# IRS Game Theory Question

Jim works as a salesman. His income is variable. In a good year (G), his income is high, so his takes liability with the IRS is 10K. In a bad year (B), his income is low and his tax liability with the IRS is 0. The IRS knows that the probability of Jim having a good year is 0.6 and the probability of him having a bad year is 0.4, but they don't know for sure which outcome has resulted for him this tax year.

In this game, first Jim decides how much income to report to the IRS. If he reports a high income (H), he pays 10K. If he reports low income (L), he pays 0. Then the IRS has to decide whether to audit Jim. If he reports high income, then they do not audit, because they automatically know they are already receiving the tax payment Jim owes. If he reports low income, then the IRS can either audit (A) or not audit (N). When the IRS audits, it costs the IRS 2k in administrative costs and also costs Jim 2k for the opportunity cost of the time spent with the auditor. If the IRS audits Jim in a bad year (B), then he owes nothing to the IRS, although they have both incurred 2k in expenses each. If the IRS audits Jim in a good year (G) then he has to pay the 10k he owes, plus they both incur the 2k in expenses.

A) Write down the extensive form of the game.

B) Write down the normal form game (i.e. the game in matrix form).

C) Determine if there is any pure strategy Nash equilibrium. If not, then find the mixed strategy Nash Equilibrium.

D) Let x equal the probability that Jim has a good year. What is the range of values of x for which Jim always reports low income in equilibrium?

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