Select whether you want 2 points or 6 points added onto your final paper grade. But there's a small catch: if more than 10% of the class selects 6 points, then no one gets any points. What would the students do?
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Observation: If 90% choose 2 points and 10% choose 6 points, nobody can gain anything by changing, so this would be an equilibrium. Also if $x%>90%$ have selected 2 points then some can change to 6 points and gain from it – John Doe May 07 '17 at 20:26
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I'd choose the 6 points... – Jacob Claassen May 07 '17 at 20:36
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I'd choose $6$, on the assumption that we are graded on a curve. If the points get cancelled, then I'm where I was. If not then I am ahead. – lulu May 07 '17 at 20:38
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@lulu and I think that more than 10% will think the same... – lesath82 May 07 '17 at 20:56
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@lesath82 so do I, but at least this way I don't disadvantage myself. Of course, if we are not grading on a curve the calculation changes. – lulu May 07 '17 at 20:58
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@lulu if the students aren't allowed to decide a common strategy, I'd say that behaving differently would be naively too optimistic towards others. So I'd do the same as you. – lesath82 May 07 '17 at 21:00
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This is a variant in the spirit of a multi-person Prisoners' dilemma. The most common answers are three. (It's hard to claim that there is a "right" answer.)
1) Randomize between "$2$" and "$6$" according to the (unique) equilibrium mixed strategies. (See comment by John Doe.)
2) Except for the case where your strategy is pivotal (playing "$6$" instead of "$2$" triggers the punishment), playing "$6$" is weakly dominant. This event is unlikely enough that one feels one should play "$6$". (See comment by Jacob Claassen.)
3) Students get together and arrange some clever scheme, with side payments and (preferably) punishment for cheaters. The smaller the class, the easier is to come up with some workable idea.
My own suggestion would be quite different. It is obvious that a grading scheme where the final score depends only on the number of people who choose one answer over another is unfair, because nobody can know which is the correct solution in advance. If the professor implements it, you can threaten him/her to call the attention of the Provost and suggest that he/she gives everybody the same grade. This will teach him/her a lesson about playing tricks with a class as game-theoretically smart as his/hers.
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