Abstract
The phenomenon of the paradoxes of the largest remainders methods has been studied by numerous authors. Nevertheless, the examples presented in their studies do not deal with the case where a party’s possible additional votes can directly lead to a loss in the party’s number of representatives. This paradox, which can be called the no-show apportionment paradox, has not previously been mentioned in the literature. It is based on the assumption that a voter’s favourite party may lose a seat if the voter votes honestly but get more seats if he or she abstains. The article gives simple examples illustrating the no-show paradox in a party-list proportional representation electoral system using the quota method of apportionment. In these situations it would be better for the party if some of their voters abstained, otherwise it would lose one or more seats.
Introduction
The voter generally assumes that a preferred party will be more likely to gain seats if the voter votes for it and, of course, less likely if that voter abstains. The voter operates under the assumption that it is impossible to do any harm if he or she votes for a specific party. P. C. Fisburn and S. J. Brams (1983) exposed a phenomenon known as the “no-show paradox”, showing that in some situations it was rational for a voter to withhold his vote to achieve a better electoral outcome for the preferred alternative.
It is well known that the largest remainders methods are non-monotone. The largest remainders methods can cause voting paradoxes known as the population paradox, Alabama paradox or new state paradox (see the seminal work on apportionment by Balinski & Young 2001). However, in the apportionment literature relating to the non-monotonicity problem of the largest remainders methods, there is no mention of the possibility that a voter can directly hurt his or her favourite political party depending on how he or she votes. In the article it will be shown that the Slovak party-list proportional representation system is not immune to the no-show paradox.
Under normal circumstances, we would expect a voting system to satisfy the straightforward criterion that giving a vote to any candidate will not make the candidate lose. It is well-known (see e.g. Doron & Kronick, 1977; Brams & Fishburn, 1984 or Nurmi, 1999, 2004), however, that many two-round voting systems do not satisfy this condition. The no-show paradox has been studied by numerous authors, e.g. H. Moulin (1988), H. Nurmi (1999), D. Lepelley and V. Merlin (2001), and J. Pérez (2001). The examples presented in their studies use Condorcet voting methods (see Moulin, 1988) or scoring run-off methods (see Nurmi, 1999; Saari, 1994). The no-show paradox arises when one or more voters are better off not voting than casting a sincere ballot. That is why the paradox is also known as the abstention paradox. The strong version of the no-show paradox occurs when the voter’s most favoured political party wins more seats if he or she does not participate in the election. If he or she does participate and vote sincerely for the favoured party, the party may lose one or more seats. H. Nurmi (1999, p. 53) emphasizes that “it is not possible that the candidate ranked first by the abstainers would be elected when they abstain and not be elected when they vote”. The no-show paradox is based on hypothetical reasoning, the same is true about the additional support paradoxes. The no-show paradox (using Felsenthal’s terminology) is a conditional paradox, “where changing one relevant datum while holding constant all other relevant data leads to a ‘surprising’ and arguably undesirable outcome” (Felsenthal, 2010, p. 3)[1].
In the next section it will be shown that it is not only two-round voting systems that are vulnerable to the additional support, or monotonicity, paradox; party-list proportional representation systems (such as that in Slovakia) that use quota methods (the largest remainders methods) to allocate mandates to the parties are vulnerable to these paradoxes too.
No-show apportionment paradox
The effect of negative voting weight may lead to a situation where a vote can have the opposite effect to what the voter intended. Under certain circumstances it can be seen to be beneficial for a voter not to cast his or her vote for the preferred party or, from the point of view of the political party, the party might gain an extra seat by not receiving so many votes. It is possible to conclude that it would be better for that political party to give away some of its votes and thereby gain a seat. The no-show paradox can also be illustrated by an example in which one voter can change the voting results in a paradoxical way.
The allocation method used in Slovakia is defined in section 68 of Act No. 180/2014, Coll. on the conditions for the execution of voting rights and on amendments and supplements to certain laws as follows:
(1) The sum of valid votes cast for the proceeding political parties or coalitions shall be divided by the number 151 (the number of seats plus one). The result of this division rounded off to a whole number is the republic electoral number.
(2) The sum of valid votes obtained by a political party or coalition shall be divided by the republic electoral number; a political party or coalition shall be allocated a number of seats equal to the number of times the republic electoral number divides into the sum of valid votes which it obtained.
(3) In the event that there is allocated one seat more than should have been, the surplus seat shall be deducted from the political party or coalition which has the smallest remainder from the division. If remainders are equal, the seat shall be deducted from the political party or coalition which obtained the fewer number of votes. If the numbers of votes are equal, the deduction shall be decided by drawing lots.
(4) In the event that not all seats are allocated or that a political party or coalition should be allocated more seats than it has candidates, then the Central Electoral Commission shall allocate such seats among other political parties or coalitions in order of their remainder from the division beginning with the highest. If remainders are equal, the seat shall be allocated to the political party or coalition which obtained the larger number of votes. If the numbers of votes are the equal, the allocation shall be decided by drawing lots. [2]
The quota (the number of votes required for a seat) is calculated using the formula Q = [V/(S+1)+0.5].[3] The number of votes for each qualifying party is then divided by the electoral quota. The result for each party usually consists of an integer part plus a fractional remainder. In the first part of the allocation process each party receives the number of seats indicated by the integer. This process almost always leaves some seats unallocated. Unallocated seats are distributed according to the largest remainder methods, i.e. the parties are ranked on the basis of the fractional remainders, and the first unallocated seat goes to the party with the highest remainder. If there are any more unallocated seats, the second one goes to the party with the second highest remainder and so on. If the remainders of two or more parties are the same, the unallocated seat goes to the party with more votes. If two or more parties obtained the same number of votes, the deadlock is decided by a lot. In other words, the allocation of seats is generally performed in two rounds. In the first round, each party obtains one seat for each whole number produced—i.e. each party i receives xi = [vi/Q]. After the first round of seat allocation, if there are any unallocated seats, the remainders of the parties are compared and the parties with the largest remainders are allocated the remaining seats. The remainder of party i is calculated by the formula ri = vi – xiQ.
In the examples presented in Tables 1 and 2, the quota is calculated using the formula Q = [V/(S+1)+0.5]. In Table 1, party A gets the majority of seats (76). Adding one more vote to party A causes party A to lose its overall majority (Table 2)[4]. One vote for party A thus has a negative impact on the voting results.
| I | vi | qi = vi/Q | [qi] | Remainder (ri) | Additional seat (adsi) | ∑si |
|---|---|---|---|---|---|---|
| A | 683 622 | 76.0002 | 76 | 0.0002 | 76 | |
| B | 269 844 | 29.9993 | 29 | 0.9993 | 29 | |
| C | 162 000 | 18.0100 | 18 | 0.0100 | 18 | |
| D | 89 945 | 9.9994 | 9 | 0.9994 | 1 | 10 |
| E | 80 950 | 8.9994 | 8 | 0.9994 | 1 | 9 |
| F | 71 959 | 7.9998 | 7 | 0.9998 | 1 | 8 |
| Total | 1 358 320 | Q = 8 995 | 147 | 3 | 150 |
| i | vi | qi = vi/Q | [qi] | Remainder (ri) | Additional seat (adsi) | ∑si |
|---|---|---|---|---|---|---|
| A | 683 623 | 75.9918 | 75 | 0.9918 | 75 | |
| B | 269 844 | 29.9959 | 29 | 0.9959 | 1 | 30 |
| C | 162 000 | 18.0080 | 18 | 0.0080 | 18 | |
| D | 89 945 | 9.9983 | 9 | 0.9983 | 1 | 10 |
| E | 80 950 | 8.9984 | 8 | 0.9984 | 1 | 9 |
| F | 71 959 | 7.9989 | 7 | 0.9989 | 1 | 8 |
| Total | 1 358 321 | Q = 8 996 | 146 | 4 | 150 |
Balinski and Young point out that
if one party increases its vote total relative to another in two successive elections it is inconceivable that the first should lose seats to the second…. This is particularly important in PR systems since otherwise it would be possible for a party to misrepresent part of its vote by giving it to a rival party and thereby gain seats (Balinski & Young, 2001, p. 89).
In the paradox described above, however, party A would still lose one seat, even when one (additional) vote is given to any other party.
The no-show paradox as discussed by P. C. Fishburn and S. J. Brams (1983), H. Nurmi (1999), and D. Lepelley and V. Merlin (2001) hinges on the existence of a third alternative. In the following examples, it will be shown that the no-show paradox can also be observed in a party-list proportional representation system where there are two parties. This can be done using a formula such as Q = [V/S], Q = [V/S+0.5], Q = [V/(S+1)+0.5] or Q = [V/(S+1)+1]. For example, if party A receives 100 200 votes and party B 500 votes, the quota is Q = [V/S] = [100 700/400] = [251.75] = 251. Party A gets 399 seats and party B one seat. In the new scenario, party A receives 7 000 more votes (107 200 votes). The quota is Q = [V/S] = [107 700/400] = [269.25] = 269. Paradoxically, in the new scenario, party A gets mores votes but fewer seats (cf. Tables 3 and 4).
| i | vi | qi = vi /Q | [qi] | Remainder (ri) | Additional seat | ∑si |
|---|---|---|---|---|---|---|
| A | 100 200 | 399.203 | 399 | 0.203 | 0 | 399 |
| B | 500 | 1.992 | 1 | 0.992 | 0 | 1 |
| 100 700 | Q = 251 | 400 | 0 | 400 |
| i | vi | qi = vi /Q | [qi] | Remainder (ri) | Additional seat | ∑si |
|---|---|---|---|---|---|---|
| A | 107 200 | 398.513 | 398 | 0.513 | 0 | 398 |
| B | 500 | 1.858 | 1 | 0.858 | 1 | 2 |
| 107 700 | Q = 269 | 399 | 1 | 400 |
The no-show paradox can also be illustrated using similar examples for other quota formulae. For example, if we use quota Q = [V/S+0.5], the no-show paradox occurs when party A receives 101 189 votes and party B receives 210 votes. The quota is calculated using the formula Q = [V/S+0.5] = [101 399/400+0.5] = [253.9975] = 253. Party A gets all the seats. The no-show paradox can be seen in the new scenario—party A receives 65 000 more votes. In that scenario the quota changes: Q = [V/S+0.5] = [166 399/400+0.5] = [416.4975] = 416, and party A gets only 399 seats and party B one seat (cf. Tables 5 and 6).
| i | vi | qi = vi/Q | [qi] | Remainder (ri) | Additional seat (adsi) | ∑si |
|---|---|---|---|---|---|---|
| A | 101 189 | 399.956 | 399 | 0.956 | 1 | 400 |
| B | 210 | 0.830 | 0 | 0.830 | 0 | 0 |
| 101 399 | Q = 253 | 399 | 1 | 400 |
| i | vi | qi = vi /Q | [qi] | Remainder (ri) | Additional seat (adsi) | ∑si |
|---|---|---|---|---|---|---|
| A | 166 189 | 399.492 | 399 | 0.492 | 0 | 399 |
| B | 210 | 0.504 | 0 | 0.504 | 1 | 1 |
| 166 399 | Q = 416 | 399 | 1 | 400 |
Conclusions
Although not everyone agrees that non-monotonicity is a huge defect in electoral systems (see e. g. Austen-Smith & Banks, 1991 or Tideman, 2006), it can be concluded (at least from a moral point of view) that there is no place for largest remainders methods in partylist proportional representation systems. The electoral law should always protect the voters’ decisions, and the impact of their votes should always be positive. In this context H. Nurmi points out that
vulnerability to the no-show paradox is a serious drawback in a voting system. After all, any reasonable voter would expect that by voting he is contributing to the possibility that his favourite wins. The realization that the very act of communicating his true preferences by voting makes the outcome worse from his point of view than it would have been had he decided not to vote all, may be demoralizing (Nurmi, 1999, p. 53).
Similar conclusions were drawn by the Federal Constitutional Court of Germany (2 BvC 1/07. 2 BvC 7/07 – published 3 July 2008) in a judgment stating
the effect of negative voting weight violates the principles of the equality and directness of elections. The provision is therefore unconstitutional insofar as the effect of negative voting weight is facilitated thereby ... The equal contribution towards success requires the contribution of each vote towards success to be equal, whatever party it was cast for. This also means that it must be able to have a positive effect for the party for which it was cast. An electoral system which is designed or at least in typical constellations permits an increase in the number of votes to lead to a loss of mandates or the electoral proposal of a party to achieve more total mandates if it itself attracts fewer votes, ... , leads to arbitrary results and to a situation in which democratic competition for the approval of the electorate appears to be paradoxical (Press release no. 68/2008 of 3 July 2008)[5].
The electoral system should prevent even the slightest possibility of the occurrence of the no-show paradox. More support should always help a party. Abstention should not be a rational choice for a voter in a party-list proportional representation system. Methods of highest averages (divisor methods) are monotonic, so they are also immune to voting paradoxes. From that point of view, there is no rational reason for using largest remainders methods in party-list proportional representation.
References
Austen-Smith, D., & Banks, J. (1991). Monotonicity in electoral systems. American Political Science Review, 85(2), 531-537.10.2307/1963173Search in Google Scholar
Balinski, M. L., & Young, H. P. (2001). Fair representation. Meeting the ideal of one man, one vote. Washington, D. C.: Brookings.Search in Google Scholar
Brams, S. J., & Fishburn, P. C. (1984). Some logical defects of the single transferable vote. In A. Lijphart & B. Grofman (Eds.), Choosing an electoral system: Issues and alternatives (pp. 147-151). New York: Praeger.Search in Google Scholar
Doron, G., & Kronick, R. (1977). Single transferable vote: An example of a perverse social choice function. American Journal of Political Science, 21(2), 303-311.10.2307/2110496Search in Google Scholar
Felsenthal, D. S. (2010). Review of paradoxes afflicting various voting procedures where one out of m candidates (m ≥ 2) must be elected. Paper presented at the Leverhulme Trust sponsored 2010 Voting Power in Practice workshop held at Chateau du Baffy, Normandy, France, 30 July – 2 August, 2010.Search in Google Scholar
Fishburn, P. C., & Brams, S. J. (1983). Paradoxes of preferential voting. Mathematics Magazine, 56(4), 207-214.10.1080/0025570X.1983.11977044Search in Google Scholar
Lepelley, D., & Merlin, V. (2001). Scoring run-off paradoxes for variable electorates. Economic Theory, 17 (1), 53-80.10.1007/PL00004103Search in Google Scholar
Moulin, H. (1988). Condorcet’s principle implies the no show paradox. Journal of Economic Theory, 45(1), 53-64.10.1016/0022-0531(88)90253-0Search in Google Scholar
Nurmi, H. (1987). Comparing voting systems. Dordrecht: D. Reidel.10.1007/978-94-009-3985-1Search in Google Scholar
Nurmi, H. (1999). Voting paradoxes and how to deal with them. Berlin – Heidelberg – New York: Springer.10.1007/978-3-662-03782-9Search in Google Scholar
Nurmi, H. (2004). Monotonicity and its cognates in the theory of choice. Public Choice, 121(1-2), 25-49.10.1007/s11127-004-4567-5Search in Google Scholar
Owen, G., & Grofman, B. (1984). To vote or not to vote: The paradox of nonvoting. Public Choice, 42, 311-325.10.1007/BF00124949Search in Google Scholar
Pérez, J. (2001). The strong no show paradoxes are a common flaw in Condorcet voting correspondences. Social Choice and Welfare, 18(3), 601-616.10.1007/s003550000079Search in Google Scholar
Pukelsheim, F. (2014). Proportional representation. Apportionment methods and their applications. Dordrecht: Springer.10.1007/978-3-319-03856-8Search in Google Scholar
Saari, D. G. (1994). Geometry of voting. Berlin – Heidelberg – New York: Springer.10.1007/978-3-642-48644-9Search in Google Scholar
Tideman, N. (2006). Collective decisions and voting: The potential for public choice. Aldershot: Ashgate.Search in Google Scholar
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