Suppose a search engine has two ad slots that it can sell. Slot a has a clickthrough

rate of $12$ and slot b has a clickthrough rate of $5.$ There are two advertisers who are

interested in these slots. Advertiser x values clicks at $5$ per click and advertiser y values

clicks at $4$ per click.

$($i$)$ Compute the socially optimal allocation and the VCG prices for it$.$

$($ii$)$ Suppose the search engine decides not to sell slot b$.$ Instead, it sells only slot a

using a sealed$-$bid, second$-$price auction. What bids will the advertisers submit for slot a$,$

who will win, and what price will they pay?

$($iii$)$ Which of these two possible procedures generate the greater revenue for the search

engine?

$($iii$)$ To check whether the result in part $($iii$)$ is general or not, consider another example.

Let there be two slots and two advertisers; let the clickthrough rates be ra for slot a and rb

for slot b$,$ with ra $>$ rb $>0$; and let the advertisers? values be vx and vy$,$ with vx $>$ vy $>0.$

Can you determine which of the two procedures generates the greater revenue for the search

engine? Explain.