Game Theory with Applications to Finance and Marketing

Show that
Π ∗ B = 17
4 , Π∗ 1 = 221
56 , Π∗ 2 = 12.
27. Before we end this note, we shall introduce a class of games with imperfect
information, called games of signal jamming. We give 5 examples
below.
28. Example 9. Consider a firm run by an owner-manager Mr. A. At
date 1, Mr. A can either costlessly exert a low effort (e = 0) or exert a
high effort (e = 1) by incurring a disutility c > 0. Mr. A’s effort choice
is unobservable to public investors. Let π 1 and π 2 be respectively the
firm’s date-1 and date-2 profits. If Mr. A exerts a low effort, the firm’s
date-1 and date-2 profits are both zero. If Mr. A exerts a high effort,
then the firm’s date-1-date-2 profits may be
(π 1 , π 2 ) =
{
(0, 5), with probability β;
(1, 0), with probability 1 − β.
The firm has no growth opportunities after date 2, and it will be liquidated
at the end of date 2. Mr. A and all investors in the financial
market are risk-neutral without time preferences.
(i) Suppose that it is common knowledge that Mr. A will never sell
shares by the end of date 2. Find a set of conditions on c and β ensuring
that Mr. A exerts a low effort in equilibrium.
(ii) Suppose that 1 > β > 1 and 0 < c < 2. Suppose that, contrary to
4
part (i), it is common knowledge that Mr. A will sell all the equity at
date 1 (without distributing any cash dividends at date 1), and that
public investors’ only date-1 information about the firm is π 1 .
• Find further conditions on c and β ensuring that Mr. A exerts a
high effort in equilibrium.
• Find further conditions on c and β ensuring that Mr. A exerts a
low effort in equilibrium.
Solution. The answer to part (i) is c > 1 + 4β.
The answer to part (ii) is c > 1 − β if we want to induce Mr. A to
exert a low effort, and this answer needs a little explanation.
33