Optimization

Question

You are bidding for an asset that you can immediately sell for 1.5 times what you pay for it but don't know exactly what the seller believes it is worth. You are told the seller's value can be accurately modeled as a random variable uniformly distributed between 375 and 1000. You can only make one bid, and the seller will only trade if your bid is higher than what they believe the asset is worth. What is your bid?

Answer

The expected profit one can expect when bidding is

$E[profit]=P(bid > price)*(\frac{3}{2}*E[price|bid > price]-bid)+P(bid \leq price)*0$

$E[profit]=\frac{bid-375}{1000-375}*(\frac{3}{2}*(375+\frac{bid-375}{2})-bid)$

$E[profit]=\frac{bid-375}{625}*\frac{1125-bid}{4}$

In order to maximize $E[profit]$ with respect to the $bid$ we can take the derivative and set equal to zero:

$\frac{\partial\ E[profit]}{\partial\ bid}=0$
$\frac{3}{5}-\frac{bid}{1250}=0$
$bid=750$