Later-no-harm criterion

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 ${\displaystyle {\tfrac {1}{2}}}$ A > B > C, and ${\displaystyle {\tfrac {1}{2}}}$ A > C > B:

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.

This can be seen with the following example with two candidates A and B and 3 voters:

Express "later" preference

Assume that the two voters supporting A (marked bold) would also approve their later preference B.

Result: A is approved by two voters, B by all three voters. Thus, B is the Approval winner.

Hide "later" preference

Assume now that the two voters supporting A (marked bold) would not approve their last preference B on the ballots:

Result: A is approved by two voters, B by only one voter. Thus, A is the Approval winner.

Conclusion

By approving an additional less preferred candidate the two A > B voters have caused their favourite candidate to lose. Thus, Approval voting doesn't satisfy the Later-no-harm criterion.

This example shows that the Borda count violates the Later-no-harm criterion. Assume three candidates A, B and C and 5 voters with the following preferences:

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.

Truncated Ballot Profile

Assume ten 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 ${\displaystyle {\tfrac {1}{2}}}$ A > B > C, and ${\displaystyle {\tfrac {1}{2}}}$ A > C > B:

Result: A is listed last on 17 ballots; B is listed last on 14 ballots; C is listed last on 19 ballots. C is listed last on the most ballots. C is eliminated, and A defeats B pairwise 33 to 17. A wins.

Adding Later Preferences

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

Result: A is listed last on 17 ballots; B is listed last on 19 ballots; C is listed last on 14 ballots. B is listed last on the most ballots. B is eliminated, and C defeats A pairwise 26 to 24. A loses.

Conclusion

The ten 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, Coombs' method doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the last place vote amongst unlisted candidates equally.

This example shows that Copeland's method violates the Later-no-harm criterion. Assume four candidates A, B, C and D with 4 potential voters and the following preferences:

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.

Truncated Ballot Profile

Assume ten 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 ${\displaystyle {\tfrac {1}{2}}}$ A > B > C, and ${\displaystyle {\tfrac {1}{2}}}$ A > C > B:

Result: There is no Condorcet winner. A is the Dodgson winner, because A becomes the Condorcet Winner with only two ordinal preference swaps (changing B > A to A > B). A wins.

Adding Later Preferences

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

Result: B is the Condorcet Winner and the Dodgson winner. B wins. A loses.

Conclusion

The ten voters supporting A decrease the probability of A winning by adding later preference B to their ballot, changing A from the winner to a loser. Thus, Dodgson's method doesn't satisfy the Later-no-harm criterion when truncated ballots are considered to apportion the possible rankings amongst unlisted candidates equally.

This example shows that the Kemeny–Young method violates the Later-no-harm criterion. Assume three candidates A, B and C and 9 voters with the following preferences:

Express later preferences

Assume that all preferences are expressed on the ballots.

The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:

The ranking scores of all possible rankings are:

Result: The ranking C > A > B has the highest ranking score. Thus, the Condorcet winner C wins ahead of A and B.

Hide later preferences

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

The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:

The ranking scores of all possible rankings are:

Result: The ranking B > C > A has the highest ranking score. Thus, B wins ahead of A and B.

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 Kemeny-Young method doesn't satisfy the Later-no-harm criterion. Note, that IRV - by ignoring the Condorcet winner C in the first case - would choose A in both cases.

Considering, that an unrated candidate is assumed to be receiving the worst possible rating, this example shows that majority judgment violates the later-no-harm criterion. Assume two candidates A and B with 3 potential voters and the following ratings:

Express later preferences

Assume that all ratings are expressed on the ballots.

The sorted ratings would be as follows:

Result: A has the median rating of "Fair" and B has the median rating of "Good". Thus, B is elected majority judgment winner.

Hide later ratings

Assume now that the voter supporting A (marked bold) would not express his later ratings on the ballot. Note, that this is handled as if the voter would have rated that candidate with the worst possible rating "Poor":

The sorted ratings would be as follows:

Result: A has still the median rating of "Fair". Since the voter revoked his acceptance of the rating "Good" for B, B now has the median rating of "Poor". Thus, A is elected majority judgment winner.

Conclusion

By hiding his later rating for B, the voter could change his highest-rated favorite A from loser to winner. Thus, majority judgment doesn't satisfy the Later-no-harm criterion. Note, that this only depends on the handling of not-rated candidates. If all not-rated candidates would receive the best-possible rating, majority judgment would satisfy the later-no-harm criterion, but not later-no-help.

If instead majority judgment ignored unrated candidates and computed the median solely from the values that the voters expressed, a voter in a later-no-harm scenario could only help candidates for whom the voter has a higher honest opinion than the society has.

This example shows that the Minimax method violates the Later-no-harm criterion in its two variants winning votes and margins. Note that the third variant of the Minimax method (pairwise opposition) meets the later-no-harm criterion. Since all the variants are identical if equal ranks are not allowed, there can be no example for Minimax's violation of the later-no-harm criterion without using equal ranks. Assume four candidates A, B, C and D and 23 voters with the following preferences:

Express later preferences

Assume that all preferences are expressed on the ballots.

The results would be tabulated as follows:

• [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

Result: C has the closest biggest defeat. Thus, C is elected Minimax winner for variants winning votes and margins. Note, that with the pairwise opposition variant, A is Minimax winner, since A has in no duel an opposition that equals the opposition C had to overcome in his victory against D.

Hide later preferences

Assume now that the four voters supporting A (marked bold) would not express their later preferences over C and D on the ballots:

The results would be tabulated as follows:

Result: Now, A has the closest biggest defeat. Thus, A is elected Minimax winner in all variants.

Conclusion

By hiding their later preferences about C and D, the four voters could change their first preference A from loser to winner. Thus, the variants winning votes and margins of the Minimax method doesn't satisfy the Later-no-harm criterion.

For example, in an election conducted using the Condorcet compliant method Ranked pairs the following votes are cast:

B is preferred to A by 51 votes to 49 votes. A is preferred to C by 49 votes to 26 votes. C is preferred to B by 26 votes to 25 votes.

There is no Condorcet winner; A, B, and C are all weak Condorcet winners and B is the Ranked pairs winner.

Suppose the 25 B voters give an additional preference to their second choice C.

The votes are now:

C is preferred to A by 51 votes to 49 votes. C is preferred to B by 26 votes to 25 votes. B is preferred to A by 51 votes to 49 votes.

C is now the Condorcet winner and therefore the Ranked pairs winner. By giving a second preference to candidate C the 25 B voters have caused their first choice to be defeated, and by giving a second preference to candidate B, the 26 C voters have caused their first choice to succeed.

Similar examples can be constructed for any Condorcet-compliant method, as the Condorcet and later-no-harm criteria are incompatible. Minimax is generally classed as a Condorcet method, but the pairwise opposition variant which meets later-no-harm doesn't actually satisfy the Condorcet criterion.

This example shows that Score voting violates the Later-no-harm criterion, and how in theory the tactical voting strategy called bullet voting could be a response.

Assume two candidates A and B and 2 voters with the following preferences:

Express later preferences

Assume that all preferences are expressed on the ballots.

The total scores would be:

Result: B is the Score voting winner.

Hide later preferences

Assume now that the voter supporting A (marked bold) would not express his later preference on the ballot:

The total scores would be:

Result: A is the Score voting winner.

Conclusion

By withholding his opinion on the less-preferred B candidate, the voter caused his first preference (A) to win the election. This both proves that Score voting is not immune to strategic voting (as no system is), and shows that Score voting doesn't satisfy the Later-no-harm criterion.

It should also be noted that this effect can only occur if the voter's expressed opinion on B (the less-preferred candidate) is higher than the opinion of the electorate about that later preference is. Thus, a later-no-harm scenario can only turn a candidate into a winner if the voter likes that candidate more than the rest of the electorate does.

This example shows that the Schulze method doesn't satisfy the Later-no-harm criterion. Assume three candidates A, B and C and 16 voters with the following preferences:

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.

Hide later preferences

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.