:Later-no-harm criterion

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.

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).Douglas Woodall (1997): [http://scorevoting.net/Woodall97.pdf Monotonicity of Single-Seat Election Rules], Theorem 2 (b)

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.

{{Main|Anti-plurality voting}}

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.

Result: A is listed last on 2 ballots; B is listed last on 3 ballots; C is listed last on 3 ballots. A is listed last on the least ballots. A wins.

; Adding Later Preferences

Now assume that the four voters supporting A (marked bold) add later preference C, as follows:

Result: A is listed last on 2 ballots; B is listed last on 5 ballots; C is listed last on 1 ballot. C is listed last on the least ballots. C wins. A loses.

;Conclusion

The four voters supporting A decrease the probability of A winning by adding later preference C to their ballot, changing A from the winner to a loser. Thus, Anti-plurality doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

|}

{{Main|Borda count}}

; Express later preferences

Assume that all preferences are expressed on the ballots.

The positions of the candidates and computation of the Borda points can be tabulated as follows:

Result: B wins with 7 Borda points.

;Hide later preferences

Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:

The positions of the candidates and computation of the Borda points can be tabulated as follows:

Result: A wins with 6 Borda points.

;Conclusion

By hiding their later preferences about B, the three voters could change their first preference A from loser to winner. Thus, the Borda count doesn't satisfy the Later-no-harm criterion.

|}

{{Main|Copeland's method}}

;Express later preferences

Assume that all preferences are expressed on the ballots.

The results would be tabulated as follows:

Result: B has two wins and no defeat, A has only one win and no defeat. Thus, B is elected Copeland winner.

;Hide later preferences

Assume now, that the two voters supporting A (marked bold) would not express their later preferences on the ballots:

The results would be tabulated as follows:

Result: A has one win and no defeat, B has no win and no defeat. Thus, A is elected Copeland winner.

;Conclusion

By hiding their later preferences, the two voters could change their first preference A from loser to winner. Thus, Copeland's method doesn't satisfy the Later-no-harm criterion.

|}

{{Main|Schulze method}}

; Express later preferences

Assume that all preferences are expressed on the ballots.

The pairwise preferences would be tabulated as follows:

Result: B is Condorcet winner and thus, the Schulze method will elect B.

Assume now that the three voters supporting A (marked bold) would not express their later preferences on the ballots:

The pairwise preferences would be tabulated as follows:

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

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

; Conclusion

By hiding their later preferences about B and C, the three voters could change their first preference A from loser to winner. Thus, the Schulze method doesn't satisfy the Later-no-harm criterion.

|}

Douglas Woodall writes:

{{blockquote|[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".Woodall, Douglas, Properties of Preferential Election Rules, [http://www.votingmatters.org.uk/ISSUE3/P5.HTM Voting matters - Issue 3, December 1994]

}}

• Later-no-help criterion
• Monotonicity criterion

{{reflist|group=note}}

• D R Woodall, "Properties of Preferential Election Rules", Voting matters, Issue 3, December 1994 [http://www.votingmatters.org.uk/ISSUE3/P5.HTM]
• Tony Anderson Solgard and Paul Landskroener, Bench and Bar of Minnesota, Vol 59, No 9, October 2002. [https://web.archive.org/web/20090204174848/http://www2.mnbar.org/benchandbar/2002/oct02/voting.htm]
• [http://rangevoting.org/BrownVsmallwood.pdf John Brown, Jr. v. W. H. Smallwood], 130 Minn. 492, 153 N.W. 953 (July 30, 1915)

{{reflist}}

{{voting systems}}

Category:Electoral system criteria