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\begin{LARGE}

\noindent
 February 12, 2003 \\
  Eric Rasmusen, Erasmuse@indiana.edu 
 



\section*{ 10   Mechanism Design   }

 
\noindent
 {\bf  10.1 The  Revelation Principle and  Moral Hazard with Hidden
Knowledge  }   
\noindent
 {\bf Moral Hazard with Hidden Knowledge}
  
   Information is complete in moral hazard games, but  in 
moral hazard with hidden knowledge    the agent, but not the
principal,  observes a move of Nature  after the game
begins.   Information is symmetric at the time of
contracting  but becomes asymmetric later.    From the
principal's point of view, agents are identical at the
beginning of the game  but   develop  private types midway
through,   depending on what they have seen. His  chief
concern is to  give them incentives to disclose   their
types later,  which gives games with hidden knowledge   a
flavor close to that of   adverse selection.  (In fact, an
alternative name for this might be {\bf 
post-contractual adverse selection.}) The agent might 
exert effort, but effort's  contractibility is  less
important when the principal does not know which effort
is appropriate because he is  ignorant of the state of the
world chosen by Nature.   The main difference   technically   
is  that if     information is symmetric at the start and only
becomes asymmetric   after a contract is signed, the 
participation constraint  is based on the agent's
expected payoffs across the different types of agent he
might become. Thus, there is just one participation
constraint even if there are eventually $n$ possible
types of agents in the model, rather than the $n$
participation constraints that would be required in a
standard adverse selection model.  



 There is more hope for obtaining efficient outcomes in  
moral hazard with hidden knowledge  than in adverse selection or
simple moral hazard.  The advantage over adverse selection  is that
information is symmetric at the time of contracting, so  neither
player can use    private information to extract surplus from the
other by choosing inefficient contract terms. The advantage over
simple moral hazard  is that the post-contractual asymmetry is with
respect to knowledge only,  which is neutral in itself, rather than
over whether the  agent exerted high effort, which  causes direct
disutility to him.

 
 

\begin{center}
 {\bf  Production Game VII: Hidden Knowledge }
 \end{center}
  {\bf Players}\\
  The principal and the agent.

 \noindent
 {\bf The Order of Play}\\
 1 The principal offers the agent a wage contract of the form
$w(q,m)$, where $q$ is output and $m$ is a message to be  sent by the
agent.\\
 2 The agent accepts or rejects the principal's offer.\\
 3 Nature chooses the state of the world $\theta$, according to
probability distribution $F(\theta)$.  The agent observes $\theta$,
but the principal does not.\\
 4 If the agent accepts, he exerts effort $e$ and sends a message
$m$, both observed by the principal.\\
 5 Output is $q(e, \theta)$.

\noindent
 {\bf Payoffs}\\
 If the agent rejects the contract,   $\pi_{agent} = \bar{U}$ and
$ \pi_{principal} = 0$.\\
 If the agent accepts the contract, $\pi_{agent}= U(e,w,\theta)$
and $\pi_{principal}= V(q - w)$.

 \bigskip
 
   The principal would like to know $\theta$ so he can tell
which effort level is    appropriate. In an ideal world he
would employ    an honest agent who always chose
$m=\theta$,   but in  noncooperative games, talk is cheap. 
Since the agent's words are worthless, the principal
must try to design a contract that either provides
incentive  for truth-telling or takes lying into
account -- he   {\bf
implements} a {\bf mechanism} to extract the agent's
information.  

 
\newpage

 
\begin{center}
 {\bf The Revelation Principle }
 \end{center}
  A principal might choose to offer a contract that induces his 
agent to lie in equilibrium, since he can take lying into account
when he designs the contract, but this complicates the analysis. Each
state of the world has a single truth, but a continuum of lies. 
Generically speaking, almost everything is false.  The following 
principle helps us simplify contract design.

 \noindent
    {\bf The Revelation Principle.} {\it For every contract
$w(q,m)$ that leads to lying (that is, to $m \neq \theta$), there is
a contract $w^*(q,m)$ with the same outcome for every $\theta$ but no
incentive for the agent to lie.}  

     Many  possible contracts  make  false messages   profitable for
the   agent because  when the state of the world is $a$ he receives a
reward of $x_1$ for the true report of $a$ and  $x_2>x_1$ for the
false report of $b$.   A contract which gives the agent the same
reward of $x_2$ regardless of whether he reports $a$ or $b$ would lead
to exactly the same payoffs for each player while giving  the agent no
incentive to lie. The revelation principle  notes that a   truth-
telling contract  like this  can always be found by imitating the
relation between states of the  world and payoffs in the equilibrium
of a contract with lying. The idea can also be applied to games in
which two  players must make reports to each other.


 Applied to concrete examples, the revelation principle
is  obvious.  Suppose we are concerned with the
effect on the moral climate of cheating on income taxes,
but anyone who makes \$70,000 a year can claim he makes
\$50,000 and we do  not have the resources
to  catch him.  The revelation principle says that we can
rewrite the tax code to set the tax to be the same for
taxpayers earning \$70,000 and  for those earning
\$50,000, and the same amount of taxes will be collected
without anyone having   incentive to lie.   Applied to
moral education, the principle says that the mother who
agrees never to punish her daughter if she tells her  all
her escapades will never hear any untruths. Clearly,
the principle's usefulness is not  so much to improve
outcomes as   to simplify contracts. The principal (and
the modeller) need only look at contracts which induce
truth-telling, so the relevant strategy space is shrunk
and we can add a third constraint  to  the incentive
compatibility and participation constraints to help
calculate the equilibrium: 

\noindent 
 (3) {\bf Truth-telling.} The equilibrium contract makes the    agent
willing
to choose $m =\theta$.

The  revelation principle says that a truth-telling equilibrium
exists, but not that it is unique.  It may well happen that the
equilibrium is a weak Nash equilibrium in which   the optimal contract
gives the agent no incentive to lie but also no incentive to tell the
truth. This is similar to the open-set problem discussed in Section
4.3;   the optimal contract may satisfy the agent's participation
constraint but makes him indifferent  between  accepting and rejecting
the contract. If agents derive the slightest utility from telling the
truth, of course, then truthtelling becomes a strong equilibrium, but
if their utility from telling the truth is really significant, it
should be made an explicit part of the model. If the utility of truth-
telling is strong enough, in fact, agency problems and the costs
associated with them  disappear.   This is one reason  why morality is
useful to business.
 
\newpage

 
\begin{center}
 {\bf NOTES FOR CLASS }
 \end{center}
Handouts: old assignments.

 In getting reference letters, editors are useful. 

 Also, this is a part of research, service to the profession. I do
comments during seminars. It is satisfying and useful.

 Getting comments is very hard. It is easier for students. People are
not as grateful as they should be. It is a major use of co-authors--
just to read the stuff.

 These next few weeks: 
 Some useful topics from my book. IOish topics. This week, mechanisms.
Important for regulatoin, especially, but also price discrimination.

 \newpage

\begin{center}
 {\bf  10.3:  Myerson Mechanism Design Example }   
 \end{center}
 
 
   A seller has 100 units of a good.

 If
it is high quality, he values it at 40 dollars per unit; if it is low
quality, at 20 dollars.

 The buyer, who cannot observe quality before
purchase, values high quality at 50 dollars per unit and low quality
at 30 dollars.


 The only way to get the seller to
truthfully reveal the quality of the good, 
 is for the buyer
to say  that if the seller admits the quality is bad, he will buy more
units than if the seller claims it is good. 

 

Who offers the contract? 

 

When it is offered?





\newpage


\begin{center}
 {\bf Myerson Trading Game I }
 \end{center}
  {\bf Players}\\
 A buyer and a seller. 

 \noindent
 {\bf The Order of Play}\\
  1  The  seller offers the buyer a contract ($  Q_H, P_H, T_H,  Q_L,
P_L, T_L$)
under which   the seller  will later   declare   his quality to be
high or low,
and  the buyer will  first pays the lump sum $T$ to the seller
(perhaps with $T<0$)
and then   buy  $Q$ units  of the 100 the seller has available, at
price $P$. \\
2 The buyer accepts or rejects the contract.\\
3  Nature chooses whether the seller's good is High quality
(probability 0.2) or low quality (probability 0.8), unobserved by the
buyer.\\
 4.  If  the contract was accepted by both sides,  the seller
declares his type to be L or H and sells at the appropriate quantity
and price as stated in the contract. \\

  

\noindent
 {\bf Payoffs}\\
 If the seller rejects the contract,  $\pi_{buyer} =  0$
$\pi_{seller \; H} =  40*100$,  and $\pi_{seller \; L} =  20*100$.
\\
 If the seller  accepts the contract and declares a type that has
price $P$ and quantity $Q$, then 
 \begin{equation} \label{e10.50a}
 \pi_{buyer| seller\; H} = -T+ (50-P)
Q\;\;\;  and\;\;\;   \pi_{buyer| seller\; L} = (30-P) Q
\end{equation}
 and 
  \begin{equation}  \label{e10.50b}
\pi_{seller \; H} =
40(100-Q) + PQ  \;\;\;  and\;\;\;  \pi_{seller \; L} =  20(100-Q) +
PQ.
\end{equation}

\newpage


\begin{center}
 {\bf CONSTRAINTS }
 \end{center}
Participation Constraint: 
   \begin{equation} \label{e10.50}
\begin{array}{ll}
0.8 \pi_b (Q_L, P_L) + 0.2 \pi_b (Q_H, P_H) &\geq  0 \\
   &\\
  0.8   (30-P_L)Q_L + 0.2 (30-P_H)Q_H&\geq  0   \\
  \end{array}
    \end{equation}

    
Incentive compatibility constraints, 
low quality:
    \begin{equation} \label{e10.51}
\begin{array}{ll}
\pi_L (Q_L, P_L)  &\geq  \pi_H (Q_H, P_H)\\
 &\\
  20 (100-Q_L) + P_L Q_L &\geq  20 (100-Q_H) + P_H Q_H,  \\
  \end{array}
    \end{equation}

 High quality:
\begin{equation} \label{e10.52}
\begin{array}{ll}
\pi_H (Q_H, P_H)& \geq \pi_H (Q_L, P_L) \\
 &\\
 30 (100-Q_H) + P_H Q_H&  \geq  30 (100-Q_L) + P_L Q_L.     \\ 
  \end{array}
    \end{equation}

 To make the contract incentive compatible,  the seller needs to set
$P_H$ greater than $P_L$, but if he does that it will be necessary to
set $Q_H$ less than $Q_L$. If he does that, then the low-quality
seller will not be irresistably tempted to pretend his quality is
high: he would be able to  sell at  a higher price, but not as  great
a quantity.

  Since $Q_H$ is being set below 100 only to make pretending to be
high-quality unattractive, there is no  reason to set $Q_L$ below 100,
so   $Q_L=100$.  The buyer will accept the contract if $P_L  \leq 30$,
so the seller should set $P_L= 30$.

\newpage
 \begin{center}
 {\bf Myerson Trading Game I, continued }
 \end{center}
  The low-quality seller's incentive compatibility constraint,
inequality (\ref{e10.51}),  will be binding, and thus becomes
 \begin{equation} \label{e10.60}
\begin{array}{ll}
\pi_L (Q_L, P_L)  &\geq  \pi_H (Q_H, P_H)\\
 &\\
  20 (100-100) + 30*100   &=  20 (100-Q_H) + P_H Q_H.   \\
  \end{array}
    \end{equation}
 Solving for $Q_H$ gives us $Q_H =   \frac{1000}{P_H-20}$, which when
substituted
into  the seller's payoff  function yields
 \begin{equation} \label{e10.57}
\begin{array}{ll}
\pi_s  &=0.8 \pi_L (Q_L, P_L) +  0.2\pi_H (Q_H, P_H)   \\
 &\\
 &=  0.8 [(20)(100-Q_L) +  P_LQ_L]   +  0.2   [(40)(100-Q_H) +
P_HQ_H]   \\
 &\\
&=  0.8 [(20)(100-100) +  30*100]   +  0.2   [(40)(100-\frac{1000}
{P_H-20}) +\\
 & 
P_H (\frac{1000}{P_H-20})]   \\
  \end{array}
    \end{equation}
 Maximizing with respect to $P_H$ subject to the constraint that $P_H
\leq 50$ (or else the buyer will turn down the contract)  yields the
corner solution of $P_H= 50$, which allows for $Q_H = 33 \frac{1}{3}$.

The participation constraint for the buyer is already binding, so we
do not need the transfers $T_L$ and $T_H$ to take away any remaining
surplus, as we might in other situations.\footnote{The transfers could
be used to adjust the prices, too. We could have $Q_L=20$ and $T_L=
1000$ in equation (\ref {e10.57}) without changing anything
important.} Thus, the equilibrium contract is
 \begin{equation} \label{e10.57a}
  (  Q_L=100, P_L=30,T_L=0,  Q_H=33 \frac{1}{3}, P_H=50, T_H=0). 
    \end{equation}
 

 
 \newpage



\begin{center}
 {\bf Myerson Trading Game II }
 \end{center}
\noindent
 {\bf The Order of Play:} 
 The same as  in Myerson Trading Game I except that the buyer makes
the
contract offer in move (1) and the seller accepts or rejects in move
(2).

  

\noindent
 {\bf Payoffs:} 
The same as in  Myerson Trading Game I.


\bigskip

The participation constraint in the buyer's mechanism design problem
is
   \begin{equation} \label{e10.53}
\begin{array}{ll}
 0.8 \pi_L (Q_L, P_L) +  0.2\pi_H (Q_H, P_H)  &\geq  0. 
  \end{array}
 \end{equation}

 The incentive compatibility constraints are just as they were before.

 As before, the mechanism will set $Q_L=100$, but it will have to make
$Q_H<100$ to deter the low-quality seller from pretending he is high-
quality. Also, $P_H\geq 40$,  or the high-quality seller will pretend
to be low-quality.

Suppose $P_H=40$. The low-quality seller's incentive compatibility
constraint   will be binding, and thus becomes
 \begin{equation} \label{e10.60a}
\begin{array}{ll}
\pi_L (Q_L, P_L)  &\geq  \pi_H (Q_H, P_H)\\
 &\\
  20 (100-100) +  P_L*100   &=  20 (100-Q_H) +40 Q_H.   \\
  \end{array}
    \end{equation}

\newpage

\begin{center}
 {\bf Myerson Trading Game II, continued }
 \end{center} Solving for $Q_H$ gives us $Q_H =    5P_L-100$,  which
when substituted into  the
buyer's payoff  function yields
 \begin{equation} \label{e10.57c}
\begin{array}{ll}
\pi_b  &=0.8 \pi_{b|L}  (Q_L, P_L) +  0.2\pi_{b|H} (Q_H, P_H)   \\
 &\\
 &=  0.8 [ (30-P_L) Q_L ]   +  0.2   [(50-P_H) Q_H ]   \\
 &\\
 &=  0.8 [ (30-P_L) 100 ]   +  0.2   [(50-40) (5P_L-100) ]   \\
 &\\
 &= 2400 -  80P_L   +    10P_L-200 = 2200-70P_L   \\
  \end{array}
    \end{equation}
 Maximizing with respect to $P_L$ subject to the constraint that $P_L
\geq 20$ (or else we would come out with  $Q_H<0$ to satisfy incentive
compatibility constraint ( \ref{e10.60}))  yields the corner solution
of $P_L=20$, which requires that $Q_H=0$.

 Would setting $P_H>40$ help? No, because that just makes it harder to
satisfy the low-quality seller's incentive compatibility constraint.
We would continue to have $Q_H=0$, and, of course, $P_H$ does not
matter if nothing is sold.  And as before, we do not need to make use
of transfers to make the participation constraint binding. Thus, the
equilibrium contract has   $P_H$ take any possible value and
 \begin{equation} \label{e10.57d}
  ( Q_L=100, P_L=20,T_L=0,  Q_H=0, T_H=0). 
    \end{equation}
 
  
  \newpage


\begin{center}
 {\bf Myerson Trading Game III }
 \end{center}
\noindent
 {\bf The Order of Play}\\
0.   Nature chooses whether the seller's good is high quality
(probability 0.2) or low quality (probability 0.8), unobserved by the
buyer.\\  
  1  The  buyer offers the seller   a contract ($  Q_H, P_H, T_H,
Q_L, P_L, T_L$)
under which   the seller  will later   declare   his quality to be
high or low,
and  the buyer will  first pays the lump sum $T$ to the seller
(perhaps with $T<0$)
and then   buy  $Q$ units  of the 100 the seller has available, at
price $P$. \\
2 The seller accepts or rejects the contract.\\
3.  If  the contract was accepted by both sides,  the seller
declares his type to be L or H and sells at the appropriate quantity
and price as stated in the contract. \\

  

\noindent
 {\bf Payoffs}:
 The same as in Myerson Trading Games I and II. 





\newpage

\begin{center}
 {\bf Myerson Trading Game III, continued }
 \end{center}The incentive compatibility constraints are unchanged
from the previous two
versions of the game, but now
the participation constraints  are different for the two types of
seller.
   \begin{equation} \label{e10.54}
\begin{array}{ll}
  \pi_L (Q_L, P_L) &\geq  0  
 \end{array}
 \end{equation}
 and 
 \begin{equation} \label{e10.55}
\begin{array}{ll}
 \pi_H (Q_H, P_H)  &\geq  0. 
  \end{array}
 \end{equation}

   Any mechanism which satisfies these two constraints would also
satisfy the single participation constraint in   MTG II, since it
says that a weighted average of the payoffs of the two sellers must be
positive. Thus, any mechanism which maximized the buyer's payoff in
MTG II would also maximize his payoff in  MTG III, if it  satisfied
the  tougher bifurcated  participation constraints.  The mechanism we
found for the game does satisfy the tougher constraints, so it is the
optimal mechanism here too.

   This is not a general feature of mechanisms.  More generally  the
optimal mechanism will not have as high a payoff when one player
starts the game with superior information, because of the extra
constraints on the mechanism.




\newpage

\begin{center}
 {\bf Myerson Trading Game IV }
 \end{center}
\noindent
 {\bf The Order of Play}:
 The same as in Myerson Trading Game III except that in (1) the seller
makes the
offer and in (2) the buyer accepts or rejects.
 

\noindent
 {\bf Payoffs}:
 The same as in Myerson Trading Games I, II, and III. 

\bigskip
 
 The incentive compatibility constraints are the same as in the
previous games, and the  participation constraint is inequality
(\ref{e10.50}), just as in Myerson Trading Game I. The big  difference
now is that unlike in the first three versions,  MTG IV has an
informed player making the contract offer. As a result, the form of
the offer can convey information, and we have  to consider out-of-
equilibrium beliefs, as in the   dynamic games of incomplete
information  in Chapter 6 (and we will see more of this  in   the
signalling models of Chapter 11). Surprisingly, however, the
importance of out-of-equilibrium beliefs does not lead to multiple
equilibria. Instead, the equilibrium contract is

 M1: $(Q_L=100, P_L=30,T_L=0,   Q_H=33 \frac{1}{3}, P_H=50, T_H=0)$,

This is part of equilibrium under the out-of-equilibrium belief that
if the seller offers any other contract, the  buyer believes the
quality is low.

This is the same  equilibrium mechanism as in MTG I.




\newpage

\begin{center}
 {\bf MTG IV, continued }
 \end{center}
    Consider two other mechanisms, M2 and M3   which satisfy the two
incentive compatibility constraints and the participation constraint,
but which are not equilibrium choices:

M2: $(Q_L=100, P_L=28, T_L=0,   Q_H=0, P_H=40, T_H=800)$.

M3: $(Q_L=100, P_L=31 \frac{3}{7}, T_L=0, Q_H= 57 \frac{1}{7}, P_H=40,
T_H=0)$.


Mechanism M2  is interesting because  the buyer expects a positive
payoff  of (30- 28) (100) = 200 if the seller is low-quality and a
negative payoff  of 800 if the seller is high-quality, for an overall
expected payoff of zero.  The contract is incentive compatible because
a low-quality seller could not increase his payoff of 28*100 by
pretending to be high-quality (he would get 20*100 + 800 instead), and
a high-quality seller would reduce his payoff of  (40*100 + 800)  if
he pretended to have low quality. Here, for the first time, we see a
positive value for the transfer $T_H$.

Under mechanism M3,    the buyer expects a negative payoff  of (30-31
$\frac{1}{7} $) (100) =  -11 $\frac{3}{7}$ if the seller is low-
quality and a positive payoff of  ($57 \frac{1}{7}) (50-40)$ =11
$\frac{3}{7}$ if  the seller is high-quality, for an overall expected
payoff of zero.  The contract is incentive compatible because a low-
quality seller could not increase his payoff of  3,142  $\frac{6}{7}$
(31 $\frac{3}{7}$) (100) by pretending to be high-quality (he would
get (57 $\frac{1}{7}$) (40)  + (42 $\frac{6}{7}$) (20) instead, which
comes to the same figure), and a high-quality seller would reduce his
payoff   if he pretended to have low quality and sold something he
valued at 40 at a price of 31 $\frac{1}{7}$.



\newpage



\begin{center}
 {\bf  What Contract Does the Seller Like?  }
 \end{center}
 M1: $(Q_L=100, P_L=30,T_L=0,   Q_H=33 \frac{1}{3}, P_H=50, T_H=0)$,

M2: $(Q_L=100, P_L=28, T_L=0,   Q_H=0, P_H=40, T_H=800)$.

M3: $(Q_L=100, P_L=31 \frac{3}{7}, T_L=0, Q_H= 57 \frac{1}{7}, P_H=40,
T_H=0)$.


Mechanism M1   maximizes the payoff of the average seller, as we found
in MTG  I, yielding the  low-quality seller a payoff of 3,000 and the
high-quality seller a payoff of  4,333 ($=(33 \frac{1}{3})(50) +   66
\frac{2}{3} (40)$), for an average payoff of 3,867.  If the seller  is
high-quality, however, he would prefer mechanism M2, which has payoffs
of 2800 and  4800 (=800+ 40(100)), for an average payoff of 3200. If
the seller  is  low-quality,  he would prefer mechanism M3, which has
payoffs of $3,142  \frac{6}{7}$  and  4000,    for an average payoff
of $3,314 \frac{6}{7}$.

Suppose  that   the seller chose M2, regardless of his type. This
could not be an equilibrium, because a low-quality seller would want
to deviate. Suppose he deviated and offered a contract almost like M1,
except that $P_L = 29.99$ instead of 30 and $P_H=49.99$ instead of 50.
This new contract would yield positive expected payoff to the buyer
whether the buyer believes the seller is low-quality or high-quality,
and so it would be accepted. It would yield higher payoff to the low-
quality seller than M2, and so the deviation would have been
profitable. If the seller chose M3 regardless of his type, a high-
quality seller could profitably deviate in the same way.

 
 \end{LARGE}

\end{document}
 
