily available.

In 2010 these include the Speaker’s constituency (since by tradition the Speaker is not opposed by major-party candidates) and one constituency won by a fourth-party (Green) candidate.

For further details pertaining to this and the other datasets, see the Appendix.

However, at the “New Labour” high tide of 1997 and 2001, Conservative second preferences for Labour exceeded Liberal second preferences for Conservatives.

In 2010, the Conservatives won a majority of seats in England, though not nationally.

Remember that there are no ties, so when (for example) the first entry that says that Labour candidates beat Liberal candidates in hypothetical straight fights in 33.6 % of the constituencies in 1992, Liberal candidates beat Labour candidates in the other 66.4 %.

Even with about 40,000 elections, ties occur in about 0.4 % of the wholly impartial profiles. The percentages reported in the following tables exclude such profiles.

In contrast, we see in the second row of Table 7 that discrepant combinations do appear with some frequency in wholly impartial profiles. This is because almost all such profiles are virtually tied with respect to first preferences (and therefore plurality status) as well as second and third preferences (and therefore anti-plurality status). The same consideration means that cycles occur in about 8.3 % of the wholly impartial profiles, though they are absent from the profiles with impartial second preferences only.

Given wholly impartial profiles, Plurality Winners beat Plurality Runners-Up, and Plurality runners-Up beat Plurality Losers, about 76 % of the time, while Plurality Winners beat Plurality Losers about 90 % of the time. These percentages (which, being based on simulations, are approximations) are fixed and intrinsic characteristics of an “impartial culture” (that, to the best of my knowledge, have never before been reported), comparable to the (likewise approximate) 8.8 % of impartial culture profiles with three alternatives that produce Condorcet cycles (Sen 1970: 164) and the 20.4 % of impartial culture profiles with two candidates that produce “election inversions” given a sufficiently large number of uniform districts (Feix et al. 2004).

The more detailed analysis of Sanders et al. (2011) estimates the 2010 distribution English seats under AV as Conservatives 277, Labour 184, and Liberals 69. The greater success of the Conservatives, and lesser success of Labour and Liberals, in these estimates probably reflects the role of the UK Independence Party. The BES survey data reported in their Table 3 show that most UKIP supporters in England indicated a Conservative second preference, so many of their votes would transfer to Conservatives under AV.

The percentages reported in Table 16 exclude ballot profiles that produce Condorcet cycles (so there is no Condorcet Winner), as well as profiles that entail ties in Condorcet relationships (found only in the wholly impartial profiles).

It should be noted that, in the bottom panel showing averages over all elections, the Gallagher Index values are indeed the averages of index values over the five elections, not the (smaller) index values that result when the formula is applied to the average deviations in that panel. (An electoral system that gives highly disproportional results in every individual election might favor different parties in different elections so that, when the index formula is applied to average vote and seat percentages, it appears to be quite proportional.)

The principal problem was identified in footnote 2.

This conjecture appears to be supported by the survey data reported in Table 2 of Sanders et al. (2011).

Undoubtedly, second preferences do differ considerably across individual constituencies, but no reliable survey data are available at the constituency level. For a possible way around this problem, see Herrmann et al. (2012).

Aldrich, J., Houck, A., Abramson, P. R., Levine, R., Scotto, T. J., & Diskin, A. (2011). Strategic voting in the 2010 U.K. election. Paper presented at the annual meeting of the American political science association, Seattle, September 1–4, 2011.

Black, D. (1948). On the rationale of group decision-making. Journal of Political Economy, 56, 23–34. CrossRef

Chamberlin, J. R., & Cohen, M. D. (1978). Toward applicable social choice theory: a comparison of social choice functions under spatial model assumptions. The American Political Science Review, 72, 1341–1356. CrossRef

Coombs, C. H. (1964). A theory of data. New York: Wiley.

Curtice, J. (2009). Recent history of second preferences. http://news.bbc.co.uk/nol/shared/spl/hi/uk_politics/10/alternative_vote/alternative_vote_june_09_notes.pdf.

Feix, M. R., Lepelley, D., Merlin, V. R., & Rouet, J.-L. (2004). The probability of conflicts in a U.S. presidential type election. Economic Theory, 23, 227–257. CrossRef

Fishburn, P. C., & Gehrlein, W. V. (1976). An analysis of simple two-stage voting systems. Behavioral Science, 21, 1–12. CrossRef

Fishburn, P. C., & Gehrlein, W. V. (1977). An analysis of voting procedures with nonranked voting. Behavioral Science, 22, 178–185. CrossRef

Gallagher, M. (1991). Proportionality, disproportionality, and electoral systems. Electoral Studies, 10, 33–51. CrossRef

Grofman, B., & Feld, S. (2004). If you like the alternative vote (a.k.a. the instant runoff), then you ought to know about the Coombs Rule. Electoral Studies, 23, 641–659. CrossRef

Herrmann, M., Munzert, D., & Selb, P. (2012). How strategic votes matter. Social Science Research Network #2127621.

Merrill, S. III (1988). Making multicandidate elections more democratic. Princeton: Princeton University Press.

Sanders, D., Clarke, H. D., Stewart, M. C., & Whiteley, P. (2011). Simulating the effects of the alternative vote in the UK general election. Parliamentary Affairs, 64, 5–23. CrossRef

Sen, A. K. (1970). Collective choice and social welfare. San Francisco: Holden Day.

Titel: The Alternative Vote and Coombs Rule versus First-Past-the-Post: a social choice analysis of simulated data based on English elections, 1992–2010
verfasst von: Nicholas R. Miller
Publikationsdatum: 01.03.2014
Verlag: Springer US
Erschienen in: Public Choice / Ausgabe 3-4/2014
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-013-0067-9

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