Player 1 (a citizen) files taxes honestly (H) or cheats (C). Player 2 (the government) chooses how much effort $a \in [0, 1]$ to invest in auditing Player 1. The players choose simultaneously.
If Player 1 chooses H, then her payoff is 0 and Player 2's payoff is $-100a^2$.
If Player 1 chooses C, then she is caught with probability $a$ (recall that $a$ is Player 2's audit effort).
If Player 1 is caught, then her payoff is -100 and Player 2's payoff is $100 - 100a^2$. If Player 1 is not caught, then her payoff is 50 and Player 2's payoff is $-100a^2$.
a.) If Player 2 believes that Player 1 chooses H, what is her optimal level of $a$?
b.) * If Player 2 believes that Player 1 chooses H with probability $p$, what is her optimal level of $a$ as a function of $p$?
c.) * What is Player 1's best-response function, as a function of $a$?
d.) * Prove that this game does not have a pure-strategy Nash equilibrium.
e.) There is a Nash equilibrium of this game in which Player 1 chooses H with probability $p \in (0, 1)$ and Player 2 chooses audit effort $a$. Find the equilibrium values of $p$ and $a$.
