Interview question: Optimal bid strategy I was asked a question during a job interview. I don't think I managed to solve it properly during the interview, so I would like someone to explain the answer. The question was as follow: * There is a good with value $S$ that is uniformly distributed on [0,1] * You are placing a bid $B$. If $B$ is larger than or equal to $S$, you receive the good * Immediately **after** your bid is placed it is determined whether you receive the good or not * Immediately after that the value of the good doubles to $2S$ How should you place the bid $B$ to maximise your expected return? (I.e. maximize $2S-B$)

Your profit is $2S-B$ when $S \lt B$ and $0$ otherwise. Integrating over $S$, we get $$\int_0^B (2S-B) dS=(S^2-BS)|_0^B=0$$ Bid whatever you like in the interval $[0,1]$ and your expected profit is the same, $0$.

This seems a surprising conclusion, so we should try some things to validate it. If you bid $0$ you will never get the object, so your profit is $0$. If you bid $1$ and the initial value is $0$, you lose $1$. If you bit $1$ and the initial value is $1$, you win $1$. It is linear in between, so your expectation at a bid of $1$ is $0$. The problem is scale invariant. If you bid $B$ you lose $B$ when the initial value is $0$ and win $B$ when you barely get the object, so your expected profit is $0$.