# Random ballot

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Random ballot, also known as random dictatorship or single stochastic vote, is a voting system in which the first preference candidate of a ballot drawn at random is elected.

When the drawn ballot is not decisive, then additional ballots are drawn and used only to resolve the indecision of previously drawn ballots.

## Properties

Random Ballot satisfies the Plurality criterion, Monotonicity criterion, Participation criterion, Later-no-harm criterion, clone independence, Favorite Betrayal criterion, and Pareto criterion.

However, Random Ballot fails the Majority criterion, Condorcet criterion, Smith criterion, and Strong Defensive Strategy criterion.

Random Ballot is strategy-proof: this follows from Gibbard's 1978 theorem.

### Example

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

Memphis wins with 42% probability, Nashville with 26%, Chattanooga 15%, and Knoxville 17%. If the Knoxville voters had instead ranked Knoxville and Chattanooga equally, then Knoxville would win with 0% probability, since it would be impossible to draw a ballot which prefers Knoxville to Chattanooga.

## Notes

Interestingly, RB is one of the only single-winner voting methods that is on average a proportional voting method when used in single-winner districts. If, for example, 30% of the voters across a nation vote for Green Party candidates, then on average 30% of the district winners will be Greens.

The concept of RB can be applied to other voting methods. For example, rated ballots could be used to give candidates a probability of winning proportional to the scores on the randomly drawn ballot.

The on-average proportional representation property can be extended into an on-average PR multiwinner method by electing a random first preference, eliminating that candidate from every ballot, and repeating for as many seats as desired. If the voters vote party list style, then the mode of the method (i.e. the winning set elected most often) is the d'Hondt outcome. This method can be derandomized using Markov chains.