David R. MacIver's Blog

Three thought experiments on majority voting

I present to you three examples in which we employ majority voting between two options (that is, we ask a population “Would you like A or B?” and we choose the option that the largest number of people preferred).

You can and should infer the obvious context for this, but I am not going to comment further on it in this post.

Tea and Cake or Death

Suppose 51% of the population vote for a bill that will result in the death of the remaining 49% of the population.

Questions

• Should the country go along with this?
• If they decide to, do the remaining 49% have a democratic obligation to accept that?
• Does the answer change if it’s 90% and 10%? 99% and 1%?
• What if the remaining 49% are merely financially ruined? Moderately inconvenienced?

Pizza or Barbecue

A group of nine friends regularly meet for dinner. Of these, five of them really like pizza and four of them really like barbecue. As good citizens of a democracy, they put this to a vote. Unsurprisingly, this results in them always having pizza.

Questions

• Is this fair?
• Suppose only four of them really like pizza, and the ninth person changes their mind regularly and thus always gets to decide where they go for dinner. Is that fair?
• Suppose the friends in question are tired of putting it to the vote each time and some of them push for a vote to go to a regular meeting place. They put the question “Should we have Pizza or Barbecue for all future group dinners?” to a vote. Pizza wins. Is that fair? Does that answer change if we have the previous 4/4/1 split?

Which president?

Fair warning: This one is by far the most complicated of the three thought experiments.

Our student mathematical society decides to elect a president. There are three candidates, Alex, Kim and Pat. We’ve read all this confusing stuff about voting theory and we can’t really decide what we like except that majority rule is clearly the best for two candidates, so we decide to reduce this to the solved problem. I pick two candidates, we vote between them and the majority winner stays in. We then vote again between them and the third remaining candidate, and the winner of that becomes president.

Note that the student body is roughly equally split between the following three preferences:

• Alex, Kim, Pat
• Kim, Pat, Alex
• Pat, Alex, Kim

As a result, the majority of people prefer Alex to Kim, the majority of people prefer Kim to Pat, and the majority of people prefer Pat to Alex.

Note that:

• If I put Alex and Kim together in the first round, then Pat is president because Alex beats Kim then Pat beats Alex.
• If I put Alex and Pat together in the first round, then Kim is president because Pat beats Alex, then faces Kim and the majority prefer Kim to Pat
• If I put Kim and Pat together in the first round, then Alex is president because Kim beats Pat in the first round and then Alex beats Kim

Questions

• How strong is the resulting president’s mandate?
• Does the answer to the previous question depend on which round they were in?
• Does the answer to the first question depend on how we chose the rounds?
• Does the answer change if instead of explicitly voting for three candidates as part of a single batch we instead have a system where a new candidate always runs against the incumbent?

Concluding Statement

Majority vote between two options is often held up as some pinacle of uncontroversial democracy where at least in this case we know what the right answer is, even though voting is complicated in general.

I hope I have convinced you that is not the case.