Wanda works as a waitress and consequently has the
opportunity to earn cash tips that are not reported by her employer to the
Internal Revenue Service. Her tip income is rather variable. In a good year
(G), she earns a high income, so her tax liability to the IRS is $5,000. In a
bad year (B), she earns a low income, and her tax liability to the IRS is $0.
The IRS knows that the probability of her having a good year is 0.6, and the
probability of her having a bad year is 0.4, but it doesn’t know for sure which
outcome has resulted for her this tax year. In this game, first Wanda decides
how much income to report to the IRS. If she reports high income (H), she pays
the IRS $5,000. If she reports low income (L), she pays the IRS $0. Then the
IRS has to decide whether to audit Wanda. If she reports high income, they do
not audit, because they automatically know they’re already receiving the tax
payment Wanda owes. If she reports low income, then the IRS can either audit
(A) or not audit (N). When the IRS audits, it costs the IRS $1,000 in
administrative costs, and also costs Wanda $1,000 in the opportunity cost of
the time spent gathering bank records and meeting with the auditor. If the IRS
audits Wanda in a bad year (B), then she owes nothing to the IRS, although she
and the IRS have each incurred the $1,000 auditing cost. If the IRS audits
Wanda in a good year (G), then she has to pay the $5,000 she owes to the IRS,
in addition to her and the IRS each incurring the cost of auditing.
(a) Suppose that Wanda has a good year (G), but she
reports low income (L). Suppose the IRS then audits her (A). What is the total
payoff to Wanda, and what is the total payoff to the IRS?
(b) Which of the two players has an incentive to bluff
(that is, to give a false signal) in this game? What would bluffing consist of?
(c) Show this game in extensive form. (Be careful about
(d) How many pure strategies does each player have in
this game? Explain your reasoning.
(e) Write down the strategic-form game matrix for this
game. Find all of the Nash equilibria to this game. Identify whether the
equilibria you find are separating, pooling, or semiseparating.
(f) Let x equal the probability that Wanda has a good
year. In the original version of this problem, we had x 5 0.6. Find a value of
x such that Wanda always reports low income in equilibrium.
(g) What is the full range of values of x for which Wanda
always reports low income in equilibrium?