Question

Mark works as a waiter, and consequently has the opportunity to earn cash tips that are not reported by his employer to the Internal Revenue Service. His tip income is rather variable. In a good year (G), he earns high income, so that his tax liability to the IRS is $5000. In a bad year (B), he earns low income, and his tax liability to the IRS is $0. The IRS knows that the probability of him having a good year is 0.6, and the probability of him having a bad year is 0.4, but it doesn't know for sure which outcome has resulted for him this tax year. In this game, first Mark decides how much income to report to the IRS. If he reports high income (H), he pays the IRS $5000. If he reports low income (L), she pays the IRS $0. Then the IRS has to decide whether to audit Mark. If he reports high income, they do not audit, because they automatically know they’re already receiving the tax payment Mark 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 $1000 in administrative costs, and also costs Mark $1000 in the opportunity cost of the time spent gathering bank records and meeting with the auditor. If the IRS audits Mark in a bad year (B), then he owes nothing to the IRS, although he and the IRS have each incurred the $1000 auditing cost. If the IRS audits Mark in a good year (G), then he has to pay the $5000 he owes to the IRS, in addition to each incurring the cost of auditing. (a) Suppose that Mark has a good year (G), but he reports low income (L). Suppose the IRS then audits him (A). What is the total payoff to Mark, and what is the total payoff to the IRS? (b) How many pure strategies does each player have in this game? Explain your reasoning. (c) Write down the normal form game matrix for this game. Find all of the Nash equilibria to this game. (d) Let x equal the probability that Mark has a good year. In the original version of this problem, we had x = 0.6. Find a value of x such that Mark always reports low income in equilibrium. (e) What is the full range of values of x for which Mark always reports low income in equilibrium?

          Mark works as a waiter, and consequently has the opportunity to
earn cash tips that are not reported by his employer to the
Internal Revenue Service. His tip income is rather variable. In a
good year (G), he earns high income, so that his tax liability to
the IRS is $5000. In a bad year (B), he earns low income, and his
tax liability to the IRS is $0. The IRS knows that the probability
of him having a good year is 0.6, and the probability of him having
a bad year is 0.4, but it doesn't know for sure which outcome has
resulted for him this tax year. In this game, first Mark decides
how much income to report to the IRS. If he reports high income
(H), he pays the IRS $5000. If he reports low income (L), she pays
the IRS $0. Then the IRS has to decide whether to audit Mark. If he
reports high income, they do not audit, because they automatically
know they’re already receiving the tax payment Mark 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 $1000 in administrative
costs, and also costs Mark $1000 in the opportunity cost of the
time spent gathering bank records and meeting with the auditor. If
the IRS audits Mark in a bad year (B), then he owes nothing to the
IRS, although he and the IRS have each incurred the $1000 auditing
cost. If the IRS audits Mark in a good year (G), then he has to pay
the $5000 he owes to the IRS, in addition to each incurring the
cost of auditing. (a) Suppose that Mark has a good year (G), but he
reports low income (L). Suppose the IRS then audits him (A). What
is the total payoff to Mark, and what is the total
payoff to the IRS?
(b) How many pure strategies does each player have in this game?
Explain your reasoning.
(c) Write down the normal form game matrix for this game. Find
all of the Nash equilibria to this game.
(d) Let x equal the probability that Mark has a good year. In
the original version of this problem, we had x = 0.6. Find a value
of x such that Mark always reports low income in equilibrium.
(e) What is the full range of values of x for which Mark always
reports low income in equilibrium?

Added by Michelle S.

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