# Later-no-harm criterion

Voting system :
| Name | Comply? |
| Plurality | Yes |
| Two-round system | Yes |
| Partisan primary | Yes |
| Instant-runoff voting | Yes |
| Minimax Opposition | Yes |
| DSC | Yes |
| Anti-plurality | No |
| Approval | N/A |
| Borda | No |
| Dodgson | No |
| Copeland | No |
| Kemeny–Young | No |
| Ranked Pairs | No |
| Schulze | No |
| Score | No |
| Majority judgment | No |
Later-no-harm is a property of some ranked-choice voting systems, first described by Douglas Woodall. In later-no-harm systems, increasing the rating or rank of a candidate ranked below the winner of an election cannot cause a higher-ranked candidate to lose. It is a common property in the plurality-rule family of voting systems.
For example, say a group of voters ranks Alice 2nd and Bob 6th, and Alice wins the election. In the next election, Bob focuses on expanding his appeal with this group of voters, but does not manage to defeat Alice—Bob's rating increases from 6th-place to 3rd. Later-no-harm says that this increased support from Alice's voters should not allow Bob to win.
Later-no-harm may be confused as implying center squeeze, since later-no-harm is a defining characteristic of first-preference plurality (FPP) and instant-runoff voting (IRV), and descending solid coalitions (DSC), systems that have similar mechanics that are based on first preference counting. These systems pass later-no-harm compliance by making sure the results either do not depend on lower preferences at all (plurality) or only depend on them if all higher preferences have been eliminated (IRV and DSC), and thus exhibit a center squeeze effect. However, this does not mean that methods that pass later-no-harm must be vulnerable to center squeezes. The properties are distinct, as Minimax opposition also passes later-no-harm.
Later-no-harm is also often confused with immunity to a kind of strategic voting called strategic truncation or bullet voting. Satisfying later-no-harm does not provide immunity to such strategies. Systems like instant runoff that pass later-no-harm but fail monotonicity still incentivize truncation or bullet voting in some situations.$^{:401}$

## Later-no-harm methods

The plurality vote, two-round system, instant-runoff voting, and descending solid coalitions satisfy the later-no-harm criterion. First-preference plurality satisfies later-no-harm trivially, by ignoring every preference after the first.

### Non-LNH methods

Nearly all voting methods not discussed above fail LNH, including score voting, highest medians, Borda count, and all Condorcet methods. The Condorcet criterion is incompatible with later-no-harm (assuming the resolvability criterion, i.e. any tie can be removed by a single voter changing their rating).
Bloc voting, which allows a voter to select multiple candidates, does not satisfy later-no-harm when used to fill two or more seats in a single district, although the single non-transferable vote does.

## Examples
### Anti-plurality

Anti-plurality elects the candidate the fewest voters rank last when submitting a complete ranking of the candidates.
Later-No-Harm 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-Harm 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.
Examples :
| # of voters | Preferences |
| 2 | A ( > B > C) |
| 2 | A ( > C > B) |
| 1 | B > A > C |
| 1 | B > C > A |
| 1 | C > A > B |
| 1 | C > B > A |
| # of voters | Preferences |
| 4 | A > C > B |
| 1 | B > A > C |
| 1 | B > C > A |
| 1 | C > A > B |
| 1 | C > B > A |

### Borda count

Examples :
| # of voters | Preferences |
| 3 | A > B > C |
| 2 | B > C > A |
| candidate | #1. | #2. | #last | computation | Borda points |
| A | 3 | 0 | 2 | 3*2 + 0*1 | 6 |
| B | 2 | 3 | 0 | 2*2 + 3*1 | 7 |
| C | 0 | 2 | 3 | 0*2 + 2*1 | 2 |
| # of voters | Preferences |
| 3 | A |
| 2 | B > C > A |
| candidate | #1. | #2. | #last | computation | Borda points |
| A | 3 | 0 | 2 | 3*2 + 0*1 | 6 |
| B | 2 | 0 | 3 | 2*2 + 0*1 | 4 |
| C | 0 | 2 | 3 | 0*2 + 2*1 | 2 |

### Copeland

Examples :
| # of voters | Preferences |
| 2 | A > B > C > D |
| 1 | B > C > A > D |
| 1 | D > C > B > A |
Pairwise election results :
| | | X | X | X | X |
| | | A | B | C | D |
| Y | A | | [X] 2 [Y] 2 | [X] 2 [Y] 2 | [X] 1 [Y] 3 |
| Y | B | [X] 2 [Y] 2 | | [X] 1 [Y] 3 | [X] 1 [Y] 3 |
| Y | C | [X] 2 [Y] 2 | [X] 3 [Y] 1 | | [X] 1 [Y] 3 |
| Y | D | [X] 3 [Y] 1 | [X] 3 [Y] 1 | [X] 3 [Y] 1 | |
| Pairwise election results (won-tied-lost): | Pairwise election results (won-tied-lost): | 1-2-0 | 2-1-0 | 1-1-1 | 0-0-3 |
| # of voters | Preferences |
| 2 | A |
| 1 | B > C > A > D |
| 1 | D > C > B > A |
Pairwise election results :
| | | X | X | X | X |
| | | A | B | C | D |
| Y | A | | [X] 2 [Y] 2 | [X] 2 [Y] 2 | [X] 1 [Y] 3 |
| Y | B | [X] 2 [Y] 2 | | [X] 1 [Y] 1 | [X] 1 [Y] 1 |
| Y | C | [X] 2 [Y] 2 | [X] 1 [Y] 1 | | [X] 1 [Y] 1 |
| Y | D | [X] 3 [Y] 1 | [X] 1 [Y] 1 | [X] 1 [Y] 1 | |
| Pairwise election results (won-tied-lost): | Pairwise election results (won-tied-lost): | 1-2-0 | 0-3-0 | 0-3-0 | 0-2-1 |

### Schulze method

Examples :
| # of voters | Preferences |
| 3 | A > B > C |
| 1 | A = B > C |
| 2 | A = C > B |
| 3 | B > A > C |
| 1 | B > A = C |
| 1 | B > C > A |
| 4 | C > A = B |
| 1 | C > B > A |
Matrix of pairwise preferences :
| | d[*,A] | d[*,B] | d[*,C] |
| d[A,*] | | 5 | 7 |
| d[B,*] | 6 | | 9 |
| d[C,*] | 6 | 7 | |

## Criticism

Douglas Woodall writes:
   [U]nder STV the later preferences on a ballot are not even considered until the fates of all candidates of earlier preference have been decided. Thus a voter can be certain that adding extra preferences to his or her preference listing can neither help nor harm any candidate already listed. Supporters of STV usually regard this as a very important property, although it has to be said that not everyone agrees; the property has been described (by Michael Dummett, in a letter to Robert Newland) as "quite unreasonable", and (by an anonymous referee) as "unpalatable".