6.12 The Tax Man: A citizen (player 1) must choose whether to file taxes honestly or to cheat. The tax man (player 2) decides how much effort to invest in auditing and can choose a [0, 1]; the cost to the tax man of investing at a level a is c(a) = 100a^2. If the citizen is honest then he receives the benchmark payoff of 0, and the tax man pays the auditing costs without any benefit from the audit, yielding him a payoff of 100a^2. If the citizen cheats then his payoff depends on whether he is caught. If he is caught then his payoff is 100 and the tax man's payoff is 100 100a^2. If he is not caught then his payoff is 50 while the tax man's payoff is 100a^2. If the citizen cheats and the tax man chooses to audit at level a then the citizen is caught with probability a and is not caught with probability (1 a). a. If the tax man believes that the citizen is cheating for sure, what is his best-response level of a? b. If the tax man believes that the citizen is honest for sure, what is his best-response level of a? c. If the tax man believes that the citizen is honest with probability p, what is his best-response level of a as a function of p? d. Is there a pure-strategy Nash equilibrium for this game? Why or why not? e. Is there a mixed-strategy Nash equilibrium for this game? Why or why not?