Choosing by means of approval-preferential voting. The revised approval choice

Abstract

We consider the problem of making a collective choice by means of approval-preferential voting. The existing proposals are briefly overviewed so as to point out several issues that leave to be desired. In particular, and following Condorcet’s last views on elections, we pay a special attention to making sure that a good option is chosen rather than aiming for the best option but not being so sure about it. We show that this goal is fulfilled in a well-defined sense by a method that we introduced in a previous paper and whose study is deepened here. This procedure, that we call path-revised approval choice, is based on interpreting the approval and paired-comparison scores as degrees of collective belief, revising them in the light of the existing implications by means of the so-called Theophrastus rule, and deciding about every option on the basis of the balance of revised degrees of belief for and against its approval. The computations rely on the path scores, which are used also in a method that was introduced by Markus Schulze in the spirit of looking for the best option. Besides dealing with the confidence in the respective results of both methods, we also establish several other properties of them, including a property of upper semicontinuity of the choice set with respect to the profile and a property of Pareto consistency (in a certain weak sense). Le mieux est le mortal ennemi du bien (Montesquieu, 1720/1755) 2 R. Camps, X. Mora, L. Saumell 1 The problem: How to make a collective choice by means of approval-preferential voting? Approval-preferential ballots give two kinds of information. On the one hand, the voter can approve or disapprove options by themselves; on the other hand, he can also express preferences between pairs of options. Naturally, both kinds of information are required to be consistent with each other. Several rules are in use or have been proposed for making a choice when the individuals vote in this way. In this section we will examine these rules as well as the different issues that they raise. This will motivate a method that we introduced in [9, § 7.2] (in a different but equivalent formulation) and whose study will be deepened here. We will refer to it and its outcomes —possibly more than one in the event of certain ties— as the path-revised approval choice(s) (in [9] we used the terms ‘goodness method’ and ‘goodness winners’). This method, which will be developed in subsequent sections, uses the preferential information, but only to the extent that it entails a revision of the approval information. This contrasts with other methods that use the preferential information to identify an option as the best or topmost one. In this connection, we will obtain a result —Theorem 3.5— that can be interpreted in the following way: in a certain well-defined sense, the confidence that a path-revised approval choice x is a good option is always greater than, or equal to, the confidence that y is preferable to x, where y is any option that is not a path-revised approval choice (but is perhaps proposed as the best or topmost option by another method). As we will see, this property is exactly in the spirit of Condorcet’s last views on elections —rather unknown and in conflict with his celebrated earlier principle— namely that a surely good option should prevail over a doubtfully best one. To our knowledge, the above-mentioned result is the first one of this kind in the social choice literature, even when the interpretation of the confidence of a decision is left open to other possibilities. Our distinction between a surely good option and a doubtfully best one can be compared with the distinction between a best option and a top-ranked one. In fact, as it is argued by H. Peyton Young in [35], the well-known rules of Borda and Condorcet-Kemény-Young can be seen as having different aims, namely and respectively, the option that is most likely to be preferable to any other, and the topmost option in the ranking that is most likely to be correct. Here we are considering a third aim, namely the option that is most likely to be preferable to a certain (possibly implicit) default option. Another difference from [35] and other works is that we will not rely on the maximum Path-revised approval choice, § 1 3 likelihood methods of standard probability theory, but on an interpretation of the pairwise scores as degrees of (collective) belief together with certain max-min methods for consistently dealing with such degrees. In contrast to most of those works, where the comparisons about different pairs of options are assumed to be independent from each other, we will take into account the existing implications, which are certainly present if the voters abide by transitivity. In this connection, our computations will rely on the path scores, which are used also in a method that was introduced by Markus Schulze in the spirit of looking for the best option [31, 32]. 1.1 The Swiss procedure An interesting example of approval-preferential voting is provided by multiple-choice referenda as they are conducted in the Swiss Confederation and its cantons. More specifically, approval-preferential voting arises in two cases: (a) popular initiatives, in which case the government can put forward a counter-proposal; and (b) the so-called ‘constructive’ referenda —in use in only a few cantons— where a proposal from the government can be followed by one or more counter-proposals from groups of voters. In these cases, the voters are asked two sets of questions. The main set is about every proposal by itself to see whether the voter approves it or not. For a proposal to be adopted it must be approved by a majority of voters. In the event of more than one proposal being approved by a majority, a choice is made on the basis of the answers to the second set of questions, where the voter is asked to express his preferences between the different proposals. In the more frequent case of two proposals, the second part reduces to a single question, namely which of the two proposals is preferred to the other. This information determines which option is chosen when both proposals satisfy the condition of being approved by a majority. This procedure was put forward in 1976 by Christoph Haab [19], and was adopted at the federal level in 1987 (see [26, p. 165]). This procedure is quite reasonable. However, it may well happen that only one proposal is approved by a majority but at the same time a majority of the voters prefer the collectively disapproved one (sic). Imagine, for instance, that the votes are as follows: 25 : a | b , 35 : b>a | , 40 : | b>a , (1) where the numbers mean quantities of voters, x>y means that x is preferred to y , and a bar indicates that the options at its left are approved and those at its right are disapproved. One easily checks that proposal a is approved by a 4 R. Camps, X. Mora, L. Saumell majority of 60%, whereas b is disapproved by a majority of 65%. Therefore, the specified procedure results in proposal a being carried through. However, one can also check that a majority of 75% expressed that they prefer b to a, which conflicts with the decision that has been adopted. The situation is very much that of a Condorcet cycle [24, §2.3]. In fact, in the present context, approving a proposal amounts to preferring it to the status quo, i.e. leaving things as they are. From now on, we will denote such a default option by 0. So in the preceding example there are three options —a, b, 0— and the collective preferences form a cycle, namely a> 0>b>a. Quite interestingly, such cycles are not just an academic possibility, but they have occurred in practice, as in the referendum that was held on the 28th November 2004 in the canton of Bern [2]. In the following we will refer to this procedure as the Swiss Procedure. In the constructive referenda one can have more than two proposals besides the status quo. In this case, it could happen that three (or more) of the proposals were approved by a majority but the collective preferences about them formed a Condorcet cycle. In the cantons of Bern and Nidwalden such situations are regulated by applying first the Copeland rule [33, p. 206–209] restricted to the set of approved options, and then, if necessary, some tiebreaking rule ([1, Art. 139.7], [29, Art. 44.3]). 1.2 The ranking approach: rank all options, including the default one, then choose Another interesting real case of approval-preferential voting is the voting procedure used by the Debian Project [16] (see also [34]). Since 1998, the votes of this organization systematically include a default option usually described as “further discussion” or “none of the above”. In this connection, it is explicitly stated that “Options which the voters rank above the default option are options they find acceptable. Options ranked below the default options are options they find unacceptable” [16, v 1.1, §A.6]. The procedure for making a choice is described in [16, Appendix A] and is sometimes called Schwartz Sequential Dropping . According to [31], it is closely related to making use of another procedure that actually ranks all the options and then choosing the top-ranked one. This ranking procedure is based on the so-called Path Scores, which will be dealt with in detail in Section 2. For the moment, it will suffice to say that it complies with the Condorcet principle, i.e. it ranks first the Condorcet winner whenever it exists. Recall that a Condorcet winner means an option that beats every other in the sense that a majority of voters prefers the former to the latter [28]. Path-revised approval choice, § 1 5 Instead of the method of path scores, one can consider any other method for selecting a best or topmost option, such as the Borda count, the method of Condorcet-Kemény-Young , or the method of Ranked Pairs. As a general reference for these and other methods, we refer the reader to [33]. The last two mentioned methods comply also with the Condorcet principle, as well as the method of the path scores. For three options, all of them amount to resolving any Condorcet cycle by dropping the weakest, i.e. less supported, of the three majoritarian views in conflict. In the case of example (1), this means dropping the view of approving a, which leads to adopting the default option 0. So these methods allow strongly supported preferences to overturn the approval information, which seems reasonable enough. 1.3 Should a small preference differential prevail over a large approval differential? However, we might be giving too much importance to preferences. Consider, for instance, the following example (from [9, eq. (109)]): 1/2 + ǫ : a>b | , 1/2− ǫ : b | a , (2) where the numbers of voters are normalized to add up to one and ǫ is a small positive quantity (for instance, 2ǫ could correspond to a single voter and the total number of voters could be one million and one). As one can see, both a and b are approved —i.e. preferred to 0— by a majority of voters; besides, a is preferred to b also by a majority. So a is a Condorcet winner, and therefore it will be chosen by the above Condorcet-compliant methods. However, the majorities in favour of a are quite slight, whereas b is approved by a whole unanimity. So, we are allowing a tiny preference differential to overcome a huge approval differential. As one can easily check, the Swiss procedure also chooses a. Examples like this suggest that one should perhaps completely forget about preferences and take into account only the approval information. However, it still seems that there should be a reasonable way to take into account the preferential information in order to make a better choice. For later reference, the Approval Choice procedure will mean simply throwing away the preferential information and choosing the most approved option. More precisely, since later on we will allow for votes where some options are neither approved nor disapproved, we understand that the approval choice procedure chooses the option that maximizes the number of approvals minus the number of disapprovals, as it is advocated in [18]. 6 R. Camps, X. Mora, L. Saumell 1.4 Condorcet’s last views on elections The point that we are leading to was formulated by Condorcet in the following way ([11, §XIII, p. 307], [27, p. 177–178], emphasis is ours): It is generally more important to be sure of electing men who are worthy of holding office than to have a small probability of electing the worthiest man. The latest works of Condorcet on voting and elections, from 1788 to his death in 1794, are indeed dominated by this idea and by the aim of being able to deal with a large number of candidates, in which case paired comparisons become rather cumbersome [14, 27]. Concerning the meaning of ‘being sure’ and ‘probability’, in another place Condorcet says the following ([10, §XIII, p. 193], [27, p. 139]): We consider a proposition asserted by 15 people, say, more probable than its contradictory asserted by only 10. The two preceding quotes from Condorcet can be viewed as referring to two different kinds of probability. In fact, the first quote can be interpreted as referring to the probability of collectively adopting a certain proposition p of the type ‘x is worthy’ or ‘x is the worthiest’ under the assumption that this proposition is true. In contrast, the second quote is definitely about the probability of p being true after knowing that it has been adopted (by a certain number of votes against the opposite of p). The first kind of probability is the subject matter of Condorcet’s celebrated jury theorem and its extensions to more than two options (see, for instance, [24] and [35]). The second kind is related to the former through Bayes theorem. However, a proposition p of the above-mentioned types does not lend itself easily to being checked for truth or falsehood. This leads to regarding its truth or falsehood as two opposite hypotheses, and to viewing its probability, whose value is obtained by means of Bayes’ theorem, as a bare degree of belief. As we have already said, in this connection we will follow a different approach where degrees of belief are dealt with in a way that does not rely on the standard probability theory. In the same spirit as example (2), Condorcet gives the following one ([14, p. 34–35], [27, p. 241]): 5 : a>c>... | ..., 4 : b>c>... | ..., (3) where he assumes a large number of candidates and, although he does not use approval bars, he explicitly says that all voters consider c worthy of the Path-revised approval choice, § 1 7 place. So the Condorcet winner a is considered the worthiest candidate by a slight majority, but c is considered worthy by unanimity. By the way, this and other examples show that Condorcet was accepting the possibility of making a choice different from the Condorcet winner. 1.5 Condorcet’s practical methods The methods that Condorcet proposed in connection with the preceding ideas are often referred to as Condorcet’s “practical” methods. In general terms, there are two of them. In both of them, the voter is required to produce an ordered list of approved candidates. Unlike proper approval voting, however, the length of this list is fixed: “It should not be too short, to give a good chance that one of the candidates will obtain a majority, [...] nor should it be too long, [so that] the voters can still complete the list without having to nominate candidates they consider unworthy” ([10, p. 203], [27, p. 143]). In his first practical proposal, formulated in 1788 [10, Article V, p. 193– 211] (translated in [27, p. 139–147]), Condorcet chooses the most approved candidate, conditioned to having obtained a majority, and the preferential information is used only in the event of ties. If no candidate has a majority of approvals, then he simply proposes to run a second round after having asked the voters to extend their lists with a certain number of additional candidates. So this proposal was very much in the spirit of approval voting. In his second and final practical proposal, formulated in 1789 and yet in 1793 (with several variations), Condorcet makes a more substantial use of the preferential information. For instance, in his last work [12] one can find the following wording (within a more complex multiround procedure): “If one candidate has the absolute majority of first votes, he will be elected. If one candidate has the absolute majority of first votes and second votes together, he will be elected. If several candidates obtain this majority, the one with the most votes will be preferred. If one candidate has the absolute majority of the three votes together, he will be elected, and if several candidates obtain this majority, the one with the most votes will be preferred” ([14, p. 41–42, §VI], [27, p. 249–250, §VI]). By the context it is clear that the number of first (resp. second) votes means the number of ballots where that candidate appears as the first (resp. second) option; here Condorcet had limited the preferential vote to three candidates, but in the following round [ibidem, §VIII] he extends this rule to preferential votes that list six candidates. Except for secondary variations, this idea spread and/or was rediscovered several times. Shortly after Condorcet’s proposal, it was adopted in Geneva, where it was analyzed in 1794 by Simon Lhuilier [25]. Later on, in the beginning of the twentieth century it was adopted by several American in8 R. Camps, X. Mora, L. Saumell stitutions, starting from the city of Grand Junction (Colorado, USA), where this method was introduced by James W. Bucklin (see [20, § 278] and [33, p. 203–206]). Another example of its use are ballroom dancing competitions, where this idea is used since 1947/48 under the name of Skating System in order to combine the rankings given by the adjudicators [15]. More recently, it was proposed again by Murat R. Sertel in 1986 under the name of Majoritarian Compromise (see [30]). On account of its origins, we will refer to this procedure as the Condorcet-Bucklin method (credit to Condorcet is already acknowledged in [20, p. 490]). The last two of the implementations that we mentioned in the preceding paragraph are not, properly speaking, about approval-preferential voting, since they assume that every voter ranks all the options and no default option is considered. By the way, in this case the Condorcet-Bucklin procedure amounts to use as main comparison criterion the median rank of each candidate, i.e. the median value of the ranks assigned to him by the different voters, and to choose the candidate that has the lowest median rank and that is ranked in this position or better by the largest number of voters. The Condorcet-Bucklin procedure is clearly aimed at making sure that the chosen option is approved by a majority. To this effect, it is essential that every vote be confined to options that the voter really approves of. Therefore, the voter should be allowed to rank as few options as he wishes, which will easily come up in practice anyway. Of course, it may happen that no option is approved by a majority, in which case it would be appropriate to choose the default option or to declare a void choice. If the votes contain any preferential information below approval, then the Condorcet-Bucklin procedure is bound to leave this information out of consideration. As one can easily check, in the case of example (1), this procedure chooses the most approved option, namely a (which is the only one that is approved by a majority). In contrast, in example (2) it does not choose the unanimously approved option b, but option a, which is approved only by a slight majority. 1.6 Approval-preferential procedures in parliamentary elections A real example where truncated rankings are used and where they can be interpreted as ordered lists of approved options are the elections to the Legislative Assemblies of Queensland and New South Wales (Australia), where every constituency elects a single representative on the basis of the (possibly) truncated rankings that are expressed by the electors [17]. Long ago, starting from 1892 in Queensland, the choice was made according to the soPath-revised approval choice, § 1 9 called Contingent Vote system, that amounts to a instant runoff between the two candidates that obtained the most first-choice votes. At present, starting from 1980 in New South Wales, the choice is made by means of the Alternative Vote system [33, p.193–195]. As it is well-known, a major flaw of these systems is their lack of monotonicity [33, p.194]. 1.7 Taking into account the preferences between non-approved options The methods of the preceding sections 1.5 and 1.6 do not take into account the preferences that a voter could have between his non-approved options. This is unfair towards the voters who do not approve at all the chosen option and would rather prefer some other non-approved option. In order to take into account all preferences one could certainly use the methods of Section 1.2 after having introduced a default option. By the way, one could include among them the Condorcet-Bucklin method for complete rankings (which in the case of (1) chooses neither a nor 0, but b!). However, as we raised in Sections 1.3 and 1.4, this approach is too preference-oriented; instead, one should give some sort of priority to the approval information. The existing proposals in this direction are essentially some more elaborated versions of the Swiss procedure that we presented in Section 1.1. One of them is the Preference Approval Voting procedure that was put forward in 2008 by Steven Brams and Remzi Sanver [3, 4]. When more than two options are approved by a majority, this procedure restricts the attention to these options and the preferential information about them is used to single out, if possible, their Condorcet winner; if this is not possible, then other rules are applied that make further use of the approval scores. A simpler possibility is the Approval Voting with a Runoff , considered in 2010 by Remzi Sanver [30]. Here, the preferences are used only to compare between the two most approved options. Anyway, in the case of (2) both these procedures keep choosing a, like the Swiss procedure, against the view expressed in Sections 1.3 and 1.4 that an overwhelming approval for an option should prevail over a slightly majoritarian preference for another. 1.8 Upper semicontinuity In this section we consider the effect of small variations in the relative profile of the vote. 10 R. Camps, X. Mora, L. Saumell By the relative profile of the vote we mean a collection of numbers uk where k runs through all possible ways of filling a ballot and uk gives the fraction of voters who expressed the view k , i.e. the absolute number of voters who expressed that view divided by the total number of voters. The reason of dividing by the total number of voters is that the outcome of the vote should not change when the numbers of voters who expressed each opinion are multiplied all of them by the same factor. Now, when the total amount of voters is very large, the fractions uk admit of very small variations. In this connection, we postulate that the dependence of the choice set on the relative profile should have the following property: every relative profile u has a neighbourhood U such that for any relative profile u in U the choice set for u is contained in that for u . In the established terminology about set functions [23], we are requiring the choice set to be an upper semicontinuous function of the relative profile. In particular, this ensures that the outcome will remain a correct choice even though reading mistakes are made in a small (enough) fraction of ballots. Such a property is quite pertinent from a practical point of view in the case of a very large number of voters. This property is easily violated by the methods that successively apply different criteria. For instance, in the case of example (2) both the Swiss procedure and the Condorcet-Bucklin one choose a for ǫ > 0 and b for ǫ ≤ 0. In order to get {a, b} for ǫ = 0 one can consider modifying these rules by replacing proper majority requirements by weak majority ones (greater than or equal to 50%). This achieves the desired result in the particular case of (2), but the problem persists in other examples. Consider, for instance, the following one: 2 + ǫ : a>e>b>c>d, 2− ǫ : b>c>a>d>e, 1 : a>b>c>d>e, 1 : b>d>c>a>e, 2 : c>d>a>b>e. (4) One can check that the Condorcet-Bucklin procedure chooses a for small positive ǫ and b for negative ǫ; at the boundary ǫ = 0 it chooses a or b depending on whether proper majority or weak majority is considered, but neither of both variants chooses {a, b} . 1.9 Dealing with ties and with incomplete information In practice, voters often do not have an opinion on some options. Besides, they may also rank equally some that they know. In order to properly deal with such possibilities, one must begin by distinguishing between them when interpreting the ballots. Path-revised approval choice, § 1 11 For instance, the Debian voting rules state that “Unranked options are considered to be ranked equally with one another” [16, §A.6.1]. However, this is really questionable. In Condorcet’s words, “When someone votes for one particular candidate, he simply asserts that he considers that candidate better than the others, and makes no assertion whatsoever about the respective merits of these other candidates. His judgement is therefore incomplete” ([10, p. 194], [27, p. 139]). A vote where two options x and y are really ranked equally with each other can be assimilated to half a vote where x is preferred to y together with half another vote with the reverse preference. In contrast, a vote that expresses neither a preference nor a tie between x and y should contribute neither to the number of voters who prefer x to y nor to the number of those who prefer y to x. In this connection, it is quite standard to take the view that “Ranked options are considered preferred to all unranked options” [16, §A.6.1]. However, in some contexts it could be more appropriate to interpret that no comparison is made between a ranked option and an unranked one. One should also be aware that not approving an option is not the same as really disapproving it. On account of all these considerations, it is certainly desirable that the ballots be designed so as to make as clear as possible what the voter really means to say (no matter whether he is being sincere or not). Besides, it is most important to clearly specify how will the ballots be interpreted. Once the information has been properly interpreted and collected, the problem remains of how should one deal with it. In fact, many existing methods assume that complete information is given, and quite often it is not clear at all how should they be extended to the general incomplete case.

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@article{Camps2014ChoosingBM, title={Choosing by means of approval-preferential voting. The revised approval choice}, author={Rosa Camps and Xavier Mora and Laia Saumell}, journal={CoRR}, year={2014}, volume={abs/1411.1367} }