Car Auction with Uniformly distributed Value

Suppose the true value $V$ of a car is uniformly distributed, between $0$ and $1000$. You can bid any amount for the car, and if you bid the true value or more, then you pay your bid and get the car. You know a very good salesman, and are confident in your ability to sell the car for $50\%$ more than its true value. What should you bid to maximize your expected profit?

My reasoning is this:

The expected value of the car is $500$ based on the distribution we are given. Therefore, on average I will be selling it for $1.5*500 =750$.

This means that if I bid $501$, then I should win more often than I lose. When I lose my payoff is $0$ and for the wins, my payoff should average out to be $750-501=249$.

I was however presented with the following solution, which contradicts mine.

let our bid $=X.$

True value of car $=V.$

Expected value $= P(\text{win})*(\text{Payoff from winning}).$

$P(\text{win})=X/1000$, since we win when our bid is above or at true value, and there are $x$ values less than or equal to our bid out of $1000.$

Payoff from winning $= 3/4(X)- X.$

$3/4\,x$ is the value I sell it for since if I win, that means that $X>V$ so $$V \sim \text{Unif}[0,X],$$ and therefore expected value of $V = 0+X/2=X/2$.

Selling price (as mentioned in question) $= 3/2*V=3/2\,(x/2)=3/4\,X.$

This suggests that the selling price will always be less than what we bid, so our expectation is negative.

Both of the above solutions make sense to me, so I'm having a hard time understanding why one is wrong; appreciate any insight.

Your reasoning has a key error: you need to think about the expectation of $V$ only conditional on winning.

"The expected value of the car is 500 based on the distribution we are given. Therefore, on average I will be selling it for 1.5∗500=750."

That's wrong. You only sell if it you win. You won't sell for 750 on average.

"This means that if I bid 501, then I should win more often than I lose."

That's true, but I'm not sure it's relevant.

"...for the wins, my payoff should average out to be 750−501=249."

This is also wrong for the same reason. If you bid 501, you only win if the true value is below 501. In those states of the world, the average true value is 250, so you'll be selling for 375.

So then consider a lower bid, say 100. Then on average the value when you win is 50, you sell for 75.