Problem Statement

Carter has come into contact with a bounty of gold. He takes it to an auction shop. The shop tells him that every bidder will place a bid uniformly between $500 and $1000. They also say they can recruit bidders for $5 per person.

Carter needs to figure out the optimal number of bidders to maximize his expected payout. What is this maximum payout?

Original Problem Link: Click here

Solution

Step 1: Modeling the Maximum Bid

Each bidder bids randomly between 500 and 1000. Now, if there are \(n\) bidders, we can assume that the bids will be evenly spaced between 500 and 1000 on average.

The expected spacing between bids is:

\[\frac{500}{n+1}\]

Thus, the expected maximum bid will be:

\[M(n) = 1000 - \frac{500}{n+1}\]

Step 2: Writing the Expected Payout

Now, Carter’s expected earnings are simply:

\[\text{Max bid} - \text{Cost of bidders}\]

Which gives us:

\[E(n) = M(n) - 5n\]

Substituting \(M(n)\):

\[E(n) = 1000 - \frac{500}{n+1} - 5n\]

Now, we need to maximize this function.

Step 3: Finding the Optimal \(n\)

We differentiate \(E(n)\) and set it to 0:

\[\frac{d}{dn} \left( 1000 - \frac{500}{n+1} - 5n \right) = 0\] \[\frac{500}{(n+1)^2} = 5\]

Solving:

\[500 = 5(n+1)^2\] \[100 = (n+1)^2\] \[n+1 = 10 \Rightarrow n = 9\]

Thus, the optimal number of bidders is \(n = 9\).

Step 4: Calculating the Maximum Expected Payout

Substituting \(n = 9\):

\[E(9) = 1000 - \frac{500}{10} - 5(9)\] \[= 1000 - 50 - 45\] \[= 905\]

Thus, the maximum expected payout Carter can get is $905.

---

💡 Did you enjoy this problem? Check out more puzzles in the Problems section!