This is a deliciously evil question to throw at a class: students were offered extra points on their final paper. They could either pick two or six as an option, however if more than 10% of the class picked six then no-one got extra points. All answers are private and confidential.
Let’s hurl ourselves into that situation. What’s the optimal choice?
One approach is to look at maximizing everyone’s points. If exactly 10% of the class chose six points, and everyone else chose two, the class’ average bonus points would be (2 * 0.9) + (6 * 0.1) or 2.4 points each. If less than 10% chose six, that value drops smoothly towards two as the percentage declines; if more chose six, the average instantly drops to zero.
But how are we going to get 10% of the class clicking on six, when students cannot communicate with one another? On the surface, that’s easy enough; just pick a random number with the same range as the class size, and pick six only if it’s less than or equal to 10% of the class size. Unfortunately, random numbers are random; that 10% threshold will actually lead to sub-optimal class averages, as about a third the time random variation will cause you to trip over the threshold. Some playing around with brute-force Monte-Carlo reveals the ideal threshold is a weird beast.
You don’t have to, though. All of the above is trying to optimize your decision based on approximately infinite attempts. This class is only going to get one shot at the problem, however, so we can’t count on the law of large numbers if the randomness don’t fall our way. Worse, the class might contain some assholes who’d take delight in robbing everyone of points.
The ideal answer here, then, is to be conservative and pick the two point option no matter what. Alas, the problem is much less diabolical than it first appears.
[HJH 2015-07-17] The professor in question has outed themselves, and in the process revealed it actually is diabolical.
Selterman says only one class — his fall 2011 group — has received the extra credit since he first implemented the exercise in 2008. But he speculates it may have merely been a fluke.
“In behavioral science, nothing is ever 100%,” he says.
Ouch. The article lists several possible reasons people take the six-point option. I find the one about fairness most compelling; if other people are getting six points instead of two, then you deserve six points too. That isn’t a rational way to approach the problem, but we’re only vaguely rational.
I do have to take a point off for neglecting to mention WEIRD, though.