I watch a dumb reality TV show which has a certain number of contestants to start with, and every week one contestant is eliminated. Here's how the elimination process works. Every week a certain number of contestants is nominated for elimination. And then viewers get to vote online on who they want to save from elimination. (A viewer can only vote to save one person.). Whoever gets the least votes gets eliminated.

I'm interested in how to vote strategically in this process. Now if I feel strongly about wanting someone not to be eliminated, it's clear what I should do: cast a vote to save them. But my question is, what should I do if I feel strongly about wanting someone to be eliminated?

I think the answer is that I should vote to save whichever contestant I expect to receive the least votes out of the contestants other than the one I want eliminated. That way, if there's a tie between that least popular contestant and the one I want eliminated, I'll break the tie and succeed in getting them eliminated. (I think it's generally true that if voting is rational, the rational way to vote is the way you would if yours was the tie-breaking vote.) Is that right according to game theory?

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    $\begingroup$ Looks about right. However, estimating who's going to be the least popular other than the one you disfavor is tricky business. See p-beauty contest for some related info. $\endgroup$ – Herr K. Jul 12 '17 at 3:44
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    $\begingroup$ There seem to be two questions that are subtly different. The first is "how do I get someone eliminated at the earliest stage possible?" The second is "how do I minimise the chance of someone finishing in first place and winning the overall competition?" My intuition for the second question is that you should vote for the strongest contender in every peirod to maximise the chance that the strongest possible opposition stays in the contests at all stages. This might mean that your target stays around for longer initially but minimises the chance of them winning in the final week. $\endgroup$ – Ubiquitous Jul 12 '17 at 17:02
  • $\begingroup$ Is this about a personal aim to achieve that goal amid such contest, or about the strategic interaction of every player, when every other player is also interested in the same as you? The former is imo trivial, as statistically you are not likely to change anything, and therefore whatever you do might not affect the result. The latter sounds more like a game theory exercise. $\endgroup$ – luchonacho Jul 13 '17 at 5:33

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