# Participation criterion

The participation criterion, also called vote or population monotonicity, is a voting system criterion that says that a candidate should never lose an election because they have "too much support." It says that adding voters who support A over B should not cause A to lose the election to B.[1]

Voting systems that fail the participation criterion exhibit the no-show paradox, and allow voters to do better by not showing up to vote.[2] Such voting systems arguably violate the principle of one man, one vote, as participation failures are cases with "one man, negative-one votes"--a vote can sometimes be worth less than nothing.

Positional and score voting methods satisfy the participation criterion, while all Condorcet methods[3][4] and highest median methods[5] can fail in some situations. Failures are especially common for runoff-based methods like IRV,[6] with one analysis estimating Louisiana's runoff system for gubernatorial elections experienced a roughly 40% failure rate.

## Common failure modes

The most common failure of the participation criterion is caused by the use of runoff systems (including instant-runoff voting).

### Incompatibility with the Condorcet criterion

When there are 3 candidates, minimax satisfies the participation criterion.[3] However, with more than 3 candidates, every electoral system is guaranteed to fail the participation criterion.[3][4] Similar incompatibilities have also been proven for set-valued voting rules.[4][7][8]

Certain conditions that are 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 if one allows ballots to include ties.[9] If the ballots must express A as their sole favorite, then there exist Condorcet methods that pass.

Another condition that is weaker than participation is half-way monotonicity, which requires that a voter cannot be better off by completely reversing their ballot. Again, half-way monotonicity is incompatible with the Condorcet criterion.[10]

## Examples

### Copeland

This example shows that Copeland's method violates the participation criterion. Assume four candidates A, B, C and D with 13 potential voters and the following preferences:

Preferences # of voters
A > B > C > D 3
A > C > D > B 1
A > D > C > B 1
B > A > C > D 4
D > C > B > A 4

The three voters with preferences A > B > C > D are unconfident whether to participate in the election.

#### Voters not participating

Assume the 3 voters would not show up at the polling place.

The preferences of the remaining 10 voters would be:

Preferences # of voters
A > C > D > B 1
A > D > C > B 1
B > A > C > D 4
D > C > B > A 4

The results would be tabulated as follows:

Pairwise election results
X
A B C D
Y A [X] 8
[Y] 2
[X] 4
[Y] 6
[X] 4
[Y] 6
B [X] 2
[Y] 8
[X] 6
[Y] 4
[X] 6
[Y] 4
C [X] 6
[Y] 4
[X] 4
[Y] 6
[X] 5
[Y] 5
D [X] 6
[Y] 4
[X] 4
[Y] 6
[X] 5
[Y] 5
Pairwise results for X,
won-tied-lost
2-0-1 1-0-2 1-1-1 1-1-1

Result: A can defeat two of the three opponents, whereas no other candidate wins against more than one opponent. Thus, A is elected Copeland winner.

#### Voters participating

Now, consider the three unconfident voters decide to participate:

Preferences # of voters
A > B > C > D 3
A > C > D > B 1
A > D > C > B 1
B > A > C > D 4
D > C > B > A 4

The results would be tabulated as follows:

Pairwise election results
X
A B C D
Y A [X] 8
[Y] 5
[X] 4
[Y] 9
[X] 4
[Y] 9
B [X] 5
[Y] 8
[X] 6
[Y] 7
[X] 6
[Y] 7
C [X] 9
[Y] 4
[X] 7
[Y] 6
[X] 5
[Y] 8
D [X] 9
[Y] 4
[X] 7
[Y] 6
[X] 8
[Y] 5
Pairwise results for X,
won-tied-lost
2-0-1 3-0-0 1-0-2 0-0-3

Result: B is the Condorcet winner and thus, B is Copeland winner, too.

#### Conclusion

By participating in the election the three voters supporting A would change A from winner to loser. Their first preferences were not sufficient to change the one pairwise defeat A suffers without their support. But, their second preferences for B turned both defeats B would have suffered into wins and made B Condorcet winner and thus, overcoming A.

Hence, Copeland fails the participation criterion.

### Instant-runoff voting

This example shows that instant-runoff voting violates the participation criterion. Assume three candidates A, B and C and 15 potential voters, two of them (in blue) unconfident whether to vote.

Preferences # of voters
A > B > C 2
A > B > C 3
B > C > A 4
C > A > B 6

#### Voters not participating

If they don't show up at the election the remaining voters would be:

Preferences # of voters
A > B > C 3
B > C > A 4
C > A > B 6

The following outcome results:

1st 2nd
A 3
B 4 7
C 6 6

Result: After A is eliminated first, B gets their votes and wins.

#### Voters participating

If they participate in the election, the preferences list is:

Preferences # of voters
A > B > C 5
B > C > A 4
C > A > B 6

The outcome changes as follows:

1st 2nd
A 5 5
B 4
C 6 10

Result: Now, B is eliminated first and C gets their votes and wins.

#### Conclusion

The additional votes for A were not sufficient for winning, but for descending to the second round, thereby eliminating the second preference of the voters. Thus, due to participating in the election, the voters changed the winner from their second preference to their strictly least preference.

Thus, instant-runoff voting fails the participation criterion.

### Kemeny–Young method

This example shows that the Kemeny–Young method violates the participation criterion. Assume four candidates A, B, C, D with 21 voters and the following preferences:

Preferences # of voters
A > B > C > D 3
A > C > B > D 3
A > D > C > B 4
B > A > D > C 4
C > B > A > D 2
D > B > A > C 2
D > C > B > A 3

The three voters with preferences A > B > C > D are unconfident whether to participate in the election.

#### Voters not participating

Assume the 3 voters would not show up at the polling place.

The preferences of the remaining 18 voters would be:

Preferences # of voters
A > C > B > D 3
A > D > C > B 4
B > A > D > C 4
C > B > A > D 2
D > B > A > C 2
D > C > B > A 3

The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:

Candidate pairs Number who prefer…
X Y X Neither Y
A B 7 0 11
A C 13 0 5
A D 13 0 5
B C 6 0 12
B D 9 0 9
C D 5 0 13

Result: The ranking A > D > C > B has the highest ranking score of 67 (= 13 + 13 + 13 + 12 + 9 + 7); against e.g. 65 (= 13 + 13 + 13 + 11 + 9 + 6) of B > A > D > C. Thus, A is Kemeny-Young winner.

#### Voters participating

Now, consider the 3 unconfident voters decide to participate:

Preferences # of voters
A > B > C > D 3
A > C > B > D 3
A > D > C > B 4
B > A > D > C 4
C > B > A > D 2
D > B > A > C 2
D > C > B > A 3

The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:

Candidate pairs Number who prefer…
X Y X Neither Y
A B 10 0 11
A C 16 0 5
A D 16 0 5
B C 9 0 12
B D 12 0 9
C D 8 0 13

Result: The ranking B > A > D > C has the highest ranking score of 77 (= 16 + 16 + 13 + 12 + 11 + 9); against e.g. 76 (= 16 + 16 + 13 + 12 + 10 + 9) of A > D > C > B. Thus, B is Kemeny-Young winner.

#### Conclusion

By participating in the election the three voters supporting A would change A from winner to loser. Their ballots support 3 of the 6 pairwise comparisons of the ranking A > D > C >B, but four pairwise comparisons of the ranking B > A > D > C, enough to overcome the first one.

Thus, Kemeny-Young fails the participation criterion.

### Majority judgment

This 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 unconfident whether to participate in the election.

#### Voters not participating

Assume 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
 ↓ Median point
A
B

 Excellent Good Fair Poor

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

Now, consider the 2 unconfident 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
 ↓ Median point
A
B

 Excellent Good Fair Poor

Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is the majority judgment winner.

#### Conclusion

By participating in the election the two voters preferring A would change A from winner to loser. Their "Excellent" rating for A was not sufficient to change A's median rating since no other voter rated A higher than "Fair". But, their "Good" rating for B turned B's median rating to "Good", since another voter agreed with this rating.

Thus, majority judgment fails the participation criterion.

### Minimax

This example shows that the minimax method violates the participation criterion. Assume four candidates A, B, C, D with 18 potential voters and the following preferences:

Preferences # of voters
A > B > C > D 2
A > B > D > C 2
B > D > C > A 6
C > A > B > D 5
D > A > B > C 1
D > C > A > B 2

Since all preferences are strict rankings (no equals are present), all three minimax methods (winning votes, margins and pairwise opposite) elect the same winners.

The two voters (in blue) with preferences A > B > C > D are unconfident whether to participate in the election.

#### Voters not participating

Assume the two voters would not show up at the polling place.

The preferences of the remaining 16 voters would be:

Preferences # of voters
A > B > D > C 2
B > D > C > A 6
C > A > B > D 5
D > A > B > C 1
D > C > A > B 2

The results would be tabulated as follows:

Pairwise election results
X
A B C D
Y A [X] 6
[Y] 10
[X] 13
[Y] 3
[X] 9
[Y] 7
B [X] 10
[Y] 6
[X] 7
[Y] 9
[X] 3
[Y] 13
C [X] 3
[Y] 13
[X] 9
[Y] 7
[X] 11
[Y] 5
D [X] 7
[Y] 9
[X] 13
[Y] 3
[X] 5
[Y] 11
Pairwise results for X,
won-tied-lost
1-0-2 2-0-1 1-0-2 2-0-1
Worst opposing votes 13 10 11 13
Worst margin 10 4 6 10
Worst opposition 13 10 11 13
• [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

Result: B has the closest biggest defeat. Thus, B is elected minimax winner.

#### Voters participating

Now, consider the two unconfident voters decide to participate:

Preferences # of voters
A > B > C > D 2
A > B > D > C 2
B > D > C > A 6
C > A > B > D 5
D > A > B > C 1
D > C > A > B 2

The results would be tabulated as follows:

Pairwise election results
X
A B C D
Y A [X] 6
[Y] 12
[X] 13
[Y] 5
[X] 9
[Y] 9
B [X] 12
[Y] 6
[X] 7
[Y] 11
[X] 3
[Y] 15
C [X] 5
[Y] 13
[X] 11
[Y] 7
[X] 11
[Y] 7
D [X] 9
[Y] 9
[X] 15
[Y] 3
[X] 7
[Y] 11
Pairwise results for X,
won-tied-lost
1-1-1 2-0-1 1-0-2 1-1-1
Worst opposing votes 13 12 11 15
Worst margin 8 6 4 8
Worst opposition 13 12 11 15

Result: C has the closest biggest defeat. Thus, C is elected minimax winner.

#### Conclusion

By participating in the election the two voters changed the winner from B to C whilst strictly preferring B to C. Their preferences of B over C and D does not advance B's minimax value since B's biggest defeat was against A. Also, their preferences of A and B over C does not degrade C's minimax value since C's biggest defeat was against D. Therefore, only the comparison "A > B" degrade B's value and the comparison "C > D" advanced C's value. This results in C overcoming B.

Thus, the minimax method fails the participation criterion.

### Ranked pairs

This example shows that the ranked pairs method violates the participation criterion. Assume four candidates A, B, C and D with 26 potential voters and the following preferences:

Preferences # of voters
A > B > C > D 4
A > D > B > C 8
B > C > A > D 7
C > D > B > A 7

The four voters with preferences A > B > C > D are unconfident whether to participate in the election.

#### Voters not participating

Assume the 4 voters do not show up at the polling place.

The preferences of the remaining 22 voters would be:

Preferences # of voters
A > D > B > C 8
B > C > A > D 7
C > D > B > A 7

The results would be tabulated as follows:

Pairwise election results
X
A B C D
Y A [X] 14
[Y] 8
[X] 14
[Y] 8
[X] 7
[Y] 15
B [X] 8
[Y] 14
[X] 7
[Y] 15
[X] 15
[Y] 7
C [X] 8
[Y] 14
[X] 15
[Y] 7
[X] 8
[Y] 14
D [X] 15
[Y] 7
[X] 7
[Y] 15
[X] 14
[Y] 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

Now, consider the 4 unconfident voters 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:

Pairwise election results
X
A B C D
Y A [X] 14
[Y] 12
[X] 14
[Y] 12
[X] 7
[Y] 19
B [X] 12
[Y] 14
[X] 7
[Y] 19
[X] 15
[Y] 11
C [X] 12
[Y] 14
[X] 19
[Y] 7
[X] 8
[Y] 18
D [X] 19
[Y] 7
[X] 11
[Y] 15
[X] 18
[Y] 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.

#### Conclusion

By participating in the election the four voters supporting A would change A from winner to loser. The clear victory of D > B was essential for A's win in the first place. The additional votes diminished that victory and at the same time gave a boost to the victory of C > D, turning D > B into the weakest link of the cycle B > C > D > B. Since A had no other victories but the one over D and B had no other losses but the one over D, the elimination of D > B made it impossible for A to win.

Thus, the ranked pairs method fails the participation criterion.

### Schulze method

This example shows that the Schulze method violates the participation criterion. Assume four candidates A, B, C and D with 25 potential voters and the following preferences:

Preferences # of voters
A > B > C > D 2
B > A > D > C 7
B > C > A > D 1
B > D > C > A 2
C > A > D > B 7
D > B > A > C 2
D > C > A > B 4

The two voters with preferences A > B > C > D are unconfident whether to participate in the election.

#### Voters not participating

Assume the two voters would not show up at the polling place.

The preferences of the remaining 23 voters would be:

Preferences # of voters
B > A > D > C 7
B > C > A > D 1
B > D > C > A 2
C > A > D > B 7
D > B > A > C 2
D > C > A > B 4

The pairwise preferences would be tabulated as follows:

Matrix of pairwise preferences
d[·, A] d[·, B] d[·, C] d[·, D]
d[A, ·] 11 9 15
d[B, ·] 12 12 10
d[C, ·] 14 11 8
d[D, ·] 8 13 15

Now, the strongest paths have to be identified, e.g. the path A > D > B is stronger than the direct path A > B (which is nullified, since it is a loss for A).

Strengths of the strongest paths
p[·, A] p[·, B] p[·, C] p[·, D]
p[A, ·] 13 15 15
p[B, ·] 12 12 12
p[C, ·] 14 13 14
p[D, ·] 14 13 15

Result: The full ranking is A > D > C > B. Thus, A is elected Schulze winner.

#### Voters participating

Now, consider the 2 unconfident voters decide to participate:

Preferences # of voters
A > B > C > D 2
B > A > D > C 7
B > C > A > D 1
B > D > C > A 2
C > A > D > B 7
D > B > A > C 2
D > C > A > B 4

The pairwise preferences would be tabulated as follows:

Matrix of pairwise preferences
d[·, A] d[·, B] d[·, C] d[·, D]
d[A, ·] 13 11 17
d[B, ·] 12 14 12
d[C, ·] 14 11 10
d[D, ·] 8 13 15

Now, the strongest paths have to be identified, e.g. the path C > A > D is stronger than the direct path C > D.

Strengths of the strongest paths
p[·, A] p[·, B] p[·, C] p[·, D]
p[A, ·] 13 15 17
p[B, ·] 14 14 14
p[C, ·] 14 13 14
p[D, ·] 14 13 15

Result: The full ranking is B > A > D > C. Thus, B is elected Schulze winner.

#### Conclusion

By participating in the election the two voters supporting A changed the winner from A to B. In fact, the voters can turn the defeat in direct pairwise comparison of A against B into a victory. But in this example, the relation between A and B does not depend on the direct comparison, since the paths A > D > B and B > C > A are stronger. The additional voters diminish D > B, the weakest link of the A > D > B path, while giving a boost to B > C, the weakest link of the path B > C > A.

Thus, the Schulze method fails the participation criterion.

## References

1. ^ Woodall, Douglas (December 1994). "Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994".
2. ^ 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.
3. ^ 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.
4. ^ a b c 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. ISBN 9781450342391.
5. ^ Markus Schulze (1998-06-12). "Regretted Turnout. Insincere = ranking". Retrieved 2011-05-14.
6. ^ Warren D. Smith. "Lecture "Mathematics and Democracy"". Retrieved 2011-05-12.
7. ^ Pérez, Joaquı´n (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.
8. ^ 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.
9. ^ 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.
10. ^ 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.