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This is obviously wrong. If you choose one million billion trillion every time you will not win 28% of the time.

You will, your opponents will choose matching numbers 28% of the time if they use the Nash-equilibrium mixed strategy.

You are right.

Choosing a bigger number makes it less likely that you will win by being smaller than the other two, but more likely that you will win by the other two knocking each other out, since there are now more numbers available for them to do so with.

I think most importantly your opponents become less likely to knock your number out. The goal is your number to win, not your opponents to lose.

but still (1) there's some probability of both players choosing a number bigger than one million billion trillion, and (2) there's some probability of both other players choosing the same number. Actually, the probability that the other both players choose 1 is pretty big (~0.21). So even if you choose a super-big number, there's a big probability of you still winning.

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