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 $$2S-B$$)

• What expected value are you trying to maximize? If you just want to win the auction then you would bid $2$. – John Douma Mar 29 at 21:28
• @JohnDouma I tried to clarify the question. You want to maximize the return of the auction. – highviolet Mar 29 at 21:33
• Assume the expected ualue is for you "gain", i.e. $0$ if you did not receive the goods, and $2S-B$ is you get it. I got expectation $0$ no matter how you place the bid $B$. Probably I did not understand the problem correctly. – Yu Ding Mar 29 at 21:34
• $\newcommand{\I}{\mathbb{I}}\newcommand{\E}{\mathbb{E}}$If you bid an amount $b$ (using lower case to avoid confusion with random variables), then your return (which is a random variable as it depends on the random $S$) is $(2S-b)\I(S\le b)$. Hence your expected profit is $$f(b):= \E[(2S-b)\I(S\le b)],$$ where $\I$ is an indicator function and $S\sim\mathsf{Unif}[0,1]$. Can you compute this expectation as a function of $b$ (use integration)? – Minus One-Twelfth Mar 29 at 21:43
• @YuDing: I agree: your expected return is $0$, no matter what you bid. Perhaps this was the answer the interviewer wanted? – TonyK Mar 29 at 21:43

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 $$0$$.
Now that we all know the expected gain is $$0$$ no matter what you bid, here's a simpler explanation "after the fact"... :)
Conditioned on you winning with bid $$B$$, the initial value of the goods is $$Uniform(0,B)$$, so the final value is $$Uniform(0,2B)$$, with an expectation of $$B$$.