The "later-no-harm criterion" criterion (sometimes referred to as "LNHarm") posits that a voter giving an additional ranking or positive rating to a less-preferred candidate should not cause a higher ranked (or rated) candidate on that volter's ballot to lose. It was published in "Voting Matters" in 1994. Its incompatibility with the Condorcet criterion was proven in "theorem 2" of Douglas Woodall's 1997 paper on the subject.
Here's a definition of the later-no-harm criterion:
A voter giving an additional ranking or positive rating to a less-preferred candidate cannot cause a more-preferred candidate to lose.
For probabilistic methods, the later-no-harm criterion is:
A voter giving an additional ranking or positive rating to a less-preferred candidate cannot reduce the probability that a more-preferred candidate wins.
Later-no-harm (usually LNH, but sometimes LNHa or LNHarm to avoid confusion with Later-no-help) is satisfied by Instant Runoff Voting, Minmax(pairwise opposition), and Douglas Woodall's Descending Solid Coalitions method. It is trivially satisfied by First-Preference Plurality and Random Ballot, since those methods do not usually regard lower preferences. Virtually every other method fails this criterion.
B is the Condorcet winner, and would win in any Condorcet method, and if using a rated method, would win if given a high enough rating by all voters. But if the A-top voters bullet vote, then they can make A the winner in several voting methods, such as most Condorcet-IRV hybrid methods and likely in the rated methods. However, notice that to pass LNH in this situation, the majority of voters who prefer B over A have to have their preferences ignored; in a method like IRV, that means that the C-top voters may have to choose between supporting C or using Favorite Betrayal to help B win. Essentially, passing LNH ensures voters never have to worry about their later preferences hurting them, but it can at times force them to lie about their higher preferences.
Later-no-harm guarantees that the method will not use a voter's lower preferences to elect a candidate who that voter likes less than the candidate that would have been elected if this voter had kept his lower preferences a secret.
As a result, voters may feel free to vote their complete ranking of the candidates, which in turn may give the election method more complete information to use to find a winner. There is a tradeoff however, in that this criterion simultaneously minimizes the amount of information that the voting method can use to find a winner.
A common criticism of LNH-failing voting reforms is that they will incentivize bullet voting to such a large degree that they will end up becoming just like FPTP. However, note that bullet voting is not always the strategically best move; if it was, FPTP would be strategyproof. In particular, any voter who would do Favorite Betrayal in FPTP likely has an incentive to support multiple candidates in LNH-failing methods.
One consequence of too many voters bullet voting is that there will be a chicken dilemma.
Criterion failure rate
It is believed that some methods fail LNH at higher rates than others, and this is used as an argument for or against them. For example, Condorcet methods are expected to fail less often than something like Score voting.
This criterion is equivalent to the criterion that the system is non-compromising in that it will never elect a compromise (i.e. a Utilitarian winner or Condorcet winner.) This is not universally desired so it cannot be claimed that this criterion is always one which would be desirable to pass. If one wants a system which can elect a compromise winner then it would be desirable to fail this criterion.
Not reflecting voter preferences
One argument against LNH is that it can result in arbitrary changes in election outcomes based on voter preferences.. If a voter's true preferences are A > B > C and he initially only ranks A, then LNH ensures that if he changes his vote to A>B, that will not change the winner from A to B. However, it might change the winner from B to C.
A counterargument would be that the example given involves two major candidates, L and R, and because the voters' pairwise preferences between the two didn't change (rather, their preference between L/R and C changed), no change should occur in who was the better of the two. This argument is an example of how those against LNH are often in favor of rated methods, where the strength of each of the voter's pairwise preferences are connected (i.e. the fact that some voters increased their support for a major candidate in relation to an irrelevant candidate is argued to mean that their strength of preference between the two major candidates ought to be weakened), rather than ranked methods, which are often based on the idea that a voter's pairwise preferences are independent and maximal (i.e. Category:Pairwise counting-based voting methods).
Though LNH is often touted as preventing voters from being hurt by indicating later preferences, in truth it only prevents voters from hurting a candidate they ranked higher when they do this. It is not known which voting methods, if any, pass the generalized form of LNH.
- Venzke, K. (2022-04-20). "Favorite Betrayal and Condorcet, and LNHarm". Election-methods mailing list archives. Retrieved 2022-04-22.
- Woodall, Douglas, Properties of Preferential Election Rules, Voting matters - Issue 3, December 1994
- Woodall, Douglas R. (1997). "Monotonicity of single-seat preferential election rules" (PDF). Discrete Applied Mathematics. Elsevier BV. 77 (1): 81–98. doi:10.1016/s0166-218x(96)00100-x. ISSN 0166-218X.
- Shentrup, Clay (2020-01-02). "Later-no-harm". Medium. Retrieved 2020-04-30.
- "Later-No-Harm Criterion". The Center for Election Science. Retrieved 2020-05-14.
So those two voters get a better result by limiting the number of candidates they rank. That is, sincerely ranking candidates after W hurt them. It’s almost as though we need to have two different criteria: voter later-no-harm, and candidate later-no-harm. The “later-no-harm” criterion is actually the latter. IRV ensures that a voter can’t harm a candidate by ranking additional less preferred candidates further down the list. But voters can still hurt themselves by doing so.