When you grade exams in a large class you inevitably face the misunderstood question dilemma. A student has given a correct answer to a question but not the question you asked.  As an answer to the question you asked it is flat out wrong. How much credit should you give?

It should not be zero. You can make this argument at two levels. First, ex post, the student’s answer reveals some understanding. To award zero points would be to equate this with writing nothing at all. That’s unfair.

You might respond by saying, tough luck, it is my policy not to reward misunderstanding the question. But even ex ante it is optimal to commit to a policy which gives at least partial credit to fortuitous misunderstanding. The only additional constraint at the ex ante stage is incentive compatibility. You don’t want to reward a student who interprets the question in a way that makes it easier and then supplies a correct answer to the easier question.

But you should reward a student who invents a harder question and answers that. And you should make it known in advance that you will do so. Indeed, taken to its limit, the optimal exam policy is to instruct the students to make up their own question and answer it, with harder questions (correctly answered) worth more than easier ones.

Incidentally I was once in a class where a certain professor asked exactly such a question.

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