Abstract
Using computer simulations based on three separate data generating processes, I estimate the fraction of elections in which sincere voting is a core equilibrium given each of eight single-winner voting rules. Additionally, I determine how often each rule is vulnerable to simple voting strategies such as ‘burying’ and ‘compromising’, and how often each rule gives an incentive for non-winning candidates to enter or leave races. I find that Hare is least vulnerable to strategic voting in general, whereas Borda, Coombs, approval, and range are most vulnerable. I find that plurality is most vulnerable to compromising and strategic exit (causing an unusually strong tendency toward two-party systems), and that Borda is most vulnerable to strategic entry. I use analytical proofs to provide further intuition for some of my key results.
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Notes
Specifically, if there are more than two candidates for a single office, and a non-dictatorial election method allows voters to rank the candidates in any order, then there must be some profile of voter preferences under which at least one voter can get a preferred result by voting insincerely. This well-known ‘Gibbard-Satterthwaite theorem’ relies in turn on the even more well-known ‘Arrow theorem’; for this, see Arrow (1951, rev. ed. 1963).
Nitzan (1985), Kim and Roush (1996), Smith (1999), Saari (1990) and Kelly (1993) all use an ‘impartial culture’ (IC) model, while Lepelley and Mbih (1994), Favardin et al. (2002), and Favardin and Lepelley (2006) use an ‘impartial anonymous culture’ (IAC) model. Tideman (2006) uses a data set consisting of 87 elections. On the other hand, Chamberlin (1985) uses both spatial model and an impartial culture model, Pritchard and Wilson use both IC and IAC, and Lepelley and Valognes (2003) use a somewhat more general model that has IC and IAC as two of its special cases.
Osborne and Slivinski (1996) focus on nomination equilibria given the plurality and runoff systems, with costs of entry and benefits of winning. Besley and Coate (1997) focus on nomination and voting equilibria given the plurality system. Moreno and Puy (2005, 2009) focus on positional voting rules, and show that plurality is unique among them in that it will elect the Condorcet winner among declared candidates when there are three or fewer potential candidates, although it will not necessarily do so when there are four or more. Another approach to strategic nomination is given by Tideman (1987), who shows that the addition or removal of ‘clone’ candidates may change the result given some voting rules, but not others.
In a result that is somewhat analogous to the Gibbard-Satterthwaite theorem, Dutta et al. (2001) show that there will in some cases be downward pressure on the number of candidates (in the present paper, I will call this an incentive for ‘strategic exit’), given any ‘unanimous’ and non-dictatorial ranked ballot voting rule. (Note that this doesn’t apply to non-ranked systems like approval or range voting.)
For example, these may be assigned randomly, according to lexicographic or anti-lexicographic order, etc., but in any case they are exogenous, and known by the voters and candidates before the election.
This system is the application to the single-winner case of proportional representation by the single transferable vote, which is often named for Thomas Hare, who introduced the idea of successively eliminating the candidate with the fewest first choice votes. See ((Hoag and Hallett 1926, pp. 162–195))
See Coombs (1964). I note that some versions of Coombs include a provision to automatically elect a candidate who holds a majority of first choice votes, but I don’t use this provision here.
Condorcet (1785) describes the pairwise comparison method and the Condorcet winner. He also observes the possibility of a majority rule cycle emerging despite transitive voter preferences—this is known as the Condorcet paradox.
Black (1958, p. 175), develops the minimax method as a possible interpretation of Condorcet’s intended proposal. Levin and Nalebuff (1995) label this method as the “Simpson-Kramer min-max rule”; presumably this is due to Simpson (1969) and Kramer (1977). (Nurmi (1999), p. 18) refers to it as “Condorcet’s successive reversal procedure”. (Tideman (2006), p. 212) refers to it as “maximin”.
This terminology was used by Blake Cretney, on his currently-defunct web site Condorcet.org.
See Duverger (1964).
To be precise, the strong incentives for compromising and strategic exit make it highly likely that there will only be two ‘viable’ candidates in any particular plurality election. The two parties that these candidates belong to may vary from jurisdiction to jurisdiction, but once two parties have been established as dominant in any given jurisdiction, it is difficult for a third party to mount an effective challenge there. Cox (1997) refers to this as ‘local bipartism’.
This is similar to the model described in Chamberlin and Cohen (1978), but without inter-dimensional covariance.
The impartial culture model is defined, for example, in Nitzan (1985).
Here, I follow Tideman and Plassmann (2011).
This process creates 931 elections with three candidates, 1553 with four candidates, 1898 with five candidates, and 1764 with six candidates.
More detailed descriptions, along with the codes themselves, are available from the author on request.
It is also possible to ask how often groups of candidates may change the result in a way that they all prefer by either simultaneously entering or simultaneously exiting (thus, looking for core equilibria instead of Nash equilibria), but I omit this additional analysis in the interest of brevity, because its results don’t differ in any particularly interesting way from the results of the analysis already described.
A margin of error of \(\pm \).0098 is the upper bound, which applies when the true probability is exactly one half. I further reduce the random error in the difference between the scores that the various voting methods receive by using the same set of randomly generated elections for each method.
Despite the similar name, this is not equivalent to the impartial culture model. Rather, the IAC model supposes that every possible division of the voters among the \(C!\) possible preference orderings is equally likely.
This was also defined by Blake Cretney on Condorcet.org. Note that it is not the same as burying, compromising, or a combination of these, as it involves shifting votes from candidate q to candidate x in order to help candidate q.
See for example Beck (1975), who shows that ties are unlikely in large two-candidate elections; it is not hard to extend this intuition to the case of a pairwise tie within a multi-candidate election.
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Appendices
Appendix I: Tables
Appendix 2: Figures
a V1, \(S=4,V=99\), b V1, I.C., \(V=9\), c V1, ANES
a Compromising, \(S=4,V=99\), b Compromising, I.C., \(V=29\), c Compromising, ANES
a Burying, \(S=4,V=99\), b Burying, I.C., \(V=9\), c Burying, ANES
a strategic exit, \(V=99,S=4\), b strategic entry, \(V=99,S=4,C_{1}=2\)
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Green-Armytage, J. Strategic voting and nomination. Soc Choice Welf 42, 111–138 (2014). https://doi.org/10.1007/s00355-013-0725-3
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DOI: https://doi.org/10.1007/s00355-013-0725-3