There are N hostages and one bad guy, who needs a password from the hostages. One by one, the bad guy asks each hostage for the password. If the hostage squeals, the game ends; otherwise, the hostage dies, and the bad guy moves on to the next hostage. There are two types of hostages: `sacrificial' types, and `selfish' types. Sacricial hostages receive payoff of -1 from dying, and -9 if the bad guy gets the password. Selfish types get -1 from dying, but don't care if the bad guy gets the password. Each hostage is sacrificial with probability 0.95, and selfish with probability 0.05.
(i) If the first hostage questioned is a sacricial type, calculate this type's payoff from not giving up the password, in terms of N.
(ii) What is the maximum number of hostages required for the sacrificial type in part (i) to not give up the password? (Hint: log xn = n log x)
(iii) Now assume that N = 50, and the probability each hostage is sacrificial is p. How large should p be to ensure that the sacricial type does not give up the password?