Later-no-harm criterion

# Later-no-harm criterion

Discussion

Encyclopedia
The later-no-harm criterion is a voting system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....

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 cannot cause a more preferred candidate to lose.

## Complying methods

Single transferable vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

(including
Instant Runoff Voting and Contingent vote
Contingent vote
The contingent vote is an electoral system used to elect a single winner, in which the voter ranks the candidates in order of preference. In an election, if no candidate receives an absolute majority of first preference votes, then all but the two leading candidates are eliminated and there is a...

), Minimax Condorcet
Minimax Condorcet
In voting systems, the Minimax method is one of several Condorcet methods used for tabulating votes and determining a winner when using preferential voting in a single-winner election...

(pairwise opposition variant which does not satisfy the Condorcet Criterion), and Descending Solid Coalitions, a variant of Woodall's Descending Acquiescing Coalitions rule, satisfy the later-no-harm criterion.

However, if a method permits incomplete ranking of candidates, and if a majority
Majority
A majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population...

of initial round votes is required to win and avoid another election, it does not satisfy Later-no-harm. A lower preference vote cast may create a majority for that lower preference, whereas if the vote was not cast, the election could fail, proceed to a runoff, repeated ballot or other process, and the favored candidate could possibly win.

## Noncomplying methods

Approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

, Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

, Range voting
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

, Schulze method
Schulze method
The Schulze method is a voting system developed in 1997 by Markus Schulze that selects a single winner using votes that express preferences. The method can also be used to create a sorted list of winners...

and Bucklin voting
Bucklin voting
Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...

do not satisfy later-no-harm. The Condorcet criterion
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...

is incompatible with later-no-harm.

When Plurality
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

is being used to fill two or more seats in a single district (Plurality-at-large) it fails later-no-harm.

The later-no-harm criterion is by definition inapplicable to any voting system in which a voter is not allowed to express more than one choice, including plurality
Plurality voting system
The plurality voting system is a single-winner voting system often used to elect executive officers or to elect members of a legislative assembly which is based on single-member constituencies...

voting, the system most commonly used in Canada, India, the UK, and the USA.

### Approval voting

For example in an election using Approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

520 voters prefer candidates in the order A>B>C and approve only candidate A. 480 voters prefer candidates in the order B>C>A and approve only candidate B. 100 voters prefer candidates in the order C>B>A and approve only candidate C.
 A 520 B 480 C 100

A is the most approved candidate and therefore the winner.

Suppose 50 of the A>B>C voters approve both candidates A and B instead of just candidate A. The result is now:
 A 520 B 530 C 100

B is now the most approved candidate and therefore the winner.

By approving an additional less preferred candidate the 50 AB voters have caused their favourite candidate to lose.

### Condorcet compliant methods

For example in an election conducted using the Condorcet
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...

compliant method Ranked pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

 49: A 25: B 26: C>B

There is no Condorcet winner and B is the Ranked pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

winner.

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

 49: A 25: B>C 26: C>B

C is now the Condorcet winner and therefore the Ranked pairs
Ranked Pairs
Ranked pairs or the Tideman method is a voting system developed in 1987 by Nicolaus Tideman that selects a single winner using votes that express preferences. RP can also be used to create a sorted list of winners....

winner.
By giving a second preference to candidate C the 25 B voters have caused their first choice to be defeated.

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 actually fails the Condorcet criterion.

### Effect of majorityMajorityA majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population... requirement

In a Single transferable vote
Single transferable vote
The single transferable vote is a voting system designed to achieve proportional representation through preferential voting. Under STV, an elector's vote is initially allocated to his or her most preferred candidate, and then, after candidates have been either elected or eliminated, any surplus or...

election with majority
Majority
A majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population...

required, and full ranking is not required, the votes are as follows (for simplicity, all voters plump for their favorites):
 40: A 39: B 21: C

If this majority failure results in a runoff between A and B, B could win.

However, suppose the B voters decide to add a second preference for A:
 40: A 39: B>A 21: C

In the Preferential voting
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...

method described as an example in Robert's Rules of Order
Robert's Rules of Order
Robert's Rules of Order is the short title of a book containing rules of order intended to be adopted as a parliamentary authority for use by a deliberative assembly written by Brig. Gen...

, elimination continues iteratively until "one pile contains more than half the ballots." So C would be eliminated, then B, and the B ballots would be counted for A, who would thereby obtain a majority
Majority
A majority is a subset of a group consisting of more than half of its members. This can be compared to a plurality, which is a subset larger than any other subset; i.e. a plurality is not necessarily a majority as the largest subset may consist of less than half the group's population...

and be elected. (If no candidate gains a majority, this will "require the voting to be repeated.") By adding a second preference vote for A, the B voters eliminated the election possibility for B.

## Commentary

Woodall writes about Later-no-harm, "... under STV [single transferable vote] 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
Michael Dummett
Sir Michael Anthony Eardley Dummett FBA D.Litt is a British philosopher. He was, until 1992, Wykeham Professor of Logic at the University of Oxford...

, in a letter to Robert Newland) as 'quite unreasonable', and (by an anonymous referee) as 'unpalatable.'"