Question

Jane wants to auction off an item, but does not know where to go to find bidders. David offers to find bidders for her, but will charge her $10 per bidder he gets to show up. Each bidder will uniformly value the item between [500. 1000). The highest bidder will win the item and pay the second-highest bidder's price (Vickrey auction). How many bidders should Jane pay David to find?

Answer 1

according to the given question, bidders bid uniformaly between 500 and 1000. For each bidder jane pays 10$ to david.

solution:

suppose there are 'n' bidders to be found,

1st bidder starts from 500.

difference of price between bidders = 500/(n-1); since the range has to be divided into (n-1) parts for n bidders to be from 500 till just before 1000. Here, i am considering the highest price to be as close to 1000 so that Jane gets as much price of the item as possible.

item price

p = price given by (n-1)^th bidder. - 10*n;

differentiating P w.r.t n to get the optimum value of n ;

equating dp/dt = 0, and solving for n, we get n = 8 as one possibel number of bidders that will increase the profir of Jane.

hope the solution is clear and please ask any query if there is some.

Thank you.