Varieties of failure of monotonicity and participation under five voting methods

Transcription

1 Theory Dec. (2013) 75:59 77 DOI /s Varieties of failure of monotonicity and participation under five voting methods Dan S. Felsenthal Nicolaus Tideman Published online: 27 April 2012 Springer Science+Business Media New York 2012 Abstract In voting theory, monotonicity is the axiom that an improvement in the ranking of a candidate by voters cannot cause a candidate who would otherwise win to lose. The participation axiom states that the sincere report of a voter s preferences cannot cause an outcome that the voter regards as less attractive than the one that would result from the voter s non-participation. This article identifies three binary distinctions in the types of circumstances in which failures of monotonicity or participation can occur. Two of the three distinctions apply to monotonicity, while one of those and the third apply to participation. The distinction that is unique to monotonicity is whether the voters whose changed rankings demonstrate non-monotonicity are better off or worse off. The distinction that is unique to participation is whether the marginally participating voter causes his first choice to lose or his last choice to win. The overlapping distinction is whether the profile of voters rankings has a Condorcet winner or a cycle at the top. This article traces the occurrence of all of the resulting combination of characteristics in the voting methods that can exhibit failures of monotonicity. Keywords Elections Non-monotonicity Participation Strategic voting Voting paradoxes Voting methods Voting procedures JEL Classification D71 D72 D. S. Felsenthal (B) School of Political Sciences, University of Haifa, Haifa, Israel msdanfl@mscc.huji.ac.il N. Tideman Department of Economics, Virginia Polytechnic Institute & State University, Blacksburg, VA 24061, USA ntideman@vt.edu

2 60 D. S. Felsenthal, N. Tideman 1 Introduction In the social choice literature, a monotonic voting method is usually defined as a method satisfying an axiom according to which if x is the unique collectively best alternative for a given profile π of individual preference rankings, and if profile π is obtained from π by moving x up in some of the rankings, leaving all else unchanged, then x should be one of the collectively best alternatives for profile π (Fishburn 1982, p. 119). 1 Of the many voting methods proposed and discussed seriously in the literature, only five methods two of which are used in practice are known to violate the monotonicity axiom and are therefore regarded as susceptible to a paradox called the More-is-Less Paradox (Fishburn and Brams 1983, p. 208). Although no single-winner voting method involving three or more candidates is paradox-free, a voting method that is susceptible to the More-is-Less paradox is widely regarded to suffer from an especially serious defect. A voting method satisfies the participation axiom if a voter never loses by joining the electorate and reporting (sincerely) his preferences (Moulin 1988, p. 55). A voting rule that fails to satisfy the participation axiom is said to be susceptible to the No-Show paradox (Fishburn and Brams 1983, p. 207). All five of the voting rules that are susceptible to the More-is-Less Paradox are also susceptible to the No-Show Paradox, 2 though different rules are susceptible in different types of circumstances. There are many other voting rules that are susceptible to the No-Show Paradox as well. In particular, Moulin (1988) shows that all Condorcet consistent voting rules are susceptible to the No-Show Paradox. In this article, however, we concentrate on the voting rules that are susceptible to both paradoxes. In this article, we identify distinctions in the circumstances in which the five voting methods can exhibit these paradoxes and determine which circumstances can arise for each method. The voting methods are: Plurality with Runoff (P-R), the Alternative Vote (AV), and the Coombs, Nanson, and Dodgson methods. The first two of these voting methods are used in practice in public elections in several countries. Under each of these five methods we either demonstrate or explain the non-existence of each variety of failure of monotonicity or participation. We hope that the explication of these paradoxes will contribute to improved evaluations of the relative merits of voting rules. This article is organized as follows. Section 2 describes briefly the five voting methods we investigate. Section 3 gives examples of four sub-types of failures of monotonicity under one or more of the investigated voting methods. In Sects. 4 and 5, we consider two categories of failures of participation. We give examples, where 1 More formally, moving x up means that all candidates other than x are in the same order after the change as before, all candidates initially below x remain below x after the change, and one or more candidates that were initially above x are below x after the change. 2 Campbell and Kelly (2002) have shown that there exists a voting method that is susceptible to the Moreis-Less paradox but not also to the No-Show paradox. However, as this method violates also the anonymity and neutrality conditions and hence does not respect majority rule nobody proposed that it should be implemented in practice. Hence we too ignore it.

3 Varieties of failure of monotonicity and participation 61 they exist under the investigated voting methods, of two sub-types of each of these failures. Section 6 summarizes. 2 Five single-winner voting methods that fail monotonicity and participation We describe briefly below the five best known single-winner voting methods that fail monotonicity and participation. 2.1 Plurality with Runoff (P-R) Under the usual version of this method, up to two voting rounds are conducted. In the first round each voter casts one vote for a single candidate. In order to win in the first round a candidate must obtain either a special plurality (usually at least 40% of the votes) or an absolute majority of the votes. If no candidate is declared the winner in the first round then a second round is conducted. In this round only the two candidates who obtained the highest numbers of votes in the first round participate, and the one who obtains the majority of votes wins. This is a very common method for electing a single candidate and is used, inter alia, for electing the President of France. For simplicity, P-R will be treated as operating on a single round of voters responses, with the responses providing full rankings of the candidates, from which the voters first choices from available candidates are taken. This simplification would affect the results only if voters using P-R wished to cast second-round ballots that were inconsistent with their votes in the first round. 2.2 Alternative Vote (AV; aka Instant Runoff Voting) This is the adaptation to the task of electing a single candidate of the multi-winner voting method known as the Single Transferable Vote (STV). STV was proposed independently by Carl George Andrae in Denmark in 1855 and by Thomas Hare in England in The use of this method for electing a single candidate was first proposed by an American, William Robert Ware, in It works as follows. All voters submit ballots that rank-order all of the candidates. In the first step, one determines whether there exists a candidate who is ranked first by an absolute majority of the voters. If such a candidate exists s/he is declared the winner. If no such candidate exists then, in the second step, the candidate who is ranked first by the smallest number of voters is deleted from all ballots and thereafter one again determines whether there is now a candidate who is ranked first by an absolute majority of the voters. The elimination process continues in this way until a candidate who is ranked first by an absolute majority of the voters is found. The AV method is used in electing the president of the Republic of Ireland, the Australian House of Representatives, as well as the mayors in some municipal elections in the US. In May 2010, a referendum was conducted in the UK to decide whether AV should be used for electing the members of the House of Commons; a majority of voters rejected this proposal.

4 62 D. S. Felsenthal, N. Tideman 2.3 The Coombs method This method was proposed by the psychologist Clyde H. Coombs in It is similar to AV except that the candidate who is eliminated in a given round under the Coombs method is the candidate who is ranked last by the largest number of voters (instead of the candidate who is ranked first by the smallest number of voters under the AV method). (cf. Coombs 1964, pp ; Straffin 1980; Coombs et al ) 2.4 The Dodgson method This method is named after the Rev. Charles Lutwidge Dodgson, aka Lewis Carroll, who referred to it implicitly in 1876 without explicitly endorsing it. It elects the Condorcet winner when one exists. 3 If the majority relation method contains a top cycle then the Dodgson method elects that candidate who can be made into a Condorcet winner by the smallest number of transpositions of adjacent candidates in the voters rankings (cf. Black 1958, pp ; McLean and Urken 1995, pp ). 2.5 The Nanson method The Nanson method is a recursive elimination method using the Borda method. 4 In the first step, one calculates each candidate s Borda score. At the end of the first step the candidates whose Borda scores do not exceed the average Borda score of the candidates in this step are eliminated from all ballots and a new Borda score is computed in the second step for each uneliminated candidate. 5 The elimination process is continued in this way until one candidate is left. If all of the uneliminated candidates have the same Borda score then one of them is elected according to a pre-determined method for breaking ties. If a (strong) Condorcet winner exists then the Nanson method elects him or her (Nanson 1883; McLean and Urken 1995, Chap. 14). 3 Examples of sub-types of non-monotonicity Most of the examples of the More-is-Less (non-monotonicity) paradox that one finds in the literature are situations where (1) the initial majority relation method 3 A Condorcet-winner is a candidate who beats all other candidates in head-to-head contests. 4 According to the Borda method, all voters submit ballots that rank-order all n candidates. Thereafter one assigns to every candidate n 1 points for every ballot in which s/he is ranked first, n 2 points for every ballot in which s/he is ranked second, and so on, and 0 points for every ballot in which s/he is ranked last. The Borda score of every candidate is the sum of points s/he received over all ballots (cf. McLean and Urken 1995, pp ). 5 Many authors state erroneously that according to the Nanson method one eliminates at the end of each round only the candidate with the lowest Borda score, rather than all candidates whose Borda score is equal to, or lower than, the average Borda score. When the social preference ordering contains a top cycle, the erroneous description of the Nanson elimination process may result, ceteris paribus, in a different outcome than that obtained under the (correct) Nanson elimination process.

5 Varieties of failure of monotonicity and participation 63 contains a top cycle and (2) the voters who moved x up in their individual rankings obtain as a result of this change the election of an alternative y that was originally ranked by them below x. But there are other types of examples that deserve to be noted. The examples of the More-is-Less paradox that are discussed in this section are distinguished from examples of the No-Show (participation) paradox discussed in Sects. 4 and 5 by the fact that the examples of the More-is-Less paradox involve a fixed number of voters. In such examples, there are always some voters whose reported rankings change as the example unfolds. We call these voters dynamic voters. We assume that all dynamic voters have the same initial rankings of the candidates and all change their rankings in the same way. It should be possible to define nonmonotonicity to include examples in which there was more than one type of dynamic voter. However, we are not aware of any change in the set of conclusions that would emerge from the relaxation of this constraint. The four sub-types of non-monotonicity discussed in this section are: 1. [M+COND+B]: The example is a failure of monotonicity [M], a Condorcet winner [COND] exists, and the dynamic voters are better off [B] from changing their reported rankings, treating their original rankings as their true rankings. 2. [M+COND+W]: As in sub-type (1) but the dynamic voters are worse off (W) from changing their rankings, treating their original rankings as their true rankings. 3. [M+CYC+B]: As in sub-type (1) but the initial majority relation method contains a top cycle (CYC). 4. [M+CYC+W]: As in sub-type (2) but the initial majority relation method contains a top cycle (CYC). The changes in the rankings of voters that occur in failures of monotonicity might come from two different sources. First, voters might change their true preferences as a result of additional information or as a result of campaigning by candidates and their supporters. Second, voters might decide to change their reported rankings strategically while their true preferences remained fixed. If the change is the result of strategic efforts, type [B] non-monotonicities represent successful strategic effort, while type [W] non-monotonicities would represent strategic efforts that backfired. If the change in reported rankings is the result of campaigning, then type [B] non-monotonicities represent situations in which the dynamic voters become even better off in terms of their old preferences but could be either better off or worse off, in terms of their new preferences, than they would have been if the candidate who rose in their rankings had continued to win, while type [W] non-monotonicities represent situations in which the dynamic voters, both by their original ranking and by their revised rankings, become worse off than they would have been if they had reported unchanged rankings. We employ the convention of assuming that the rankings first reported by the dynamic voters represent their true preferences, so that we can specify whether the change in their ranking makes them better off or worse off.

6 64 D. S. Felsenthal, N. Tideman 3.1 Non-monotonicity under P-R and AV An example of sub-type [M+COND+B] Suppose there are 43 voters whose rankings of three candidates, a, b, and c, areas follows 7 : 7 a b c 9 a c b 14 b c a 13 c a b Here there is a transitive ranking by the majority relation method, so there is a Condorcet winner, namely c. But with these rankings, under P-R and AV c will be eliminated in the first round and a will beat b in the second round and thus become the ultimate winner. Now suppose that 5 out of the 14 voters whose ranking is b c a (who are not happy with the prospect that a will be elected if all voters vote sincerely), decide to change their ranking (strategically) to a b c (thereby increasing a s support). As a result of this change b (rather than c) will be eliminated in the first round and c (the Condorcet winner) will beat a in the second round thereby demonstrating the vulnerability of P-R (as well as AV) to strategic non-monotonicity. (Note that the five dynamic voters benefited from strategically not representing their true preferences in their rankings) An example of sub-type [M+COND+W] Suppose there are 17 voters whose rankings of three candidates, a, b, and c, areas follows: 3 a b c 2 a c b 4 b a c 2 b c a 4 c a b 2 c b a Here there is a transitive ranking by the majority relation method of a b c., i.e., a is the Condorcet winner. Under P-R (as well as under AV) a will be eliminated after the first round and b will beat c in the second round and hence b becomes the ultimate winner. 6 When there are only three candidates the elimination process and final outcome under P-R and AV are the same. Miller (2012) has recently specified the precise conditions under which non-monotonicity arises in three-candidate elections under AV. 7 When the notation a b is used with respect to a voter it means that the voter ranks candidate a ahead of candidate b. When it is used with respect to a set of voters, it means that a majority of the voters rank a ahead of b.

7 Varieties of failure of monotonicity and participation 65 Now suppose that, ceteris paribus, the two voters whose ranking is c b a change it to b c a, thereby increasing b s support. As a result of this change c (rather than a) is eliminated in the first round, and a (the Condorcet winner) beats b in the second round and becomes the ultimate winner. Note that the dynamic voters became worse off An example of sub-type [M+CYC+B] Suppose there are 43 voters whose rankings of three candidates, a, b, and c, areas follows: 8 a b c 8 a c b 14 b c a 13 c a b Here the majority relation method is cyclical [a b c a]. Under P-R (as well as under AV) c will be eliminated after the first round and a will beat b in the second round and hence a will become the ultimate winner. Now suppose that, ceteris paribus, two of the voters whose ranking is b c a change it to a b c thereby increasing a s support. As a result of this change b (rather than c) is eliminated in the first round, and c beats a in the second round and becomes the ultimate winner. Note that the dynamic voters became better off An example of sub-type [M+CYC+W] Suppose there are 17 voters whose rankings of three candidates, a, b, and c, areas follows: 6 a b c 2 b a c 4 b c a 5 c a b The majority relation method contains a top cycle [a b c a]. In the first round c is eliminated, and a beats b in the second round and becomes the ultimate winner. Now suppose that the two voters with ranking b a c change their ranking to a b c, thereby increasing their support of a. As a result of this change candidate b will be eliminated in the first round and c will beat a in the second round and become the ultimate winner. Note that here the two dynamic voters were harmed by this change because they obtained their least favored alternative. 3.2 Non-monotonicity under the Coombs method There are no [W] examples under the Coombs method. This is so because the definition of non-monotonicity involves moving the original winner up. In order to affect the

8 66 D. S. Felsenthal, N. Tideman outcome under the Coombs method, the least favored candidate of the dynamic voters must be changed. If the least favored candidate is changed and the winner moves up, the winner must have been in last place. If the winner is the last place candidate of the dynamic voters, then there is no way to change the outcome and make them worse off. Thus there are no [W] examples for the Coombs method An example of sub-type [M+COND+B] Suppose that 45 voters have to elect under the Coombs method one out of three candidates, a, b, or c, and that their rankings of these three candidates are as follows: 1 a b c 10 a c b 11 b a c 11 b c a 10 c a b 2 c b a There is a transitive ranking by the majority relation method of b c a, i.e., b is the Condorcet winner. However, since none of the candidates is ranked first by an absolute majority of the voters, one deletes according to the Coombs method the candidate who is ranked last by the largest number of voters. In the above example this candidate is b, the Condorcet winner. After the deletion of b, candidate c is ranked first by an absolute majority of the voters and is elected. Now suppose that, ceteris paribus, the 11 voters whose ranking is b a c (who are not happy with the prospect that c will be elected) are motivated to increase c s support by changing their ranking to b c a. Candidate b is still the Condorcet winner but as a result of this change, a (rather than b) will be eliminated first under the Coombs method, and thereafter b will be elected thus making the dynamic voters better off An example of sub-type [M+CYC+B] This example is due to Nurmi (1999, Table 6.2, p. 58). Suppose that 100 voters have to elect under the Coombs method one out of three candidates, a, b,orc, and that their rankings of these three candidates are as follows: 40 a b c 15 b c a 30 c a b 15 c b a The majority relation method is cyclical (a b c a). Since none of the candidates is ranked first by an absolute majority of the voters, one deletes according to the Coombs method the candidate who is ranked last by the largest number of voters. In the above example this candidate is c.afterc is deleted a beats b and thereby becomes the ultimate winner.

9 Varieties of failure of monotonicity and participation 67 Now suppose that the 15 voters whose ranking is c b a change it to c a b thereby increasing a s support. As a result b will be eliminated under the Coombs method and thereafter c will be elected thus making the dynamic voters better off. 3.3 Non-monotonicity under the Nanson method As the Nanson method is Condorcet-efficient there are no [COND] examples under this method An example of sub-type [M+CYC+B] Suppose there are 36 voters who must elect one out of four candidates, a, b, c, or d, under the Nanson method and whose rankings of these candidates, as well as the resultant Borda scores of the four candidates, are as follows: 1 a b c d 2 c a b d 1 a b d c 1 c a d b 2 a c b d 3 c b a d 2 a c d b 2 c b d a 1 a d b c 1 c d a b 2 a d c b 2 c d b a 2 b a c d 1 d a b c 2 b a d c 1 d a c b 1 b c a d 0 d b c a 1 b c d a 2 d b a c 2 b d a c 1 d c a b 1 b d c a 2 d c b a Each number in the body of the matrix of paired comparisons, below, is the number of voters who rank the candidate listed at the left of the row ahead of the candidate listed at the top of the column. Thus, for example, the number 16 in row a and column b denotes that 16 voters rank a ahead of b in their ballots, and therefore the complementary number 20 appears in row b and column a. The Borda scores of the candidates can be computed as the sums of the rows of the matrix: a b c d Sum a b c d Total 216 The sum of Borda scores of all four candidates is 216, 8 hence the average Borda score is 54, that is, 216/4. According to the Nanson method, one eliminates at the end of every counting round those candidates whose Borda scores is equal to or smaller than the average of the scores of all candidates participating in this round. Hence only 8 Note that the sum of the Borda scores of all candidates can also be obtained by multiplying the number of voters (36 in this example) by the number of paired comparisons among the candidates (6 in this example).

10 68 D. S. Felsenthal, N. Tideman candidate d is eliminated after the first round. So in the second counting round, we have: 4 a b c 7 a c b 8 b a c 3 b c a 5 c a b 9 c b a which generates the following matrix of paired comparisons and Borda scores: a b c Sum a b c Total 108 Here the sum of Borda scores of all three candidates is 108, hence their average Borda score is 36, that is, 108/3. So according to the Nanson method one eliminates at the end of the second counting round both candidates a and b thus candidate c becomes the ultimate winner. Now suppose that, ceteris paribus, the voter whose ranking is a b c d who is not happy with the prospect that candidate c may be elected is motivated to increase his support of candidate c by changing his ranking to a c b d. As a result of this change, the Borda scores of candidates b and c change to 54 and 59, respectively, while the Borda scores of the remaining two candidates, as well as the sum of all Borda scores and the average Borda score, remain the same. So now both candidate b and candidate d are eliminated after the first counting round. In the second counting round the (revised) Borda scores of candidates a and c are 19 and 17, respectively, so candidate a becomes the ultimate winner thus the voter who changed his ranking is better off An example of sub-type [M+CYC+W] This example is due to Nurmi (1999, Table 6.3, pp ). Suppose there are 100 voters whose rankings of four candidates, a, b, c, and d, are as follows: 20 a b d c 12 a c b d 5 a c d b 12 b a c d 21 b d c a 30 c a d b The majority relation method is cyclical [a b c a] d. The Borda scores of the candidates can be derived from the sum of the lines in the following paired comparisons matrix:

11 Varieties of failure of monotonicity and participation 69 a b c d Sum a b c d Total 600 The sum of Borda scores of all four candidates is 600, and the average score is 600/4, or 150. So according to the Nanson method, one eliminates at the end of the first counting round candidate d, and the re-computed Borda scores for candidates a, b, and c, are 116, 86, and 98, respectively. Since the average of these scores is 100, a wins. Now suppose that the 12 voters with ranking b a c d change it to a b c d thereby increasing a s support. Then the Borda scores of a, b, c, and d are, 207, 139, 157, and 97, respectively. As the sum and average of these scores remain the same (600 and 150, respectively), candidates b and d are eliminated in the first round and thereafter c wins hence the dynamic voters are worse off. 3.4 Non-monotonicity under the Dodgson method As the Dodgson method is Condorcet-efficient, there are no [COND] examples under this method. It is not possible to cause a Condorcet winner to lose by moving that candidate up in the voters rankings. There are also no [B] examples. The reason is that under the Dodgson method if the dynamic voters raise the original winner in their rankings in order to effect non-monotonicity, the indirect benefit, if any, goes to the candidates who were originally ranked below the original winner in the dynamic voters ranking, who are now able to beat the candidates who had been ranked above the original winner in their initial ranking through fewer interchanges of adjacent candidates. But if the initial ranking of the dynamic voters is their true ranking, then they cannot prefer over the original winner any candidate that they had ranked below the original winner in their initial ranking, and since these are the only candidates other than the original winner who can benefit from an upward movement of the original winner, there are no [B] examples. So if non-monotonicity can be demonstrated under the Dodgson method, the dynamic voters must be worse off. Therefore one can only demonstrate under the Dodgson method an example of non-monotonicity of type [M+CYC+W]. Such an example must include at least four candidates (see Fishburn 1982, p. 132) An example of sub-type [M+CYC+W] This example is adapted from Fishburn (1977, p. 478). Suppose there are 100 voters who are divided into four groups, who must elect one out of five candidates a, b, c, d, e, under the Dodgson method, and whose rankings of the candidates are as follows:

12 70 D. S. Felsenthal, N. Tideman Group G1 42 b a c d e G2 26 a e c b d G3 21 e d b a c G4 11 e a b d c The majority relation method has a top cycle: [b a e b] c d. It can be depicted in the following matrix of paired comparisons: a b c d e a b c d e For candidate a to become the Condorcet winner at least 14 voters who are either in group G1 or group G3 must invert b a in their rankings to a b, i.e., a total of 14 pair inversions. For candidate b to become the Condorcet winner at least 9 voters who are either in group G2 or G3 or G4 must move b up two notches in their rankings, i.e., a total of 18 inversions. For candidate e to become the Condorcet winner at least 19 voters in group G2 must invert a e in their ranking to e a, i.e., a total of 19 inversions. As the number of inversions needed to make a the Condorcet winner is smallest, a is elected under the Dodgson method. Now suppose that, ceteris paribus, the 11 voters in group G4 increase their support of candidate a by changing their rankings from e a b d c to a e b d c. The results of this change are depicted in the following paired comparisons matrix: a b c d e a b c d e From this matrix it is possible to see that despite the increase in a s support it would still take at least 14 persons from group G1 to invert in their rankings b a to a b in order for a to become the Condorcet winner, whereas now for b to become the Condorcet winner only 9 voters in group G4 would have to invert e b to b e in their rankings. So as the number of changes needed for b to become the Condorcet winner is smallest, b would be elected under the Dodgson method thereby making the members of group G4 worse off. 4 Type [P-BOT] paradoxes In this section, we replace the fixed electorate of the traditional definition of nonmonotonicity with an electorate that increases in such a way as to show a paradox that is closely related to the More-is-Less (non-monotonicity) paradox. The paradox of this section arises if one of the candidates, say candidate c, who has not been elected originally, may be elected if, ceteris paribus, the electorate is increased as

13 Varieties of failure of monotonicity and participation 71 a result of additional voters whose bottom-ranked candidate is c join the electorate, and consequently these additional voters are worse off. A voting rule that exhibits this paradox is said to violate the axiom of participation and to exhibit the No-Show paradox. 9 Although many voting methods are vulnerable to this paradox, only two of the five voting methods investigated in this article are vulnerable to this paradox, namely P-R and AV. 10 So under these two methods we shall demonstrate two versions of P-BOT: [P-BOT+COND+W] and [P-BOT+CYC+W]. Here [COND] means that a Condorcet winner exists initially regardless of whether s/he is initially elected. 4.1 The [P-BOT] paradox under P-R and AV An example of sub-type [P-BOT+COND+W] This example is due to Felsenthal and Maoz (1992, Example 5, p. 119). Suppose there are 19 voters whose rankings of three candidates, a, b, and c, are as follows: 4 a b c 1 a c b 2 b a c 4 b c a 3 c a b 5 c b a The ranking by majority relation method is b c a, i.e., b is a Condorcet winner. Under the P-R and AV methods, candidate a is eliminated after the first round and b is elected in the second round. As candidate c has not been elected, suppose now that, ceteris paribus, two additional voters whose ranking is a b c join the electorate (thereby further downgrading c). As a result b is eliminated in the first round, and c is elected in the second round in violation of the participation axiom. 9 This paradox, which is closely related to the More-is-Less paradox of non-monotonicity, was originally suggested by Fishburn and Brams (1983, p. 207) who called it the No-Show Paradox because it displays a situation in which the additional voters who joined the electorate would have been better off if they had not participated in the election. Another type of violation of the Participation axiom, which perhaps should also be called No-Show paradox, is exemplified in the next section. 10 The Coombs method is not susceptible to the [P-BOT] type of non-monotonicity because under this method one eliminates sequentially the candidates who are ranked last by the largest number of voters. So if candidate c was not elected originally, then c can certainly not be elected under the Coombs method if additional voters who rank c last join the electorate. For similar reasons, the Nanson method is also not susceptible to the [P-BOT] type of non-monotonicity. This is so because under the Nanson method one eliminates sequentially the candidate(s) whose Borda score is equal to, or lower than, the average Borda score. So relative to the other candidates c s Borda score is decreased further if additional voters who rank c last join the electorate. The Dodgson voting method too is not susceptible to the [P-BOT] type of nonmonotonicity because if, ceteris paribus, additional voters who rank c last join the electorate implies that more inversions in the voters rankings will be needed in order for c to become a Condorcet winner, which decreases further the chances of c to be elected.

14 72 D. S. Felsenthal, N. Tideman An example of sub-type [P-BOT+CYC+W] Suppose there are 102 voters whose rankings of three candidates, a, b, and c, areas follows: 20 a b c 13 a c b 13 b a c 21 b c a 35 c a b The majority relation method is cyclical (a b c a). Under the P-R and AV methods candidate a is eliminated after the first round and b is elected in the second round. As candidate c has not been elected, suppose now that, ceteris paribus, two additional voters whose ranking is a b c join the electorate (thereby further downgrading c). As a result b is eliminated in the first round and c is elected in the second round in violation of the participation axiom. 5 Non-monotonicity with a variable electorate of type [P-TOP] In this section, we replace [P-BOT] with [P-TOP]. According to this paradox, which is also closely related to the More-is-Less (non-monotonicity) paradox, if a candidate, say candidate x, has been elected initially, then it is possible that another candidate, y, will be elected if, ceteris paribus, additional voters whose top-ranked candidate is x join the electorate. 11 Of the five investigated voting methods, two of them (P-R and AV) are not vulnerable to the [P-TOP] paradox. 12 Since under the remaining three investigated voting methods the additional voters are inevitably harmed by the [P-TOP] paradox, only one or both of the following two sub-types of the [P-TOP] paradox exist: [P-TOP+COND+W] and/or [P-TOP+CYC+W] Woodall (1997) calls this paradox either Mono-Add-Plump or Mono-Add-Top depending whether the additional voters list only x in their ranking lists or whether they also list below x one or more of the remaining candidates. However, as we stated in footnote 9, this paradox too should be called a No-Show paradox because it also displays a situation where the additional voters would have been better off if they had not participated in the election. 12 Under these two methods, the candidate(s) who are top-ranked by the fewest number of voters are eliminated sequentially and the candidate who is top-ranked by an absolute majority of the voters is elected. Hence if x was elected initially under these methods then, ceteris paribus, x will be elected a-fortiori if additional voters whose top-rank is x join the electorate. However, Fishburn and Brams (1983,pp ) demonstrate that STV (in an instance where two out of four candidates must be elected) is susceptible to the [P-TOP] paradox. 13 Because a Condorcet winner, if one exists, may not be elected under the Coombs method, both sub-types of the [P-TOP] paradox can be exemplified under these methods. However, since a Condorcet winner, when one exists, is always elected under the Nanson and the Dodgson voting methods, only the [P-TOP+CYC+W] sub-type of this paradox can be demonstrated under these methods.

15 Varieties of failure of monotonicity and participation The [P-TOP] paradox under the Coombs voting method An example of sub-type [P-TOP+COND+W] Suppose there are 42 voters who must elect one out of four candidates, a, b, c, ord, under the Coombs method, and that their rankings of the candidates are as follows: 7 a c d b 6 a d b c 3 b a c d 7 b c a d 9 b c d a 4 c a d b 6 d a b c Here a is the Condorcet winner. Since none of the candidates is ranked first by an absolute majority of the voters, candidate c is eliminated in the first round under the Coombs method, candidate b is eliminated in the second round, and thereafter candidate a is elected. Now suppose that, ceteris paribus, three additional voters join the electorate whose ranking is a c b d. Although the number of voters who rank a first has now increased, still none of the candidates is ranked first by an absolute majority of the voters. So according to the Coombs method candidate d is eliminated in the first counting round, candidate a is eliminated in the second counting round, whereupon candidate b is elected. Thus not only is candidate a harmed by receiving additional top-rank support, but also the voters who provided this additional support are harmed An example of sub-type [P-TOP+CYC+W] In the first part of Example 3.2.2, candidate a was elected under the Coombs method. Now suppose that, ceteris paribus, a group of 11 voters with ranking a c b joins the electorate. Although the number of voters who rank a first has now increased, still none of the candidates is ranked first by an absolute majority of the voters. So according to the Coombs method candidate b is eliminated, whereupon candidate c is elected. Thus not only is candidate a harmed by receiving additional top-rank support, but also the voters who provided this additional support are harmed. 5.2 The [P-TOP] paradox under the Nanson voting method An example of sub-type [P-TOP+CYC+W] This example is due to Hannu Nurmi (private communications to the first-named author on and ). Suppose there are 18 voters who must elect one out of four candidates, a, b, c, and d, under the Nanson method, and that their rankings of the candidates are as follows:

16 74 D. S. Felsenthal, N. Tideman No. of Voters Ranking 5 a b d c 5 b c d a 6 c a d b 1 c b a d 1 c b d a The majority relation method contains a top cycle [a b c a] d. The Borda scores of candidates a, b, c, and d, are 28, 29, 34, and 17, respectively, and the average Borda score is 27. Hence according to the Nanson method, candidate d is eliminated in the first count. The Borda scores of candidates a, b, and c, inthe second counting round are 16, 17, and 21, respectively, and the average Borda score is 18. Consequently both candidates a and b are eliminated and therefore candidate c wins. Now suppose that, ceteris paribus, one additional voter whose ranking is c b d a joins the electorate, thereby increasing the number of voters who rank c at the top of their ranking. The Borda scores of candidates a, b, c, and d, are now 28, 31, 37, and 18, respectively, and the average Borda score is Hence according to the Nanson method, candidates a and d are eliminated in the first count, and candidate b beats c in the second counting round and is elected. Thus not only is candidate c harmed by receiving additional top-rank support, but also the voter who provided this additional support is harmed. 5.3 The [P-TOP] paradox under the Dodgson method An example of sub-type [P-TOP+CYC+W] In the first part of Example candidate a was elected under the Dodgson method. Now suppose that, ceteris paribus, a group (G5) of 10 voters with ranking a b e c d joins the electorate. As a result, we obtain the following paired comparison matrix: a b c d e a b c d e From this matrix, it is possible to see that despite the increase in a s support it would still take at least 9 voters from group G1 to invert in their rankings b a to a b in order for a to become the Condorcet winner, whereas now for b to become the Condorcet winner only 4 voters in group G4 would have to invert e a to a e in their rankings, and thereafter to invert e b to b e i.e., a total of 8 inversions. So as the number of inversions needed for b to become the Condorcet winner is smallest, b would be elected under the Dodgson method thereby making the (additional) members of group G5 worse off.

17 Varieties of failure of monotonicity and participation 75 Table 1 Summary Sub-type Method P-R/AV Coombs Nanson Dodgson [M+COND+B] Yes Yes No No [M+COND+W] Yes No No No [M+CYC+B] Yes Yes Yes No [M+CYC+W] Yes No Yes Yes [P-BOT+COND+W] Yes No No No [P-BOT+CYC+W] Yes No No No [P-TOP+COND+W] No Yes No No [P-TOP+CYC+W] No Yes Yes Yes 6 Summary and conclusions Table 1 summarizes the types of examples provided under each of the investigated five voting methods. Yes means we provided an example, and No means we explained why no such example exists. On the basis of this table we can say that, a-priori, of the five investigated voting methods P-R and AV both of which are used in practice in public elections are vulnerable to more sub-types of the More-is-Less (non-monotonicity) paradox when the electorate remains fixed than the other three methods, and they are the only methods that are vulnerable to the [P-BOT] paradox. On the other hand, both P-R and AV are not at all vulnerable to the [P-TOP] paradox, while the Coombs method is more vulnerable to this paradox than either the Nanson method or the Dodgson method. Some of the sub-types of non-monotonicity or the sub-types of its closely related paradoxes have either been totally ignored in the literature, or at least not explored in examples under one or more of the investigated methods. We therefore hope that our article sheds some new light on those paradoxes afflicting the investigated voting methods. As stated at the outset, although all proposed single-winner voting methods involving three or more candidates suffer from some defect, it is generally agreed among social choice theorists that a voting method that is susceptible to non-monotonicity suffers from a particularly serious defect. So why are some of these methods actually used? The answer is probably that, if instances of non-monotonicity arise in actual elections, voters or analysts would generally not know that the outcome of the election exemplified some type of non-monotonicity or a closely related paradox, because they would generally not have access to all voters ballots (and hence would not be able to verify how all other voters ranked the competing candidates). 14 We therefore wonder whether, for example, there would have been a public outcry in France to 14 Potthoff (2011) suggests in a recent article how to construct public opinion polls which may identify the Condorcet winner, if one exists, and thus enable some voters who would not otherwise vote (strategically) for the Condorcet winner to do so and thereby obtain an outcome they may prefer.

18 76 D. S. Felsenthal, N. Tideman abolish the P-R voting method in electing its president if, following the runoff of a particular presidential election, it would have been reliably ascertained that, ceteris paribus, presidential candidate x would have been elected if fewer voters had voted for x in the first round. 15 In any case, it would be interesting to verify in future research the a-posteriori likelihood of the various sub-types of non-monotonicity and closely related paradoxes occurring under the five investigated methods, either by gaining access to the ballots cast in actual elections conducted under one or more of these methods, or by means of suitable simulations. Acknowledgments The authors are grateful for comments made by an anonymous reviewer. References Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press. Campbell, D. E., & Kelly, J. S. (2002). Non-monotonicity does not imply the no-show paradox. Social Choice and Welfare, 19, Coombs, C. H. (1964). A theory of data. New York: Wiley. Coombs, C. H., Cohen, J. L., & Chamberlin, J. R. (1984). An empirical study of some election systems. American Psychologist, 39, Felsenthal, D. S., & Maoz, Z. (1992). Normative properties of four single-stage multi-winner electroral procedures. Behavioral Science, 37, Fishburn, P. C. (1977). Condorcet social choice functions. SIAM Journal of Applied Mathematics, 33, Fishburn, P. C. (1982). Monotonicity paradoxes in the theory of voting. Discrete Applied Mathematics, 4, Fishburn, P. C., & Brams, S. J. (1983). Paradoxes of preferential voting. Mathematics Magazine, 56, Gierzynski, A., Hamilton, W., & Smith, W. (2009). Burlington Vermont 2009 IRV mayor election: Thwarted-majority, non-monotonicity & other failures (oops). Range Voting Organization. Accessed August 23, 2011, from McLean, I. & Urken, A. B. (Eds. & trans). (1995). Classics of social choice. Ann Arbor: University of Michigan Press. Miller, R. N. (2012). Monotonicity failure in IRV elections with three candidates. Paper presented at the world meeting of the public choice societies, Miami, March 8 12, Accessed March 15, 2012, from Moulin, H. (1988). Condorcet s principle implies the no show paradox. Journal of Economic Theory, 45, Nanson, E. J. (1883). Methods of elections. Transactions and Proceedings of the Royal Society of Victoria, 19, In I. McLean & A. B. Urken (Eds. & trans), Classics of social choice (pp ). Ann Arbor: University of Michigan Press, Nurmi, H. (1999). Voting paradoxes and how to deal with them. Heidelberg: Springer. 15 A display of non-monotonicity under the Alternative Vote method has actually occurred and reported recently in the March 2009 mayoral election in Burlington, Vermont. Among the three biggest vote getters, the Republican got the most first-place votes, the Democrat the fewest, and the Progressive won after the Democrat was eliminated. Yet if many of those who ranked the Republican first had ranked the Progressive first, the Republican would have been eliminated and the Progressive would have lost to the Democrat. In March 2010, Burlington replaced the Alternative Vote for electing its mayor with the Plurality with Runoff method which is also susceptible to non-monotonicity. See the detailed report on this election in Gierzynski et al. (2009). Smith (2007) analyses the Irish 1990 presidential election which was conducted under AV and concludes (p. 3) that it seems clearest to regard this election as suffering from a hybrid form of non-monotonicity (type II) and no-show paradoxes.

19 Varieties of failure of monotonicity and participation 77 Potthoff, R. F. (2011). Condorcet polling. Public Choice, 148, Smith, W. D. (2007). The Irish 1990 presidential election, won by Mary Robinson: Success or failure for instant runoff voting (ITV)? Range Voting Organization. Accessed August 23, 2011, from Straffin, P. D. (1980). Topics in the theory of voting. Boston: Birkhäuser. Woodall, D. R. (1997). Monotonicity of single-seat preferential election methods. Discrete Applied Mathematics, 77,