### The mathematics of the Million Second Quiz

*The Million Second Quiz*is a quiz event taking place in the United States on the NBC TV network, online and in person at the moment. It consists of a series of quiz bouts between a champion and a series of challengers, taking place around the clock over the course of a million seconds, or about eleven and a half days.

The champion earns a nominal $10 per second while they remain the champion, whether the quiz bouts are in progress or not, until they are defeated by a challenger. Defeated champions only convert their nominal prize into an actual payout if they are the reigning champion at the end of the million seconds or if they are one of the four most successful defeated champions along the way, and there is set to be an extra competition at the end of the million seconds to pay out an extra bonus to one of them.

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The quiz bouts during the live televised program follow what I consider to be a reasonably interesting structure. They last for either three hundred or four hundred seconds, as announced in advance, and are made up of a series of multiple-choice questions with four answers. The two contestants are asked to identify the correct answer from among the four each time within a five-second time limit. Contestants earn points for correct answers and the contestant with more points at the end of the quiz wins the bout.

Questions started in the first hundred seconds have a base value of one point, questions in the second hundred seconds have a base value of two points and so on. Both contestants independently answer the same question and earn the base value if their answer is correct.

However, as an alternative to answering the question, either contestant may press their "doubler" button at any time. This pauses the bout. The doubling contestant's opponent then has five seconds to answer the question. If the opponent answers correctly, they score double the base value; if they answer incorrectly, the doubling contestant scores double the base value.

That said, the doubling contestant's opponent can go on to "double back" and return the question to the original doubling contestant. The original doubling contestant then has no choice but to answer the question within a further five seconds. If they answer correctly, they score four times the base value; if they answer incorrectly, their opponent scores four times the base value.

So there are two interesting gameplay decisions alongside trying to answer the questions:

1) Should I double a question?

2) If my opponent doubles a question to me, should I double back?

I think these are worthy of a little Expected Value analysis. I haven't seen anyone perform this analysis yet, so I shall go ahead and do so. In summary, the mathematics confirms some intuitive predictions about what optimal strategy

*might*seem to be and extends this by formalising the parameters used to make the decision.

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So in conclusion, in order to decide whether to double or not, you must work out whether your opponent is likely to think you know it or not.

If you think your opponent is likely to think you don't know it, you should assume that they will double back and you should choose to double it if you have a 58% chance of being right, or less (as low as 42%) if you think your opponent knows it.

If you think your opponent is likely to think you do know it, you should assume that they will answer and you should choose to double it if your opponent has less than a 67% chance of being right, or less (as low as 33%) if you know it for sure.

If your opponent doubles a question to you, whether or not to double back depends twice as much on whether you think your opponent knows it than on whether you know it. Even if you are almost certain about the answer, you should choose to double it back if your opponent is guessing completely by chance.

I would not expect these conclusions to be considered counter-intuitive or surprising at all, but the maths underpinning them interests me. Additionally, I do not think it realistic to be able to calculate exact probabilities within the timespan of a few seconds given by the show, though only a general sense is required, and I think that that is realistic. It is far more important to be able to answer the questions correctly, particularly the crucial ones, than anything else!

(With thanks to K. for improvements to a draft of this.)

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