The later-no-help 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 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
Two-round system, Single transferable vote (including traditional forms of Instant Runoff Voting and Contingent vote), Approval voting, Borda count, Range voting, Bucklin voting, and Majority Judgment satisfy the later-no-help criterion.
When a voter is allowed to choose only one preferred candidate, as in plurality voting, later-no-help can either be considered satisfied (as the voter's later preferences can not help their chosen candidate) or not applicable.
Noncomplying methods
All Minimax Condorcet methods (including the pairwise opposition variant), Ranked Pairs, Schulze method, Kemeny-Young method, Copeland's method, Nanson's method, and Descending Solid Coalitions, a variant of Woodall's Descending Acquiescing Coalitions, 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 A > B > C, and A > C > B:
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:
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 A > B > C, and A > C > 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:
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.