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\section*{10 Signalling} 

 \noindent
 June 30, 1993.april 19, 1999. 
 
\subsection{The Informed Player Moves First: Signalling} % 10.1
 
 \noindent
   Signalling is a way for an agent to communicate his type under
adverse selection.  The signalling contract specifies a wage that
depends on an observable characteristic---the signal--- which the
agent chooses for himself after Nature chooses his type. Figures 7.1d
and 7.1e showed the extensive forms of two kinds of models with
signals.  If the agent chooses his signal before the contract is
offered, he is signalling to the principal. If he chooses the signal
afterwards, the principal is screening him.  Not only will it become
apparent that this difference in the 
 order of moves is important, it will also be seen that signalling costs must
differ between agent types for signalling to be useful, and the
outcome is often inefficient. 

 We begin with signalling models in which workers choose education
levels to signal their abilities. Section 10.1 lays out the
fundamental properties of a signalling model, and section 10.2 shows
how the details of the model affect the equilibrium. Section 10.3
steps back from the technical detail to more practical considerations
in applying the model to education.  Section 10.4 turns the game into
a screening model.  Section 10.5 switches to diagrams and applies
signalling to new stock issues to show how two signals need to be
used when the agent has two unobservable characteristics.



  Spence (1973) introduced the idea of signalling in the context of
education. We will construct a series of models which formalize the
notion that education has no direct effect on a person's ability to be
productive in the real world but useful for demonstrating his ability
to employers.  Let half of the workers have the type ``high ability''
and half ``low ability,'' where ability is a number denoting the
dollar value of his output.  Output is assumed to be a
noncontractible variable and there is no uncertaintyIf output is contractible, it should be in the contract, as
we have seen in chapter 7.  Lack of uncertainty is a simplifying assumption, imposed so that the contracts are functions only of the signals rather than a combination of the signal and the output.


Employers do not observe the worker's ability, but they do know the
distribution of abilities, and they observe the worker's education.
  To
simplify, we will specify that the players are one worker and two
employers.  The employers compete profits down to zero and the worker
receives the gains from trade. The worker's strategy is his education
level and his choice of employer. The employers' strategies are the
 contracts they offer giving wages as functions of education
level.  The key to the model is that the signal, education, is less
costly for workers with higher ability.  

In the first four variants of the game, workers choose their education levels
before employers decide how pay should vary with education.  

\begin{center}
\noindent
 {\bf Education I}
\end{center}
 {\bf Players}\\
  A worker and two employers.


\noindent
 {\bf Order of Play}\\
 (0) Nature chooses the worker's ability $a \in \{2,5.5\}$, the $Low$
and $High$ ability each having probability 0.5. The variable $a$ is
observed by the worker, but not by the employers.\\
 (1) The worker chooses education level $s \in \{0,1\}$.\\
 (2) The employers each offer a wage contract $w(s)$.\\
 (3) The worker accepts a contract, or rejects both of them.\\
 (4) Output equals $a$.

\noindent
 {\bf Payoffs}\\
 The worker's payoff is his wage minus his cost of education, and the
employer's is his profit.\\
  $ \pi_{worker} = w - 8s/a$ if the worker accepts contract $w$.\\
$ \pi_{worker} = $ $\left\{
  \begin{tabular}{ll}$w - 8s/a$ if the worker accepts contract $w$.\\
 0  &  if the worker rejects both contracts.\\
\end{tabular}
\right.$

    $ \pi_{employer} = $ $\left\{
  \begin{tabular}{ll}
  $a - w$ & for the employer whose contract is accepted.\\
 0  & for the other employer.
\end{tabular}
\right.$

 The payoffs assume that education is 
 more costly for a worker if his ability takes a lower value, which is what
  permits separation to occur.  As in any
hidden knowledge game, we must think about both pooling and
separating equilibria. Education I has both. In the pooling
equilibrium, which we will call PE 1.1, both types of workers pick
zero education and the employers pay the zero-profit wage of 3.75 (=
[2+5.5]/2) regardless of the education level.

\begin{quotation}
\noindent
\begin{tabular}{cl}
{\bf Pooling Equilibrium 1.1} & $\left\{
  \begin{tabular}{l}
 $ s(Low)= s(High) = 0$\\
  $ w(0)=w(1) = 3.75$\\
  $Prob(a=Low|s=1) = 0.5$
\end{tabular}
 \right.$\\ 
  (PE 1.1) & \\
\end{tabular}
\end{quotation}

 PE 1.1 needs to be specified as a perfect Bayesian equilibrium
rather than simply a Nash equilibrium because of the importance of
the interpretation that the uninformed player puts on
out-of-equilibrium behavior. The equilibrium needs to specify the
employer's beliefs when he observes $s=1$, since that is never
observed in equilibrium. In PE 1.1, the beliefs are passive
conjectures (see section 6.2): employers believe that a worker who
chooses $s=1$ is $Low$ with the prior probability, 0.5.  Given this
belief, both types of workers realize that education is useless, and
the model reaches the unsurprising outcome that workers do not bother
to acquire unproductive education.
 
 Under other beliefs, the pooling equilibrium breaks down.  Under the
belief $Prob(a=Low|s=1) = 0$, for example, employers believe that any
worker who acquired education is a $High$, so pooling is not Nash
because the $High$ workers are tempted to deviate and acquire
education.  This leads to the separating equilibrium for which
signalling is best known, in which the high-ability worker acquires
education to prove to employers that he really has high ability.

\begin{quotation}
\noindent 
\begin{tabular}{cl}
 {\bf Separating Equilibrium 1.2} & $\left\{
\begin{tabular}{l}
  $s(Low) = 0, s(High) =1$ \\
                           $ w(0) = 2, w(1) = 5.5 $\\ 
 \end{tabular}
 \right.$\\
 (SE 1.2) & \\
\end{tabular}
\end{quotation}

 
 Following the method used in chapters 7 and 8, we will show that SE 1.2
is a perfect Bayesian equilibrium by using the standard constraints which
an equilibrium must satisfy.  A pair of separating contracts must
maximize the utility of the $High$s and the $Low$s subject to two constraints:  (a) the particpation constraints that the firms can offer the contracts
without making losses; and (b) the self-selection constraints that the
$Low$s are not attracted to the $High$ contract, and the $High$s are
not attracted by the $Low$ contract.  The participation constraints
for the employers require that
 \begin{equation} \label{e10.1}
 w(0)     \leq a_L = 2 \;\; {\rm and} \;\; w(1) \leq
a_H=5.5.  
 \end{equation}
  Competition between the employers makes the expressions in
(\ref{e10.1}) hold as equalities.  The self-selection
constraint of the $Lows$ is \begin{equation} \label{e10.2}
  U_L (s = 0) \geq U_L(s = 1), 
 \end{equation} 
 which in Education I is 
\begin{equation} \label{e10.3}
 w(0) - 0 \geq w(1) - \frac{8}{2}.  
\end{equation} 
 Since in SE 1.2 the separating wage of the $Low$s is 2 and the
separating wage of the $High$s is 5.5 from (\ref{e10.1}), the self-
selection constraint (\ref{e10.3}) is satisfied.

   The self-selection  constraint of the $High$s is 
\begin{equation} \label{e10.4}
U_H(s = 1) \geq U_H(s=0), 
\end{equation} 
 which in Education I is
 \begin{equation} \label{e10.5}
  w(1) - \frac{8}{5.5} \geq w(0) - 0.  
 \end{equation} 
   Constraint (\ref{e10.5}) is satisfied by SE 1.2.

 There is another conceivable pooling equilibrium for Education
I, in which $ s(Low)= s(High) = 1$, but this turns out not to be an equilibrium, however, because the $Low$s
would deviate to zero education. Even if such a deviation caused the
employer to believe they were low-ability with probability 1 and
reduce their wage to 2, the low-ability workers would still prefer
to deviate, because 
 \begin{equation} \label{e10.0a}
    U_L (s = 0)= 2  \geq U_L(s = 1) = 3.75 - \frac{8(1)}{2}.  
  \end{equation}
  Thus, a pooling equilibrium with $s=1$ would violate incentive
compatibility for the $Low$ workers. 


Notice that we do not need to worry about a nonpooling constraint for
this game, unlike in the case of the games of chapter 9. One might think that
because employers compete for workers, competition between them might
result in their offering a pooling contract that the high-ability
workers would prefer to the separating contract.  The reason this
does not matter is that the employers do not compete by offering
contracts, but by reacting to workers who have acquired education.
That is why this is signalling and not screening: the employers
cannot offer contracts in advance that change the workers' incentives
to acquire education.  



 We can test the equilibrium by looking at the best responses.  Given
the worker's strategy and the other employer's strategy, an employer
must pay the worker his full output or lose him to the other
employer.  Given the employers' contracts, the $Low$ has a choice
between the payoff 2 (=$2-0$) for ignorance and 1.5 ($=5.5-8/2$) for
education, so he picks ignorance.  The $High$ has a choice between
the payoff 2 ($=2 - 0$) for ignorance and 4.05 ($=5.5 - 8/5.5$,
rounded) for education, so he picks education.

 Unlike the pooling equilibrium, the separating equilibrium does not
need to specify beliefs. Either of the two education levels might be
observed in equilibrium, so Bayes' Rule always tells the employers
how to interpret what they see. If they see that an agent has
acquired education, they deduce that his ability is $High$ and if
they see that he has not, they deduce that it is $Low$.  A worker is
free to deviate from the education level appropriate to his type, but
the employers' beliefs will continue to be based on equilibrium
behavior. 
   If a $High$ worker deviates by choosing $s=0$ and tells the
employers he is a $High$ who would rather pool than separate, the
employers disbelieve him and offer him the $Low$ wage of 2 that is
appropriate to $s=0$, not the pooling wage of 3.75 or the $High$ wage
of 5.5.

 Separation is possible because education is more costly for workers
if their ability is lower.  If education cost the same for both types
of worker, education would not work as a signal, because the
low-ability workers would imitate the high-ability workers. This
requirement of different signalling costs is known as the {\bf
single-crossing property}, since when the costs are depicted
graphically, as in section 10.4, the indifference curves of the two
types intersect a single time.  

  A strong case can be made that the beliefs required for the pooling
equilibria are not sensible.  Harking back to the equilibrium
refinements of section 6.2, recall that one suggestion (from Cho \&
Kreps [1987]) is to inquire into whether one type of player could not
possibly benefit from deviating, no matter how the uninformed player
changed his beliefs as a result.  Here, the $Low$ worker could never
benefit from deviating from PE 1.1.  Under the passive conjectures
specified, the $Low$ has a payoff of 3.75 in equilibrium versus
$-0.25$ ($=3.75-8/2$) if he deviates and becomes educated. Under the
belief that most encourages deviation--that a worker who deviates is
$High$ with probability one--the $Low$ would get a wage of 5.5 if he
deviated, but his payoff from deviating would only be 1.5 ($=5.5 -
8/2$), which is less than 2.  The more reasonable belief seems to be
that a worker who acquires education is a $High$, which does not
support the pooling equilibrium.

 The nature of the separating equilibrium lends support to the claim
that education {\it per se} is useless or even pernicious, because it
imposes social costs but does not increase total output.  While we
may be reassured by the fact that Professor Spence himself thought it
worthwhile to become Dean of Harvard College, the implications are
disturbing and suggest that we should think seriously about how well
the model applies to the real world.  We will do that in section
10.3. For now, note that in the model, unlike most real-world
situations, information about the agent's talent has no social value,
because all agents would be hired and employed at the same task even
under full information.  Also, if side payments are not possible, SE
1.2 is second-best efficient in the sense that a social planner could
not make both types of workers better off.  Separation helps the
high-ability workers even though it hurts the low-ability workers. 

\subsection{Variants on the Signalling Model of Education  } %  
 
  Although Education I is a curious and important, it does not exhaust the
implications of signalling which can be discovered in simple models.
This section will start with Education II, which will show an
alternative to the arbitrary assumption of beliefs in the perfect
Bayesian equilibrium concept. Education III will be the same as
Education I except for its  different parameter value, and will have two
pooling equilibrium rather than one separating and one pooling
equilibrium. Education IV    will allow a continuum of education
levels, and will unify Education I and Education III by showing
how all of their equilibria and more can be obtained in a model with
a less restricted strategy space. 

 
\bigskip
  \noindent
 {\bf Education II: Modelling Trembles so Nothing is Out of
Equilibrium}

\noindent
   The pooling equilibrium of Education I required the modeller to
specify the employers' out-of-equilibrium beliefs.  An equivalent
model constructs the game tree to support the beliefs  instead of
introducing them via the equilibrium concept.  This approach was
briefly mentioned in connection with the game of PhD Admissions in section 6.2.
The advantage is that the assumptions on beliefs are put in the rules
of the game along with the other assumptions. So let us replace
Nature's move  in Education I and modify the payoffs as follows. 
\begin{center}
{\bf Education II}
\end{center}

    (0) Nature chooses worker ability $a \in \{2,5.5\}$, each ability
having probability 0.5. ($a$ is observed by the worker, but not by
the employer.)  With probability 0.001, Nature endows a worker with
free education. 
$\ldots$ 

 \noindent
  {\bf Payoffs} \\
 $\pi_{worker} =$ $\left\{
  \begin{tabular}{ll}
 $ w  - 8s/a$ & if the worker accepts contract $w$ (ordinarily)\\
 $ w  $ & if the worker accepts contract $w$ (with free education) \\
$0$ & if the worker does not accept a contract\\
 \end{tabular}
 \right.$\\


   With probability $0.001$ the worker receives free education
regardless of his ability.  If the employer sees a worker with
education, he knows that the worker might be one of this rare type,
in which case the probability that the worker is $Low$ is 0.5. Both
$s=0$ and $s=1$ can be observed in any equilibrium and Education
II has almost the same two equilibria as Education I, without
the need to specify beliefs.  The separating equilibrium did not
depend on beliefs, and remains an equilibrium. What was Pooling
Equilibrium 1.1 becomes ``almost'' a pooling equilibrium--- almost
all workers behave the same, but the small number with free education
behave differently. The two types of greatest interest--- the $High$
and the $Low$--- are not separated, but the ordinary workers are
separated from the workers whose education is free. Even that small
amount of separation allows the employers to use Bayes' Rule and
eliminates the need for exogenous beliefs.  

\bigskip
 \noindent
 {\bf Education III: No Separating Equilibrium, Two Pooling
Equilibria}

\noindent
   Let us next modify Education I by changing the possible worker
abilities from $\{2, 5.5\}$ to $\{2, 12\}$.  The separating
equilibrium vanishes, but a new pooling equilibrium emerges.  In
equilibria PE 3.1 and PE 3.2, both pooling contracts pay the same
zero-profit wage of 7 ($=[2+12]/2$), and both types of agents acquire
the same amount of education, but the amount depends on the
equilibrium.


\begin{quotation}
\noindent
\begin{tabular}{cl}
 {\bf Pooling Equilibrium 3.1 }
 & $\left\{
 \begin{tabular}{l}
  $ s(Low)=s(High) = 0$\\
                              $ w(0)=w(1)=7$\\
                            $Prob(a=Low|s=1) = 0.5$ (passive conjectures) \\
\end{tabular}
 \right.$\\
 (PE 3.1) & \\
 \end{tabular}



\bigskip

\noindent 
\begin{tabular}{cl}
 {\bf Pooling Equilibrium 3.2 } & $\left\{
 \begin{tabular}{l}
  $s(Low)=s(High) = 1$\\
                                 $ w(0) = 2, w(1)=7$\\
                                $Prob(a=Low|s=0) = 1$ \\
 \end{tabular}
 \right.$\\
 (PE 3.2) & \\ 
 \end{tabular}
\end{quotation}


  PE 3.1 is similar to the pooling equilibrium in Education I and
II, but PE 3.2 is inefficient. Both types of workers receive the
same wage, but they incur the education costs anyway.  Each type is
frightened to do without education because the employer would pay him
not as if his ability were average, but as if he were known to be
$Low$.

 Examination of PE 3.2 shows why a separating equilibrium no longer exists.  Any separating equilibrium would require $w(0) = 2$ and $w(1) = 7$, but this is the contract that leads to PE 3.2.  The self-selection and zero-profit constraints cannot be satisfied simultaneously, because the $Low$ type is willing to acquire $s = 1$ to obtain the high wage.


 It is not surprising that information problems create inefficiencies
in the sense that first-best efficiency is lost. Indeed, the surprise
is that in some games with asymmetric information, such as Broadway
Game I in section 7.4, the first-best can still be achieved by
tricks such as boiling-in-oil contracts. More often, we discover that
the outcome is second-best efficient: given the informational
constraints, a social planner could not alter the equilibrium without
hurting some type of player. PE 3.2 is not even second-best
efficient, because PE 3.1 and PE 3.2 result in the exact same wages
and allocation of workers to tasks.  The inefficiency is purely a
problem of unfortunate expectations, like the inefficiency from
choosing the dominated equilibrium in Ranked Coordination. 



  PE 3.2 also illustrates a fine point of the definition of pooling,
because although the two types of workers adopt the same strategies,
the equilibrium contract offers different wages for different
education.  The implied threat to pay a low wage to an uneducated
worker never needs to be carried out, so the equilibrium is still
called a pooling equilibrium.  Notice that perfectness does not rule
out threats based on beliefs. The model imposes these beliefs on the
employer, and he would carry out his threats, because he believes
they are best responses. The employer receives a higher payoff under
some beliefs than under others, but he is not free to choose his
beliefs.

Following the approach of Education II, we could eliminate PE 3.2
by adding an exogenous probability 0.001 that either type is
completely unable to buy education. Then no behavior is never
observed in equilibrium and we end up with PE 3.1 because the only
rational belief is that if $s=0$ is observed, the worker has equal
probability of being $High$ or being $Low$. To eliminate PE 3.1 requires
less reasonable beliefs; for example, a probability of 0.001 that a
$Low$ gets free education together with a probability of 0 that a
$High$ does.


These first three games illustrate the basics of signalling: (a)
separating and pooling equilibria both may exist, (b)
out-of-equilibrium beliefs matter, and (c) sometimes one perfect
Bayesian equilibrium can Pareto dominate others. These results are
robust, but Education IV    will illustrate some dangers of using
simplified games with binary strategy spaces instead of continuous and
unbounded strategies.  So far education has been limited to $s=0$ or
$s=1$; Education IV    allows it to take greater or intermediate
values. 


\bigskip
 \noindent
 {\bf Education IV: Continuous Signals and Continua of Equilibria}


Let us now return to Education I, with one change that    education $s$  take 
  any level on the continuum between 0 and infinity.  

 The game now has continua of pooling and separating equilibria which
differ according to the value of education chosen.  In the pooling
equilibria, the equilibrium education level is $s^*$, where each
$s^*$ in the interval $ [0,\overline{s}]$ supports a different
equilibrium. The out-of-equilibrium belief most likely to support a
pooling equilibrium is 
 $Prob(a=Low|s \neq s^*) =1$, so let us use this to find the value of
$\overline{s}$, the greatest amount of education that can be
generated by a pooling equilibrium.  The equilibrium is PE 4.1, where
$s^* \in [0, \overline{s} ]$.
 
\begin{quotation}
\noindent
\begin{tabular}{cl}
{\bf Pooling Equilibrium 4.1} & $\left\{
  \begin{tabular}{l}
 $ s(Low)= s(High) = s^*$\\
  $ w(s^*) = 3.75$\\
  $ w(s \neq s^*) = 2$\\
   $Prob(a=Low|s \neq s^*) = 1$
\end{tabular}
 \right.$\\ 
  (PE 4.1) & \\
\end{tabular}
\end{quotation}

The critical value $\overline{s}$ can be discovered from the
incentive compatibility constraint of the $Low$ type, which is
binding if $s^* = \overline{s}$. The most tempting deviation is to
zero education, so that is the deviation that appears in the
constraint.  
   \begin{equation} \label{e10.1a}
U_L(s = 0)=2 \leq  U_L(s = \overline{s})= 3.75  - \frac{8  \overline{s}}{2}. 
   \end{equation}
 Equation (7) yields $\overline{s} = \frac{7}{16}$. Any
value of $s^*$ less than $\frac{7}{16}$ will also support a pooling
equilibrium. Note that the incentive-compatibility constraint of the
$High$ type is not binding. If a $High$ deviates to $s=0$, he, too, will
be thought to be a $Low$, so 
   \begin{equation} \label{e10.1b} U_H(s = 0)=2 \leq U_H(s =
\frac{7}{16} )= 3.75 - \frac{8 \overline{s}}{5.5} \approx 3.1. 
   \end{equation}

  In the separating equilibria, the education levels chosen in
equilibrium are 0 for the $Low$s and $s^*$ for the $High$s, where
each $s^*$ in the interval $ [ \overline{s},\overline{\overline{s}}]$
supports a different equilibrium. 
 A difference from the case of separating equilibria in games with binary
strategy spaces is that now there are possible out-of-equilibrium
actions even in a separating equilibrium. The two types of workers
will separate to two education levels, but that leaves an infinite
number of out-of-equilibrium education levels. As before, let us use
the most extreme belief for the employers' beliefs after observing an
out-of-equilibrium education level: that $Prob(a=Low|s \neq s^*) =1$.
     The equilibrium is SE 4.2, where $s^* \in [ \overline{s},s] $.
 
\begin{quotation}
\noindent
\begin{tabular}{cl}
{\bf Separating Equilibrium 4.2} & $\left\{
  \begin{tabular}{l}
 $ s(Low)= 0 ,\;\;\;  s(High) = s^*$\\
  $ w(s^*) =  5.5$\\
  $ w(s \neq s^*) = 2$\\
   $Prob(a=Low|s \not\in \{0,s^*\}) = 1$
\end{tabular}
 \right.$\\ 
  (SE 4.2) & \\
\end{tabular}
\end{quotation}

The critical value $\overline{s}$ can be discovered from the
incentive-compatibility constraint of the $Low$, which is binding if
$s^* = \overline{s}$.  
   \begin{equation} \label{e10.9}
U_L(s = 0)=2 \geq  U_L(s = \overline{s})= 5.5  - \frac{8  \overline{s}}{2}. 
   \end{equation}
 Equation (9) yields $\overline{s} = \frac{7}{8}$. Any
value of $s^*$ greater than $\frac{7}{8}$ will also deter the $Low$
workers from acquiring education.  If the education needed for the
wage of 5.5 is too great, the $High$ workers will give up on
education too. Their incentive compatibility constraint requires that

   \begin{equation} \label{e10.1d} U_H(s = 0)=2 \leq U_H(s =
\overline{\overline{s}} )= 5.5 - \frac{8
\overline{\overline{s}}}{5.5} . 
   \end{equation}
 Equation (9) yields $\overline{\overline{s}} =
\frac{77}{32}$. $s^*$ can take any lower value than $\frac{77}{32}$
and the $High$s will be willing to acquire education. 

 The big difference from Education I is that Education IV    has
Pareto-ranked equilibria. Pooling can occur not just at zero
education but at positive levels, as in Education III, and the
pooling equilibria with positive education levels are all Pareto
inferior. Also, the separating equilibria can be Pareto ranked, since
separation with $s^*= \overline{s} $ dominates separation with $s^*=
\overline{\overline{s}}$. Using a binary strategy space instead of a
continuum conceals this problem. 

Education IV    also shows how restricting the strategy space can
alter the kinds of equilibria that are possible. Education III
had no separating equilibrium because at the maximum possible signal,
$s=1$, the $Low$s were still willing to imitate the $High$s.
Education IV    would not have any separating equilibria either if
the strategy space were restricted to allow only education levels
less than $\frac{7}{8}$.  Using a bounded strategy space eliminates
possibly realistic equilibria. 

This is not to say that models with binary strategy sets are always
misleading. Education I is a fine model for showing how
signalling can be used to separate agents of different types; it
become misleading only when used to reach a conclusion such as ``If a
separating equilibrium exists, it is unique''.  As with any
assumption, one must be careful not to narrow the model so much as to
render vacuous the question it is designed to answer.  


  

 
 \subsection{ General Comments on Signalling in Education} %10.3

\noindent
{\bf Signalling and Similar Phenomena}

\noindent
         The distinguishing feature of signalling is that the agent's
action, although not directly related to output, is useful because it
is related to ability.  For the signal to work, it must be less
costly for an agent with higher ability. Separation can occur in
Education I because when the principal pays a greater wage to
educated workers, only the $High$s, whose utility costs of education
are lower, are willing to acquire it. That is why a signal works
where a simple message would not: actions speak louder than words.
 
   Signalling is outwardly similar to other solutions to adverse
selection.  The high-ability agent finds it cheaper than the
low-ability one to build a reputation, but the reputation-building
actions are based directly on his high ability. In a typical
reputation model he shows ability by producing high output period
after period.  Also, the nature of reputation is to require several
periods of play, which signalling does not.  

  Another form of communication is possible when some observable
variable not under the control of the worker is correlated with
ability.  Age, for example, is correlated with reliability, so an
employer pays older workers more, but the correlation does not arise
because it is easier for reliable workers to acquire the attribute of
age. Because age is not an action chosen by the worker, we would not
need game theory to model it.

\bigskip 
 \noindent
 {\bf Problems in Applying Signalling to Education}

\noindent
 On the empirical level, the first question to ask of a signalling
model of education is, ``What is education?'' For operational
purposes this  means, ``In what units is education measured?''  Two
possible answers are ``years of education'' and ``grade point
average.''  If the sacrifice of a year of earnings is greater for a
low-ability worker, years of education can serve as a signal.  If
less intelligent students must work harder to get straight As, then
grade-point-average can also be a signal.

         Layard \& Psacharopoulos (1974) give three rationales for rejecting  signalling as an
important motive for education. First, dropouts get
as high a rate of return on education as those who complete degrees,
so the signal is not the diploma, although it might be the years of
education.  Second, wage differentials between different education
levels rise with age, although one would expect the signal to be less
important after the employer has acquired more observations on the
worker's output.  Third, testing is not widely used for hiring,
despite its low cost relative to education.  Tests are
available,  but unused: students commonly take tests like the American
SAT whose results they could credibly communicate to employers, and
their scores correlate highly with subsequent grade point average.  One
would also expect an employer to prefer to pay an 18-year-old low
wages for four years to determine his ability, rather than waiting
to see what grades he gets as a history major. 


\bigskip
\noindent
{\bf Productive Signalling}

\noindent
 Even if education is largely signalling, we might not want to close
the schools.  Signalling might be wasteful in a pooling equilibrium
like PE 3.2, but in a separating equilibrium it can be second-best
efficient for at least three reasons. First, it allows the employer
to match workers with jobs suited to their talents.  If the only jobs
available were ``professor'' and ``typist,'' then in a pooling
equilibrium, both $High$ and $Low$ workers would be employed, but they
would be randomly allocated to the two jobs. Given the principle of
comparative advantage, typing might improve, but I think, pridefully,
that research would suffer. 

 Second, signalling keeps talented workers from moving to jobs where
their productivity is lower  but their talent is known.  Without
signalling, a talented worker might leave a corporation and
start his own company, where he would be less productive but better
paid. The naive observer would see that corporations hire only one
type of worker ($Low$), and imagine  there was no welfare loss.

   Third, if ability is endogenous--- moral hazard rather than
adverse selection--- signalling encourages workers to acquire ability.
  One of my teachers said that you always understand your
next-to-last econometrics class. Suppose that solidly learning
econometrics increases the student's ability, but a grade of A is not
enough to show that he solidly learned the material.  To signal his
newly acquired ability, the student must also take ``Time Series,''
which he cannot pass without a solid understanding of  econometrics.  ``Time Series''
might be useless in itself, but if it did not exist, the students
would not be able to show he had learned basic 
 econometrics.





 \subsection{ The Informed Player  Moves Second: Screening}  

  In screening games, the informed player moves second, which means
that he moves in response to contracts offered by the uninformed
player.  Having the uninformed player make the offers is important
because his offer conveys no information about himself, unlike in a
signalling model.  



\begin{center}
 {\bf Education V: Screening with a Discrete Signal}
 \end{center}
  {\bf Players}\\
  A worker and two employers.

 
\noindent
 {\bf Order of Play}\\
 (0) Nature chooses worker ability $a \in \{2,5.5\}$, each ability
having probability 0.5.\\
 Employers do not observe ability, but the worker does.\\ 
 (1) Each employer offers a wage contract $w(s)$.\\
 (2) The worker chooses education level $s \in \{0,1\}.$\\
 (3) The worker accepts a contract, or rejects both of them.\\
 (4) Output equals $a$.

 \noindent
 {\bf Payoffs} \\
 $ \pi_{worker} =$   $\left\{ 
\begin{tabular}{ll}
$w - \frac{8s}{ a}$ if the worker accepts contract $w$.\\
 0 &if the worker rejects both contracts.\\
\end{tabular}
\right.$

 $ \pi_{employer} =$   $\left\{ 
\begin{tabular}{ll}
$ a - w$ & for the employer whose contract is accepted.\\
 0 & for the other employer.\\
\end{tabular}
\right.$
\bigskip

     Education V    has no   pooling equilibrium, because if one
employer tried to offer the zero profit pooling contract, $w(0)=3.75$,
the other employer would offer $w(1) = 5.5$ and draw away all the
$High$s. The unique equilibrium is

\noindent
\begin{tabular}{cl}
 {\bf Separating Equilibrium 5.1} & $\left\{ 
 \begin{tabular}{l}
$s(Low) = 0, s(High) = 1$\\
      $ w(0) =2, w(1) = 5.5$\\
 \end{tabular}
  \right\$\\
 (SE 5.1) & \\
\end{tabular}

 Beliefs do not need to be specified in a screening model. The
uninformed player moves first, so his beliefs after seeing the move
of the informed player are irrelevant. The informed player is fully
informed, so his beliefs are not affected by what he observes. This
is much like simple adverse selection, in which the uninformed player
moves first, offering a set of contracts, after which the informed
player chooses one of them.  The modeller does not need to refine
perfectness in a screening model, although he might be tempted to
abandon it altogether in favor of the reactive or Wilson equilibrium
concepts described in section 8.5. The similarity between adverse
selection and screening is strong enough that Education V    would
not have been out of place in chapter 9, but it is presented here
because the context is so similar to the signalling models of
education.

Education VI      allows a continuum of education levels, in a game
otherwise the same as Education  V.


\begin{center}
 {\bf Education VI: Screening with a Continuous Signal}
 \end{center}
 {\bf Players}\\
 A worker and two employers.

 
 \noindent
 {\bf Order of Play} \\
  (0) Nature chooses worker ability $a \in \{2,5.5\}$, each ability
having probability 0.5.\\
 Employers do not observe ability, but the worker does.\\ 
 (1) Each employer offers a wage contract $w(s)$.\\
 (2) The worker choose education level $s \in [0, 1].$\\
 (3) The worker chooses a contract, or rejects both of them.\\
 (4) Output equals $a$.

\noindent
  {\bf Payoffs.}\\
  $ \pi_{worker} =  \left\{ 
\begin{tabular}{ll}
$w - 8s/a$  if the worker accepts contract $w$.\\
0 & if the worker rejects both contracts.\\
\end{tabular}
 \right.$
 $ \pi_{employer} =  \left\{ 
\begin{tabular}{ll}
$ a - w$ & for the employer whose contract is accepted.\\
 0 & for the other employer.\\
\end{tabular}
\right.$
\bigskip

 Pooling equilibria generally do not exist in screening games with
continuous signals, and sometimes separating equilibria in pure
strategies do not exist either---recall Insurance Game III from
section 9.4.  Education VI, however, does have a separating Nash
equilibrium, with a unique equilibrium path.

\noindent
\begin{tabular}{cl}
 {\bf Separating Equilibrium 6.1} & $\left\{
 \begin{tabular}{l}
  $s(Low) = 0, s(High) =0.875$\\
                            $w =\left\{ \begin{tabular}{ll}
                                 2 & if $s < 0.875$\\
                                5.5 & if $s \geq 0.875$
                                \end{tabular} \right.$\\
  \end{tabular}
 \right.$\\
 (SE 6.1) & \\
\end{tabular}

 In any separating contract, the $Low$s must be paid a wage of 2 for
an education of 0, because this is the most attractive contract that
breaks even.  The separating contract for the $High$s must maximize
their utility subject to the constraints discussed in Education
I. When the signal is continuous, the constraints are especially
useful to the modeller for calculating the equilibrium.  The
participation constraints for the employers require that 
 \begin{equation} \label{e10.11}
 w(0) \leq a_L=2 \;\; {\rm and} \;\; w(s^*) \leq a_H=5.5,
\end{equation}
 where $s^*$ is the separating value of education that we are trying
to find. Competition turns (11) into equalities.  The
self selection constraint for the low-ability workers is 
 \begin{equation} \label{e10.12}
 U_L (s = 0) \geq U_L(s = s^*),
\end{equation}
 which in Education VI    is 
 \begin{equation} \label{e10.13}
w(0) - 0 \geq w(s^*) - \frac{8s^*}{2}. 
\end{equation}
   Since the separating wage is 2 for the $Low$'s and 5.5 for the
$High$'s, constraint (13) is satisfied as an equality if $s^* =
0.875$, which is the crucial education level in SE 6.1. 
 \begin{equation} \label{e10.14}
 U_H (s = 0)= w(0) \leq U_H(s = s^*) = w(s^*) - \frac{8s^*}{5.5}.
\end{equation}
 If $s^* = 0.875$, inequality (10.\ref{e10.14}) is true, and it would
also be true for higher values of $s^*$. Unlike the case of the
continuous-strategy signalling game, Education IV, however, the
equilibrium contract in Education VI    is unique, because the
employers compete to offer the most attractive contract that
satisfies the participation and incentive compatibility constraints.
The most attractive is the separating contract that Pareto dominates
the other separating contracts by requiring the relatively low separating signal of
$s^*=0.875.$

 Similarly, competition in offering attractive contracts rules out
pooling contracts. The nonpooling constraint, required by competition
between employers, is
 \begin{equation} \label{e10.15}
 U_H(s = s^*) \geq U_H(pooling), \end{equation} 
 which, for Education
VI, is, using the most attractive possible pooling contract, 
 \begin{equation} \label{e10.16}
w(s^*) - \frac{8 s^*}{5.5}  \geq 3.75.
\end{equation}
  Since the payoff of $High$s in the separating contract is 4.23 ($=
5.5 - 8 \cdot 0.875/5.5$, rounded), the nonpooling constraint is
satisfied.


\bigskip
\noindent
 {\bf No Pooling Equilibrium in Education VI}

 \noindent
  Education VI    lacks a pooling equilibrium, which would require
the outcome $\{s= 0, w(0) = 3.75\}$, shown as $C_1$ in figure 10.1.
If one employer offered a pooling contract requiring more than zero
education (such as the inefficient PE 3.2), the other employer could
make the more attractive offer of the same wage for zero education.
The wage is 3.75 to ensure zero profits.  The rest of the wage
function--- the wages for positive education levels--- can take a
variety of shapes, so long as the wage does not rise so fast with
education that the $High$s are tempted to become educated.

 But no equilibrium has these characteristics. In a Nash equilibrium,
no employer can offer a pooling contract, because the other employer
could always profit by offering a separating contract paying more to
the educated. One such separating contract is $C_2$ in figure 10.1,
which pays 5 to workers with an education of $s = 0.5$ and yields a
payoff of 4.89 ($= 5 - [8\cdot 0.5]/5.5$, rounded) to the $High$s and
3 ($= 5 - 8\cdot 0.5/2$) to the $Low$s.  Only $High$s prefer $C_2$
to the pooling contract $C_1$, which yields payoffs of 3.75 to both
$High$ and $Low$, and if only $High$s accept $C_2$, it yields positive
profits to the employer.

\begin{center}
 {\bf Figure 10.1}  Education VI:  No Pooling Nash Equilibrium 
 \end{center}

\epsfysize=3in

 
\epsffile{/Users/erasmuse/AAANewChapters/Figures/f9.1.eps}



  Nonexistence of a pooling equilibrium in screening models without
continuous strategy spaces is a general result.  The linearity of the
curves in Education VI    is special, but in any screening model the
$Low$'s would have greater costs of education, which is equivalent to
steeper indifference curves.  This is the {\bf single-crossing
property} alluded to in Education I.  Any pooling equilibrium
must, like $C_1$, lie on the vertical axis where education is zero
and the wage equals the average ability. A separating contract like
$C_2$ can always be found to the northeast of the pooling contract,
between the indifference curves of the two types, and it will yield
positive profits by attracting only the $High$s. 


\bigskip
     \noindent
 {\bf Education VII: No Nash Equilibrium }

\noindent
  In Education VI    we showed that screening models have no pooling
equilibria.  In Education VII    the parameters are changed a little
to eliminate even the separating equilibrium. Let the proportion of
$High$s be 0.9 instead of 0.5, so the zero-profit pooling wage is
5.15 ($ = 0.9[5.5] + 0.1[2]$) instead of 3.75.  Consider the
separating contracts $C_3$ and $C_4$, shown in figure 10.2,
calculated in the same way as SE 5.1.  The pair $(C_3, C_4)$ is the
most attractive pair of contracts that separates $High$'s from $Low$'s
by satisfying constraint (7). $Low$ workers accept contract $C_3$,
obtain $s=0$, and receive a wage of 2, their ability.  $High$s accept
contract $C_4$, obtain $s=0.875$, and receive a wage of 5.5, their
ability.  Education is not attractive to $Low$'s because the $Low$
payoff from pretending to be $High$ is 2 $(= 5.5 - 8\cdot 0.875/2$),
no better than the $Low$ payoff of 2 from $C_3$ $(=2 - 8 \cdot 0/2).$

\begin{center}
 {\bf Figure 10.2} Education VII:  No Nash equilibrium 
\end{center}

\epsfysize=3in

 
\epsffile{/Users/erasmuse/AAANewChapters/Figures/f9.2.eps}



  The wage of the pooling contract $C_5$ is 5.15, so that even the
$High$s strictly prefer $C_5$ to $(C_3,C_4)$.  But our reasoning that
no pooling equilibrium exists is still valid; some contract $C_6$
would attract all the $High$s from $C_5$. No Nash equilibrium in pure
strategies exists, either separating or pooling.

\bigskip
\noindent
{\bf Screening and Adverse Selection}

\noindent
  We also encountered a nonexistence problem in the adverse selection
games of section 9.4, where, just as in screening, the informed player
takes no action until after the uninformed player has offered a set
of contracts. Screening models behave much the same way as simple
adverse selection models, in contrast to signalling models.

    Compare Education VI, the first screening model of education,
with the Rothschild \& Stiglitz (1976) Insurance Game III of
section 9.4. Both have contracts which give something of value in
exchange for costs incurred to communicate the informed players'
private information.  In Education VI    the contracts specified a
higher wage if the worker acquired more education.  In Insurance
Game III the contracts specified a lower premium if the customer
accepted more risk by accepting a higher coinsurance rate. In both
models, the two types of informed player separate, picking two
different levels of the costly variable.  The outcomes are only
second-best efficient because in Insurance Game III the $Safe$
type is not fully insured and in Education VI    the $High$ type
incurs the cost of education. Insurance Game III specifies two
distinct contracts, as opposed to the $w(s)$ function of Education
V, but since the worker only picks one of two education levels in
equilibrium, the difference is more apparent than real. The fact that
Education VI    is a game of certainty and Insurance Game III is
not is also unimportant.

   What then is the difference between the two games?  The main
difference is that in Education VI, it is possible to conceive of  
education apart from the wage contract, whereas in Insurance Game III
the signal is communicated by the choice of the insurance contract.
The inefficiency is more striking in the screening game, where the
cost of communication is distinct from the act of accepting a
particular contract.


\bigskip
 \noindent
  {\bf Wilson Equilibrium and Reactive Equilibrium}

\noindent
  As in Insurance Game III, it is possible to go beyond Nash
equilibrium to find an equilibrium of some other kind for Education
VII.  Is it reasonable to say that a pooling equilibrium could
always be broken by a contract which draws away the $High$'s?  After
the $High$s departed, the old pooling contract, which would still
soak up all the $Low$s, would be unprofitable.  In figure 10.2, if
$C_5$ is withdrawn after $C_6$ is offered, the $Low$s prefer $C_6$ to
the zero they obtain from unemployment, and $C_6$ becomes a pooling
contract.  This is irrelevant to the question of whether $C_5$ is a
Nash equilibrium, but it might lead one to doubt the wisdom of the
equilibrium concept.

  Under the concept of the {\bf Wilson equilibrium } from section
9.5, the pooling equilibrium is legitimate, because an employer
thinking about introducing the new equilibrium-breaking contract
would realize that the new contract would be unprofitable once the
old contract was withdrawn.  {\bf Reactive equilibrium} can also be
applied, and generates a separating equilibrium.  Under its
reasoning, the separating equilibrium cannot be broken by a pooling
contract, because the pooling contract would in turn be broken by a
second separating contract.  The Wilson $C_5$ and the reactive
$(C_3,C_4)$ are the two clear candidates for equilibrium in figure
10.2.  Alternately, we could restructure the model so that the worker
moves first--- the assumption in sections 10.1 and 10.2.  While
avoiding the existence problem, that introduces the need to think
about out-of-equilibrium beliefs, and it really models a different
situation, in which workers cannot change their education in response
to employers' contracts.


\bigskip
 \noindent
  {\bf A Summary of the Education Models}
   
\noindent
 Because of signalling's complexity, most of this chapter has been
devoted to elaboration of the education model. We began with
Education I, which showed how with two types and two signal
levels the perfect Bayesian equilibrium could be either separating or
pooling. Education II  took the same model and replaced the
specification of out-of-equilibrium beliefs with an additional move
by Nature, while Education III changed the parameters in
Education I to increase the difference between types and toshow how
signalling could continue with pooling. Education IV    changed
Education I by allowing an continuum of education levels, which
resulted in a continuum of inefficient equilibria, each with a
different signal level. After a purely verbal discussion of how to
apply signalling models, we looked at screening, in which the
employer moves first.  Education V    was a screening reprise of
Education I, while Education VI    broadened the model to allow
a continuous signal, which eliminates pooling equilibria. Education
VII modified the parameters of Education VI    to show that
sometimes no pure-strategy Nash equilibrium exists at all.

 Throughout it was implicitly assumed that all the players were risk
neutral. Risk neutrality is unimportant, because there is no
uncertainty in the model and the agents bear no risk.  If the workers
were risk averse and they differed in their degrees of risk aversion,
the 
  contracts could try to use the difference to support a separating
equilibrium because willingness to accept risk might act as a signal.
If the principal were risk averse he might offer a wage less than the
average productivity in the pooling equilibrium, but he is under no
risk at all in the separating equilibrium, because it is fully
revealing. The models are also games of certainty, and this too is
unimportant.  If output were uncertain, agents would just make use of the expected payoffs rather than the raw payoffs  and very
little would change.

We could extend the education models further--- allowing more than
two levels of ability would be a high priority--- but instead, let us
turn to the financial markets and look graphically at a model with
two continuous characteristics of type and two continuous signals.


  \subsection{  Two Signals: Game of Underpricing New Stock Issues} % 9.5

\noindent
 One signal might not be enough when there is not one but two
characteristics of an agent that he wishes to communicate to the
principal.  This has been generally analyzed in Engers (1987), and
multiple signal models have been especially popular
in financial economics, with models of warranty issue by Matthews \&
Moore (1987) and of the role of investment bankers in new stock issues by
Hughes (1986). We will use a model of initial public offerings of
stock as the example in this section.  
 
Empirically, it has been found that companies consistently issue
stock at a price so low that it rises sharply in the days after the
issue, an abnormal return estimated to average 11.4 percent (Copeland
\& Weston [1988], p. 377). The game of Underpricing New Stock Issues tries to
explain this using the percentage of the stock retained by the
original owner and the amount of underpricing as two signals.  The
two characteristics being signalled are the mean of the value of the
new stock, which is of obvious concern to the potential buyers, and
the variance, the importance of which will be explained later. 

 
\begin{center}
 {\bf Underpricing New Stock Issues}\\
 (Grinblatt \& Hwang [1989])
 \end{center}
 {\bf Players}\\
  The entrepreneur and many investors.

 
\noindent
 {\bf Order of Play}\\
 (See Figure 2.3a for a time line.)\\
 (0) Nature chooses the expected value ($\mu$) and variance
($\sigma^2$) of a share of the firm using some distribution $F$.\\
 (1) The entrepreneur retains fraction $\alpha$ of the stock and
offers to sell the rest at a price per share of $P_0$.\\
  (2) The investors decide whether to accept or reject the offer.\\
 (3) The market price becomes $P_1$, the investors' estimate of
$\mu$. \\
 (4) Nature chooses the value $V$ of a share using some distribution
$G$ such that $\mu$ is the mean of $V$ and $ \sigma^2$ is the variance. With probability $\theta$, $V$ is revealed to the
investors and becomes the market price. \\ 
 (5) The entrepreneur sells his remaining shares at the market price.

 \noindent
  {\bf Payoffs} \\
 \begin{tabular}{lll}
        $\pi_{entrepreneur}$&$ =$&$ U( [1-\alpha]P_0 + \alpha [\theta
V + (1-\theta) P_1] ),$ where $U'>0$ and $U''<0$.\\
 $\pi_{investors}$&$ =$& $(1-\alpha)(V - P_0) + \alpha(1- \theta) (V
- P_1)$.
  \end{tabular}
 

  The entrepreneur's payoff is the utility of the value of the shares
he issues at $P_0$ plus the value of those he sells later at the
price $P_1$ or $V$. The investors' payoff is the true value of the
shares they buy minus the price they pay.
 
 Underpricing New Stock Issues subsumes the simpler model of
Leland \& Pyle (1977), in which $\sigma^2$ is common knowledge and if
the entrepreneur chooses to retain a large fraction of the shares,
the investors deduce that the stock value is high. The one signal in
that model is fully revealing because holding a larger fraction
exposes the undiversified entrepreneur to a larger amount of risk,
which he is unwilling to accept unless the stock value is greater
than investors would guess without the signal.

   If the variance of the project is high, that also increases the
risk to the undiversified entrepreneur, which is important even
though the investors are risk neutral and do not care directly about
the value of $\sigma^2$.  Since the risk is greater when variance is
high, the signal $\alpha$ is more effective and retaining a smaller
amount allows the entrepreneur to sell the remainder at the same
price as a larger amount for a lower-variance firm. Even though the
investors are diversified and do not care directly about
firm-specific risk, they are interested in the variance because it
tells them something about the effectiveness of entrepreneur-retained
shares as a signal of share value.  Figure 10.3 shows the signalling
schedules for two variance levels.  

\begin{center}
{\bf Figure 10.3:} How the Signal Changes with the Variance
\end{center}

\epsfysize=3in

 
\epsffile{/Users/erasmuse/AAANewChapters/Figures/f9.3.eps}



    In the game of Underpricing New Stock Issues, $\sigma^2$ is not known to
the investors, so the signal is no longer fully revealing. An
$\alpha$ equal to 0.1 could mean either that the firm has a low value
with low variance, or a high value with high variance.  But the
entrepreneur can use a second signal, the price at which the stock is
issued, and by observing $\alpha$ and $P_0$, the investors can deduce
$\mu$ and $\sigma^2$.

I will use specific numbers for concreteness.  The entrepreneur could signal that the stock has the high mean value, $\mu = 120$, in two ways:  (a) retaining a high percentage, $\alpha = 0.4$, and making the initial offering at a high price of $P_0 = 90$, or (b) retaining a low percentage, $\alpha = 0.3$, and making the initial offering at a low price, $P_0 = 80$.  Figure 10.4 shows the different combinations of initial price and fraction retained that might be used.  If the stock has a high variance, he will want to choose behavior (b), which reduces his risk.  Investors deduce that the stock of anyone who retains a low percentage and offers a low price actually has $\mu = 120$ and a high variance, so stock offered at the price of 80 rises in price.  If, on the other hand, the entrepreneur retained $\alpha = .3$ and offered the high price $P_0 = 90$, investors would conclude that $/mu$ was lower than 120, but the variance was low also, so the stock would not rise in price.  The low price conveys the information that this stock has a high mean and high variance rather than a low mean and low variance. 

\begin{center}
{\bf Figure 10.4:} Different Ways to Signal a Given $\mu$.
\end{center}

\epsfysize=3in

 
\epsffile{/Users/erasmuse/AAANewChapters/Figures/f9.4.eps}



  This model explains why new stock is issued at a low price. The
entrepreneur knows that the price will rise, but only if he issues it
at a low initial price to show that the variance is high. The price
discount shows that signalling by holding a large fraction of stock
is unusually costly, but he is nonetheless willing to signal. The
discount is costly because he is selling stock at less than its true
value, and retaining stock is costly because he bears extra risk, but
both are necessary to signal that the stock is valuable.




 \subsection {  Signal Jamming} 

   
    This book has examined a number of models in which an informed player tries to convey information to an uninformed player by some means or other--- by entering into an incentive contract, or by signalling. Sometimes, however, the informed party has the opposite problem:  his  natural behavior  would convey  his private information but he wants to keep  it secret. This happens, for example, if one firm is informed about its poor ability to compete successfully, and it wants to conceal this information from a rival. The informed player may then engage in costly actions, just as in signalling, but now the costly action will be {\bf signal jamming}  (a term   coined in 
  Fudenberg
\& Tirole [1986c]):
 preventing information from appearing rather than generating information. 


 The model  I will use  to illustrate  signal jamming  is one of my own,    the Rasmusen (1997) limit pricing model.\footnote{ ``Signal Jamming and Limit Pricing: A Unified Approach,'' in  {\it Public Policy and Economic Analysis}, Moriki Hosoe and Eric Rasmusen, editors, Fukuoka, Japan: Kyushu University Press, 1997. } \footnote{Limit pricing can be explained in a variety of ways; notably, as a way for the incumbent to signal that he   has   low enough  costs  that rivals  would regret entering, as in problem 6.2 and Milgrom \& Roberts [1982a]).}   Limit pricing refers to the practice of keeping prices low to deter entry.   Here, the explanation for limit pricing  will be signal jamming: by keeping profits low, the incumbent keeps it unclear to the rival whether the market  is big enough to  accommodate two firms profitably.  In the model, the incumbent can control    $S$, a  public signal of  the size of the market.  In Rasmusen (1997), this signal is the price that the incumbent charges, but it could equally well represent the incumbent's choice of advertising or capacity.  The reason the signal is important is that the entrant must decide whether to enter based on his belief as to the probability that the market is large enough to support two firms profitably. 

\newpage
 \begin{large} 
\begin{center}
 {\bf  ``Limit Pricing as Signal Jamming''}
 \end{center}

  {\bf Players}\\
   The incumbent and the rival. 

 \noindent
 {\bf Order of Play}\\
 (0)   Nature chooses the market size $M$ to be $M_{Small}$ with
probability $\theta$ and $M_{Large}$ with probability $(1-\theta)$,
observed only by the incumbent. \\
  (1) The incumbent chooses   the signal $S$ to equal $s_0$ or $s_1$
for the first period if the market is small, $s_1$ or $s_2$ if it is
large.  This results in    monopoly profit $\mu f(S)-C$, where $\mu >1$.   Both players observe the value of $S$. \\
  (2)  The rival decides whether to be $In$ or
$ Out$ of the market. \\
   (3) If the rival chooses $In$, each player incurs cost $C$ in the second period and they each earn the duopoly profit  $M-C$.  Otherwise, the  incumbent earns $\mu M-C$.  

\noindent
 {\bf Payoffs}\\
 If the rival does not enter, the payoffs are  $\pi_{incumbent} =  (\mu  f(S)-C ) + (  \mu M - C) $ and $\pi_{rival} = 0$.\\
 If the rival does   enter, the payoffs are  $\pi_{incumbent} =  (\mu  f(S)-C) + (M - C) $ and $\pi_{rival} = M -C$.\\
 Assume that  $f(s_0) < f(s_1) = M_{Small}< f(s_2)=M_{Large}$,  $  M_{Large}-C >0$, and $M_{Small} - C<0$. 
   
   Thus, if the incumbent chooses $s_1$,  its profit will  equal the maximum profit from a small market, even if the market is really large, but if it chooses $s_2$, its profit will  be the maximum value for a large market-- but that choice will reveal that the market is large.  The duopoly profit in a large market is large enough to sustain two firms, but the duopoly profit in a small market will result in losses for both firms. 
   
   \newpage
 
  There are four equilibria, each appropriate to a different parameter region   in     Figure 14.2.  If the  parameter $\mu$, which shows the value to  being a monopoly,  is small enough, a nonstrategic equilibrium exists,  in which the incumbent simply maximizes profits in each period separately. This equilibrium is:  
 ( E1:  NONSTRATEGIC.  $s_2|Large$, $s_1|Small$,  $Out|s_0$, $Out|s_1$, $In|s_2$). The 
  incumbent's equilibrium payoff in a large market is
$ \pi_I (s_2|Large ) = (\mu M_{Large}-C) + (M_{Large}-C),   
$ compared with the deviation payoff of 
$ \pi_I (s_1|Large) = (\mu M_{Small}-C) + (\mu M_{Large}-C)$.     
  The incumbent has no incentive to deviate if   
 $ \pi_I (s_2|Large ) - \pi_I (s_1|Large)= (1+\mu)M_{Large} -  \mu (M_{small}+M_{Large})  \geq 0$,    
  which is equivalent to 
  \begin{equation} \label{e23}
  \mu    \leq \frac{M_{Large}}{M_{small}}, 
 \end{equation}    
  as shown in Figure 14.2.  
     The rival will not deviate,   because
the incumbent's choice fully reveals the size of the market.
 
 
\newpage

 Signal jamming occurs if monopoly profits are somewhat higher, and if the rival would refrain from entering the market unless he decides it is more profitable than his prior beliefs would indicate.  The equilibrium is 
 (E2: PURE SIGNAL-JAMMING. $s_1|Large$, $s_1|Small$,
$Out|s_0$, $Out|s_1$, $In|s_2$ ).  The rival's strategy is the
same as in E1, so the incumbent's optimal behavior remains the same, and he chooses $s_1$ if the opposite of condition (14.\ref{e23}) is true. As for the rival, if he   stays out, his second-period payoff is 0, and if he   enters, its expected value is $   \theta (M_{small}-C) + (1-\theta)(M_{Large}-C)$.  
 Hence,  as shown in Figure 14.2, he will follow the equilibrium behavior of  staying out  if  
   \begin{equation}\label{e26}
 \theta \geq \frac{M_{Large}-C}{M_{Large}-M_{small}}.
\end{equation}

 The intuition behind the signal-jamming equilibrium is straightforward. The incumbent  knows he will attract entry if he  fully exploits the market when it is large,  so he purposely dulls his efforts to conceal whether the market is large or small. If potential entrants place are unwilling to enter without positive information that the market is large, the incumbent can thus deter entry. 

 Signal jamming shows up in   other contexts.   A wealthy man may refrain from  buying  a big house, landscaping his front yard, or wearing expensive clothing in order to avoid being a target for  thieves or for  political leaders in search of wealthy victims to tax or loot.     A cabinet  with shaky  support may  purposely  take risky  assertive positions  because greater caution might  hint to his rivals that his position was insecure and induce them to campaign actively against him.  A general may advance  his troops even when he is outnumbered, because  to go on the defensive would  provoke the enemy to attack.\footnote{xxx I'll want real examples of all of these. Read Herodotus again.  }   Note,however, that in each of these examples it is key that the uninformed player decide not to act aggressively if he  fails to acquire any information.     

 \newpage

 A mixed form of signal jamming occurs if the probability of a small market is unlikely, so  if the signal of first-period revenues was jammed completely, the rival would enter anyway. This equilibrium is 
  (E3:  MIXED SIGNAL-JAMMING. ($ s_1|Small$, $
s_1|Large$ with probability $\alpha$, $ s_2|Large$ with probability
$(1-\alpha)$, $Out|s_0$,
 $In|s_1$ with probability
$\beta$, $Out|s_1$ with probability
$(1-\beta)$, $In|s_2$).    If the incumbent played
$ s_2|Large$ and $s_1|Small$,   the rival would interpret $s_1$ 
as indicating a small market---an interpretation which would give the
incumbent incentive to play $s_1|Large$. But if the incumbent always
plays $ s_1$,  the rival would enter even after
observing $ s_1$, knowing there was a  high probability  that the
market   was really large. Hence, the equilibrium must be in mixed
strategies, which is  equilibrium E3, or the incumbent must convince the rival to stay out by playing $s_0$, which  
is  equilibrium E4. 

  For the rival to mix, he   must
be indifferent between the second-period payoffs of 
 $ \pi_E (In|s_1) =   \frac{\theta}{\theta + (1-\theta)\alpha} (M_{small}-C) + 
 \frac{ (1-\theta)\alpha}{\theta + (1-\theta)\alpha} (M_{Large}-C) $
  and 
 $ \pi_E (Out|s_1) = 0$.
 Equating these two payoffs and solving for $\alpha$ yields 
 $ \alpha  =\left( \frac{\theta}{1-\theta} \right) \left( \frac{C-M_{small} }{M_{Large}-C }
\right),  
 $  which is always non-negative, but   avoids equalling one
only if condition (14.\ref{e26}) is false. 
    
For
the incumbent to mix when the market is large, he must be indifferent
between
$ \pi_I (s_2|Large) =  (\mu M_{Large}-C) + (M_{Large}-C)
$ and
$ \pi_I (s_1|Large) = (\mu M_{small}-C)  +  \beta (M_{Large}-C)  + (1-\beta)(\mu M_{Large}-C).
$ Equating these two payoffs and solving for $\beta$ gives 
 $ \beta  = \frac{ \mu M_{small}-M_{Large}  }{(\mu -1) M_{Large}}, 
$ which is   strictly less than one,  and which is non-negative
 if  condition  (14.\ref{e26}) is false.  
 
 If the market
is small, the incumbent's alternative payoffs are the equilibrium payoff of 
 $ \pi_I (s_1|Small) = (\mu M_{small}-C)  +  \beta (M_{small}-C)  + (1-\beta)(\mu M_{small}-C)$ and the deviation payoff of
$ \pi_I (R_0|Small) = (M R_0-C) + (MM_{small}-C). 
$ The difference is 
  \begin{equation} \label{e35}
  \pi_I (s_1|Small)-\pi_I (R_0|Small)  = [MM_{small} + \beta M_{small} + (1-\beta)MM_{small} ]    - [M R_0 +MM_{small}] 
    \end{equation}
Expression (14.\ref{e35}) is non-negative under either of two conditions.  The first is if  $R_0$ is small enough; that is, if 
 \begin{equation} \label{e35a}
 R_0 \leq M_{small}\left(  1- \frac{M_{small}}{M_{Large}}   \right). 
 \end{equation}
  The second is if  $M$ is no greater than some amount $Z^{-1}$ defined  so that 
 \begin{equation} \label{e36}
  M \leq  \left(   \frac{M_{small}}{M_{Large}} - 1 + \frac{R_0 }{ M_{small}}  \right)^{-1} = Z^{-1}.
 \end{equation}
        If condition (14.\ref{e35a}) is false, then $ Z^{-1}  > \frac{M_{Large}}{M_{small}} $, because $ Z  < \frac{M_{small}}{M_{Large}} $ and  $Z >0$.  Thus, we can draw region E3 as it is shown in Figure 14.2.  

\newpage
   
       It follows  that if condition
  (14.\ref{e36}) is replaced by its converse, 
 the unique equilibrium is for the incumbent to choose
$s_0|Small$, and the equilibrium is  (E4:  SIGNALLING.  $s_0|Small$, $s_2|Large$,  $Out|s_0$, $In|s_1$, $In|s_2$).  Passive conjectures will support this pooling signalling equilibrium, as will the out-of-equilibrium belief that  
if the rival observes $s_1$, he believes the market is large with
probability $\frac{(1-\theta)\alpha}{\theta + (1-\theta) \alpha}$, as
in equilibrium E3.  

 The signalling equilibrium is also an equilibrium for other
parameter regions outside of E4, though less reasonable beliefs are required.  Let the out-of-equilibrium belief be
$Prob(Large|s_1) = 1$.  The equilibrium payoff is 
 $ \pi_I (s_0|Small) = (\mu  f(s_0)-C) + (\mu M_{small}-C)$   
 and the deviation payoff is 
 $ \pi_I (s_1|Small) = (\mu M_{small}-C) + (M_{small} -C)$.  
 The signalling equilibrium remains an equilibrium so long as 
  $  \mu  \geq  \frac{M_{small}}{f(s_0)}$.  
        
The signalling equilibrium is an interesting one, because  it turns the asymmetric information problem full circle. The informed player wants to conceal his private information by  costly signal jamming if the information is 
$Large$, so when the information is $Small$, the player must  signall at some cost   that he is not signal jamming. If E4 is the equilibrium, the incumbent is hurt by the possibility of signal jamming; he would much prefer a simpler world in which it was illegal or nobody considered the possibility.  This is often the case: strategic behavior can help a player in some circumstances, but  given that the other players know he might be behaving strategically,  everyone would prefer a world  in which  everyone  is honest and  non-strategic. 
  

Homework problem: 
   What happens to the equilibria of the signal jamming  model if  $s_0$ is not a feasible choice? 




\newpage



\begin{small}




\noindent
{\bf  Notes}

   
\bigskip
\noindent
 {\bf N10.1} {\bf The Informed Player Moves First: Signalling} % 10.2
 \begin{itemize}
  \item
 The term ``signalling'' was introduced by Spence (1973). The games
in this book take advantage of hindsight to build simpler and more
rational models of education than in his original article, which used
a rather strange equilibrium concept: a strategy profile from
which no worker has incentive to deviate and under which the
employer's profits are 0. Under that concept, the firm's
incentives to deviate are irrelevant.

$\;\;\;$ The distinction between signalling and screening has been
attributed to Stiglitz \& Weiss (1989). The literature has shown wide
variation in the use of both terms, and ``signal'' is such a useful
word that it is often used in models that have no signalling of the
kind discussed in this chapter.

\item 
   One convention sometimes used in signalling models is to call the
signalling player (the agent), the {\bf sender} and the player
signalled to (the principal), the {\bf receiver}. 
 
   

\item
 The applications of signalling are too many to properly list. A few
examples are the use of prices in C.  Wilson (1980) and Stiglitz
(1987), the payment of dividends in Ross (1977), bargaining (section
11.5), and greenmail (section 15.2).  Banks (1990) has written a
short book surveying signalling models in political science.
Empirical papers include Layard \& Psacharopoulos (1974) on education
and Staten \& Umbeck (1986) on occupational diseases. 

\item
 Legal bargaining is one area of application for signalling.  See
Grossman \& Katz (1983).  Reinganum (1988) has a nice example of the
value of precommitment in legal signalling. In her model, a
prosecutor who wishes to punish the guilty and release the innocent
wishes, if parameters are such that most defendants are guilty, to
commit to a pooling strategy in which his plea bargaining offer is
the same whatever the probability that a particular defendant would be
found guilty.

\item
  The peacock's tail may be a signal. Zahavi (1975) suggests that a
large tail may benefit the peacock because, by hampering him, it
demonstrates to potential mates that he is fit enough to survive even
with a handicap.



\item
\noindent
{\bf Advertising}

  Advertising is a natural application for signalling. The literature
includes Nelson (1974), written before signalling was well-known,
Kihlstrom \& Riordan (1984) and Milgrom \& Roberts (1986).  I will
briefly describe a model based on Nelson's.  Firms are one of two
types, low-quality or high-quality.  Consumers do not know that a
firm exists until they receive an advertisement from it, and they do not know
its quality until they buy its product. They are unwilling to pay
more than zero for low quality, but any product is costly to produce.
This is not a reputation model, because it is finite in length and
quality is exogenous. 

$\;\;\;$          If the cost of an advertisement is greater than the profit from one
sale, but less than the profit from repeat sales, then high rates of
advertising are associated with high product quality.  A firm with
low quality would not advertise, but a firm with high quality would.

$\;\;\;$ The model can work even if consumers do not understand the
market and do not make rational deductions from the firm's incentives, so it
does not have to be a signalling model.  If consumers react passively
and sample the product of any firm from whom they receive an
advertisement, it is still true that the high-quality firm advertises
more, because the customers it attracts become repeat customers.  If
consumers do understand the firms' incentives, signalling reinforces
the result.  Consumers know that firms which advertise must have high
quality, so they are willing to try them. This understanding is
important, because if consumers knew that 90 percent of firms were
low-quality but did not understand that only high-quality firms
advertise, they would not respond to the advertisements whichthey received. This
should bring to mind section 6.2's game of PhD Admissions.



\item
  If there are just two workers in the population, the model is
different depending on whether:

\noindent
 (1) Each is $High$ ability with objective probability 0.5, so possibly
both are $High$ ability; or\\
(2) One of them is $High$ and the other is $Low$, so only the subjective
probability is 0.5.
 
 The outcomes are different because in case (2) if one worker
credibly signals he is $High$ ability, the employer knows the other one
must be $Low$ ability.






 \end{itemize}


 
{\bf Problems}
\bigskip


    {\bf 10.1: Is Lower Ability Better?} 

Change Education I so
that the two possible worker abilities are $a \in \{1,4\}$.

 (10.1a)  What are the equilibria of this game?  What are
the payoffs of the workers (and the payoffs averaged across workers) in each
equilibrium?           

   \hspace*{16pt} (10.1b) Apply the Intuitive Criterion (see N6.2). Are the
equilibria the same?          

\hspace*{16pt} (10.1c)  What happens to the equilibrium worker payoffs if the high-ability is 5 instead of 4?


  \hspace*{16pt} (10.1d) Apply the Intuitive Criterion to the new
game. Are the equilibria the same? 

 \hspace*{16pt} (10.1e) Could it be that a rise in the maximum
ability reduces the average worker's payoff? Can it hurt all the
workers?   




  
  
 {\bf 10.2: Productive Education and Nonexistence of Equilibrium.}

Change Education I so that the two equally likely abilities are
$a_L=2$ and $a_H = 5$ and education is productive: the payoff of the
employer whose contract is accepted is $\pi_{employer} = a + 2s - w$.
The worker's utility function remains $U = w - \frac{8s}{a}$.

   \hspace*{16pt} (10.2a) Under full information, what are the wages
for educated and uneducated workers of each type, and who acquires
education?           

 \hspace*{16pt} (10.2b) Show that with incomplete information the
equilibrium is unique (except for    beliefs and
wages out of equilibrium) but   unreasonable.  
   

   
 
 

  {\bf 10.3: Price and Quality.}

 Consumers have prior beliefs that
Apex produces low-quality goods with probability 0.4 and high-quality
with probability 0.6.  A unit of output costs 1 to produce in either
case, and it is worth 10 to the consumer if it is high- quality and 0
if low-quality. The consumer, who is risk neutral, decides whether to
buy in each of two periods, but he does not know the quality until he
buys. There is no discounting. 

 \hspace*{16pt}(10.3a) What is Apex' price and profit if it must
choose one price, $p^*$, for both periods? 
  

 \hspace*{16pt} (10.3b) What is Apex' price and profit if it can
choose two prices, $p_1$ and $p_2$, for the two periods, but it
cannot commit ahead to $p_2$?   

\hspace*{16pt} (10.3c) What is the answer to part (b) if the discount
rate is $r = 0.1$?    

 \hspace*{16pt} (10.3d) Returning to $r=0$, what if Apex can commit
to $p_2$?  

 \hspace*{16pt} (10.3e) How do the answers to (a) and (b) change if
the probability of low quality is 0.95 instead of 0.4? (There is a
twist to this question.) 
   
  
  
 
 
 
 {\bf 10.4: Signalling with a Continuous Signal.}

 Suppose that with
equal probability a worker's ability is $a_L=1$ or $a_H=5$, and the
worker chooses any amount of education $y \in [0,\infty)$. Let
$U_{worker}=w - \frac{8y}{a}$ and $\pi_{employer} = a-w$.


  \hspace*{16pt} (10.4a) There is a continuum of pooling equilibria,
with different levels of $y^*$, the amount of education necessary to
obtain the high wage.  What education levels, $y^*$, and wages,
$w(y)$, are paid in the pooling equilibria, and what is a set of
out-of-equilibrium beliefs that supports them? What are the incentive
compatibility constraints?     
    
  

 \hspace*{16pt} (10.4b) There is a continuum of separating
equilibria, with different levels of $y^*$.  What are the education
levels 
  and wages in the separating equilibria? Why are out-of-equilibrium
beliefs needed, and what beliefs support the suggested equilibria?
What are the self selection constraints for these equilibria?  
  
      
   
 \hspace*{16pt} (10.4c) If you were forced to predict one equilibrium
which will be the one played out, which would it be?   
 
   
 
  {\bf 10.5: Advertising.} Brydox introduces a new shampoo which is
actually very good, but is believed by consumers to be good with only a
probability of 0.5.  A consumer would pay 10 for high quality and 0
for low quality, and the shampoo costs 6 per unit to produce.  The
firm may spend as much as it likes on stupid TV commercials showing
happy people washing their hair, but the potential market consists of
100 cold-blooded economists who are not taken in by psychological
tricks. The market can be divided into two periods. 

\hspace*{16pt} (10.5a) If advertising is banned, will Brydox go out
of business?   
  
 

  \hspace*{16pt} (10.5b) If there are two periods of consumer
purchase, and consumers discover the quality of the shampoo if they
purchase in the first period, show that Brydox might spend
substantial amounts on stupid commercials.  
 

 \hspace*{16pt} (10.5c) What is the  minimum  and maximum that
Brydox might spend on advertising, if it spends a positive amount?  







 \end{small}

 


\end{document}



