# Later-no-help criterion

The later-no-help criterion (or LNHe, not to be confused with LNH) is a voting system criterion formulated by Douglas Woodall. The criterion is satisfied if, in any election, a voter giving an additional ranking or positive rating to a less-preferred candidate can not cause a more-preferred candidate to win. Voting systems that fail the later-no-help criterion are vulnerable to the tactical voting strategy called mischief voting, which can deny victory to a sincere Condorcet winner.

## Complying methods

Approval, instant-runoff, highest medians, and score all satisfy the later-no-help criterion. Plurality voting satisfies it trivially (as plurality only applies to the top-ranked candidate). Descending Solid Coalitions also satisfies later-no-help.

## Noncomplying methods

All Minimax Condorcet methods, Ranked Pairs, Schulze method, Kemeny-Young method, Copeland's method, and Nanson's method do not satisfy later-no-help. The Condorcet criterion is incompatible with later-no-help.

## Checking Compliance

Checking for failures of the Later-no-help criterion requires ascertaining the probability of a voter's preferred candidate being elected before and after adding a later preference to the ballot, to determine any increase in probability. Later-no-help presumes that later preferences are added to the ballot sequentially, so that candidates already listed are preferred to a candidate added later.

## Examples
### Anti-plurality

Anti-plurality elects the candidate the fewest voters rank last when submitting a complete ranking of the candidates.
Later-No-Help can be considered not applicable to Anti-Plurality if the method is assumed to not accept truncated preference listings from the voter. On the other hand, Later-No-Help can be applied to Anti-Plurality if the method is assumed to apportion the last place vote among unlisted candidates equally, as shown in the example below.

#### Truncated Ballot Profile

Assume four voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ A > C > B:
| # of voters | Preferences |
| 2 | A ( > B > C) |
| 2 | A ( > C > B) |
| 4 | B > A > C |
| 3 | C > B > A |
Result: A is listed last on 3 ballots; B is listed last on 2 ballots; C is listed last on 6 ballots. B is listed last on the least ballots. B wins. A loses.

#### Adding Later Preferences

Now assume that the four voters supporting A (marked bold) add later preference C, as follows:
| # of voters | Preferences |
| 4 | A > C > B |
| 4 | B > A > C |
| 3 | C > B > A |
Result: A is listed last on 3 ballots; B is listed last on 4 ballots; C is listed last on 4 ballots. A is listed last on the least ballots. A wins.

#### Conclusion

The four voters supporting A increase the probability of A winning by adding later preference C to their ballot, changing A from a loser to the winner. Thus, Anti-plurality fails the Later-no-help criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

### Coombs' method

Coombs' method repeatedly eliminates the candidate listed last on most ballots, until a winner is reached. If at any time a candidate wins an absolute majority of first place votes among candidates not eliminated, that candidate is elected.
Later-No-Help can be considered not applicable to Coombs if the method is assumed to not accept truncated preference listings from the voter. On the other hand, Later-No-Help can be applied to Coombs if the method is assumed to apportion the last place vote among unlisted candidates equally, as shown in the example below.

#### Truncated Ballot Profile

Assume four voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ A > C > B:
| # of voters | Preferences |
| 2 | A ( > B > C) |
| 2 | A ( > C > B) |
| 4 | B > A > C |
| 4 | C > B > A |
| 2 | C > A > B |
Result: A is listed last on 4 ballots; B is listed last on 4 ballots; C is listed last on 6 ballots. C is listed last on the most ballots. C is eliminated, and B defeats A pairwise 8 to 6. B wins. A loses.

#### Adding Later Preferences

Now assume that the four voters supporting A (marked bold) add later preference C, as follows:
| # of voters | Preferences |
| 4 | A > C > B |
| 4 | B > A > C |
| 4 | C > B > A |
| 2 | C > A > B |
Result: A is listed last on 4 ballots; B is listed last on 6 ballots; C is listed last on 4 ballots. B is listed last on the most ballots. B is eliminated, and A defeats C pairwise 8 to 6. A wins.

#### Conclusion

The four voters supporting A increase the probability of A winning by adding later preference C to their ballot, changing A from a loser to the winner. Thus, Coombs' method fails the Later-no-help criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

### Copeland

This example shows that Copeland's method violates the Later-no-help criterion. Assume four candidates A, B, C and D with 7 voters:

#### Truncated preferences

Assume that the two voters supporting A (marked bold) do not express later preferences on the ballots:
| # of voters | Preferences |
| 2 | A |
| 3 | B > A |
| 1 | C > D > A |
| 1 | D > C |
The results would be tabulated as follows:
Pairwise election results :
| | | X | X | X | X |
| | | A | B | C | D |
| Y | A | | [X] 3 [Y] 3 | [X] 2 [Y] 5 | [X] 2 [Y] 5 |
| Y | B | [X] 3 [Y] 3 | | [X] 2 [Y] 3 | [X] 2 [Y] 3 |
| Y | C | [X] 5 [Y] 2 | [X] 3 [Y] 2 | | [X] 1 [Y] 1 |
| Y | D | [X] 5 [Y] 2 | [X] 3 [Y] 2 | [X] 1 [Y] 1 | |
| Pairwise election results (won-tied-lost): | Pairwise election results (won-tied-lost): | 2-1-0 | 2-1-0 | 0-1-2 | 0-1-2 |
Result: Both A and B have two pairwise wins and one pairwise tie, so A and B are tied for the Copeland winner. Depending on the tie resolution method used, A can lose.

#### Express later preferences

Now assume the two voters supporting A (marked bold) express later preferences on their ballot.
| # of voters | Preferences |
| 2 | A > C > D |
| 3 | B > A |
| 1 | C > D > A |
| 1 | D > C |
The results would be tabulated as follows:
Pairwise election results :
| | | X | X | X | X |
| | | A | B | C | D |
| Y | A | | [X] 3 [Y] 3 | [X] 2 [Y] 5 | [X] 2 [Y] 5 |
| Y | B | [X] 3 [Y] 3 | | [X] 4 [Y] 3 | [X] 4 [Y] 3 |
| Y | C | [X] 5 [Y] 2 | [X] 3 [Y] 4 | | [X] 1 [Y] 3 |
| Y | D | [X] 5 [Y] 2 | [X] 3 [Y] 4 | [X] 3 [Y] 1 | |
| Pairwise election results (won-tied-lost): | Pairwise election results (won-tied-lost): | 2-1-0 | 0-1-2 | 2-0-1 | 1-0-2 |
Result: B now has two pairwise defeats. A still has two pairwise wins, one tie, and no defeats. Thus, A is elected Copeland winner.

#### Conclusion

By expressing later preferences, the two voters supporting A promote their first preference A from a tie to becoming the outright winner (increasing the probability that A wins). Thus, Copeland's method fails the Later-no-help criterion.

### Dodgson's method

Dodgson's' method elects a Condorcet winner if there is one, and otherwise elects the candidate who can become the Condorcet winner after the fewest ordinal preference swaps on voters' ballots.
Later-No-Help can be considered not applicable to Dodgson if the method is assumed to not accept truncated preference listings from the voter. On the other hand, Later-No-Help can be applied to Dodgson if the method is assumed to apportion possible rankings among unlisted candidates equally, as shown in the example below.

#### Truncated Ballot Profile

Assume ten voters (marked bold) submit a truncated preference listing A > B = C by apportioning the possible orderings for B and C equally. Each vote is counted ${\tfrac {1}{2}}$ A > B > C, and ${\tfrac {1}{2}}$ A > C > B:
| # of voters | Preferences |
| 5 | A ( > B > C) |
| 5 | A ( > C > B) |
| 10 | B > A > C |
| 2 | C > B > A |
| 1 | C > A > B |
Pairwise Contests :
| | Against A | Against B | Against C |
| For A | | 11 | 20 |
| For B | 12 | | 15 |
| For C | 3 | 8 | |
Result: B is the Condorcet winner and the Dodgson winner. A loses.

#### Adding Later Preferences

Now assume that the ten voters supporting A (marked bold) add later preference C, as follows:
| # of voters | Preferences |
| 10 | A > C > B |
| 10 | B > A > C |
| 2 | C > B > A |
| 1 | C > A > B |
Pairwise Contests :
| | Against A | Against B | Against C |
| For A | | 11 | 20 |
| For B | 12 | | 10 |
| For C | 3 | 13 | |
Result: There is no Condorcet winner. A is the Dodgson winner, because A becomes the Condorcet Winner with only two ordinal preference swaps (changing B > A to A > B). A wins.

#### Conclusion

The ten voters supporting A increase the probability of A winning by adding later preference C to their ballot, changing A from a loser to the winner. Thus, Dodgson's method fails the Later-no-help criterion when truncated ballots are considered to apportion the possible rankings amongst unlisted candidates equally.

### Ranked pairs

For example, in an election conducted using the Condorcet compliant method Ranked pairs the following votes are cast:
| 28: A | 42: B>A | 30: C |
A is preferred to C by 70 votes to 30 votes. (Locked) B is preferred to A by 42 votes to 28 votes. (Locked) B is preferred to C by 42 votes to 30 votes. (Locked)
B is the Condorcet winner and therefore the Ranked pairs winner.
Suppose the 28 A voters specify second choice C (they are burying B).
The votes are now:
| 28: A>C | 42: B>A | 30: C |
A is preferred to C by 70 votes to 30 votes. (Locked) C is preferred to B by 58 votes to 42 votes. (Locked) B is preferred to A by 42 votes to 28 votes. (Cycle)
There is no Condorcet winner and A is the Ranked pairs winner.
By giving a second preference to candidate C the 28 A voters have caused their first choice to win. Note that, should the C voters decide to bury A in response, B will beat A by 72, restoring B to victory.
Similar examples can be constructed for any Condorcet-compliant method, as the Condorcet and later-no-help criteria are incompatible.