How to score a second try on a multiple choice test

Question

Suppose a multiple choice question has $n$ answers. You allow a student $m$ tries. The student gets the correct answer on the $k$-th try. What is a "fair" score?

Answer 1

More context is needed to establish what is "fair". To illustrate this, take an extreme example. Say the question is "What is $1+1$?", and $99$ ridiculous answers are provided, along with the correct one, and $50$ tries are allowed. One student needs $10$ attempts to get it right, while another needs $40$. It's hard to argue that the first student has demonstrated more knowledge than the second one. In this case, the only reasonable scoring system would be $1$ point (or whatever) for a correct first answer and no credit otherwise. The situation would be different if the question were hard and the answers were plausible but required graded levels of knowledge to eliminate (or confirm) them. For these reasons, I don't think that the posted question, as it stands, can be given a mathematical answer.

Answer 2

You could have a scoring system where the problem is worth $n$ points. You can guess as many options (choices) as you want. You get $1$ point for each correctly chosen correct option and $1$ point for each nonchoice of each incorrect option.

Thus in your scenario you would score $1 +(n-k)$ (one correct choice of the correct option; and $n-k$ correct non-choices of incorrect options).

So the score would be $n+1-k$ out of a possible $n$ points.