$4.$ Modeling strategic situations: Tax audits with heterogeneous taxpayers. Recall the tax audit game from Lecture $11.$ There, we interpreted the mixed strategy of player $2,$ the taxpayer, as describing the fractions of a population of taxpayers choosing to be Honest or to Cheat on their taxes. In this interpretation, we imagined all taxpayers in the population to have the same payoffs, and the reason they made different choices $($ H or C $)$ was dictated by the logic of mixed strategy NE$.$

In this problem, you will explore a version of the game with a population of taxpayers who have different propensities to cheat on their taxes. Specifically, there is a population of size $1$ of taxpayers. The taxpayers differ with respect to the payoff from being honest, x$,$ which is distributed uniformly on the interval $\backslash ([0,6]\backslash ).$ Think of taxpayers being indexed by x $.$ The payoffs of taxpayer x from being honest $($H$)$ and from cheating $($C$)$ are as follows:

There is a single auditor who must choose between auditing $($A$)$ or not $($N$).$ The auditor knows that the payoff from being honest, x$,$ is distributed uniformly from $0$ to $6$ in the population, but they cannot observe the x of any individual taxpayer. Thus, their choice between A and N applies equally to all taxpayers. For example, if they play A $,$ then all taxpayers are audited. Or if they randomize and play A with probability p$,$ then every taxpayer has probability p of being audited. The auditor's payoffs are as given in the above payoff matrix. For example, if all taxpayers play $\backslash ($ H $\backslash )$ and the auditor plays N $,$ then the auditor's payoff is $2.$ Or if fraction q of taxpayers play H and the auditor plays N $,$ then the auditor's payoff is q $*2(1-$q$)*0.$

$($a$)$ Suppose that the auditor plays A with probability p$.$ How does a taxpayer's best response depend on their payoff from being honest, x$?$ What fraction of taxpayers will report honestly $($i$.$e$.,$ play H$)?$