Problem $5(2$ points$)$

A search engine has three ad slots, Slot $1,$ Slot $2$ and Slot $3.$ Slot $1$ has a click$-$through rate of $6,$ Slot $2$ has a click$-$through rate of $4$ and Slot $3$ has a click$-$through rate of $1.$ Three advertisers, A$,$ B$,$ and C$,$ are interested in the slots. A values clicks at $5$ per click, advertiser B values clicks at $3$ per click and advertiser C values clicks at $1$ per click.

$($a$)$ What is the socially optimal allocation of slots to advertisers? point$)$

$($b$)$ What are the VCG prices for the slots? $(0\mathrm{.}5$ point$)$

$($c$)$ Can you think of applications of sponsored search markets beyond those discussed in the book? $(1$ point$)$