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Constitutional Political Economy

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Open Access 04.02.2023 | Original Paper

A Dodgson-Hare synthesis

verfasst von: James Green-Armytage

Erschienen in: Constitutional Political Economy | Ausgabe 3/2023

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Abstract

In 1876, Charles Dodgson (better known as Lewis Carroll) proposed a committee election procedure that chooses the Condorcet winner when one exists and otherwise eliminates candidates outside the Smith set, then allows for re-votes until a Condorcet winner emerges. The present paper discusses Dodgson’s work in the context of strategic election behavior and suggests a “Dodgson-Hare” method: a variation on Dodgson’s procedure, for use in public elections. This method allows for candidate withdrawal and employs Hare’s plurality-loser-elimination method to resolve persistent cycles. Given plausible assumptions about how candidates decide whether to withdraw when there is a cycle, Dodgson-Hare outperforms Hare, Condorcet-Hare, and 12 other voting rules in a series of spatial-model simulations that count how often each rule is vulnerable to coalitional manipulation. In the case of a one-dimensional spatial model, all coalitional voting strategies that are possible under Condorcet-Hare can be undone in Dodgson-Hare, by the withdrawal of candidates who have incentive to withdraw.

Vorheriger Artikel Implications of strategic position choices by candidates

Nächster Artikel Choosing among the Variety of proposed Voting Reforms

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For both the text of Dodgson’s pamphlets and their historical context, I rely on Black (1958).

These committees decided things like academic appointments, and at one time the design of a new belfry.

Green-Armytage (2015) described a similar idea (referring to it as a “Dodgsonesque procedure”), but only in brief.

Hoag and Hallett (1926, 162–95) indicate that Hare began advocating this provision (as a refinement of the single transferable vote) in 1865.

For more detail about the other two pamphlets, see Section A2 of the online Appendix to this paper.

Dodgson wrote, “If the issues cannot be all arranged in one cycle, but form a cycle and a set of issues each of which is separately beaten by each of the cycle, it shall be formally moved that this cycle be retained and all other issues struck out, and, if none object, this shall be done.” This definition aligns closely with Smith (1973).

For example, by Nurmi (1983), and by Bartholdi et al. (1989).

This metamorphosis of a Dodgson score into determinative Dodgson election rule occurs in Fishburn (1973), who wrote under the header “Dodgson’s Inversion Method”: “Our final function is based on C. L. Dodgson's idea of taking inversions in the orders in D, and will therefore be referred to as Dodgson's function.” Nurmi (1983) defined “Dodgson’s method” equivalently, and Bartholdi et al. (1989) referred to the same as “the Dodgson voting scheme.”.

Their proofs established that individual voters could sometimes get a preferred outcome by reporting false preferences; this implies in turn that groups of voters can also sometimes get a mutually preferred outcome by reporting false preferences. In large elections, it is unlikely that any individual will have an opportunity to change the result by voting strategically; hence, in selecting a rule for deciding an election with many voters, vulnerability to coalitional strategy (rather than individual strategy) is the more pressing concern.

Here I assume that voters may submit complete rankings, but plurality uses only their first preferences.

Green-Armytage (2014) defined “burying” and “compromising” strategies while attributing both terms to Blake Cretney, as they appeared in essays on his (no longer active) web site Condorcet.org during the early 2000s.

See Duverger (1964), and note that the two dominant parties may vary from district to district within the same nation.

Green-Armytage (2014) provides evidence that plurality is particularly susceptible to both the compromising strategy and to strategic exit: defined as a candidate exiting a race in order to change the outcome to one they prefer. Both of these results suggest that plurality has an especially strong tendency to reduce the number of parties to two.

Here I define Borda, approval, and range voting as the following rules which share the feature that the candidate with the most points wins. In Borda, a first-choice vote is worth N − 1 points (where N is the number of candidates), a second-choice vote is worth N − 2 points, a third-choice vote is worth N − 3 points, and so on. In approval voting, each voter may assign each candidate either one point or zero points. In range voting, each voter may assign each candidate any score within a predetermined closed interval.

Green-Armytage (2014) provides evidence that Borda, approval, and range voting are vulnerable to strategic voting with unusually high frequency, and also that Borda has an unusual vulnerability to strategic entry, defined as a candidate entering a race to change the outcome to one they prefer, without winning. Hence, whereas strategic nomination in plurality tends to reduce the number of candidates, strategic nomination in Borda would most likely increase the number of candidates, leading to unforeseen consequences.

Define runoff as the rule by which voters rank the candidates, the two candidates with the most first-choice votes proceed to a runoff election, and the candidate with the most votes in the runoff election wins. Define Hare as the rule by which voters rank the candidates, and the candidate with the fewest first-choice votes is eliminated (and deleted from the rankings) in each of a series of rounds, until one candidate remains.

Green-Armytage (2014) provides evidence that Hare is rarely vulnerable to strategic voting, but both Hare and runoff are quite vulnerable to strategic exit. The latter result fits with the intuition from Example 4: That is, even if the CBA voters do not compromise in order to elect the Condorcet winner, candidate C (who is losing anyway) could achieve the same result by withdrawing before the race, so that B is elected. Since all of C’s supporters prefer B to A, it is likely that C will as well, in which case C has both the means and the motive to exit the race.

Defining P as the pairwise matrix and thus P_XY as the number of voters who rank candidate X above Y, define minimax as the rule by which the winner is the candidate Y with the smallest maximum value of P_XY.

See Tideman (1987) and Schulze (2003) for the definitions of ranked pairs and beatpath, respectively.

That is, both strategic exit and entry. Intuitively, whenever the voting rule is Condorcet-consistent and there is a sincere Condorcet winner, no candidate will have an incentive to exit or enter strategically.

That is, voting rules that elect the Condorcet winner when one exists, and otherwise use Hare’s method of successively eliminating the plurality loser. Green-Armytage (2011) defines and compares four such rules, which are equivalent in the three-candidate case.

See for example Chamberlin (1985), Lepelley and Valognes (2003), Favardin and Lepelley (2006), Tideman (2006), Green-Armytage (2014), and Green-Armytage et al. (2016).

Define Smith-Hare as a rule that eliminates all candidates outside the Smith set, then conducts a Hare tally.

In the latter case, if candidates’ decisions to withdraw depend on whether doing so produces a winner who is ideologically closer to them, more “central” candidates (relative to the other candidates) are more likely to win. For example, in a spatial model with a three-candidate cycle among A, B, and C, if C is more distant from both A and B than they are from each other, then candidate C will not win Dodgson-Hare.

For proofs of all formal propositions in this paper, see Section A1 of the online Appendix.

For example (supposing for simplicity that candidates are not also voters), place candidate A and 45 voters at (0,0) in a two-dimensional space, B and 10 voters at (0,1), C at (2,0), 10 voters at (1.125,1), and 35 voters at (2,1).

Section A3 of the online Appendix provides descriptions of the algorithms used to determine whether coalitional manipulation is possible.

Bartholdi, J., Tovey, C. A., & Trick, M. A. (1989). Voting schemes for which it can be difficult to tell who won the election. Social Choice and Welfare, 6(2), 157–165.CrossRef

Black, D. (1958). The theory of committees and elections. The University Press.

Chamberlin, J. (1985). An investigation into the relative manipulability of four voting systems. Behavioral Science, 30(4), 195–203.CrossRef

Durand, F., Mathieu, F., & Noirie, L. (2016). Can a Condorcet rule have a low coalitional manipulability? European Conference on Artificial Intelligence, 285, 707–715.

Favardin, P., & Lepelley, D. (2006). Some further results on the manipulability of social choice rules. Social Choice and Welfare, 26, 485–509.CrossRef

Fishburn, P. (1973). The theory of social choice. Princeton University Press.

Gibbard, A. (1973). Manipulation of voting schemes: A general result. Econometrica, 41, 587–601.CrossRef

Green-Armytage, J. (2011). Four Condorcet-Hare hybrid methods for single-winner elections. Voting Matters, 29, 1–14.

Green-Armytage, J. (2014). Strategic voting and nomination. Social Choice and Welfare, 42(1), 111–138.CrossRef

Green-Armytage, J. (2015). Direct voting and proxy voting. Constitutional Political Economy, 26(2), 190–220.CrossRef

Green-Armytage, J., Tideman, T. N., & Cosman, R. (2016). Statistical evaluation of voting rules. Social Choice and Welfare, 46(1), 183–212.CrossRef

Hoag, C., & Hallett, G. (1926). Proportional representation. MacMillan.

Lepelley, D., & Valognes, F. (2003). Voting rules, manipulability and social homogeneity. Public Choice, 116, 165–184.CrossRef

Nagel, J. (2007). The Burr dilemma in approval voting. Journal of Politics, 69(1), 43–58.CrossRef

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Satterthwaite, M. (1975). Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10, 187–217.CrossRef

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Titel: A Dodgson-Hare synthesis
verfasst von: James Green-Armytage
Publikationsdatum: 04.02.2023
Verlag: Springer US
Erschienen in: Constitutional Political Economy / Ausgabe 3/2023
Print ISSN: 1043-4062
Elektronische ISSN: 1572-9966
DOI: https://doi.org/10.1007/s10602-023-09392-2

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