Participation criterion

# Participation criterion

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The participation criterion is a voting system criterion. It is also known as the "no show paradox". It has been defined as follows:
• In a deterministic framework, the participation criterion says that the addition of a ballot, where candidate A is strictly preferred to candidate B, to an existing tally of votes should not change the winner from candidate A to candidate B.
• In a probabilistic framework, the participation criterion says that the addition of a ballot, where each candidate of the set X is strictly preferred to each other candidate, to an existing tally of votes should not reduce the probability that the winner is chosen from the set X.

Plurality voting, 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...

, 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...

, and 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...

all satisfy the participation criterion. All Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...

s, 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...

, and IRV
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

fail.

Voting systems that fail the participation criterion allow a particularly unusual strategy of not voting to, in some circumstances, help a voter's preferred choice win.

The participation criterion for voting systems is one example of a rational participation constraint
Participation constraint (mechanism design)
In game theory, and particularly mechanism design, participation constraints or rational participation constraints are said to be satisfied if a mechanism leaves all participants at least as well off as they would have been if they hadn't participated....

for social choice mechanisms
Mechanism design
Mechanism design is a field in game theory studying solution concepts for a class of private information games...

in general.

## Quorum requirements

The most common failure of the participation criterion is not in the use of particular voting systems, but in simple yes or no measures that place quorum
Quorum
A quorum is the minimum number of members of a deliberative assembly necessary to conduct the business of that group...

requirements. A public referendum
Referendum
A referendum is a direct vote in which an entire electorate is asked to either accept or reject a particular proposal. This may result in the adoption of a new constitution, a constitutional amendment, a law, the recall of an elected official or simply a specific government policy. It is a form of...

, for example, that required majority approval and a certain number of voters to participate in order to pass would fail the participation criterion, as a minority of voters preferring the "no" option could cause the measure to fail by simply not voting rather than voting no. In other words, the addition of a "no" vote may make the measure more likely to pass. A referendum that required a minimum number of yes votes (not counting no's), by contrast, would pass the participation criterion.

### Two-round system

An example of a two-round election that may well have failed the participation criterion is the Louisiana
Louisiana
Louisiana is a state located in the southern region of the United States of America. Its capital is Baton Rouge and largest city is New Orleans. Louisiana is the only state in the U.S. with political subdivisions termed parishes, which are local governments equivalent to counties...

gubernatorial election in 1991. The votes in the first round were as follows:

Edwin W. Edwards 523,096

David Duke
David Duke
David Ernest Duke is a former Grand Wizard of the Knights of the Ku Klux Klan an American activist and writer, and former Republican Louisiana State Representative. He was also a former candidate in the Republican presidential primaries in 1992, and in the Democratic presidential primaries in...

491,342

Buddy Roemer
Buddy Roemer
Charles Elson "Buddy" Roemer III is an American politician who served as the 52nd Governor of Louisiana, from 1988 to 1992. He was elected as a Democrat but switched to the Republican Party on March 11, 1991...

410,690

Other candidates 124,127

Edwards and Duke advanced to the run-off, which was won by Edwards with 1,057,031 votes to Duke's 671,009.

Now suppose there were 80,653 voters who voted Duke first and preferred Roemer to Edwards. If they had stayed home instead of voting, then the first round would have been as follows:

Edwin W. Edwards 523,096

Buddy Roemer 410,690

David Duke 410,689

Other candidates 124,127

The runoff would have been Roemer vs. Edwards. Pre-election polls had suggested that in a run-off between those two candidates, Roemer would have won, and the 80,653 voters would have got their second-choice rather than their third-choice candidate. Thus, these 80,653 voters would have achieved a more desired result by staying at home than by voting.