Question $18$

Batman and Robin are trying to rescue Catwoman from the Joker. They suspect that the Joker has taken her to one of his two hideouts: A large cave $($C$),$ or a deserted two$-$story building $($B$).$ After some deliberation Batman and Robin conclude that the Joker is equally likely to be at either location but if he is in the deserted building, then with probability $0\mathrm{.}7$ he will be in his laboratory on the second floor $($B$2)$ and with probability of $0\mathrm{.}3$ he will be downstairs $($B$1).$ Batman and Robin believe that the Joker injected Catwoman with a slow acting poison and wish to rescue her as soon as possible in order to give her the antidote before it is too late.

Batman and Robin have two alternatives:

A$1-$ Search the building first

A$2-$ Search the cave first.

The rescue times under each alternative are as follows:

$\backslash$table$[[,$If Joker and Catwoman are in$],[$Alternative$,$B$1,$B$2,$C$],[$A$1,5$ minutes,$15$ minutes,$50$ minutes$],[$A$2,50$ minutes,$60$ minutes,$10$ minutes$]]$

a$-$ Find the best strategy for Batman and Robin. What will be the expected rescue time under the optimal strategy?

b$-$ A well$-$dressed stranger suddenly appears and offers "partial" perfect information: He only will say whether or not Joker is in the cave. Should Batman and Robin be willing to talk to him for $10$ minutes to get his information?

c$-$ Suppose the well$-$dressed man offered to provide complete perfect information: He will say exactly where the joker is$.$ How long should Batman and Robin be willing to talk to him in that case? $($Hint: calculate EVPI$)$

d$-$Batman and Robin chose not to talk to the well$-$dressed stranger but instead they consider a test: Robin will go to the deserted building and listen outside for ghoulish laughter. If the Joker is in downstairs then the probability that Robin will hear laughter is $0\mathrm{.}8.$ If the Joker is in the laboratory upstairs, however, Robin will not hear him laughing with probability $0\mathrm{.}6.$ Should they make this test before deciding whether to search the cave or the building if it takes $15$ minutes $($in addition to the search times$)?$