In social choice, a no-show paradox is a voting pathology in which a candidate loses an election because they have "too many" supporters.[1][2] More formally, a no-show paradox occurs when adding voters who prefer Alice to Bob causes Alice to lose the election to Bob.[3] Voting systems that do not exhibit the no-show paradox are satisfy the participation criterion.[4] In systems that fail the participation criterion, a voter can end up effectively disenfranchised by the electoral system, because turning out to vote would make the result worse. Such voters are sometimes referred to as having negative vote weights, particularly in the context of German constitutional law, where courts have ruled negative vote weights violate the principle of one man, one vote.[5][6][7]
The criterion can also be justified as a weaker form of strategyproofness: while it is impossible for honesty to always be a perfect strategy (by Gibbard's theorem), the participation criterion guarantees honesty will always be an effective, rather than actively counterproductive, strategy (i.e. a voter may safely cast a sincere vote, which is guaranteed to improve the outcome).[1] However, no-show paradoxes are generally described as undesirable because they expose a logical contradiction in the underlying system, rather than grounds of strategic voting.[8]
Positional methods and score voting satisfy the participation criterion. All deterministic voting rules that satisfy pairwise majority-rule[1][9] can fail in situations involving four-way cyclic ties, though such scenarios are empirically rare, and the randomized Condorcet rule is not affected by the pathology. More notably, ranked choice voting (RCV) and the two-round system fail the participation criterion with high frequency in competitive elections, typically as a result of a center squeeze.[2][3][10]
The no-show paradox is similar to, but not the same as, perverse response. Perverse response happens when voters who are already in the electorate cause a candidate to win by increasing that candidate's rating, or vice versa. For example, under ranked-choice voting, moving a candidate from first- to last-place on a voter's ballot may cause them to win.[11]
Noncompliant methods
editRanked-Choice Voting
editThe most common cause of no-show paradoxes is the use of instant-runoff (often called ranked-choice voting in the United States). In instant-runoff voting, a no-show paradox can occur even in elections with only three candidates, and occur in 50%-60% of all 3-candidate elections where the results of IRV disagree with those of plurality.[10][3]
A notable example is given in the 2009 Burlington mayoral election, the United States' second instant-runoff election in the modern era, where Bob Kiss won the election as a result of 750 ballots ranking him in last place.[12]
An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, say that a group of pro-Top voters joins the election, making the electorate more supportive of the Top party, and more strongly opposed to the Bottom party. This increase in the number of voters who rank Bottom last causes the Center candidate to lose to the Bottom party:
| More-popular Bottom | Less-popular Bottom | |||||
|---|---|---|---|---|---|---|
| Round 1 | Round 2 | Round 1 | Round 2 | |||
| Top | 
|
+6 | Top | 31 | 46 | |
| Center | 30 | 55 
|
Center | |
||
| Bottom | 39 | 39 | Bottom | 39 | 54
| |
Thus the increase in support for the Top party allows it to defeat the Center party in the first round. This makes the election an example of a center-squeeze, a class of elections where instant-runoff and plurality have difficulty electing the majority-preferred candidate.[13]
Condorcet methods
editWhen there are at most 3 major candidates, Minimax Condorcet and its variants (such as Ranked Pairs and Schulze's method) satisfy the participation criterion.[14] However, with more than 3 candidates, Hervé Moulin proved that every deterministic Condorcet method can sometimes fail participation.[14][15] Similar incompatibilities have also been proven for set-valued voting rules.[15][16][17]
Studies suggest such failures may be empirically rare, however. One study surveying 306 publicly-available election datasets found no participation failures for methods in the ranked pairs-minimax family.[18]
Certain conditions weaker than the participation criterion are also incompatible with the Condorcet criterion. For example, weak positive involvement requires that adding a ballot in which candidate A is one of the voter's most-preferred candidates does not change the winner away from A. Similarly, weak negative involvement requires that adding a ballot in which A is one of the voter's least-preferred does not make A the winner if it was not the winner before. Both conditions are incompatible with the Condorcet criterion.[19]
In fact, an even weaker property can be shown to be incompatible with the Condorcet criterion: it can be better for a voter to submit a completely reversed ballot than to submit a ballot that ranks all candidates honestly.[20]
Quota rules
editProportional representation systems using largest remainders for apportionment (such as STV or Hamilton's method) allow for no-show paradoxes.
In Germany, situations where a voter's ballot has the opposite of its intended effect (e.g. a vote for a party or candidate causes them to lose) are called Negatives Stimmgewicht (lit. 'negative voting weights'). An infamous example occurred in the 2005 German federal election, when an article in Der Spiegel laid out how CDU voters in Dresden's 2nd district would have to vote against their own party if they wished to avoid losing a seat in the Bundestag.[5] This led to a lawsuit by electoral reform organization Mehr Demokratie and Alliance 90/The Greens, joined by the neo-Nazi National Democratic Party of Germany, who argued the election law was undemocratic.[21]
The Federal Constitutional Court agreed with the plaintiffs, ruling that negative vote weights violate the German constitution's guarantee of equal and direct suffrage. The majority wrote that:[6][22]
A seat allocation procedure that allows an increase in votes to lead to a loss of seats, or results in more seats being won if [proportionally] fewer votes are cast for it, contradicts the meaning and purpose of a democratic election[...]
Such nonsensical relationships between voting and electoral success not only impair the equality of the right to vote and the equal opportunities of the parties, but also the principle of a popular election, as it is no longer apparent to the voter how their vote results in the success or failure of a candidate.[...]
Negative vote weights cannot be accepted as constitutional on the premise that they cannot be predicted or planned, and thus can hardly be influenced by the individual voter. To what extent this is true can be set aside, as such arbitrary results make a mockery of the democratic competition for support from the electorate.
The ruling forced the Bundestag to abandon its old system of ignoring overhang seats at the state level, and instead adopt a new system of compensation involving leveling seats.[21]
Quorum requirements
editA common cause no-show paradoxes is the use of a quorum. For example, if a public referendum requires 50% turnout to be binding, additional "no" votes may push turnout above 50%, causing the measure to pass. A referendum that instead required a minimum number of yes votes (e.g. >25% of the population voting "yes") would pass the participation criterion.[23]
Many representative bodies have quorum requirements where the same dynamic can be at play. For example, the requirement for a two-thirds quorum in the Oregon Legislative Assembly effectively creates an unofficial two-thirds supermajority requirement for passing bills, and can result in a law passing if too many senators turn out to oppose it.[24] Deliberate ballot-spoiling strategies have been successful in ensuring referendums remain non-binding, as in the 2023 Polish referendum.
Examples
editMajority judgment
editThis example shows that majority judgment violates the participation criterion. Assume two candidates A and B with 5 potential voters and the following ratings:
| Candidates | # of
voters | |
|---|---|---|
| A | B | |
| Excellent | Good | 2 |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The two voters rating A "Excellent" are unsure whether to participate in the election.
Voters not participating
editAssume the 2 voters would not show up at the polling place.
The ratings of the remaining 3 voters would be:
| Candidates | # of
voters | |
|---|---|---|
| A | B | |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The sorted ratings would be as follows:
| Candidate |
| |||||||||
| A |
| |||||||||
| B |
| |||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Poor". Thus, A is elected majority judgment winner.
Voters participating
editNow, consider the 2 voters decide to participate:
| Candidates | # of
voters | |
|---|---|---|
| A | B | |
| Excellent | Good | 2 |
| Fair | Poor | 2 |
| Poor | Good | 1 |
The sorted ratings would be as follows:
| Candidate |
| |||||||||
| A |
| |||||||||
| B |
| |||||||||
|
Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is the majority judgment winner.
Condorcet methods
editThis example shows how Condorcet methods can violate the participation criterion when there is a preference paradox. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:
| Preferences | # of voters |
|---|---|
| A > D > B > C | 8 |
| B > C > A > D | 7 |
| C > D > B > A | 7 |
This gives the pairwise counting method:
X Y
|
A | B | C | D |
|---|---|---|---|---|
| A | 14 8
|
14 8
|
7 15
| |
| B | 8 14
|
7 15
|
15 7
| |
| C | 8 14
|
15 7
|
8 14
| |
| D | 15 7
|
7 15
|
14 8
|
|
| Pairwise results for X, won-tied-lost |
1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
| Pair | Winner |
|---|---|
| A (15) vs. D (7) | A 15 |
| B (15) vs. C (7) | B 15 |
| B (7) vs. D (15) | D 15 |
| A (8) vs. B (14) | B 14 |
| A (8) vs. C (14) | C 14 |
| C (14) vs. D (8) | C 14 |
Result: A > D, B > C and D > B are locked in (and the other three can't be locked in after that), so the full ranking is A > D > B > C. Thus, A is elected ranked pairs winner.
Voters participating
editNow, assume an extra 4 voters, in the top row, decide to participate:
| Preferences | # of voters |
|---|---|
| A > B > C > D | 4 |
| A > D > B > C | 8 |
| B > C > A > D | 7 |
| C > D > B > A | 7 |
The results would be tabulated as follows:
X Y
|
A | B | C | D |
|---|---|---|---|---|
| A | 14 12
|
14 12
|
7 19
| |
| B | 12 14
|
7 19
|
15 11
| |
| C | 12 14
|
19 7
|
8 18
| |
| D | 19 7
|
11 15
|
18 8
|
|
| Pairwise results for X, won-tied-lost |
1-0-2 | 2-0-1 | 2-0-1 | 1-0-2 |
The sorted list of victories would be:
| Pair | Winner |
|---|---|
| A (19) vs. D (7) | A 19 |
| B (19) vs. C (7) | B 19 |
| C (18) vs. D (8) | C 18 |
| B (11) vs. D (15) | D 15 |
| A (12) vs. B (14) | B 14 |
| A (12) vs. C (14) | C 14 |
Result: A > D, B > C and C > D are locked in first. Now, D > B can't be locked in since it would create a cycle B > C > D > B. Finally, B > A and C > A are locked in. Hence, the full ranking is B > C > A > D. Thus, B is elected ranked pairs winner by adding a set of voters who prefer A to B.
See also
editReferences
edit- ^ a b c Moulin, Hervé (1988-06-01). "Condorcet's principle implies the no show paradox". Journal of Economic Theory. 45 (1): 53–64. doi:10.1016/0022-0531(88)90253-0.
- ^ a b Doron, Gideon; Kronick, Richard (1977). "Single Transferrable Vote: An Example of a Perverse Social Choice Function". American Journal of Political Science. 21 (2): 303–311. doi:10.2307/2110496. ISSN 0092-5853. JSTOR 2110496.
- ^ a b c Ray, Depankar (1986-04-01). "On the practical possibility of a 'no show paradox' under the single transferable vote". Mathematical Social Sciences. 11 (2): 183–189. doi:10.1016/0165-4896(86)90024-7. ISSN 0165-4896.
- ^ Woodall, Douglas (December 1994). "Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994".
- ^ a b Pukelsheim, Friedrich (2014). Proportional representation : apportionment methods and their applications. Internet Archive. Cham; New York : Springer. ISBN 978-3-319-03855-1.
- ^ a b dpa (2013-02-22). "Bundestag beschließt neues Wahlrecht". Die Zeit (in German). ISSN 0044-2070. Retrieved 2024-05-02.
- ^ BVerfG, Urteil des Zweiten Senats vom 03. Juli 2008 - 2 BvC 1/07, 2 BvC 7/07 -, Rn. 1-145. 121 BVerfGE - 2 BvC 1/07, 2 BvC 7/07 - (BVerfG 03 July 2008). ECLI:DE:BVerfG:2008:cs20080703.2bvc000107
- ^ Holliday, Wesley H.; Pacuit, Eric (2023-08-29). "Split Cycle: a new Condorcet-consistent voting method independent of clones and immune to spoilers". Public Choice. 197 (1–2): 1–62. doi:10.1007/s11127-023-01042-3. ISSN 0048-5829.
Of course, a method not satisfying participation will incentivize some strategic non-voting, as the voters in question will have an incentive not to vote (sincerely). But again, all voting methods incentivize strategic behavior[...] By contrast, we are troubled by failures of positive or negative involvement, as this shows that the method responds in the wrong way to unequivocal support for (resp. rejection of) a candidate.
- ^ Brandt, Felix; Geist, Christian; Peters, Dominik (2016-01-01). "Optimal Bounds for the No-Show Paradox via SAT Solving". Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems. AAMAS '16. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 314–322. arXiv:1602.08063. ISBN 9781450342391.
- ^ a b McCune, David; Wilson, Jennifer (2024-04-07). "The Negative Participation Paradox in Three-Candidate Instant Runoff Elections". arXiv:2403.18857 [physics.soc-ph].
- ^ Fishburn, Peter C.; Brams, Steven J. (1983-01-01). "Paradoxes of Preferential Voting". Mathematics Magazine. 56 (4): 207–214. doi:10.2307/2689808. JSTOR 2689808.
- ^ Graham-Squire, Adam T.; McCune, David (2023-06-12). "An Examination of Ranked-Choice Voting in the United States, 2004–2022". Representation: 1–19. arXiv:2301.12075. doi:10.1080/00344893.2023.2221689.
- ^ Laslier, Jean-François; Sanver, M. Remzi, eds. (2010). Handbook on Approval Voting. Studies in Choice and Welfare. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 2. doi:10.1007/978-3-642-02839-7. ISBN 978-3-642-02838-0.
By eliminating the squeezing effect, Approval Voting would encourage the election of consensual candidates. The squeezing effect is typically observed in multiparty elections with a runoff. The runoff tends to prevent extremist candidates from winning, but a centrist candidate who would win any pairwise runoff (the "Condorcet winner") is also often "squeezed" between the left-wing and the right-wing candidates and so eliminated in the first round.
- ^ a b Moulin, Hervé (1988-06-01). "Condorcet's principle implies the no show paradox". Journal of Economic Theory. 45 (1): 53–64. doi:10.1016/0022-0531(88)90253-0.
- ^ a b Brandt, Felix; Geist, Christian; Peters, Dominik (2016-01-01). "Optimal Bounds for the No-Show Paradox via SAT Solving". Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems. AAMAS '16. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems: 314–322. arXiv:1602.08063. ISBN 9781450342391.
- ^ Pérez, Joaquín (2001-07-01). "The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences". Social Choice and Welfare. 18 (3): 601–616. CiteSeerX 10.1.1.200.6444. doi:10.1007/s003550000079. ISSN 0176-1714. S2CID 153489135.
- ^ Jimeno, José L.; Pérez, Joaquín; García, Estefanía (2009-01-09). "An extension of the Moulin No Show Paradox for voting correspondences". Social Choice and Welfare. 33 (3): 343–359. doi:10.1007/s00355-008-0360-6. ISSN 0176-1714. S2CID 30549097.
- ^ Mohsin, F., Han, Q., Ruan, S., Chen, P. Y., Rossi, F., & Xia, L. (2023, May). Computational Complexity of Verifying the Group No-show Paradox. In Proceedings of the 2023 International Conference on Autonomous Agents and Multiagent Systems (pp. 2877-2879).
- ^ Duddy, Conal (2013-11-29). "Condorcet's principle and the strong no-show paradoxes". Theory and Decision. 77 (2): 275–285. doi:10.1007/s11238-013-9401-4. hdl:10379/11267. ISSN 0040-5833.
- ^ Sanver, M. Remzi; Zwicker, William S. (2009-08-20). "One-way monotonicity as a form of strategy-proofness". International Journal of Game Theory. 38 (4): 553–574. doi:10.1007/s00182-009-0170-9. ISSN 0020-7276. S2CID 29563457.
- ^ a b Die Beschlüsse des Bundestages am 29. und 30. September. In: bundestag.de, abgerufen am 25. Juli 2012. Das Gesetz trat am 3. Dezember 2011 in Kraft: Neunzehntes Gesetz zur Änderung des Bundeswahlgesetzes (19. BWahlGÄndG). In: buzer.de, abgerufen am 25. Juli 2012.
- ^ BVerfG, Urteil des Zweiten Senats vom 03. Juli 2008 - 2 BvC 1/07, 2 BvC 7/07 -, Rn. 1-145. 121 BVerfGE - 2 BvC 1/07, 2 BvC 7/07 - (BVerfG 03 July 2008). ECLI:DE:BVerfG:2008:cs20080703.2bvc000107
- ^ Aguiar-Conraria, Luís; Magalhães, Pedro C. (2009-09-19). "Referendum design, quorum rules and turnout". Public Choice. 144 (1–2): 63–81. doi:10.1007/s11127-009-9504-1. ISSN 0048-5829.
- ^ Shumway, Julia (2023-06-14). "Oregon Democrats float new proposal to change quorum laws as Senate Republican walkout drags on • Oregon Capital Chronicle". Oregon Capital Chronicle. Retrieved 2024-08-15.
Further reading
edit- Woodall, Douglas R, "Monotonicity and Single-Seat Election Rules" Voting matters, Issue 6, 1996