Fall 2011 Strategic Practice 9: Section 3 (Conditional Expectation) - Question 4
You are given an amazing opportunity to bid on a mystery box containing a mystery prize! The value of the prize is completely unknown, except that it is worth at least nothing, and at most a million dollars. So the true value V of the prize is considered to be Uniform on [0,1] (measured in millions of dollars). You can choose to bid any amount b (in millions of dollars). You have the chance to get the prize for considerably less than it is worth, but you could also lose money if you bid too much. Specifically, if $b<\frac{2}{3}V$, then the bid is rejected and nothing is gained or lost. If $b\geq \frac{2}{3}V$ then the bid is accepted and your net payoff is V - b (since you pay b to get a prize worth V). What is your optimal bid b (to maximize the expected payoff)?
Solution: Condition on all the information. It is crucial in the above calculation to use E(V | V =< 1.5b) rather than E(V) = 1/2. The solution is negative except at b = 0, so the optimal bid is 0. See iTunes for full solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."