Jane street interview question

Q. Suppose you are given the opportunity to bid for a treasure chest, which you know with 100% confidence to be priced anywhere between $0-$1000. If you bid equal to or above the price, you win the treasure chest (at the cost of your bid). If you bid below the price, you do not earn the treasure chest. Now, also suppose you have a friend who is willing to buy the treasure chest from you for one and a half times the price of the treasure chest (should you obtain the chest). What should your bid be?

Let’s say we bid at a price ‘y’.

The payoff function at bid price y i.e. f(y)->

1.5*x -y (for y ≥x)

0 (for x>y)

So Expected payoff at bid price y is => E(f(y))

=> f(y,x).p(x) =>

integral from x = 0 to x = y of [(1.5*x — y)*(1/1000)*dx] (We know x is from U(0,1000))

=> (0.75*y² — y²)*(1/1000) = -y²/4000

So no matter what the bid price, the expected payoff is going to be ≤ 0.

This is because in 1.5*x — y the negative region from x = 0 to x = 2/3 y overpowers the positive region from x = 2/3y to x = y.