VCG and GSP Auctions can be useful in a variety of settings, including in allocating

resources. Consider the following situation: You work in a hospital, assigning rooms

to patients. When you do this, you need to consider the amenities that each room

has $($measured by an "amenities score" from $0$ to $1,$ where $0$ means the room is

barebones and $1$ means the room has all possible amenities$)$ and the importance of

these amenities to the patient's recovery $($measured by a "recovery quotient," where

a lower number means that amenities have less of an impact on the patient's

recovery and a higher number means amenities have more impact$).$

Today you need to assign three patients each to a different room:

Patient $1$ has a recovery quotient of $60$

Patient $2$ has a recovery quotient of $50$

Patient $3$ has a recovery quotient of $30$

You have three rooms open:

Room $1$ has an amenities score of $0\mathrm{.}7$

Room $2$ has an amenities score of $0\mathrm{.}3$

Room $3$ has an amenities score of $0\mathrm{.}2$

a$.$ A patient's valuation is their level of recovery in a given room, which can be

calculated as:

level of recovery $=$ patient's recovery quotient $**$ room's amenities score

If your goal is to maximize the total level of recovery $($i$.$e$.,$ have a socially

optimal outcome$),$ which patients would get assigned to which rooms?

b$.$ The hospital charges the patient the auction price for their stay. In a VCG

auction, how much is each patient charged? Show all of your work.

c$.$ Since being healthier means fewer follow$-$up medical costs, the patient's

payoff is level of recovery $-$ price. In a VCG auction, what is each patient's

payoff? Show all of your work.

d$.$ It turns out that the recovery quotients are calculated using a survey. A

patient advocate wants to help Patient $2$ save money $($on room costs $+$

follow$-$up medical costs$),$ so they're trying to coach the patient to answer in a

way that changes their recovery quotient and gets them a higher payoff. Is it

possible for Patient $2$ to retake the survey and get a higher payoff than in part

$($c$)?$ If yes, give an example of a different recovery quotient the patient could

get and show how, with this recovery quotient, the patient's payoff increases.

If no$,$ explain why not.

e$.$ In a GSP auction, how much is each patient charged and what is their payoff?

Show all of your work.

f$.$ Is it possible for Patient $2$ to retake the survey and get a higher payoff than in

part $($e$)?$ If yes, give an example of a different recovery quotient the patient

could get and show how, with this recovery quotient, the patient's payoff

increases. If no$,$ explain why not.

g$.$ In parts $($d$)$ or $($f$),$ if the patient could improve their payoff, did they do so by

increasing or decreasing their level of recovery? Could they have taken the

opposite strategy as well $($i$.$e$.,$ if they improved payoff by decreasing their

level of recovery could they also have improved payoff by increasing their

level of recovery$)?$ Explain.