Random ballot

The term random ballot or random dictatorship refers to a randomized electoral system where the election is decided on the basis of a single randomly-selected ballot. A closely-related variant is called random serial (or sequential) dictatorship, which repeats the procedure and draws another ballot if multiple candidates are tied on the first ballot.

Random dictatorship was first described in 1977 by Allan Gibbard, who showed it to be the unique social choice rule that treats all voters equally while still being strategyproof in all situations. Its application to elections was first described in 1984 by Akhil Reed Amar.

While rarely, if ever, suggested as a serious electoral system, the rule is often used as a tiebreaker.

Single-winner
Satisfied criteria include:


 * Decisive: there is no possibility of a tied vote, assuming some selected voter has expressed a preference.
 * Strategyproof: there is never any advantage to tactical voting.

Failed criteria include:


 * Determinism: the results depend on chance.
 * Majority-rule: even if a single candidate has support from a majority in every subelection, that candidate may lose.

For multimember bodies
If the random ballot is used to select the members of a multi-constituency body, it can serve to retain the attractive features of both first past the post and proportional representation.

As the winner of each ballot is chosen randomly, the party with the largest vote share is most likely to get the greatest number of candidates. In fact, as the number of ballots grows, the percentage representation of each party in the elected body will get closer and closer to their actual proportion of the vote across the entire electorate. At the same time, the chance of a randomly selected highly unrepresentative body diminishes.

For example, a minority party with 1% of the vote might have a 1/100 chance of getting a seat in each ballot. In a 50-person assembly, the probability of a majority for this party being chosen by random ballot is approximately (using the binomial distribution CDF)


 * $$ \sum_{k=0}^{24} {50 \choose k} \left(\frac{99}{100}\right)^k\left(\frac{1}{100}\right)^{50-k} = I_{1/100}(26, 25) \approx 10^{-38}, $$

This is a vanishingly small chance, which negates the possibility of small parties winning majorities due to random chance.

At same time, the random ballot preserves a local representative for each constituency, although this individual may not have received a majority of votes of their constituents.

Prevalence
There are no examples of the random ballot in use in practice, but it has been used as a thought experiment, and it is occasionally used in real life as a tiebreaker for other methods.

Randomness in other electoral systems
There are various other elements of randomness (other than tie-breaking) in existing electoral systems:

1. Randomly ordering candidates on a list. It is often observed that candidates who are placed in a high position on the ballot-paper will receive extra votes as a result, from voters who are apathetic (especially in elections with compulsory voting) or who have a strong preference for a party but are indifferent among individual candidates representing that party (when there are two or more). For this reason, many societies have abandoned traditional alphabetical listing of candidates on the ballot in favour of either ranking by the parties (e.g., the Australian Senate), placement by lot, or rotation (e.g., Hare-Clark STV-PR system used in Tasmania and the Australian Capital Territory). When candidates are ordered by lot on the ballot, the advantage of donkey voting can be decisive in a close race.

2. Randomly selecting votes for transfer. In some single transferable vote (STV) systems of proportional representation, an elected candidate's surplus of votes over and above the quota is transferred by selecting the required number of ballot papers at random. Thus, if the quota is 1,000 votes, a candidate who polls 1,200 first preference votes has a surplus of 200 votes that s/he does not need. In some STV systems (Ireland since 1922, and Australia from 1918 to 1984), electoral officials select 200 ballot-papers randomly from the 1,200. However, this has been criticised since it is not replicable if a recount is required. As a result, Australia has adopted a variant of fractional transfer, a.k.a. the "Gregory method", by which all 1,200 ballot-papers are transferred but are marked down in value to 0.1666 (one-sixth) of a vote each. This means that 1,000 votes "stay with" the elected candidate, while the value of the 1,200 ballot-papers transferred equals only 200 votes.

3. Randomly selecting winners. This method is called sortition: rather than choosing ballots, it chooses candidates directly by lot, with no input from the voters (except perhaps a nominating or screening process). This is not the same as random ballot, since random ballot is weighted in favor of candidates who receive more votes. Random ballot would behave identically to random winner only if all candidates received the same number of votes.