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\noindent \textbf{Answers to Odd-Numbered Problems, 4th Edition of Games and
Information, Rasmusen }

\begin{center}
 {\bf   CHAPTER 10: Mechanism Design in Adverse Selection and in Moral Hazard
with Hidden Information}
 \end{center}

\noindent     December 29, 2003. 21 November 2005. 

\noindent
  This appendix contains answers to the odd-numbered problems in the fourth
edition of \textit{Games and Information} by Eric Rasmusen, which I am working
on now and perhaps will come out in 2006. The answers to the even-numbered
problems are available to instructors or self-studiers on request to me at
Erasmuse@indiana.edu.

Other books which contain exercises with answers include Bierman \& Fernandez
(1993), Binmore (1992), Fudenberg \& Tirole (1991a), J. Hirshleifer \& Riley
(1992), Moulin (1986), and Gintis (2000). I must ask pardon of any authors from
whom I have borrowed without attribution in the problems below; these are the
descendants of problems that I wrote for teaching without careful attention to
my sources.

\newpage

\begin{center}
 {\bf PROBLEMS FOR CHAPTER 10: Mechanism Design in Adverse Selection and in
Moral Hazard with Hidden Information}
 \end{center}

\noindent
  \textbf{10.1. Unravelling} \\
  An elderly prospector owns a gold mine worth an amount $\theta$ drawn from the
uniform distribution $U[0,100]$ which nobody knows, including himself. He will
certainly sell the mine, since he is too old to work it and it has no value to
him if he does not sell it. The several prospective buyers are all risk neutral.
The prospector can, if he desires, dig deeper into the hill and collect a sample
of gold ore that will reveal the value of $\theta$. If he shows the ore to the
buyers, however, he must show genuine ore, since an unwritten Law of the West
says that fraud is punished by hanging offenders from joshua trees as food for
buzzards.

\begin{enumerate}

 \item[(a)]
 For how much can he sell the mine if he is clearly too feeble to have dug into
the hill and examined the ore? What is the price in this situation if, in fact,
the true value is $\theta=70$?

\noindent
\underline{\textit{Answer}.} The price is 50 -- the expected value of the
uniform distribution from 0 to 100. Even if the mine is actually worth $ \theta=
70 $, the price remains at 50.

\item[(b)]
 For how much can he sell the mine if he can dig the test tunnel at zero cost?
Will he show the ore? What is the price in this situation if, in fact, the true
value is $\theta=70$?

\noindent
\underline{\textit{Answer}}. The expected price is 50. Unravelling occurs, so
\textrm{he will show the ore}, and the buyer can discover the true value, which
is 50 on average. If the true value is $\theta=70$, the buyer receives 70.

\item[(c)]
 For how much can he sell the mine if, after digging the tunnel at zero cost and
discovering $\theta$, it costs him an additional 10 to verify the results for
the buyers? What is his expected payoff?

\noindent
\underline{\textit{Answer}.} He shows the ore iff $\theta \in [20,100] $. This
is because if the minimum quality ore he shows is $b$, then the price at which
he can sell the mine without showing the ore is $\frac{b}{2} $. If $ b=20$ and
the true value is 20, then he can sell the mine for 10, and showing the ore to
raise the price to 20 would not increase his net profit, given the display cost
of 10.

With probability 0.2, his price is 10, and with probability 0.8, it is an
average price of 60 but he pays 10 to display the ore. Thus, the prospector's
expected payoff is 42 (= $0.2(10) +0.8 (60- 10)= 2+40 = 42.)$

\item[(d)]
 Suppose that  with probability 0.5 digging the test tunnel costs 5 for    the
prospector, but with probability 0.5 it costs  him 120.  Keep in mind that the
0-100 value of the mine is  net of the buyer's digging cost. Denote the
equilibrium price that buyers will pay for the mine after the prospector
approaches them without showing ore by $P$. What is the buyer's posterior belief
about the probability it costs 120 to dig the tunnel, as a function of $P$?
Denote this belief by $B(P)$  (Assume, as usual, that all these parameters are
common knowledge, although only the prospector learns whether the cost is
actually 0 or 120.)

\noindent
\underline{\textit{Answer}.}   The prospector will  decide not to show any ore
after digging  a test tunnel if  the  value is less than $P$. This   happens
with probability   $P/100$ after a tunnel is dug.

 The buyers' prior belief that the prospector did not dig a tunnel was  .5, but
if the prospector approaches them without any ore, that lack of ore is relevant
information, to be incorporated into the posterior. Using Bayes's Rule,
$Prob(cost=120|no;  sample)$ equals
   \begin{equation}  \label{e101d}
\begin{array}{ l}
 \frac{  Prob(cost=120)Prob( no;  sample|cost=120)}{ Prob( no;  sample) }\\
  \\
  = \frac{  Prob(cost=120)Prob( no;  sample|cost=120)}{  Prob( no;  sample|cost=
120)Prob(cost=120)  +  Prob( no;  sample|cost=5)Prob(cost=5)     } \\
   \\
  = \frac{  .5  (1)}{   .5  (1)  +  \frac{P}{100} (1)      } \\
   \\
  = \frac{ 1}{   1 +  \frac{P}{50}        } \\
\end{array}
 \end{equation}

\item[(e)]
 What is the prospector's expected payoff in the conditions of part (d) if (i)
the tunnel costs him  120,  or (ii) the tunnel costs him 5?


\noindent
\underline{\textit{Answer}.}  The answer depends on the  equilibrium value of
$P$.
 The  expected payoff to a buyer  if he buys the mine without seeing any sample
ore is
\begin{equation}  \label{e102d}
\begin{array}{ll}
\pi_b (no\; sample) &= -P + Prob(cost=120|no;  sample)  \left( \frac{ 100- 0}{2}
\right)  \\
 & +  Prob(cost=5|no;  sample) \left( \frac{P- 0}{2} \right)  \\
 &\\
  &= -P + \left( \frac{ 1}{   1 +  \frac{P}{50}        } \right)  \left( 50
\right) + \left(1-  \frac{ 1}{   1 +  \frac{P}{50}        } \right)  \left(
\frac{P }{2} \right)  \\
  &\\
  &= -P + \left( \frac{ 1}{   1 +  \frac{P}{50}        } \right)  \left( 50
\right) + \left(  \frac{ \frac{P}{50} }{   1 +  \frac{P}{50}        } \right)
\left( \frac{P }{2} \right)  \\
 &\\
  &= -P + \left( \frac{ 50}{  50+P       } \right)  \left( 50 \right) + \left(
\frac{ P }{   50+P       } \right)  \left( \frac{P }{2} \right)  \\
\end{array}
 \end{equation}
 because with probability $\left( \frac{ 50}{  50+P        } \right)$ the
prospector did not dig a tunnel and the expected value is 50, and with the
remaining  probability  he did dig a tunnel, but its value was less than $P$ and
so he did not show any ore.

Equating the buyer's payoff to 0 because buyers compete profits down to 0 yields
\begin{equation}  \label{e103d}
  \begin{array}{l}
   -P + \left( \frac{ 50}{  50+P       } \right)  \left( 50 \right) + \left(
\frac{ P }{   50+P       } \right)  \left( \frac{P }{2} \right) =0  \\
 \\
   -50P  -P^2  +   2500 +  \frac{P^2 }{2}  =0  \\
\\
  - \frac{P^2 }{2} -50P     +  2500  =0  \\
\\
 P = \frac{ - (-50) \pm \sqrt{ (-50)^2 - 4 ( - \frac{1 }{2}) (2500)   }  }{2 (
- \frac{1 }{2})} \\
    \\
 P = -  50 \pm \sqrt{ 2500 + 5000 }  \\
    \\
 P \approx -  50 \pm  86.60  = 36.60\\
    \\
\end{array}
 \end{equation}


 (i) With probability  .5, the prospector will not dig the tunnel,  because the
cost of 120 for digging is greater than the greatest possible value of the gold,
which is 100.  He will show no ore, as a result, but he will still  receive  the
price of  36.60, which will be his payoff.

(ii) With  probability  .5 he  does dig the tunnel at a cost of 5, finding some
ore. He then refuses to disclose that ore    with probability $.3660$, for a
price of $36.60$, and discloses with probability $1-.3660 $, for an average
price of $\frac{36.60+100}{2} = 68.30.  $  His  expected payoff from digging the
costless tunnel is  $ -5+ ( .366  )(36.60 ) + (.634 )(68.30)  \approx -5+  13.40
+ 43.30= 51.70. $


   The prospector's overall ex ante  expected payoff   is thus $.5 (36.60 )+.5 (
51.70) = 44.15.$








 \end{enumerate}

% -----------------------------------------------

\bigskip
 \noindent
\textbf{10.3. Agency Law }\\
  Mr. Smith is thinking of buying a custom-designed machine from either Mr.
Jones or Mr. Brown. This machine costs 5,000 dollars to build, and it is useless
to anyone but Smith. It is common knowledge that with 90 percent probability the
machine will be worth 10,000 dollars to Smith at the time of delivery, one year
from today, and with 10 percent probability it will only be worth 2,000 dollars.
Smith owns assets of 1,000 dollars. At the time of contracting, Jones and Brown
believe there is there is a 20 percent chance that Smith is actually acting as
an ``undisclosed agent'' for Anderson, who has assets of 50,000 dollars.

Find the price be under the following two legal regimes: (a) An undisclosed
principal is not responsible for the debts of his agent; and (b) even an
undisclosed principal is responsible for the debts of his agent. Also, explain
(as part [c]) which rule a moral hazard model like this would tend to support.

\noindent
\underline{\textit{Answer}.} (a) The zero profit condition, arising from
competition between Jones and Brown, is
 \begin{equation}  \label{e1}
-5000 + .9P + .1 (1000) =0,
 \end{equation}
    because Smith will only pay for the machine with probability  .9, and
otherwise will default and only pay up to his wealth, which is 1,000. This
yields $P
\approx 5,444$.

\noindent
(b) If Anderson is responsible for Smith's debts, then Smith will pay the 5,000
dollars. Hence, zero profits require
\begin{equation}
 \label{e2} -5000 + .9P + .1 (.2) P + .1(.8)(1000) =0,
 \end{equation} which yields $P \approx 5,348$.

\noindent
(c) Moral hazard tends to support rule (b). This is because it reduces
bankruptcy and the agent will be more reluctant to order the machine when there
is a high chance it is unprofitable. In the model as constructed, this does not
arise, because there is only one type of agent, but more generally it would,
because there would be a continuum of types of agents, and some who would buy
the machine under rule (b) would find it too expensive under rule (a).

Even in the model as it stands, rule (a) leads to the inefficient outcome that a
machine worth 2,000 to Smith is not give to Smith. Rather, he pays his wealth
and lets the seller keep the machine, which is inefficient since the machine
really is worth 2000 to Smith.

This is a question about zero-profit prices. Guessing would have been a good
idea here: it is very intuitive that the price would always be above \$5,000,
and that it would be higher if the principal never had to cover the agent's
debts. You should be able to tell that $P>10,000$ is impossible, because Smith
would never pay it. Also, the sellers compete, so it is their profits that
provide a participation constraint, not the benefit to the buyer.

% -----------------------------------------------



\noindent
{\bf  10.5.    The Groves Mechanism  }\\
 A new computer costing 10 million dollars  would benefit existing
Divisions 1, 2, and 3 of a company with 100 divisions.  Each
divisional manager knows the benefit to his division (variables
$v_i,  i = 1,...,3$), but nobody else does, including the company
CEO.     Managers maximize the   welfare of their own divisions.
What    dominant strategy mechanism   might  the CEO   use to induce
the managers to tell the truth  when they report their valuations?
Explain why this mechanism will induce truthful reporting, and
denote the reports by   $x_i,  i = 1,...,3$.   (You may assume that
any  budget transfers to and from the divisions in this mechanism
are permanent-- that the divisions will not get anything back later
if the CEO collects more  payments than he gives, for example.)

\noindent
  \underline{ {\it Answer}.}  Let    Division 1 pay $(10 -  x_2 - x_3)$,
Division 2 pay
$(10 -  x_1 - x_3)$, and Division 3 pay $(10 -  x_1 - x_2)$ if the
computer is bought,  where that payment could be negative,  and  buy
the computer if $x_1 + x_2 + x_3  \geq 10$.

Manager  i's  report does not affect its  payment except by
affecting whether the computer is bought.   Let us take the case of
Manager  1 for concreteness.    His payoff is   $v_1 -  (10 -  x_2 -
x_3)$ if the computer is bought and 0 otherwise.   He therefore
wants the  computer to be bought  if  and only if     $v_1 + x_2 +
x_3 \geq 10$.  By reporting  $x_1 = v_1$, he achieves exactly that
outcome-- the computer is bought only when he wants it to be bought.
If  the other two divisions overreport, he wants the computer to be
bought because the mechanism will  make him pay less than $x_1$, and
if they underrport, he wants it not to be bought, because the
mechanism will make him pay more than $x_1$.






%---------------------------------------------------------------


\bigskip
\noindent
{  \bf 10.7. Selling Cars }\\
  A car dealer must pay \$10,000 to the manufacturer for each car he
adds to his inventory. He faces three buyers. From the point of view of
the dealer, Smith's valuation is uniformly distributed between \$11,000
and \$21,000, Jones's   is between \$9,000 and \$11,000, and Brown's
is between \$4,000 and \$12,000.  The dealer's policy  is to make a
separate  take-it-or-leave-it offer to each customer, and he is smart
enough to avoid making different offers to customers who could resell to
each other.  Use the notation that the maximum valuation is $
\overline{V}$ and the range of valuations is $ R $.




\begin{enumerate}

  \item[(a)]
What will the offers be?

\noindent
  \underline{ {\it Answer}.} Let us use units of
thousands of dollars.  The expected profit from a customer with
maximum valuation $\overline{V}>10$ and range of valuations $R$ is,
if price $P$ is charged:
 \begin{equation}  \label{e119}
  \begin{array}{ll}
  \pi(P; V,R) &= \int^{\overline{V}}_P \frac{P-10}{R}dV \\
   & \\
     & = \left.\left(\frac{PV}{R} - \frac{10V}{R} \right)
\right|^{\overline{V}}_P\\
      & \\
      & = \frac{\overline{V}P}{R} - \frac{10 \overline{V}}{R}
-\frac{P^2}{R} + \frac{10P}{R}.\\
     \end{array}
      \end{equation}
  Maximizing profit with respect to $P$ yields the first order
condition
   \begin{equation}  \label{e120}
 \frac{d\pi(P; V,R)}{dP} =\frac{\overline{V}}{R} -\frac{2P}{R} +
\frac{10}{R} = 0,
  \end{equation}
   so
  \begin{equation}  \label{e120a}
     P^* = \frac{\overline{V}}{2}+ 5.
   \end{equation}

  Note that the optimal price does not depend on $R$, the range of
possible valuations.  That is because what $R$ determines is the
probability that the customer's value is greater than \$10,000, but  if
it is greater than \$10,000, the seller's optimal price is determined by
the possible values between \$10,000 and $\overline{v}$, and $R$ is
irrelevant.


 Applying (\ref{e120a}) to the specific customers: {  \rm Smith will be
offered $P= \frac{21}{2}+5 = \$15, 500$, Jones will be offered  $P=
\frac{11}{ 2} + 5 = \$10,500$,  and Brown  will be offered  $P =
\frac{12}{ 2} + 5 = \$11,000.$ } Moreover,  Brown probably values the
car less than Jones, but because of the higher probability that he
values it more than \$10,000, he will end up paying more if he buys at
all.

 \item[(b)]
  Who is most likely to buy a car? How does this compare with    the
outcome  with perfect price discrimination under full information?   How
does it compare with the   outcome   when the dealer charges \$10,000 to
each customer?

\noindent
  \underline{ {\it Answer}.}  Smith will  buy with
probability 0.55, which is $     {\frac{21-15.5}{21-11}  }$.
    Jones will buy with
probability 0.25.
    Brown  will  buy with
probability
     0.125.
  Thus, {  \rm Smith is the buyer most likely to buy.}

   Whether the dealer charges \$10,000 or uses perfect price
discrimination, the outcome is the same as far as allocative
efficiency: Smith buys with probability 1, Jones buys with
probability 0.5, and Brown  buys with probability 0.25.



  \item[(c)]
  What happens to the equilibrium prices if with probability 0.25
each buyer has a valuation of \$0, but the probability distribution
remains otherwise the same?   What happens to the equilibrium expected
profit?

\noindent
  \underline{ {\it Answer}.} {  \rm The prices are the same as in part
(a)}. If a buyer values the car at less than \$10,000, it is irrelevant
what his value may be, since it is unprofitable to sell to him anyway.
Only the part of his distribution above \$10,000 matters to the seller's
strategy. Note that this has the same flavor as the analysis of
auctions, where a bidder's strategy is conditioned on his having the
highest valuation, since if he does not, he will generally lose the
auction anyway and his bid is irrelevant.

 The   equilibrium expected profit, however, drops to 0.75 of its former
level, since now with probability 0.25 there is no sale for the new
reason that $v=0$.

  Another way to look at this is to think about two moves by Nature. In
the first move, Nature chooses between $v=0$ and $v>0$, with
probabilities 0.25 and 0.75. In the second move, if $v>0$ Nature decides
how much greater it is. Suppose the seller observes the first of these
moves. If he sees that $v=0$, he is indifferent among all prices greater
than \$10,000, since he will not sell a car anyway. If he sees that $v>
0$, he is in exactly the  situation of part (a), so he will choose the
same prices as we found there.

  \item[(d)]
  What happens to the equilibrium price the seller offers to seller
Jones  if with probability 0.25
Jones  has a valuation of \$30,000, but with probability 0.75 his
valuation is uniformly distributed between
  \$9,000 and \$11,000 as before?  Show the relation between price and
profit on a rough graph.

\noindent
  \underline{ {\it Answer}.} Start by deriving the seller's expected
profit from Jones. If the price is below $P=30$, profit is
 \begin{equation}  \label{e119c}
  \begin{array}{ll}
  \pi(P; V,R) &= 0.75 \int^{\overline{V}}_P \frac{P-10}{R}dV  + 0.25 (P-
10)\\
   & \\
     & =0.75  \left.\left(\frac{PV}{R} - \frac{10V}{R} \right)
\right|^{\overline{V}}_P + 0.25 (P-10)\\
      & \\
      & = 0.75 \frac{\overline{V}P}{R} - \frac{10 \overline{V}}{R}
-\frac{P^2}{R} + \frac{10P}{R} + 0.25 (P-10).\\
     \end{array}
      \end{equation}
  Maximizing profit with respect to $P$ yields the first order
condition
   \begin{equation}  \label{e120z}
 \frac{d\pi(P; V,R)}{dP} = 0.75 \left( \frac{\overline{V}}{R} -\frac{2P}
{R} +
\frac{10}{R} \right)  +0.25  =  0,
  \end{equation}
  which solves, given  that $\overline{V} = 11$ and $R=2$,  to
  \begin{equation}  \label{e120d}
     P  =   10 \; \frac{5}{6}.
   \end{equation}
 This, however, is not the profit-maximizing price! The problem is that
this is just a local maximum, not a global maximum, for profit.
 It is a maximum, because the second derivative is  $\frac{d^2\pi(P; V,
R)}{dP^2}  =  \frac{0.75 (2)}{R} <0$, but that just means that it is the
profit-maximizing price given that the price is no greater than \$11,
000.
 Suppose, however, that the seller gives up on selling the car during
the 75\% of the time that Jones has a value between \$9,000 and \$11,
000, and raises his price to $P= \$30,000$. His expected  profit will
then rise from the roughly  \$ (0.26) 11,000 to  exactly \$ (0.25) 30,000.

Figure A10.1 is a rough graph of profits:

 

\includegraphics[width=150mm]{a10-1.jpg}

\begin{center}
{\bf Figure A10.1: Prices and Profits}
\end{center}


 \end{enumerate}

\end{document}


