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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 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.

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.

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.

• Approval voting
  • Combined approval voting
  • Unified primary
• Borda count
• Bucklin voting
• Condorcet methods
  • Copeland's method
  • Dodgson's method
  • Kemeny–Young method
  • Minimax Condorcet method
  • Nanson's method
  • Ranked pairs
  • Schulze method
• Exhaustive ballot
• First-past-the-post voting
• Instant-runoff voting
  • Coombs' method
  • Contingent vote
  • Supplementary vote
• Simple majoritarianism
• Plurality
• Positional voting system
• Score voting
• STAR voting
• Two-round system
• Graduated majority judgment

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• MMP
• Additional member system
• Alternative vote plus
• Mixed single vote
• Mixed ballot transferable vote
• Scorporo
• Vote linkage mixed system

• Single non-transferable vote
• Limited voting
• Cumulative voting
• Satisfaction approval voting

• Condorcet winner criterion
• Condorcet loser criterion
• Consistency criterion
• Independence of clones
• Independence of irrelevant alternatives
• Independence of Smith-dominated alternatives
• Later-no-harm criterion
• Majority criterion
• Majority loser criterion
• Monotonicity criterion
• Mutual majority criterion
• Participation criterion
• Plurality criterion
• Resolvability criterion
• Reversal symmetry
• Smith criterion
• Seats-to-votes ratio

• Ballot
• Election threshold
• First-preference votes
• Liquid democracy
• Spoilt vote
• Sortition
• Unseating

• Comparison of voting systems
• Voting systems by country