Help Solving this Probability Expectation Problem

Question

You have the option to bid on a House which can be Priced anywhere between $[1-1000]. If you bid equal or above the price, you can sell it for 1.5 times its original price. Should you bid or not and if yes, how much?

Is this a simple Probability expectation problem similar to the dice roll game?

If that were the case, since the house can be priced anywhere between 1-1000 and assuming each value has equal probability, the expected value would be 500.5. So multiplying that to 1.5 would be 750.75. So a bid should ideally be any value less than that?

Answer 1

I assume that if your bid is at or above the original price, you pay the amount of your bid to get the house.

If you bid $1000,$ then you will certainly get the house. In that case, assuming the original price of the house is uniformly distributed either over the integers $1$ to $1000$ or over the real numbers in $[1,1000]$, you have correctly calculated the expected resale price.

Since you pay $1000$ in that case, you expect a loss.

Now suppose you bid less in order to avoid that loss. For example, suppose you bid $500.$ In that case, if the original price is greater than $500,$ you don’t get the house and you neither gain nor lose anything. If you get the house, the most its original price can possibly be is $500,$ and the most you can possibly sell it for is $750.$ Do you think the expected amount you can get by reselling the house is $750.75$ in that case?

For any bid less than $1000,$ it does not matter what the expected resale price of the house is; what matters is the chance you will get the house and the expected resale price conditioned on the event that the original price is equal to or less than your bid.

Another way to approach the problem is to suppose you make a certain bid $b,$ and work out, for each $x$ in $[1,1000],$ how much you gain or lose if the original price is $x.$ Keep in mind that when $x>b$ your gain is zero.

Answer 2

$\newcommand{\I}{\mathbb{I}}\newcommand{\E}{\mathbb{E}}$Hints: Let the house price be $X$. Then $X\sim\mathrm{Uniform}[1,1000]$. Say you bid a value $b$. Then your profit $P$ will be: $$P = (1.5X-b)\I(X \le b).$$ (Make sure you can explain why this is! (I'm assuming you are only allowed to buy the house if you bid at least the original house price.) Also note that it is clear that you will only ever bid some $1\le b\le1000$ if you decide to bid ($b$ must be at least $1$ in order to stand a chance of being able to get the house).)

Here $\I(\cdot)$ is an indicator function. Now, can you evaluate $\E[P]$ (which is a function of $b$), and then figure out what $b \in [1,1000]$ maximises this (assuming your goal is to maximise your expected profit)? Is the maximum expected profit positive?