# Later-no-harm criterion

Voting system
Name Comply?
Two-round system Yes
Single transferable vote Yes
Instant-runoff voting Yes
Contingent vote Yes
Minimax Condorcet Yes
Anti-plurality No
Approval voting No
Borda count No
Dodgson's method No
Copeland's method No
Kemeny–Young method No
Ranked Pairs No
Schulze method No
Range voting No

The later-no-harm criterion 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 lose.

## Complying methods

Two-round system, Single transferable vote, Instant Runoff Voting, Contingent vote, Minimax Condorcet (a pairwise opposition variant which does not satisfy the Condorcet Criterion), and Descending Solid Coalitions, a variant of Woodall's Descending Acquiescing Coalitions rule, satisfy the later-no-harm criterion.

When a voter is allowed to choose only one preferred candidate, as in plurality voting, later-no-harm can be either considered satisfied (as the voter's later preferences can not harm their chosen candidate) or not applicable.

## Noncomplying methods

Approval voting, Borda count, Range voting, Majority Judgment, Bucklin voting, Ranked Pairs, Schulze method, Kemeny-Young method, Copeland's method, and Nanson's method do not satisfy later-no-harm. The Condorcet criterion is incompatible with later-no-harm (assuming the discrimination axiom, according to which any tie can be removed by some single voter changing her rating).[1]

Plurality-at-large voting, which allows the voter to select up to a certain number of candidates, doesn't satisfy later-no-harm when used to fill two or more seats in a single district.

## Checking Compliance

Checking for satisfaction of the Later-no-harm 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 decrease in probability. Later-no-harm 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 least number of 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.

### Approval voting

Since Approval voting does not allow voters to differentiate their views about candidates for whom they choose to vote and the later-no-harm criterion explicitly requires the voter's ability to express later preferences on the ballot, the criterion using this definition is not applicable for Approval voting.

However, if the later-no-harm criterion is expanded to consider the preferences within the mind of the voter to determine whether a preference is "later" instead of actually expressing it as a later preference as demanded in the definition, Approval would not satisfy the criterion. Under Approval voting, this may in some cases encourage the tactical voting strategy called bullet voting.

### 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-Harm 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-Harm 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.

### 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 least number of ordinal preference swaps on voters' ballots.

Later-No-Harm 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-Harm can be applied to Dodgson if the method is assumed to apportion possible rankings among unlisted candidates equally, as shown in the example below.

## Criticism

Woodall, author of the Later-no-harm 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,[2] 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".[3]

Warren Smith writes that the Later-no-harm criterion is "a silly criterion" saying that "objectively, LNH is not even a desirable property with honest voters". He argues that rating a candidate higher should allow the possibility of that candidate winning if the candidates collective ratings are high enough.[4] The Center for Election Science, founded by Smith, also voices their opinion that the name itself is "misleading" and raises the concern that while "a voter can't harm a candidate by ranking additional less preferred candidates, .. voters can still hurt themselves by doing so. This begs the question of whether the later-no-harm criterion actually has value."[5]