The Borda Count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia

Abstract

This article examines strategic elements of voter behaviour in parliamentary elections where the voting method is a scoring rule other than plurality: the Borda Count, which is used for the election of ethnic minorities in Slovenia, and the Dowdall rule, which is used in the Pacific island state of Nauru in multi-seat districts. After first examining the general properties of scoring rules, and generating theoretical differences between the two rules, we look at empirical evidence from Nauru and Slovenia. This casts a doubt on predictions based simply on a voting rule's mathematical properties and on the accuracy of assumptions of sincere rank ordering.

本文研究了议会选举投票者行为的策略因素。议会选举中的投票方法是一个有别于多数法的计分法则：斯洛文尼亚少数族群选举使用的波达计数法；太平洋岛屿国家瑙鲁多席位选区使用的斗多尔法则。作者首先分析了计分法则的一般属性，并对两种法则做了理论区分。然后研究了瑙鲁和斯洛文尼亚的实证资料。作者对仅仅基于投票规则数学属性的预测以及有关纯粹排行假设的精确性表示了质疑。

Keywords:

• electoral systems
• Borda Count
• preferential voting systems
• Nauru
• Slovenia

Notes

¹The country's Secretary for Justice, an Irishman, Desmond Dowdall, devised the system in 1971. The Nauru government in 1971 indicated a preference that its system be described as the ‘Dowdall system’ instead of the previously used term ‘exhaustive ballot system’ (cited in Supreme Court of Nauru 1977).

²A similar method was also used (until 2002) for the parliamentary selection of nominees for presidential elections in another small Pacific Island nation, Kiribati (Reilly 2001a, 2001b, 2002; Reilly and Gratschew 2001; Van Trease 1993).

³The reasons for adopting the Borda Count in Slovenia were connected to its potential for generating election of a more moderate candidate:There are two or three interest groups in each of the two national minorities (Italian, Hungarian). If plurality or majority system were used, these groups would confront each other and one would win over the other. With Borda, usually the winner is the person who is most acceptable to all and who is not an extremist. (Jure Toplak, personal communication, 28 February 2013)

⁴The Condorect winner, also known as the majority winner, is that alternative, if any, which can defeat all others in a paired contest (or, for an even number of voters, it is that alternative which cannot be defeated by any other alternative in a paired contest). When it exists, many authors (Black 1958) regard the Condorcet winner as the normatively desired choice. Black, however, proposed that the Borda winner be chosen when no Condorcet winner existed. Nanson (1882) proposed a complex scheme based on the Borda rule that has the property that it always yields the Condorcet winner when one exists (see McLean and Urken 1995: 57–60; Tangian 2013: 203).

⁵There has been a large body of empirical work measuring strategic voting in real-world single-seat plurality contests in countries such as Canada and the UK, and for some elections, including party primaries, in the USA, but little empirical work has been done measuring strategic voting in multi-seat systems (see, however, Irwin and Van Holsteyn 2012).

⁶For example, if there were three alternatives and three voters, then if the weight vector for the scoring rule were (1, ¾, 0) and Voter 1 ranked the alternatives abc, Voter 2 ranked the alternatives abc, and Voter 3 ranked the alternatives cba, the scores corresponding to each alternative would be a = 2 (2 × 1 + 1 × 0), b = 9/4 (3 × ¾) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of bac. For the same example, if the weights were (1, ½, 0), the scores corresponding to each alternative would be a = 2 (2 × 1 + 1 × 0), b = 1.5 (3 × ½) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of abc. For the same example, if the weights were (1, ¼, 0), the scores corresponding to each alternative would be a = 2 (2 × 1 + 1 × 0), b = ¾ (3 × ¼) and c = 1 (1 × 1 + 2 × 0), which gives us the ranking of acb.

⁷In its original 1971 formulation, the Nauru system used the fractions 1, ½, ⅓, etc., but this was subsequently converted to a decimal basis. For mathematical tractability in deriving exact analytic results, we begin with the fractional representation, and then report results as decimals.

⁸A similar non-uniqueness result is found when we look at weight vectors for weighted voting rules (Felsenthal and Machover 1998).

⁹As noted earlier, there are many alternative ways to assign Borda weights, and Borda himself used a different scheme from the ‘modern’ instantiation of the Borda Count (Emerson 2013).

¹⁰Reilly (2002: 154, 366) and Reilly and Gratschew (2001: 699) report use of the AV in the 26 January 1968 election. They draw on the Nauru Supreme Court's 1977 decision which mentions the 1965 Electoral Act, which provided for use of the AV both in single-member by-elections and, at general elections, in the two- and four-member constituencies (Supreme Court of Nauru 1977). Available electoral data for 1968 records quotas, however, which are clearly STV quotas calculated for each constituency. The records also note that the election was conducted in accordance with Australia's Commonwealth Electoral Act 1918–66. Since independence came on 31 January 1968, that is, four days after the election, we believe that the election was conducted under the Australian law using STV, perhaps because of an awareness that this is a better method of handling elections in constituencies where M>1 than via usage of multi-member AV. Australia had a negative experience with use of multi-member AV during the interwar years (Lijphart 1997). The Dowell system was introduced by way of the Electoral (Electoral System) Regulations 1971, gazetted 22 January. The new system was used for the general elections of 23 January 1971 (Supreme Court of Nauru 1977).

¹¹In 2010, a second election was held in June in an effort to resolve an impasse generated by a 9 versus 9 MP split between government and opposition (Le Roy 2010).

¹²Prior to 2000, the threshold was three parliamentary seats. The system works like a ‘closed list’ system at the level of the district. The parties field single candidates in each of the 88 districts (11 for each of the 8 districts), but voters effectively select a party by choosing a candidate.Mandates are first allocated within each constituency to the eligible candidate lists using the Droop quota. After the first allocation, the overall proportional calculation of the number of mandates each party is entitled to on a nationwide basis is done, using the d'Hondt method. The mandates not already distributed are then allocated to candidate lists among all the constituencies, in the order of the highest remainder of votes in proportion to the quota used in each constituency. Mandates within a list are assigned to specific candidates on the basis of the percentage of the vote each received in his/her district. That is, candidates within a list are ranked on the basis of the percentage of votes they received in comparison to the overall total of valid votes in their respective districts. The overall equality of the vote is ensured in the Slovenian system because mandates are awarded to parties proportionately to their nationwide vote total. However, because the allocation of mandates to candidates within a party list is done on the basis of the percentage of the vote received by each candidate, the voting weight of voters in a small district can be greater than that of voters in a large district in allocating mandates within a party list. The Constitutional Court decided in 2000 that this did not violate the principle of equality of the vote. (OSCE 2012: 4)

¹³The Office for Democratic Institutions and Human Rights concludes that such dual voting rights diverge from principles ‘regarding equality of the vote and are at odds with international good practice’ (OSCE 2012: 1).

¹⁴In 2008, unranked candidates were scored with ones rather than zeros.

¹⁵§73 of the National Assembly Elections Act 1992 (amended 2000) provides that ‘A voter may vote for only one candidate’ (Republic of Slovenia 2000), but there is some debate about whether this applies solely to the PR districts or also to the minority districts.

¹⁶Borda elections in Slovenia are disconnected from regular party competition while Nauru has no political parties, only loose and regularly changing parliamentary factions.

¹⁷Ben Reilly has remarked that ‘the Nauruan system [is] much more majoritarian than a standard Borda count’ (2002: 365). Here we make that statement much more precise.