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Social Choice and Welfare

Erschienen in:

01.10.2014

A comparison of theoretical and empirical evaluations of the Borda Compromise

verfasst von: William V. Gehrlein, Florenz Plassmann

Erschienen in: Social Choice and Welfare | Ausgabe 3/2014

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Abstract

The Borda Compromise states that, if one has to choose among five popular voting rules that are not Condorcet consistent, one should always give preference to the Borda rule over the four other rules. We assess the theoretical as well as the empirical support for the Borda Compromise. We find that, despite considerable differences between the properties of the theoretical framework and the characteristics of two sets of observed ranking data, all three analyses provide considerable support for the Borda Compromise.

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See, for example, Chamberlin and Featherston (1986), Regenwetter et al. (2006) and Tideman and Plassmann (2012)

We describe the five voting rules in Sect. 3.

Gehrlein (2006) and Gehrlein and Lepelley (2010) survey the extensive theoretical and empirical literature on Condorcet’s paradox and the Condorcet efficiency of different rules.

Plassmann and Tideman (2014) estimate, for 14 voting rules, the frequencies of occurrence of 12 voting paradoxes and of ties. Gehrlein and Lepelley (2010) derive theoretical probability representations for combinations of some of these voting paradoxes and some of these voting rules.

For example, Felsenthal (2012) shows that, while the Plurality rule, the Negative Plurality rule, and the Borda rule do not always elect a Condorcet winner if one exists, these rules are not vulnerable to the reinforcement paradox (which occurs when the same candidate wins in two separate electoral districts if the ballots are evaluated separately but the candidate does not win if the ballots are evaluated jointly). Felsenthal (2012) also shows that eight popular rules that always elect a Condorcet winner if one exists are nevertheless vulnerable to the reinforcement paradox.

The Impartial Anonymous Culture condition states that all possible voting situations (regardless of the value of \(k\)) are equally likely to be observed.

This is mathematically equivalent to asking each voter to submit one vote for the candidate he likes least, in which case the candidate with the fewest votes wins. Our description of the voting rule accounts for the possibility that voters feel more comfortable to vote for two candidates instead of against one candidate. Effectively, neither PR nor NPR require voters to report complete preference rankings of the candidates.

See, for example, Dodgson (1884, pp. 29–30) and Black (1958, p. 182) for concerns about obtaining complete rankings from voters. In an election with three candidates, the procedure for PER elects the same candidate as the Alternative Vote. However, the procedure for NPER does not always elect the same candidate as the Coombs rule. The Coombs rule declares a candidate as winner if this candidate is ranked first by a majority of the voters, and it eliminates candidates only if no candidate receives a majority of first-rank votes. But if such a majority-rule winner is also ranked last by the most voters, then NPER eliminates this candidate and determines the winner among the two remaining candidates.

For example, the French National Assembly uses two-stage elections for situations in which no candidate receives a majority of the votes in the first stage. In such a case, each member of the electorate is asked to cast a vote, in the second stage, for one of the candidates who received at least 12.5 % of the votes. The winner is the candidate who receives a plurality of the votes in the second round. Concerns about two-stage voting rules are raised in Condorcet (1994, pp. 174–175).

See Gehrlein and Lepelley (2012).

The appendix is available at “https://www.researchgate.net/publication/259755770_Appendix_to_A_Comparison_of_Theoretical_and_Empirical_Evaluations_of_the_Borda_Compromise?” Unlike in the two other data sets that contain thermometer scores, many voters ranked only a subset of the candidates in many of the ERS elections. Hence the ERS data have somewhat different characteristics than the two primary data sets that we analyze in this paper.

There are ways of accommodating equal ratings—for example, one can resolve ties randomly or record equal fractions of one vote for each of the tied ratings. Although we found that accommodating tied ratings in these ways made no notable difference to our results, we discarded tied responses nevertheless to avoid any suspicion that the tied ratings may have driven our results.

Determining the Condorcet efficiency for each of the five voting rules requires that we resolve any ties among the candidates consistently across rules. We achieve this by randomly choosing, for each voting situation, one of the six strict rankings as tie-breaking ranking. Whenever we encounter a tie between two or more candidates during the evaluation of a voting rule for this voting situation, we resolve the tie in favor of the candidate ranked higher (highest in case of a three-way tie) in this ranking.

We use the subscript N to distinguish \(f_\mathbf{N}\) and \(\pi _\mathbf{N} \) from other density functions and parameters vectors that we define below. Thus the subscript N is a label, and it does not denote a variable.

The voting situation of a two-candidate election consists of a vector r of length 2, and the associated density function \(f_\mathbf{P} (\mathbf{p},\pi _\mathbf{P} )\) is defined on the unit 1-simplex—a line from (0,1) to (1,0). In an election with \(n\) candidates, the density function \(f_\mathbf{P} (\mathbf{p}, \pi _\mathbf{P} )\) is defined on the unit (\(n! - 1\))-simplex.

The parameters that we use to simulate the PB voting situations imply that \(f_\mathbf{N} (\mathbf{r},\pi _\mathbf{N} )\) is the density function of the multinomial distribution, for which the limiting frequency of voting cycles equals zero. In contrast, the parameters of the ANES model imply that \(f_\mathbf{N} (\mathbf{r},\pi _\mathbf{N} )\) is the density function of the multinomial-Dirichlet distribution, which has a positive limiting frequency of cycles (see Tideman and Plassmann 2013). Thus it is not surprising that we observe a positive number of cycles in the simulated voting situations that correspond to the ANES voting situations and no cycles in the simulated voting situations that correspond to the PB voting situations.

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

Chamberlin JR, Featherston F (1986) Selecting a voting system. J Politics 48:347–369CrossRef

de Condorcet M (1994) On the form of elections (1789). In: McLean I, Hewitt F (eds) Condorcet: foundations of social choice and political theory. Edward Elgar Press, Hants, pp 169–189

Daunou PCF (1991) A paper on elections by ballot (1803). In: Sommerlad F, McLean I (eds) The Political Theory of Condorcet II, University of Oxford Working Paper, pp 235–279

Dodgson CL (1884) The principles of parliamentary representation. Harrison and Sons Publishers, London

Felsenthal DS (2012) Review of paradoxes afflicting procedures for electing a single candidate. In: Felsenthal D, Machover M (eds) Electoral systems: paradoxes, assumptions, and procedures. Springer, Berlin, pp 19–91CrossRef

Felsenthal DS, Machover M (1992) After two centuries, should Condorcet’s voting procedure be implemented? Behav Sci 37:250–274CrossRef

Felsenthal DS, Machover M (1995) Who ought to be elected and who is actually elected? An empirical investigation of 92 elections under three procedures. Elect Stud 14:145–169CrossRef

Fishburn PC (1974) Paradoxes of voting. Am Polit Sci Rev 68:537–546

Gehrlein WV (2006) Condorcet’s paradox. Springer, Berlin

Gehrlein WV (2011) Strong measures of group coherence and the probability that a pairwise majority rule exists. Qual Quant 45:365–374CrossRef

Gehrlein WV, Lepelley D (2010) Voting paradoxes and group coherence: the Condorcet efficiency of voting rules. Springer, Berlin

Gehrlein WV, Lepelley D (2012) The value of research based on simple assumptions about voters’ preferences. In: Felsenthal D, Machover M (eds) Electoral systems: paradoxes, assumptions, and procedures. Springer, Berlin, pp 173–200CrossRef

Plassmann F, Tideman TN (2011) How to predict the frequency of voting events in actual elections. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1911286. Mimeo

Plassmann F, Tideman TN (2014) How frequently do different voting rules encounter voting paradoxes in three-candidate elections? Soc Choice Welf 42:31–75

Regenwetter M, Grofman B, Marley A, Tsetlin I (2006) Behavioral social choice. Cambridge University Press, Cambridge

Saari DG (2001) Decisions and elections: explaining the unexpected. Cambridge University Press, Cambridge

Tideman TN, Plassmann F (2012) Modeling the outcomes of vote-casting in actual elections. In: Felsenthal D, Machover M (eds) Electoral systems: paradoxes, assumptions, and procedures. Springer, Berlin, pp 217–251CrossRef

Tideman TN, Plassmann F (2013) Developing the aggregate empirical side of computational social choice. Ann Math Artif Intell 68:31–64

Titel: A comparison of theoretical and empirical evaluations of the Borda Compromise
verfasst von: William V. Gehrlein
Florenz Plassmann
Publikationsdatum: 01.10.2014
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 3/2014
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-014-0798-7

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