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\begin{LARGE}
\begin{center}
{\bf 11 Signalling}
\end{center}
\end{LARGE}
\noindent
10 November 2005. Eric Rasmusen, Erasmuse@indiana.edu. Http://www.rasmusen.org.
\bigskip
\noindent
{\bf 11.1: The Informed Player Moves First: Signalling}
\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 1d and 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 11.1 lays out the fundamental properties of a
signalling model, and Section 11.2 shows how the details of the model affect the
equilibrium. Section 11.3 steps back from the technical detail to more practical
considerations in applying the model to education. Section 11.4 turns the game
into a screening model. Section 11.5 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. Section 11.6 addresses the rather
different idea of signal jamming: strategic behavior a player uses to cover up
information rather than to disclose it.
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 uncertainty. If 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 The 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} = $ $\left\{ \begin{tabular}{ll}$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 Pooling
Equilibrium 1.1, both types of workers pick zero education and the employers pay
the zero-profit wage of 3.75 regardless of the education level (3.75= [2+5.5]
/2).
\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.$\\ & \\ \end{tabular}
\end{quotation}
Pooling Equilibrium 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 Pooling
Equilibrium 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, which is 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 $\\ \end{tabular} \right.$ \\
& \\
\end{tabular} \end{quotation}
Following the method used in Chapters 7 to 10, we will show that Separating
Equilibrium 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 participation 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(1) }{2}. \end{equation} Since in Separating Equilibrium 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 (1)}{5.5} \geq w(0) - 0. \end{equation} Constraint (\ref{e10.5})
is satisfied by Separating Equilibrium 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, 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 for ignorance (=$2-0$)
and 1.5 for education ($=5.5-8/2$), so he picks ignorance. The $High$ has a
choice between the payoff 2 for ignorance ($=2 - 0$) and 4.05 for education
($=5.5 - 8/5.5$, rounded), 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's 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 were to 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 the {\bf single-crossing property} that we have seen in Chapter 10. When the
costs are depicted graphically, as they will be in Figure 1, 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 Pooling Equilibrium 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
later. 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, Separating Equilibrium 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.
Separation can also occur even if the signal of education has zero or
negative cost for the high-ability workers, so long as it has positive cost for
the low-ability workers. In such a case, however, the signalling is
nonstrategic; it merely happens that the high-ability worker's efficient level
of education is too high for low-ability workers to want to imitate them, and
employers therefore deduce both natural and acquired ability from education.
\bigskip
\noindent
{\bf 11.2: Variants on the Signalling Model of Education }
Although Education I is a curious and important model, it does not exhaust
the implications of signalling. 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 values, 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}
\noindent {\bf The Order of Play}\\
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 of $s=1$. \\
$\ldots$
\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 (this might model idiosyncratic reasons someone might be educated that
are unrelated to job prospects). 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's Rule and eliminates the need for exogenous beliefs.
More formally, the pooling equilibrium would have both abilities of workers
choosing $s=0$ unless a worker was one of the few endowed with automatic $s=1$.
Out-of-equilibrium, if the employer observed $s=1$, he could use Bayes's Rule:
\begin{equation}\label{e200}
\begin{array}{ll}
Prob (a=Low|s=1) &= \frac{Prob (s=1|a=L) Prob (L)}{Prob (s=1|a=L) Prob (L) +
Prob (s=1|a=H) Prob (H)}\\
& \\
& = \frac{(0.001)(0.5)}{(0.001)(0.5) + (0.001)(0.5)}\\
& \\
& = 0.5\\
\end{array}
\end{equation}
Normal workers would not deviate from $s=0$ because it would not increase the
employer's estimate of their ability.
\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 Pooling Equilibria 3.1 and
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.$\\ \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.$\\ \end{tabular} \end{quotation}
Pooling Equilibrium 3.1 is similar to the pooling equilibrium in Education
I and II, but Pooling Equilibrium 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 Pooling Equilibrium 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 Pooling Equilibrium 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. Pooling Equilibrium 3.2 is not even
second-best efficient, because Pooling Equilibrium 3.1 and Pooling Equilibrium
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.
Pooling Equilibrium 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 Pooling
Equilibrium 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 Pooling Equilibrium 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 Pooling Equilibrium 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 $s$ can take
any level on the continuum between 0 and infinity.
\begin{center} \noindent
{\bf Education IV} \end{center}
{\bf Players}\\
A worker and two employers.
\noindent
{\bf The 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 $s \in [0,\infty$.\\
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} = $ $\left\{ \begin{tabular}{ll}$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 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
Pooling Equilibrium 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.$\\ \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 (\ref{e10.1a}) 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 Separating Equilibrium 4.2, where $s^* \in [
\overline{s},\overline{\overline{s}}]$.
\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.$\\
\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 (\ref{e10.9}) 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 (\ref{e10.1d}) 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 {\it Education I} is that {\it Education IV } has
Pareto-ranked equilibria. Pooling can occur not just at zero education but at
positive levels, as in {\it 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.
{\it Education IV} also shows how restricting the strategy space can alter
the kinds of equilibria that are possible. {\it 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. {\it 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 becomes 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.
\bigskip
\noindent
{\bf 11.3: General Comments on Signalling in Education}
\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 Pooling
Equilibrium 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.
\bigskip
\noindent
{\bf 11.4: 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 The 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.$
\noindent
$ \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. $\\
\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. 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 The 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.$
\noindent $ \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.$\\ \end{tabular} \right.$\\ \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 the inequalities in (\ref{e10.11}) 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 (\ref{e10.13}) is satisfied as an equality if $s^* = 0.875$, which
is the crucial education level in Separating Equilibrium 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 (\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 Screening: Education VI}
\noindent
The screening game
Education VI lacks a pooling equilibrium, which would require the outcome
$\{s= 0, w(0) = 3.75\}$, shown as $C_1$ in Figure 1. If one employer offered a
pooling contract requiring more than zero education (such as the inefficient
Pooling Equilibrium 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 1, which pays 5 to workers with an
education of $s = 0.5$ and yields a payoff of 4.89 to the $High$s ($= 5 -
[8\cdot 0.5]/5.5$, rounded) and 3 to the $Low$s ($= 5 - 8\cdot 0.5/2$). 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.
\includegraphics[width=150mm]{fig11-01.jpg}
\begin{center}
{\bf Figure 1: Education VI: No Pooling Equilibrium in a Screening Game }
\end{center}
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 Pure-Strategy Equilibrium in a Screening Game }
\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 in pure strategies. 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 2, calculated in the same way as Separating Equilibrium 5.1. The pair
$(C_3, C_4)$ is the most attractive pair of contracts that separates $High$s
from $Low$s. $Low$ 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 $C_3$ $(=2 - 8 \cdot 0/2).$
\includegraphics[width=150mm]{fig11-02.jpg}
\begin{center}
{\bf Figure 2: Education VII: Neither Separating nor Pooling Pure-Strategy
Equilibria in a Screening Game }
\end{center}
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 Should the Modeller Worry about Multiple Equilibria, or Nonexistence of a
Pure-Strategy Equilibrium?}
There is no reason why the modeller should be more worried about nonexistence
of an equilibrium in pure strategies, per se, in a signalling model. As far
back as Chapter 3 we looked at games in which the only realistic equilibrium is
in mixed strategies. In that chapter, and also in Chapter 1 and in Chapter 6
(on dynamic games with incomplete information) we also discussed multiple
equilibria, and why we should not be unduly worried about them either-- the main
reason being that under the logic of Nash equilibrium, which is self-fulfilling
expectations, we should indeed expect different expectations to yield different
outcomes.
It is in coordination models and signalling models that multiple equilibria
have created the most anxiety among economists. For coordination models, there
has been hope that cheap talk and focal points can often narrow down our
predictions to a single outcome. For signalling models, the various refinements
discussed in Chapter 6 have been proposed, but with unenthusiastic reception.
For both coordination games and signalling games, however, there is another way
out: changing the order of moves. If moves are not simultaneous, and if, in the
signalling game, it is not the informed player but the uninformed player who
moves first, the outcome is much more likely to be both uniquely determined and
efficient. Because the outcome is more likely to be efficient, it will also
often be the case that the players would like the moves to be sequential, or
that evolution will cause such a game to more often observed.
Consider what happens if the principal makes the
offer of a particular reward for a particular signal level--a screening model
instead of a signalling model. That simple change will eliminate pareto-
dominated contracts, because the uninformed player will not offer them.
As a specific example, suppose job candidates can heavily research
the employer before an interview, and this works as a signal
because less serious applicants are less willing to expend
that effort.
One possibility is a pooling equilibrium in which both good
and bad applicants signal by researching. The equilibrium
is maintained by the fear that if an applicant does {\it
not} he will be thought to be the bad type. This is
inefficient, because the employer, in the end, gets no
information.
An employer, therefore, would like to switch to a
screening game. He would say to applicants that they should
not research the company, because he knows that researching is not a
good signal. In response, neither type of applicant would research. And
the company would profit by reducing the salary it offers,
acquiring some of the social savings from less researching.
(A ``no presents'' rule at a birthday party is another example).
Note, however, that just because you observe all
applicants researching, an apparent pooling equilibrium, it
might not be the case that the equilibrium really is
pooling. It could be that the analyst's data has a self-
selection problem: given that researching is necessary to
have a chance at the job, only the good types apply-- and
they all research it. Be careful in concluding that a costly
signal really is not separating out the types.
A second, related, inefficient equilibrium is when the equilibrium is separating but the signal is inefficiently high. In equilibrium, job applicants must be able to recite a list of the past four presidents of the company, but even requiring a list of two presidents would separate the good applicants from the bad. In response, the company could say before the interview that applicants need only know the history of the company for the past five years, and that it considers excess knowledge a sign of desperation on the part of the applicant.
A third, different kind of equilibrium is pooling where no signalling occurs: applicants believe that employers are unimpressed by their knowledge of company history and employers are unimpressed because only a blundering applicant would bother to research it. Such an equilibrium need not be inefficient. It might be that the cost of the signal is not worth the efficiency gains from separation. But it might be inefficient: the gains from hiring the most enthusiastic applicant, one who could stand to spend ten hours researching the history of an accounting firm merely to slightly increase his probability of getting a job there, might be worth the cost. If so, the employer could break out of the inefficient pooling equilibrium by announcing that applicants will be rated partly on their knowledge of company history. The pooling equilibrium would then be too delicate to survive.
A fourth kind of equilibrium is inefficient separation-- an equilibrium where signalling occurs and separate good types of applicants from bad, but the cost to the applicants is not worth the social benefit. The good type of applicant may benefit, but the bad type loses even more, since the benefit is a pure transfer of who gets the job and this is achieved at a cost, the cost of the research.
Inefficient separation of this kind is more robust than the first three kinds of inefficient equilibria. The reason is that if one employer deviates and announces that it will no longer give applicants credit for their knowledge of company history, the result would simply be that they would attract even more low-quality applicants, while their high-quality applicants would desert them for employers who would respond to the research signal. If, however, there were only one employer, that employer could successfully pursue such a policy. It would change the signalling game to a screening game, and announce a reduction in the wage to the applicants' reservation level but also ignore knowledge of company history in interviews. This is an interesting efficiency advantage of a monopsony.
Thus, very often, if not always, the timing of moves in a signalling game is flexible enough that in the metagame in which the players can choose the order of the moves, they will choose to convert it to a screening game. In the end, this will often eliminate all but one equilibrium outcome, and often the remaining outcome will be efficient, or at least pareto-optimal.
\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 to show how signalling could continue with pooling.
Education IV changed Education I by allowing a 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.
\bigskip
\noindent
{\bf Ways to Communicate}
We have by now seen a variety of ways one player can convey information to another. Following the Crawford-Sobel model of Chapter 10, let us call these players the Sender and the Receiver. Why might the Receiver believe the Sender's message? The reason depends on the setting.
\bigskip
\noindent
{\bf 1. Cheap Talk Games. } The Sender's message is costless and there is no penalty for lying.
In a cheap talk game, messages have no direct impact on payoff functions. If the Receiver ignores the message, the Sender’s payoff is unaffected by the message. If the Receiver changes his action in response, though, that might affect the Sender. Usually, these are coordination games, where the Sender's preference for the action the Receiver will choose in response to the message is the same as the Receiver's, or at least correlated with it. Chapter 10's Crawford-Sobel model is an example of imperfect correlation, in which cheap talk results in some information being exchanged, even though the message is coarse and the Sender cannot convey precisely what he knows.
\bigskip
\noindent
{\bf 2. Truthful Announcement Games. } The Sender's message is costless. He may be silent instead of sending a message, but if he sends a message it must be truthful.
Here, sending the message is costless if it is truthful, as with cheap talk, but sending a message is costly if it is false--- infinitely costly in the limiting case. The Sender's type usually varies from bad to good in these models, and they are subject to the ``unravelling'' illustrated in Chapter 10's example, since silence indicates bad news. As discussed there, unravelling might only be partial, for a variety of reasons such as some exogenous probability that the Sender actually is unable to send even a truthful message. One variant on this kind of game is when the Sender has committed to the truth, perhaps via a mechanism negotiated with the Receiver, converting a cheap talk game into a truthful announcement game.
\bigskip
\noindent
{\bf 3. Auditing Games. } The Sender's message might or might not be costly. The Receiver may audit the message at some cost and discover if the Sender was lying.
We saw examples of this in Chapter 3, with commitment to auditing. A separate class of models is when the Receiver cannot commit to auditing, in which case there will be an equilibrium in mixed strategies, with the Sender sometimes truthful, sometimes not, and the Receiver sometimes auditing, sometimes not. An example is Rasmusen (1993), in which the Sender is a lobbyist or protester who sends a costly message to the Receiver, a government official. That cost is wasted if the Receiver audits and finds the message is a lie, even if there is no additional punishment of the kind found in truthful announcement games. Another example is Rasmusen (2001b), in which the Sender's message is a purportedly pareto-improving contract clause he writes and the Receiver decides whether to read carefully or not to discover whethe the clause actually benefits only the Sender and should be rejected.
\bigskip
\noindent
{\bf 4. Mechanism Games. } The Sender's message might or might not be costly. Before he sends it, he commits to a contract with the Receiver, with their decisions based on what they can observe and enforcement based on what can be verified by the courts.
Mechanisms were the topic of Chapter 10. Commitment is the key. Mechanisms work out most simply if they are chosen before the Sender receives his private information, since otherwise the choice of mechanism may itself convey information.
\bigskip
\noindent
{\bf 4. Signalling Games. } The Sender’s message is costly when he lies, and more costly when he lies than when he tells the truth. He sends it before the Receiver takes any action.
Signalling is the main topic of this chapter, Chapter 11.
The Sender’s type usually varies from bad to good in these models. The single-crossing property is crucial--- that if the Sender's type is better, it is cheaper for him to send a message that his type is good. It might well be costless for the Sender to send a truthful message (or even have negative costs), because what matters is the difference in costs between a truthful message and a false one. Usually in these models the message is an indirect one, conveyed by the choice of some action such as years of education seemingly unrelated to communication.
\bigskip
\noindent
{\bf 5. Expensive-Talk Games. } The Sender's message is costly, but the cost is the same regardless of his type. There is no penalty for lying.
The difference between expensive talk and signalling is that in expensive talk, the single-crossing property is not satisfied.
Truthful communication might still work, for reasons akin to those in the Cheap-Talk Game, if the high-type Sender has a greater desire than the low type for the Receiver to adopt a high response. Chapter 6's PhD Admissions Game is an example. The student who hates economics has no incentive to pay the cost of applying for the PhD program. As with cheap talk and signalling, expectations are crucial because of multiple equilibria.
\bigskip
\noindent
{\bf 6. Screening Games. } The Sender's message is costly when he lies, and more costly when he lies than when he tells the truth. He sends it in response to an offer by the Receiver.
We have discussed screening games here in Chapter 11. If the Receiver can commit to his response to a signal, this is a mechanism game. If he cannot, it is an expensive talk game, where the Receiver's offer is made as part of an equilibrium in which the Sender's making the offer influences the Receiver's expectations of what the Sender will do.
\bigskip
In the rest of this chapter, we will look at more specialized models related to signalling.
\bigskip
\noindent
{\bf *11.5: Two Signals: The Game of Underpricing New Stock Issues}
\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, for example, the multiple signal
model used to explain 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 The Order of Play}\\
(See Figure 3a of Chapter 2 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 3 shows the signalling schedules for two variance levels.
\includegraphics[width=150mm]{fig11-03.jpg}
\begin{center}
{\bf Figure 3: How the Signal Changes with the Variance}
\end{center}
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, $\alpha = 0.1$, and
making the initial offering at a low price, $P_0 = 80$. Figure 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 $\alpha = .1$ and offered the high price $P_0 =
90$, investors would conclude that $\mu$ was lower than 120 but that 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.
\includegraphics[width=150mm]{fig11-04.jpg}
\begin{center}
{\bf Figure 4: Different Ways to Signal a Given Company Value }
\end{center}
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 none the less
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.
\bigskip
\noindent
{ \bf *11.6 Signal Jamming and Limit Pricing} (New Model for the 4th edition)
This chapter 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.
Limit pricing refers to the practice of keeping prices low to deter entry.
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]. 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 previous editions I used a model from
Rasmusen (1997) as an illustration, but in this edition I will use a simpler
model that is still able to convey the idea of signal jamming. The market will
be one with inelastic demand up to a reservation price. Entry would split that
inelastic quantity evenly between the firms, and the price would fall. What is
crucial is that before entry the entrant does not know the reservation price,
and hence does not know how far the price would fall after entry. The incumbent
may or may not find it worthwhile to keep his price low to prevent the entrant
from learning something about the reservation price.
\begin{center}
{\bf Limit Pricing as Signal Jamming }
\end{center}
\noindent
{\bf Players}\\
The incumbent and the rival.
\noindent
{\bf The Order of Play}\\
0 Nature chooses the reservation price $v$ using the continuous density $h(v)$
on the support $[c, d]$, observed only by the
incumbent. (The parameter $c$ is marginal cost.) \\
1 The incumbent chooses the first-period price $p_1$, generating sales of $q
$ if $p_1 \leq v$ and $ 0$ otherwise.The variables $p_1$ and $q$ are observed by both
players. \\
2 The rival decides whether to enter at cost $F$ or to stay out.\\
3 If the rival did not enter, the incumbent chooses the second period price,
$p_{2,monopoly}$, generating sales of $q $ if $p_{2m} \leq v$ and $ 0$
otherwise.\\
If the rival did enter, the duopoly price is $p_{2}(v)$, with $p_{2} \geq c$ and
$\frac{d p_{2}}{dv} >0$. \\
Let us make more specific assumptions to avoid caveats later.
Assume that the second-period duopoly price is halfway between marginal cost and
the monopoly price, so
\begin{equation} \label{e108}
p_2(v) = \frac{c+ v }{2},
\end{equation}
the distribution of reservation prices is uniform, so
\begin{equation} \label{e109}
h(v) = \frac{1}{d-c},
\end{equation}
that $c=0$, and that $ d > 6.4F/q$. We will defer using these special assumptions until midway through our exposition, however.
\noindent
{\bf Payoffs}\\
If the rival does not enter and $p_1$ and $p_{2m}$ are no greater than
$v$, the payoffs are
\begin{equation} \label{e100}
\begin{array}{ll}
\pi_{incumbent}& = (p_1-c) q + (p_2-c) q \\
& \\
\pi_{rival} & = 0.\\
\end{array}
\end{equation}
If the rival does enter and $p_1$ are no greater than $v$, the payoffs
are
\begin{equation} \label{e101}
\begin{array}{ll}
\pi_{incumbent}& = (p_1-c) q + (p_{2}(v)-c) \left( \frac{q}{2}\right)
\\
& \\
\pi_{rival}& = -F + (p_{2}(v)-c) \left( \frac{q}{2} \right).\\
\end{array}
\end{equation}
The incumbent needs to trade off a high price in the first period against the
possibility of inducing entry. As always, work back from the end.
The incumbent's choice of $p_{2,monopoly}$ is easy: choose $p_{2,monopoly}=v$,
since there is no threat of future entry.
The rival's expected payoff from entry depends on his beliefs about $v$,
which in turn depend upon which of the multiple equilibria of this game is being
played out.
If the incumbent were nonstrategic, he would charge $p_1= v$, maximizing his
first-period profit. The rival would deduce the value $v$ and would enter in
the second period if $p_1 $ exceeded the critical value $v^* =
c + \frac{4F}{q}$, a value which yields zero profits because
\begin{equation} \label{e102}
{\displaystyle -F + \left[ \left(\frac{c + \frac{4F}{q}}{2}-c\right)\left( \frac{q}{2}
\right) \right] =0.}
\end{equation}
If the incumbent is willing to charge a lower $p_1$, though,
and accept lower first-period profits, he may be able to deter entry,
and if he can, the higher second-period profits can more than make up for his
sacrifice. Thus, nonstrategic behavior is not an equilibrium for this game.
Instead, consider the strategy profile below.
\noindent
{\bf A Limit Pricing Equilibrium } \\
{\bf Incumbent:} $p_2 =v$. For particular values $a$ and $b$ to be specified later,
\begin{equation} \nonumber
p_1 = \left\{
\begin{array} {lll}
v & if & v b \\
\end{array}
\right.
\end{equation}
\noindent
{\bf Rival:} Enter if $p_1 >a$. Otherwise, stay out. If the rival observes $p_1 \in (a,b]$, his out-of-equilibrium belief is that $ v= \frac{h(v)}{
\int_{p_1}^b h(v) dv} $, the expected value of $v$ if it lies between $p_1$ and $b$. \\
\noindent
The rival's equilibrium payoff if he enters is
\begin{equation} \label{e103}
\begin{array}{ll}
\pi_{rival, enter}& = {\displaystyle -F + (Ep_{2}(v)-c) \left( \frac{q}{2}
\right)}\\
& \\
& ={\displaystyle -F + \left( \int_a^b p_{2}(v) \left( \frac{h(v)}{
\int_a^b h(v) dv} \right) dv -c \right) \left( \frac{q}{2}\right),}
\end{array}
\end{equation}
where the density for the expectation $Ep_{2}(v)$ is $ \frac{h(v)}{
\int_a^b h(v) dv} $ instead of just $h(v)$ because it is conditional on
$v$ being between $a$ and $b$, rather than $v$ taking any of its possible
values.
The payoff in (\ref{e103}) equals zero in equilibrium for the entry-deterring
values of $a$ and $b$.
\noindent
If the incumbent deters entry, his payoff is
\begin{equation} \label{e104}
\pi_{incumbent} (no\; entry) = (p_1-c) q + (v-c)q
\end{equation}
and if he does not it is
\begin{equation} \label{e105}
{\displaystyle \pi_{incumbent} (entry) = (v-c) q + (p_{2}(v)-
c) \left( \frac{q}{2}\right). }
\end{equation}
The incumbent's advantage from limit pricing is the difference between these when $p_1=a$, which is
\begin{equation} \label{e106}
{\displaystyle [(a-c) q + (v-c)q ] -
\left[(v-c) q + (p_{2 }(v)-c) \left( \frac{q}{2} \right) \right]. }
\end{equation}
This advantage is declining in $v$, and $b$ is defined as the value at which it equals zero. Choosing $v=b$ in expression (\ref{e106}), equating to zero, and solving for $a$ yields
\begin{equation} \label{e107}
a = \frac{c+ p_2(b) }{2}.
\end{equation}
We now have two equations (equation [\ref{e107}] and $\pi_{rival}=0$ using
expression [\ref{e103}]) for two unknowns ($a$ and $b$). Now let us return to the special assumptions in equations (\ref{e108}) and (\ref{e109}). They
Given our specific assumptions on $h(v)$ and $p_2(v)$, $p_2(b) = b/2$ and $h(v) =
1/d$, so from (\ref{e107}),
\begin{equation} \label{e110}
a = \frac{b }{4}.
\end{equation}
Since $\int_a^b h(v)dv = \int_a^b (1/d)dv= \frac{b}{d} - \frac{a}{d}$,
\begin{equation} \label{e111}
\begin{array}{ll}
\pi_{rival}& = {\displaystyle -F + \left( \int_a^b \left( \frac{v}{2}
\right) \left( \frac{ \frac{1}{d}}{\frac{b}{d} - \frac{a}{d}} \right) dv
\right) \left( \frac{q}{2}\right)} \\
& \\
& = {\displaystyle -F + \bigg|_{v=a}^b \left( \frac{v^2}{4 (b-a)}\right)
\left( \frac{q}{2}\right)} \\
& \\
& = {\displaystyle -F + \left( \frac{b^2}{4 (b-a)} - \frac{a^2}{4(b-a)}
\right) \left( \frac{q}{2}\right)} \\
& \\
& = {\displaystyle -F + \left( b^2 - \frac{b^2}{16} \right) \left(
\frac{q}{8(b- \frac{b}{4})}\right)} \\
\end{array}
\end{equation}
so
\begin{equation} \label{e112}
b = \left( \frac{ 8}{5} \right) \left( \frac{ 4F}{q} \right),
\end{equation}
where $b 6.4F/q$. From equation (\ref{e110}),
\begin{equation} \label{e113}
a = \left( \frac{ 2}{5} \right) \left( \frac{ 4F}{q} \right).
\end{equation}
It is useful to compare the limits $a$ and $b$ with $v^*$, the value of $v$ for which entry
yields a payoff of zero to the rival. That value, which we found above using
equation (\ref{e102}), is
\begin{equation} \label{e113a}
v^* = \frac{ 4F}{q}.
\end{equation}
Thus, $a < v^* < b$. This makes sense. The values of $a$ and $b$ were chosen
so that, in effect, $v^*$ was the expected value of $v$ on $[a,b]$, and the
rival would be deterred even though he did not know the exact value of $v$.
If $a \geq v^*$, then the expected value of $v$ on $[a,b]$ would be greater than
$v^*$, and the rival would feel safe in entering if the incumbent charged $p_1=
a$. Thus, to use limit pricing, the incumbent must charge strictly less than the
monopoly price appropriate if the duopoly market would yield zero
profits to an rival. This kind of signal jamming reduces the information that reaches the rival compared to nonstrategic behavior, since the rival learns the precise value of $v$ only if $v$ is less than $a$ or greater than $b$.
Note, however, that limit pricing would not work if this were a truthful announcement game, in which the incumbent could remain silent but could not lie to the rival. The incumbent would then tell the rival the value of $v$ whenever it was $v^*$ or less--- which means that limit pricing for values in the interval $[v^*, b]$ would no longer be effective.
%-----------------------------------------------
\bigskip
\noindent
{\bf *11.7. Countersignalling} (New in the 4th edition)
Feltovich,
Harbaugh \& To (2002) constructed a model to explain a situation commonly
observed: that although the worst types in a market do not signal, often the
best types do not signal either, and their very lack of signalling can be a sign
of their confidence in their high quality ({\bf countersignalling}). What is
crucial in such situations is that the highest-quality types have an alternative
way to convey their quality to the market besides costly signalling. We will
illustrate that here with a model of banking.
\begin{center}
{\bf Countersignalling}
\end{center}
\noindent
{\bf Players}\\
Banks and depositors.
\noindent
{\bf The Order of Play}\\
0 Nature chooses the solvency $\theta_i$ of each bank on a continuum of
length 1, using cumulative distribution $F(\theta)$ on the support $[-10,10]$.
\\
1 Bank $i$ chooses to spend $s_i$ on its building, where it must spend at
least $\bar{s} =4$ to operate at all, and otherwise must exit. \\
2 Depositors on a continuum of length $D=1$ observe $\hat{\theta}=
\theta_i+u_i$, where the $u_i$ values are chosen independently and take values
of $-5, 0$, and $+5$ with equal probability. \\
3 Each depositor chooses a bank.\\
\noindent
{\bf Payoffs}\\
Bank $i$'s payoff if it attracts depositors is
\begin{equation} \label{e114}
\pi_{i} = \frac{D}{B} - \left( \frac{s_i}{10+\theta_i} \right),
\end{equation}
where $B$ is the interval of banks that attract depositors. If the bank
attracts no depositors, its payoff is $-\left( \frac{s_i}{10+\theta_i} \right)
$.
\noindent
A depositor's payoff is 1 if he picks a bank with quality at or above 0 and
$0$ if he picks quality below 0.
\bigskip
\noindent
{\bf Equilibrium} \\
Banks with solvency $\theta \in [-10,0]$ choose $s=0$. They do not signal,
and attract no depositors.\\
Banks with solvency $\theta \in [0, 5)$ choose $s= Max\{4, s^*= \frac{10D}
{1-F(0)} \}$ and attract depositors. \\
Banks with solvency $\theta \in [5,10]$ choose $s=4$, and attract
depositors. \\
Depositors choose banks with either $\hat{\theta} \geq 5$ or $s \geq s^*=
\frac{10D}{1-F(0)}$ or both.\\
In this equilibrium, $B = 1 - F(0)$ because depositors choose the banks
which are solvent. If $\hat{\theta} \geq 5$, they can be sure the bank is
solvent, because the minimum possible true level of $\theta$ is 0, with $u=+5$.
Otherwise, they rely on the signal. No out-of-equilibrium beliefs need to be
specified, because in equilibrium the choice $s-s^*$ might be observed in
conjunction with any value of $\hat{\theta}$ within $[-5, 10]$, and if
$\hat{\theta}< -5$ the depositors know that $\theta<0$.
The self-selection constraint requires that banks with $\theta<0$ not signal.
Thus, the crucial signal level, $s^*$, requires that $\pi_{0} (s=0) = \pi_{0}
(s=s^*)$, so
\begin{equation} \label{e115}
0 = \frac{D}{B} - \left( \frac{s^*}{10+0} \right)
\end{equation}
Solving equation (\ref{e115}) yields
\begin{equation} \label{e115a}
s^* = \frac{10D}{1-F(0)}.
\end{equation}
Entry is also unprofitable for all $\theta \leq 0$ if the equilibrium
signal is $Max\{4, s^*\} =4$, entry is also unprofitable, or the equilibrium
signal is $Max\{4, s^*\} = s^*$ and a bank chooses to enter with $s=4 $ and
attract no depositor.
\noindent
The payoff of a bank in $ \theta=[0,5)$ if it enters and signals $s^*$ is
\begin{equation} \label{e115b}
\pi(s=s^*) = \frac{D}{B} - \left( \frac{s^*}{10+ \theta} \right).
\end{equation}
This is at least as high as the zero payoff from not entering, because
substituting in for $B$ and $s^*$ yields
\begin{equation} \label{e115c}
\pi(s=s^*) = \frac{D}{ 1-F(0)} - \frac{ \frac{10D}{1-F(0)}}{10+ \theta}
\geq \frac{D}{ 1-F(0)} - \frac{10D }{10 [1-F(0)] } = 0.
\end{equation}
If $Max\{4, s^*\} =4$, then the bank's payoff is also non-negative:
\begin{equation} \label{e115c1}
\pi(s=4) = \frac{D}{ 1-F(0)} - \left( \frac{ 4}{10+ \theta} \right) \geq
\frac{D}{ 1-0} - \left( \frac{4 }{10} \right) = 0.
\end{equation}
This last equality is true because $D =1 \geq 0.4$.
The mid-quality banks could also pretend to be high-quality banks by entering,
not signalling and hoping to have a positive measurement error $u$. The payoff
from that would be composed of a 2/3 probability of attracting no customers and
a 1/3 probability of attracting $D/B$:
\begin{equation} \label{e115d}
\begin{array}{ll}
\pi(s=4) = \left( \frac{2}{3} \right) \left( 0 \right) + \left( \frac{1}{3}
\right) \left( \frac{D}{B} \right) - \left( \frac{4}{10+ \theta}\right)
\\
\end{array}
\end{equation}
This is no more than the payoff from $s= s^*$ in (\ref{e115b}) if
\begin{equation} \label{e115e}
\begin{array}{ll}
\pi(s=2) = \left( \frac{2}{3} \right) \left( 0 \right) + \left( \frac{1}{3}
\right) \left( \frac{D}{ 1-F(0) } \right) - \left( \frac{4}{10+ \theta}
\right) \leq \pi(s=s^*)= \left( \frac{D}{ 1-F(0)} \right) - \left( \frac{
\frac{10D}{1-F(0)}}{10+ \theta} \right),
\end{array}
\end{equation}
which is equivalent to
\begin{equation} \label{e115f}
\left( \frac{10D}{10+\theta} \right) - \left( \frac{4 (1-F(0))}
{10+\theta} \right) \leq \left( \frac{2}{ 3 } \right)D.
\end{equation}
Inequality (\ref{e115f}) is true because the left-hand side is largest if
$\theta=0$ and $F(0)=0$, in which case it becomes $D - 0.4 \leq (2/3)D$, which
is true if $D=1$.
The payoff of the high-quality banks in $ \theta=[5,10]$ is highest from entry
without signalling, when it is $\pi(\theta) = D/B - 4/(10+\theta)$, which is
positive since $D>0.4$ and $\theta >5$ for this group. Signalling $s^*$ merely
adds costs, while not entering yields a payoff of zero.
As always, there also exists a nonsignalling perfect bayesian equilibrium, in
which depositors ignore any signal and only choose banks with $\hat{\theta}\geq
5$. And there exist other, inefficient, equilibria in which the crucial signal
level is $s^* > \frac{10D}{1-F(0)}$. You might also wish to investigate what
happens if the number of consumers, $D$ takes values other than 1.
What is interesting in this model is that contrary to previous models, it is
not the highest-quality players that signal, but the middle-quality ones. The
highest-quality players have no need to signal, because their quality is so
high that information about it reaches the market by a different means-- here,
the imperfect observation $\hat{\theta}$. The ``countersignal'' is the absence
in the highest quality players of a normal signal. The crucial feature of
countersignalling models that gives rise to this is that signalling not be
the only way in which information reaches the market.
Another effect that can
arise in countersignalling models (though it does not in the simple one here) is that the absence of a signal can have a positive,
additional, effect on the uninformed player's estimate of the informed
player's quality. In our banking game, that can't happen because if the noisy
observation is $\hat{\theta} \geq 5$, depositors have the highest possible
opinion of the bank's safety: that it is solvent with probability one. Thus, the
high-quality banks abstain from signalling simply to save money, not because the
countersignal actually helps them.
Suppose, though, that
we added new depositors to the model, an amount small enough to be
measure zero so they would not affect the equilibrium, and these new depositors
had the special features that: (a) they could not observe even $\hat{\theta}$, and
(b) their payoffs were 0 not just for insolvent banks but for any bank with
$\theta <7$. These depositors would observe only that some banks have $s=4$
and some have $s=s^*$. Since they would be looking for especially high-quality banks,
they would actually prefer to choose a bank with the lower signal value of $s=
4$, because they could be sure that if $s=s^* $ then $\theta <7$. Thus, having
a smaller building acts as a countersignal, an indicator that this bank has such
high quality that it does not need to be ostentatious.
This informational value of the countersignal is what prompted the title of Feltovich, Harbaugh \& To (2002): ``Too Cool for School? Signalling and Countersignalling''. Low-quality high-school students might do badly in their schoolwork because doing well is too costly a signal. Mid-quality students would do better, to separate themselves from the low-quality students. But high-quality students might also do badly in their schoolwork, as a form of countersignalling. An observable indicator of quality might already show them to be high-quality (``cool''), so they don't need the signal. But they might suppress their schoolwork even though they really would prefer to do well, because a student who does his schoolwork would be considered mid-quality, not high-quality. In such a case, the countersignal is actually costly-- the student is refraining from schoolwork purely for strategic reasons. This is not quite signal jamming, because the high-quality students is trying to escape from pooling rather than hide in it, but it shares with signal jamming the idea of costly suppression of an observable variable.
The term ``countersignalling'' is new, but the idea that the more desirable type might signal less goes as far back as Part III, Article III of Adam Smith's {\it Wealth of Nations}, where he lays out a theory of morality and wealth, noting that the poor are often more rigorous than the rich in their morality. Most of his theory is based on the higher utility cost of vice to the poor rather than on information, but he also discusses the observable effects of vice. He suggests that for the urban poor, in the anonymity of the city, belonging to a strict church can signal their morality, while for rich people, the very indulgence in vice can signal wealth, since high expenditure would quickly ruin someone of moderate means. Countersignalling will be a fruitful area for empirical work to test whether it explains signalling by moderate types only in various situations. One fascinating study of quality and information provision is Jin \& Leslie (2003), a study of restaurant responses to government hygiene grade cards for restaurants in Los Angeles, cards whose posting in restaurant windows was optional and later was compulsory.
%-----------------------------------------------
\newpage
\begin{small}
\noindent
{\bf Notes}
\bigskip
\noindent
{\bf N11.1} {\bf The Informed Player Moves First: Signalling}
\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 combination from which no worker has incentive
to deviate and under which the employer's profits are zero. 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. Where confusion
might arise, the word ``indicator'' might be better for an informative variable.
\item The applications of signalling are too many to properly list. A few
examples are the use of prices in Wilson (1980) and Stiglitz (1987), the payment
of dividends in Ross (1977) and bargaining under asymmetric information
(Section 12.5). 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. Riley
(2001) surveys the signalling literature, and Rochet \& Stole (2003)
surveys multi-dimensional screening.
\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. By now there is an
entire literature on signalling in the natural world; see
John Maynard-Smith and David Harper's 2004 book, {\it Animal Signals.}
\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, high rates of
advertising are associated with high product quality. Only firms with high quality would advertise.
$\;\;\;$ 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 which they 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}
\newpage
\noindent
{\bf Problems}
\bigskip
\noindent
{\bf 11.1. Is Lower Ability Better?} (medium)\
Change Education I so that the two possible worker abilities are $a \in \{1,4\}
$.
\begin{enumerate} \item[(a)] What are the equilibria of this game? What are
the payoffs of the workers (and the payoffs averaged across workers) in each
equilibrium?
\item[(b)] Apply the Intuitive Criterion (see N6.2). Are the equilibria
the same?
\item[(c)] What happens to the equilibrium worker payoffs if the high
ability is 5 instead of 4?
\item[(d)] Apply the Intuitive Criterion to the new game. Are the
equilibria the same?
\item[(e)] Could it be that a rise in the maximum ability reduces the
average worker's payoff? Can it hurt all the workers? \end{enumerate}
\bigskip \noindent
{\bf 11.2. Productive Education and Nonexistence of Equilibrium} (hard)\\ 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
$U = w - \frac{8s}{a}$.
\begin{enumerate}
\item[(a)] Under full information, what are the wages for educated and
uneducated workers of each type, and who acquires education?
\item[(b)] Show that with incomplete information the equilibrium is unique
(except for beliefs and wages out of equilibrium) but unreasonable.
\end{enumerate}
\bigskip
\noindent
{\bf 11.3. Price and Quality } (medium)\\
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.
\begin{enumerate} \item[(a)]
What is Apex's price and profit if it must choose one price, $p^*$, for both
periods?
\item[(b)]
What is Apex's price and profit if it can choose two prices, $p_1$ and $p_2$,
for the two periods, but it cannot commit ahead to $p_2$?
\item[(c)] What is the answer to part (b) if the discount rate is $r = 0.1$?
\item[(d)] Returning to $r=0$, what if Apex can commit to $p_2$?
\item[(e)] 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.)
\end{enumerate}
\bigskip \noindent
{\bf 11.4. Signalling with a Continuous Signal } (hard)\\
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 $\pi_{employer} = a-w$.
\begin{enumerate}
\item[(a)] 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?
\item[(b)]
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?
\item[(c)]
If you were forced to predict one equilibrium which will be the one played out,
which would it be?
\end{enumerate}
\bigskip \noindent
{\bf 11.5: Advertising } (medium) \\
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 11 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 110 cold-blooded economists who are not taken in by
psychological tricks. The market can be divided into two periods.
\begin{enumerate}
\item[(a)] If advertising is banned, will Brydox go out of business?
\item[(b)] 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.
\item[(c)] What is the minimum and maximum that Brydox might spend on
advertising, if it spends a positive amount?
\end{enumerate}
\bigskip \noindent
{\bf 11.6. Game Theory Books } (easy)\\
In the Preface, I explain why I listed
competing game theory books by saying, ``only an author quite confident that his
book compares well with possible substitutes would do such a thing, and you
will be even more certain that your decision to buy this book was a good
one.''
\begin{enumerate}
\item[(a)] What is the effect of on the value of the signal if there is a
possibility that I am an egotist who overvalues his own book?
\item[(b)] Is there a possible non strategic reason why I would list competing
game theory books?
\item[(c)] If all readers were convinced by the signal of providing the list
and so did not bother to even look at the substitute books, then the list
would not be costly even to the author of a bad book, and the signal would fail.
How is this paradox to be resolved? Give a verbal explanation.
\item[(d)]
Provide a formal model for part (c).
\end{enumerate}
\bigskip
\noindent
{\bf 11.7 A Continuum of Pooling Equilibria } (medium)\\
Suppose that with equal probability a worker's ability is $a_L=1$ or $a_H=5$,
and that the worker chooses any amount of education $y \in [0,\infty)$. Let
$U_{worker}=w - \frac{8y}{a}$ and $\pi_{employer} = a-w$.
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 self-
selection constraints?
%---------------------------------------------------------------
\bigskip \noindent
{\bf 11.8. Signal Jamming in Politics} (hard) \\
A congressional committee has
already made up its mind that tobacco should be outlawed, but it holds televised
hearings anyway in which experts on both sides present testimony. Explain why
these hearings might be a form of signalling, where the audience to be
persuaded is congress as a whole, which has not yet made up its mind. You can
disregard any effect the hearings might have on public opinion.
\bigskip \noindent {\bf 11.9. Salesman Clothing } (medium)\\
Suppose a
salesman's ability might be either $x= 1$ (with probability $\theta$) or $x=4$,
and that if he dresses well, his output is greater, so that his total output is
$x + 2s$ where $s$ equals 1 if he dresses well and 0 if he dresses badly. The
utility of the salesman is $U= w - \frac{8s}{x}$, where $w$ is his wage.
Employers compete for salesmen.
\begin{enumerate}
\item[(a)] Under full information, what will the wage be for a salesman with
low ability?
\item[(b)] Show the self selection contraints that must be satisfied in a
separating equilibrium under incomplete information.
\item[(c)]
Find all the equilibria for this game if information is incomplete.
\end{enumerate}
\bigskip
\noindent
\textbf{11.10. Crazy Predators } (hard) (adapted from Gintis [2000], Problem 12.10)\\
Apex has a monopoly in the market for widgets, earning profits of $m$ per
period, but Brydox has just entered the market. There are two periods and no
discounting. Apex can either $Prey$ on Brydox with a low price or accept
$Duopoly$ with a high price, resulting in profits to Apex of $-p_a$ or $d_a$ and
to Brydox of $-p_b$ or $d_b$. Brydox must then decide whether to stay in the
market for the second period, when Brydox will make the same choices. If,
however, Professor Apex, who owns 60 percent of the company's stock, is crazy,
he thinks he will earn an amount $p^*> d_a$ from preying on Brydox (and he does
not learn from experience). Brydox initially assesses the probability that Apex
is crazy at $\theta$.
\begin{enumerate}
\item[(a)] Show that under the following condition, the equilibrium will be
separating, i.e., Apex will behave differently in the first period depending on
whether the Professor is crazy or not:
\begin{equation} \label{e15.a1} -p_a +m < 2d_a \end{equation}
\item[(b)] Show that under the following condition, the equilibrium can be
pooling, i.e., Apex will behave the same in the first period whether the
Professor is crazy or not: \begin{equation} \label{e15.a2} \theta \geq
\frac{d_b}{p_b+d_b} \end{equation}
\item[(c)]
If neither two condition (\ref{e15.a1}) nor (\ref{e15.a2}) apply, the
equilibrium is hybrid, i.e., Apex will use a mixed strategy and Brydox may or
may not be able to tell whether the Professor is crazy at the end of the first
period. Let $\alpha$ be the probability that a sane Apex preys on Brydox in the
first period, and let $\beta$ be the probability that Brydox stays in the market
in the second period after observing that Apex chose $Prey$ in the first period.
Show that the equilibrium values of $ \alpha $ and $\beta$ are:
\begin{equation} \label{e15.a3}
\alpha = \frac{ \theta p_b }{(1-\theta)d_b} \end{equation} \begin{equation}
\label{e15.a4} \beta =\frac{-p_a +m -2d_a}{m-d_a} \end{equation}
\item[(d)] Is this behavior related to any of the following phenomenon?--
Signalling, Signal Jamming, Reputation, Efficiency Wages.
\end{enumerate}
\bigskip \noindent
{\bf 11.11. Monopoly Quality } (medium)\\
A consumer faces a monopoly. He initially believes that the probability that
the monopoly has a high-quality product is H, and that a high-quality monopoly
would be able to send him an advertisement at zero cost. With probability (1-
H), though, the monopoly has low quality, and it would cost the firm A to send
an ad. The firm does send an ad, offering the product at price P. The
consumer's utility from a high-quality product is $X>P$, but from a low
quality product it is 0. The production cost is $C$ for the monopolist
regardless of quality, where $C < P- A$. If the consumer does not buy the
product, the seller does not incur the production cost.
You may assume that the high-quality firm always sends an ad, that the consumer
will not buy unless he receives an ad, and that $P$ is exogenous.
\begin{enumerate}
\item[(a)] Draw the extensive form for this game.
\item[(b)] What is the equilibrium if H is sufficiently high?
\item[(c)] If H is low enough, the equilibrium is in mixed strategies. The
high-quality firm always advertises, the low quality firm advertises with
probability M, and the consumer buys with probability N. Show using Bayes
Rule how the consumer's posterior belief $R$ that the firm is high-quality
changes once he receives an ad.
\item[(d)] Explain why the equilibrium is not in pure strategies if H is too
low (but $H$ is still positive).
\item[(e)] Find the equilibrium probability of M. (You don't have to figure
out N.)
\end{enumerate}
%---------------------------------------------------------------
\newpage
\begin{center}
{\bf Signalling Marriageability: A Classroom Game for Chapter 11}
\end{center}
Each student is a Man or a Woman with a random and secret wealth level $W$ distributed between 0 and 100. The wealth levels if there are ten people are 0, 10, 20, 30, 50, 60, 80, 90, 100, and 100. The instructor will choose from these values, repeating them if there are more than ten students and omitting some values if the number of students is not divisible by ten.
A person's wealth level is secret, because the society has a taboo on telling your wealth level to someone else. Nonetheless, everybody is very interested in wealth because everyone's objective is to marry someone with high wealth.
Each year, each student first simultaneously writes down how much to spend on clothes that year. Then, in whatever order it happens, students pick someone else to pair up with temporarily. Both actions are publicly revealed-- you may show people your scoresheet. In the fifth year, the pairings become permanent: marriage.
Your payoff is a concave function the original wealth minus clothing expenditures of you and your spouse (if you have one). Clothing has no value in itself. Thus, if $i$ is married to $j$ his payoff is, letting $C_i$ and $C_j$ denote the cumulative clothing purchases over the five periods,
$$
U_i = log (W_i - C_i + W_j- C_j)
$$
Table 1 shows some of the possible payoffs from this function.
\begin{center}
\begin{tabular} {|l| l l l l l l l l l l|}
\hline
& & & & & & & & & & \\
Remaining Family Wealth & 0 & 1 & 2 & 5 & 10 & 25 & 50 &100 &150 &200 \\
& & & & & & & & & & \\
\hline
& & & & & & & & & & \\
Utility & -$\infty$ & 0 & 0.7 & 1.6 & 2.3 &3.2 & 3.9 &4.6 & 5.0 & 5.3 \\
& & & & & & & & & & \\
\hline
\end{tabular}
{\bf Table 1: Marriage Values}
\end{center}
The first time you play the game, the only communication allowed is ``Will you pair with me?'' and ``Yes'' or ``No''. These pairing are not commitments, and can be changed even within the period.
If there is time, the game will be played over with new wealths and with unlimited communication.
\end{small}
\end{document}