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\noindent   21 July  2005. Eric Rasmusen, Erasmuse@indiana.edu.  
 
 

This is a section that was in the third edition of {\it Games and Information}. In the fourth edition, I have kept only about a third of it.



 \noindent 
{\bf  10.3:  Myerson Mechanism Design  }

 Now let's look at another example, a classic one from Myerson. 

Myerson (1991) uses a trading example in Sections 6.4  and 10.3 of his
book to illustrate mechanism design.  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. For efficiency, all of the good should
be transferred from the seller to the buyer. The only way to get the
seller to truthfully reveal the quality of the good, however, 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. Let us see
how this works out.

  Depending on who offers the contract and when it is offered, various
games result.  We will start with one in which the seller makes the
offer, and does so before he knows whether his quality is high or low.

\begin{center} 
{\bf Myerson Trading Game I: Moral Hazard, Seller Offers} 
\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 $m$ to be high or low,  and  the buyer   will first pay  the
lump sum $T_l$  or $T_h $ to the seller (perhaps with the lump sum
being negative) and then   buy  $q_l$ or $q_h$   units of the 100 the
seller has available, at price $p_l$ or $p_h$. The contract is $ \{
w(m) = q(m)p(m) +T(m) , q(m) \}$. Zero is paid if the wrong output is
delivered.  \\
 2 The buyer accepts or rejects the contract.\\ 
3  Nature chooses whether the seller's type $s$ of  good is High
quality (probability 0.2) or Low  (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 buyer rejects the contract,  $\pi_{buyer} =  0$, $\pi_{seller
\; H} =  40*100$,  and $\pi_{seller \; L} =  20*100$. \\
    If the buyer accepts the contract and  the seller declares a type
that has price $p$,   quantity $q$, and transfer $T$,  then
 \begin{equation}   \label{e25}%  \label{e10.50a} 
\pi_{buyer| seller\; H} = -T+ (50-P) q\;\;\;  and\;\;\;   \pi_{buyer|
seller\; L} = -T+ (30-p) q
\end{equation}
 and
 \begin{equation}   \label{e26}%   \label{e10.50b}
 \pi_{seller \; H} =T+ 40(100-Q) + pq  \;\;\;  and\;\;\;  \pi_{seller
\; L} =  T+ 20(100-q) + pq.
 \end{equation}

 \bigskip

 The seller wants to design a contract subject to two sets of
constraints. First, the buyer must accept the contract. Thus, the
participation constraint is\footnote{Another kind of participation
constraint would apply if the buyer had the option to reject
purchasing anything, after accepting the contract and hearing the
seller's type announcement.  That would not make a difference here. }
 \begin{equation}  \label{e27}%   \label{e10.50}
  \begin{array}{ll}
  0.8 \pi_{buyer| seller\; H} (q_l, p_l, T_l) + 0.2 \pi_{buyer|
seller\; L} (q_h, p_h, T_h) &\geq  0 \\
 &\\
 0.8   [-T_l +(30-p_l)q_l] + 0.2 [-T_h +(30-p_h)q_h]  &\geq  0   \\ 
\end{array}
   \end{equation}

  There  might also  be a participation constraint for the seller
himself, because it might be that even when he designs the contract
that maximizes his payoff, his payoff is no higher than when he
refuses to offer a contract. He can always offer the  acceptable (if
vacuous) null  contract,  $(q_l=0, p_l=0,T_l=0,  q_h=0, p_h=0, T_h=0)
$, however, so we do not need to write out the seller's participation
constraint separately.

  Second, the seller must design a contract that will induce  himself
to tell the truth later once he discovers his type. This is, of course
a bit unusual--- the seller is like a principal designing a contract
for himself as agent. That is why things are  different in this
chapter than in the chapters on   moral hazard.   What is happening is
that the seller is trying to sell not just a good, but a contract, and
so he must make the contract attractive to the buyer. Thus, he faces
incentive compatibility constraints: one for when he is low quality,
 \begin{equation}  \label{e28}%   \label{e10.51}
 \begin{array}{ll}
 \pi_{seller \; L} (q_l, p_l, T_l)  &\geq  \pi_{seller \; L} (q_h,
p_h, T_h)\\
 &\\
 20 (100-q_l) + p_l q_l +T_l &\geq  20 (100-q_h) + p_h q_h+T_h,  \\
  \end{array} 
\end{equation}
 and  one for when he has high quality,
  \begin{equation} \label{e29}% \label{e10.52} 
 \begin{array}{ll}
 \pi_{seller \; H} (q_h, p_h, T_h)& \geq \pi_{seller \; H} (q_l, p_l,
T_l) \\
 &\\
 40 (100-q_h) + p_h q_h+T_h&  \geq  40 (100-q_l) + p_l q_l +T_l. \\ 
\end{array}
 \end{equation}
  There is not just one incentive compatibility constraint, but two,
one for each type, something different from moral hazard.

 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 $q_l$. 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 $p_l= 30$.   The low-quality seller's
incentive compatibility constraint, inequality (\ref{e28}), will be
binding, and thus becomes
 \begin{equation}   \label{e29}%  \label{e10.60}
 \begin{array}{ll} 
\pi_{seller \; L} (q_l, p_l,T_l)  &\geq  \pi_{seller \; L} (q_h, p_h,
T_h)\\
&\\ 20 (100-100) + 30*100  +0  &=  20 (100-q_h) + p_h q_h +0.   \\
 \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{e30}%   \label{e10.57}
 \begin{array}{ll}
 \pi_s  &=0.8 \pi_{seller \; L} (q_l, p_l, T_l) +  0.2\pi_{seller \;
H} (q_h, p_h, T_h)   \\
 &\\ &=  0.8 [(20)(100-q_l) +  p_lq_l +T_l]   +  0.2   [(40)(100-q_h)
+ p_hq_h +T_h]   \\
 &\\ 
&=  0.8 [(20)(100-100) +  30*100+0]   +  0.2   [(40)(100-\frac{1000}
{p_h-20}) + p_h (\frac{1000}{p_h-20})+0]   \\
 \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 {e30}) without changing anything important.}
Thus, the equilibrium contract is
  \begin{equation}  \label{e31}%   \label{e10.57a}
\begin{array}{ll}
   q_l=100, p_l=30,T_l=0\\
 \\
   q_h=33 \frac{1}{3}, p_h=50, T_h=0 .
\end{array}
 \end{equation}

This mechanism will not work if further offers can be made after the
end of the game.  The mechanism is not  first-best efficient; if the
seller is high-quality, then he only sells  $33 \frac{1}{3}$ units to
the buyer instead of all 100, even though both realize that the
buyer's value is 50 and the seller's is only 40. If they could agree
to sell the remaining $66 \frac{2}{3}$  units, then the mechanism
would not be incentive compatible in the first place, though, because
then the low-quality seller would pretend to be high-quality, first
selling  $33 \frac{1}{3}$  units and then selling the rest. The
importance of commitment is a general feature of mechanisms.

\bigskip

What if it is the buyer who makes the offer?

\begin{center}
 {\bf Myerson Trading Game II: Moral Hazard, Buyer Makes the Offer }
 \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{e32}%  \label{e10.53}
 \begin{array}{ll}
 0.8 \pi_{seller \; L} (q_l, p_l, T_l) +  0.2\pi_{seller \; H} (q_h,
p_h, T_h)  &\geq  0.
 \end{array} 
\end{equation}

 The incentive compatibility constraints are just as they were before,
since the buyer has to design a mechanism which makes the seller
truthfully reveal his type.

 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, inequality (\ref{e28}),  will be binding, and thus becomes
 \begin{equation}  \label{e33}%   \label{e10.60a} 
\begin{array}{ll}
 \pi_{seller \; L} (q_l, p_l, T_l)  &\geq  \pi_{seller \; H}(q_h, p_h,
T_h)\\
 &\\
 20 (100-100) +  p_l*100  +0  &=  20 (100-q_h) +40 q_h+0.   \\
 \end{array}
 \end{equation} 
 Solving for $q_h$ gives us $q_h =    5p_l-100$,  which when
substituted into  the buyer's payoff  function yields
\begin{equation}  \label{e34}%   \label{e10.57c}
 \begin{array}{ll} \pi_b  &=0.8 \pi_{b|L}  (q_l, p_l, T_l) +
0.2\pi_{b|H} (q_h, p_h, T_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{e29}))  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{e35}%   \label{e10.57d} 
\begin{array}{l}
 ( q_l=100, p_l=20,T_l=0\\
 \\
 q_h=0, T_h=0).\\
 \end{array}
  \end{equation}

\bigskip

 In the  next version of the game, we will continue to let the buyer
make the offer, but he makes it at a time when the seller already
knows his type. Thus, this will be an adverse selection model.

\begin{center}
 {\bf Myerson Trading Game III: Adverse Selection, The Buyer Makes the
Offer }
\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.

\bigskip

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{e36}%   \label{e10.54}
 \begin{array}{ll}
 \pi_L (q_l, p_l, T_l) &\geq  0 
\end{array}
 \end{equation} 
and 
\begin{equation}   \label{e37}%  \label{e10.55}
 \begin{array}{ll} \pi_H (q_h, p_h, T_h)  &\geq  0.
 \end{array}
 \end{equation}

   Any mechanism which satisfies these two constraints would also
satisfy the single participation constraint in   Myerson Trading Game
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 Myerson Trading Game II would also maximize his
payoff in  Myerson Trading Game 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.

\bigskip

 In the last of our versions of this game, the seller makes the offer,
but after he knows his type.

\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 {\it Myerson Trading Game I}. The big
difference now is that unlike in the first three versions,  Myerson
Trading Game 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 Myerson Trading Game I.
It is interesting to compare it to two other mechanisms, M2 and M3,
which satisfy the two incentive compatibility constraints and the
participation constraint, but which are not equilibrium choices:
\footnote{xxx M2-- hwo about an oo belief that a deviator is a HIGH
type? Then no deviation is profitable.}

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 $\frac{1}{7}$.

In Myerson Trading Game IV, unlike the previous versions of the game,
the particular mechanism chosen in equilibrium is not necessarily the
one  that the player who offers the contract likes best.  Instead, an
informed offeror--- here, the seller--- must worry that his offer
might make the uninformed receiver believe the offeror's type is
undesirable.

Mechanism M1   maximizes the payoff of the average seller, as we found
in Myerson Trading Game  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. Similarly, if the seller chose M3 regardless of his type,
a high- quality seller could profitably deviate in the same way.

The {\it Myerson Trading Game} is a good introduction to the flavor of
the algebra in mechanism design problems. For more on this game, in a
very different style of presentation, see Sections 6.4 and Chapter 10
of Myerson (1991).  We will next go on to particular economic
applications of mechanism design.\footnote{xxx Think abvout risk
sharing too, with risk aversion. Then it makes a big difference when
the seller learns his type.}

 
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