Question #$4$: Cheating in Sports $[20$ Points$]$

Sydney McLaughlin is an Olympic and World Track Champion in the $400$m hurdles. Suppose that

McLaughlin$$s performance without using steroids will generate an initial wealth of $$7600.$ The use of

steroids would greatly improve athletic performance. If McLaughlin were to use steroids and not be

caught she would be break multiple world records and be a multiple Olympic gold medalist. As a result

she would gain valuable corporate sponsorships. Assume that the benefit that she gets from cheating

would be an extra $$3000.$ Thus McLaughlin$$s wealth if she cheats and is not caught $($WB$)$ would be

$$10,600.$ On the other hand, if McLaughlin uses steroids and is caught she would be banned from the

sport for $3$ years. She would lose corporate sponsors and prize money at future track events. Assume that

the cost to McLaughlin if she is caught cheating is $$7000.$ Thus McLaughlin$$s wealth if she cheats and is

caught $($WC$)$ is $$600.$ Assume that the probability that McLaughlin takes steroids and will be caught is p

$=0\mathrm{.}18.$ Assume that the probability that McLaughlin can take steroids without being caught $(1-$p$)=0\mathrm{.}82.$

Assume McLaughlin has the following utility function: U$($W$)=7$W$0\mathrm{.}5.$

$($a$)$ Suppose McLaughlin decides to take steroids. What would be her expected wealth, E$($W$)?[2$ Points$]$

$($b$)$ Suppose McLaughlin decides to take steroids. What would be her expected utility, E$($U$)?$ Round

your answer to $2$ decimal places. $[6$ Points$]$

$($c$)$ Based on your answer in Part $($b$)$ would Sydney McLaughlin cheat and take steroids? Show your

work. $[6$ Points$].$

$($d$)$ Suppose the IAAF $($the governing body of track and field$)$ is concerned about pervasive doping in

track. They therefore decide to increase their anti$-$doping staff by $30\%$ which increases the likelihood of

catching track athletes who take steroids. Assume that the probability that McLaughlin takes steroids and

will be caught increases to p $=0\mathrm{.}34.$ Conversely, the probability that she takes steroids and is not caught

decreases to $(1-$p$)=0\mathrm{.}66.$ Re$-$do Parts $($b$)($c$).[6$ Points$]$