Q.6.16 - A First Course in Probability

Question

You and three other people are to place bids for an object, with the high bid winning. If you win, you plan to sell the object immediately for \(10,000. How much should you bid to maximize your expected profit if you believe that the bids of the others can be regarded as being independent and uniformly distributed between \)7,000 and $10,000 thousand dollars? 

Step-by-Step Solution

Answer

We should bid for $9,250 in order to maximise our predicted profit.

Step 1:To find

A bid for an object is placed in order to maximise the predicted profit.

Step 2: Explanation

For an object involving four people, bids are placed, and the highest bidder wins.

After winning the bid, the object is promptly sold for $10,000.

The other three people's bids are unrelated and uniformly distributed between $7,000 and $10,000.

Let's say we put X bucks into the bid.

Only if we win the bid would the other three people's bids be lower than ours.

Since the bids are independent

The probability for winning the bid

$\prod _{i=1,2,3}P\left(U_i<x\right)={\left(\frac{x-7}{4}\right)}^3$

Step 3: Calculation

Now, We need to know our profit. if we win,

Since the object to be sold immediately for $10,000,

We make the profit of (10-x).

If we lose, We don't have any profit.

Thus, The expected profit,

$E\left(P\right(x\left)\right)=(10-x){\left(\frac{x-7}{4}\right)}^3$

By differentiating the above equation and simplifying it

Then we have

$\frac{d}{dx}E\left(P\right(x\left)\right)=\frac{(x-7{)}^2(4x-37)}{64}=0$

Th roots are x= 7

And

$x=\frac{37}{4}=9.25$

We should put the bid for $ 9,250 because we need to maximise our earnings.