Electoral Tiebreaking by Lottery

A recent article on electoral tiebreaking by lottery. Like a lot of these articles (at least, those that don’t regard lotteries as some kind of communist plot), it takes essentially the Churchillian position–a coin toss is the worst way to break a tie, except for all the others:

When a State Election Can Be Literally Determined by a Coin Toss

It’s one of the weirder traditions of American democracy: In many states, if a race is tied, a “game by lot” — cards, straws, or most often, a coin toss — determines who goes to the house and who goes home. Months of campaigning, committee assignments, the fortunes of careers, the possibility of political change — it all comes down, like possession in a football game, to heads or tails.

Allowing chance to enter the core of a democratic system seems counterintuitive, although it’s widely recognized today as an electoral tiebreak. In fact, the roots of election by lottery stretch back to ancient Athens. (Modern-day Americans aren’t the first people to be wary of the method; it was also used by sorcerers to predict the future. “Sorcery” comes from the Latin sors, meaning “lot.”) More recently, coin tosses have broken ties in New York, Illinois, Wisconsin, Ohio, Missouri, Washington, Florida, Minnesota and New Hampshire. South Dakota and Arizona have used card games. In Virginia, the winner has been chosen from a hat.

8 Responses

  1. I’m always slightly annoyed by people making a fuss about this.

    There is so much statistical noise in elections that a coin toss in the vanishing unlikely case of ties really makes no difference at all in who is more likely to win the elections. There are so many variables which can be considered random, if the weather is bad some people will choose to stay at home on voting day, so will they if the get ill; watching the TV at a certain moment might mean hearing the decisive argument from a given candidate. Meeting some old friend by chance might mean being convinced to vote in a certain way. In the case of a perfect tie recounting the votes would almost certainly give a different result. From this point of view any tiebreaker would be acceptable: use alphabetical order do decide the winner, or let the Pope decide. Thinking it matters in my view means not having a good understanding of how probability affect the world. Of course the method chosen might have some symbolic value, but nothing else.

    And to be sure this hasn’t much to do with sortition. The tiebreaking doesn’t. But the fact that there is statistical noise anyway does: it shows how sortition does’t really introduce randomness, it just makes it clearly measurable.


  2. I wonder if there isn’t a case for deciding all elections within a certain margin by coin toss, not just ties (say, within 100 votes). The argument is 1) that the candidate with the most support should win, 2) elections are an instrument for measuring candidate support, 3) one must have sufficient confidence in the instrument before relying on the distinctions it makes, and 4) if the vote is close enough, this confidence is not justified. (I think Edgeworth’s argument regarding examinations was similar to this.) The critical step, of course, if #3. Not sure if that could be justified–will have to give Edgeworth a close look.


  3. Let me extend peterstone’s concept: what if all positions are filled by a weighted coin toss/lottery in which the weights correspond to the votes garnered by the candidates. This has some interesting properties. It could combine the benefits of single member districts (clear responsibility) with the benefits of proportional representation (over time, groups should expect to hold the seat in proportion to the composition of the electorate). In the situation where every citizen sought the office, it would correspond to sortition, but would allow citizens who did not wish to hold the seat to delegate their power to representatives.


  4. The proposal you describe is a version of lottery voting, sometimes called the Random Dictator rule. In lottery voting, everyone votes, and then one vote is drawn at random to select the winner. Interestingly, lottery voting has a number of desirable properties; in particular, it’s strategy-proof (i.e., nobody ever has an incentive to vote for their second choice for fear of “throwing away their votes”). The relevant paper (very hard, mathematically speaking) is

    Manipulation of Schemes that Mix Voting with Chance
    Allan Gibbard
    Vol. 45, No. 3 (Apr., 1977) (pp. 665-681)


  5. 4) if the vote is close enough, this confidence is not justified

    Which is to say is to say, when the results are close, the winner is determined more by chance than as a consequence of greater support. I don’t see how tossing a coin would improve on this.

    Lottery voting seems interesting, I had thought myself about something quite alike. I specially like the fact that it is just a generalization of sortition.


  6. @peterstone: Thank you for directing me to this paper and the related terms.

    @Fela: The relationship between sortition and lottery voting may indicate a useful way of thinking about voting systems that include a random element.

    Let there be a population of k citizens, anyone of which can both vote for and hold a single leadership position. Each citizen casts exactly one ballot. Let F be a function that maps the votes into the probability of each of the k citizens being elected.

    Plurality elections: F is the function that gives probability 1 to the citizen with the most votes.

    Coin flips for close elections (say within a votes) have an F that is piecewise: zero up to threshold, gives all candidates within the margin equal probability, and then assigns probability 1 to a candidate if he/she gets more than a votes than the nearest competitor.

    The random dictator F assigns probability proportional to the number of votes received by each citizen. (And if everyone votes for his or herself, this is sortition as we noted).

    These forms could be interestingly combined or altered: such as allowing a weighted lottery within a threshold or using non-linear rates of change in candidate probabilities (e.g., an exact tie would be a 50/50 coin flip, but winning 60% of the vote would get you only p = 0.55).


  7. Nice, I like that kind of modelling. The next step would be formalize useful properties of voting systems described this way, similar to these kind of properties for deterministic voting:
    Does Gibbard does anything of the sort in his paper? I should read it, as soon as I’ve some free time (could be a while…)


  8. Basically, yes. Gibbard shows that some form of lottery voting is the only procedure that is strategy proof and weakly paretian (i.e., if everyone prefers x to y, then x wins). I think that’s right–it’s been a while since I read the paper. But basically, it’s a part of the literature of social choice theory, the study of the properties of voting rules. (Gibbard also produced the Gibbard-Satterthwaite Theorem, which is the central result on strategy-proofness of voting rules.)


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