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    {\bf     Signal Jamming and Limit Pricing: A Unified Approach},   in  {\it  
Public Policy and Economic Analysis}, eds., Moriki Hosoe and Eric
Rasmusen, Fukuoka, Japan: Kyushu University Press, 1997. 
\\
             \end{large}
                      
                     October 16, 1996  \\
                    \bigskip                     Eric Rasmusen  
                      
                    {\it Abstract} 
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		    \begin{small}

                     
		       In signal jamming, an rival uses observed
profits to predict profitability, but those profits can be
manipulated by a rival firm. In the present model, the size of the
market is known to the incumbent, who is one of two firms that might
occupy it.  The potential rival observes profits, which can be
manipulated by the incumbent. Depending on the monopoly premium and
the prior probability that the market is large, the equilibrium may
be pooling in pure or mixed strategies, or separating, which are
similar to the signal-jamming and signalling equilibria of Fudenberg
\& Tirole (1986) and Milgrom \& Roberts (1982a) respectively. In
contrast to the common result that strategic behavior encourages
innovation even though it introduces current distortions, in this
model the possibility of strategic behavior can either encourage or
discourage entry into markets as yet unserved by any firm.
 
A chapter in: {\it Public Policy and Economic Analysis}, eds., Moriki Hosoe and
Eric Rasmusen, Fukuoka, Japan: Kyushu University Press, forthcoming. 
 
 
            
          \noindent 
\hspace*{20pt}	  	  Indiana University
School of Business ({\bf not the economics department}), Rm. 450,   
  10th Street  and Fee Lane,
  Bloomington, Indiana, USA 47405-1701.
  Office: (812) 855-9219.  Fax: (812) 855-8679. Internet:
Erasmuse@indiana.edu.\\ 
      Draft:  7.1 (Draft 1.1, February 1991).
  
 \vspace{ 10pt}
 
 I would like to thank Kyle Bagwell, David Hirshleifer, Steven
Postrel, Daniel Spulber, and seminar participants at the University
of Colorado, Erasmus University, Texas A \& M, the Wharton School,
Yale SOM, and for helpful comments, and George Michaelides for
research assistance.  Much of this work was completed while the
author was Olin Faculty Fellow at Yale Law School and on the faculty
of UCLA's Anderson Graduate School of Management.  

Forthcoming in: {\it   Public Policy and Economic Analysis},  eds., Moriki Hosoe 
and Eric Rasmusen, Fukuoka, Japan: Kyushu University Press. 


            \end{small}
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\newpage


\noindent
 {\bf 1. Introduction}

 EMPIRICAL MOTIVATION HERE     
   Every once in a while I think of an invention and have a fantasy
of starting up a business. David Friedman, for example, suggested to
me that neckties should be sold in different lengths for different
sized people. Let's think about the trouble of trying to implement
that suggestion. THe cost would be relatively easy to determine. I
would buy my neckties from some manufacturer, made to my designs, and
Icould get bids on that. My costs would be common knowledge
intheindustry, as, indeed, most firms' costs would be. 

Demand is another matter. The reason the market does not exist now is
because nobody else thinks it is big enough to be profitable.  I
think differnetly, and I will find out. Suppose I am right, nad the
market is profitable. What is my next big problem?

My big problem is entry by competitors. At best, they could enter and
force me to share the profits. More likely, they have lower costs
than me, and better marketing, and they will wipe me out. So my
number one problem is to prevent entry. I can't do this credibly by
pretneding to be irratoinal, or bankrupting my competitors, etc. But
I might be able to do so by persuading them  that the market is not
big enough for two firms to operate in. 


 So,I must not appear too profitable. I may purposely keep my sales
and profits small to make the markt appaer unprofitable, even though
my continued existence in the market will convey SOME information.


Note that my competitors can observe a lot of things and still not
know whether the market is large or not. They can certainly see my
prices.They migt be albe to see my output and my profits. BUt even
seeing all of these does not necessarily tell them whether themarket
couldhold two firms. 


There are two ways this could work out. Maybe my competitors can
roughly see how well I am doing even if they do not operate
themselves.Then my porfit-reducng tactic will be LIMIT PRICING.

Or, maybe my competitors have to enter to get a feel for the
market.Then my profit-reducing tactic will bePREDATORY PRICNG,though
reallyI may still be pricng above cost.


 This is a model of predatory or limit pricing, depending on whether
the rival needs to enter or not to discover themarket conditions.


 I will model this situation. Many models of predation and limit pricing exist. 
In brief, the model below differs by being based on demand uncertainty and by 
involving    signalling and signal jamming in the same model. A survey of the 
literature appears below in Section 3. 

  
      Section 2 will lay out the limit pricing model and find the equilibrium. 
Section 3 will interpret the results and discuss the literature on predation and 
limit pricing.  Section 4 will take the
limit pricing model a step back to the source of the incumbent's
informational advantage and his original entry decision, to help
 answer the question of whether  strategic entry
deterrence encourages innovation or not.  Section 5 concludes. 



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

\newpage    


\bigskip
\noindent
{\bf 2. Model I:  Limit Pricing } 

There are two firms, an incumbent and an rival.  Each firm
incurs fixed cost $C$ per period that it is active in the market and
earns a net operating revenue of $R$ per period if both firms are in
the market. If the incumbent is alone, its revenue is $RM$, with $M
>1$.\footnote{If the product is homogeneous, $M>2$ is appropriate,
but the model allows for heterogeneous products, in which case the
industry's duopoly revenue might be greater than the monopoly
revenue.} The market is $Small$ with probability $\theta$ and $Large$
with probability $1-\theta$. 
  In each period the incumbent chooses $R$ to be either $R_0$ or
$R_1$ if the market is small and $R_1$ or $R_2$ if it is large, where
$R_0<R_1<R_2$.  Assume that $R_2-C >0$, $R_1-C <0$, and $R_1M -C>0$,
so a large market can support two firms profitably but a small market
can only support one.  The payoffs for each firm is the sum of the
profits from operations in the two periods. Both firms are risk
neutral and do not discount future profits. 

 The incumbent is already operating in the market in period 1 and by
assumption will remain with probability one.\footnote{xxx fix this:  Otherwise, 
the
two firms are in a ``Chicken'' game, each vying to be the sole
survivor if the market is small. The expected payoffs for each firm
in the second-period Chicken subgame would equal zero in the
symmetric mixed-strategy equilibrium. A limit pricing model based on
this would not behave very differently from Model II below if an
entry fee were also included, because the rival would stay out if
the market were small to avoid paying the entry fee in exchange for
an expected subgame payoff of zero.  A model like Model I, however,
in which the rival begins in the market but is ignorant of its
size, would result in the paradox of the rival unintentionally
driving out the incumbent.  The incumbent, knowing that the market
was small, would exit before incurring the losses of the first
period, since the expected subgame payoff would be zero, but the
rival, not knowing the market was small, would have no such
incentive to exit.  }
 The rival cannot observe the size of the market directly  and must try to 
deduce it from $R$. Having made his estimate, he decides whether to be in or out 
of the market in the second period. 
 
  
   
   The variable $R$ is a convenient way to model  an imperfect indicator of 
market size that is correlated with the
incumbent's profits.  The aim is to analyze as  simply as possible a    market 
in which the rival cannot rely on public information to determine the market's 
size.  It could be that the rival         
 observes both price and quantity, but still cannot determine the
market size; to observe one price-quantity combination is to observe
just one point on the demand curve, and what the rival cares about
is the point on the demand curve that would be reached in
nonstrategic duopoly competition. Observing the incumbent's monopoly
or strategic duopoly price and quantity does not pin down what the
nonstrategic duopoly profit would be, especially if other features
besides price influence demand.  If the rival observes that the
incumbent's price and profits are moderate, this might be either
because the market is truly small or because the incumbent is
charging less than the monopoly price in a large market. Whether a
price is high is a relative matter, and the rival cannot tell
whether it is high relative to what the market could bear unless he
knows the size of the market to begin with; it is even possible that
the monopoly price might be higher in a smaller market.  The three
revenue levels represent two extremes that reveal the market size,
$R_0$ and $R_2$, and a moderate revenue that that is uninformative,
$R_1$. The assumption that there exists a revenue level $R_0$ that
definitively reveals a small market is important; what happens if it
is relaxed is discussed below.  The assumption that the monopoly
revenue is $M$ times the duopoly revenue, regardless of whether the
market is large or small, may seem arbitrary. Its justification is
purely heuristic: this assumption permits a single variable, $M$ to
be used to parametrize the value of being a monopoly, rather than
requiring two variables, one for each size of market. 
  


 
 

  
\noindent
 {\bf The  Order of Play in Model I}

\begin{enumerate}

 \item
 Nature chooses the market to be $Small$ with
probability $\theta$ and $Large$ with probability $(1-\theta)$,
observed only by the incumbent. 
 \item
 The incumbent chooses $R$ to equal $R_0$ or $R_1$
for the first period if the market is small, $R_1$ or $R_2$ if it is
large. 
  \item
 The players both observe the incumbent's
first-period profits.
 \item
The rival decides whether to be $In$ or
$ Out$ of the market. 
\item
 The incumbent chooses $R$ to equal $R_0$ or $R_1$
for the second period if the market is small, $R_1$ or $R_2$ if it is
large. 
  \item
The incumbent and rival collect their
second-period profits, which equal $R-C$ apiece if both are in the
market, $RM-C$ and 0 if the incumbent is alone.
  \end{enumerate}
   
   The equilibrium takes one of four types, depending on the parameter
values.  Parameter $M$, which measures the value of being a monopoly
instead of a duopoly, is what matters most to the incumbent's
strategy, since it reflects the benefits of entry deterrence.
Parameter $\theta$, which measures the prior probability that the
market is small, is what matters most to the rival's strategy,
since it reflects the probability that an apparently small market
truly is small.  Figure 1 shows which parameter values are associated
with which equilibria.\footnote{Please note that  in some cases $\frac{R_1}{R_0} 
> Z^{-1}$, even though the opposite is drawn on the diagram. }   In every 
equilibrium the incumbent will
choose $R_2 $ in the second period if the market is $Large$ and $R_1
$ if it is $Small$, since there is no point in reducing profits once
the rival has made his decision, so this decision will be dropped
from the equilibrium description. Given this behavior, $In|R_2$ and
$Out|R_0$ will be dominant for the rival, since $R_2$ and $R_0$
definitely communicate the size of the market.

\pagebreak
\epsfysize=5in

\epsffile{jam1a.eps} 
 

\begin{verse}
 PROPOSITION 1: The four possible equilibria of the limit pricing
model are\\
  \hspace*{.2in} (E1) NONSTRATEGIC.  $R_2|Large$, $R_1|Small$,  $Out|R_0$, 
$Out|R_1$, $In|R_2$.\\
 \hspace*{.2in} (E2 ) PURE SIGNAL-JAMMING. $R_1|Large$, $R_1|Small$,
$Out|R_0$, $Out|R_1$, $In|R_2$.\\
 \hspace*{.2in} (E3) MIXED SIGNAL-JAMMING. ($ R_1|Small$, $
R_1|Large$ with probability $\alpha$, $ R_2|Large$ with probability
$(1-\alpha)$, $Out|R_0$,
 $In|R_1$ with probability
$\beta$, $Out|R_1$ with probability
$(1-\beta)$, $In|R_2$.\\
 \hspace*{.2in} (E4) SIGNALLING.  $R_0|Small$, $R_2|Large$, $Out|R_0$,  
$In|R_1$, $In|R_2$. 
 \end{verse}

 \noindent
PROOF: There are four equilibria to consider.   

\noindent 
 \hspace*{.2in} (E1) NONSTRATEGIC.  $R_2|Large$, $R_1|Small$,  $Out|R_0$, 
$Out|R_1$, $In|R_2$.\\
  The incumbent's equilibrium payoff in a large market is
$ \pi_I (R_2|Large ) = (M R_2-C) + (R_2-C),   
$ compared with the deviation payoff of 
$ \pi_I (R_1|Large) = (MR_1-C) + (MR_2-C)$.     
  The incumbent has no incentive to deviate if   
 $ \pi_I (R_2|Large ) - \pi_I (R_1|Large)= (1+M)R_2 - M(R_1+R_2)  \geq 0$,    
  which is equivalent to 
  \begin{equation} \label{e23}
  M     \leq \frac{R_2}{R_1}.
 \end{equation}    
 Inequality (\ref{e23}) is  a necessary   condition for the equilibrium to be  
nonstrategic.
   The rival will not deviate from equilibrium, because
the incumbent's choice fully reveals the type of market, and under the 
assumptions that $R_2-C >0$ and $R_1-C <0$, remaining in the market is only 
profitable if it is large. 



\noindent 
 \hspace*{.2in} (E2 ) PURE SIGNAL-JAMMING. $R_1|Large$, $R_1|Small$,
$Out|R_0$, $Out|R_1$, $In|R_2$.\\
  The rival's strategy is the
same as in E1, so the incumbent's optimal behavior remains the same:  for the 
 incumbent to choose $R_1$,  the converse of (\ref{e23}) must be true, and  
\begin{equation} \label{e24}
  M     \geq \frac{R_2}{R_1}.
 \end{equation}  
  If the rival  stays out, his second-period payoff is 0. If he  enters, its 
expected value is $
  \theta (R_1-C) + (1-\theta)(R_2-C)$.  
 Hence, he will follow the equilibrium behavior of staying out if  
   \begin{equation}\label{e26}
 \theta \geq \frac{R_2-C}{R_2-R_1}.
\end{equation}
 Conditions (\ref{e24}) and (\ref{e26}) are the necessary conditions for 
equilibrium E2. 


\noindent 
\hspace*{.2in} (E3) MIXED SIGNAL-JAMMING. ($ R_1|Small$, $
R_1|Large$ with probability $\alpha$, $ R_2|Large$ with probability
$(1-\alpha)$, $Out|R_0$,
 $In|R_1$ with probability
$\beta$, $Out|R_1$ with probability
$(1-\beta)$, $In|R_2$.\\
 If $
  M     >  \frac{R_2}{R_1}$
but $
  \theta  < \frac{R_2-C}{R_2-R_1}$,   neither E1 nor E2         remain  as 
equilibria.  If the incumbent played
$ R_2|Large$ and $R_1|Small$,   the rival would interpret $R_1$ 
as indicating a small market---an interpretation which would give the
incumbent incentive to play $R_1|Large$. But if the incumbent always
plays $ R_1$,  the rival would enter even after
observing $ R_1$, knowing there was a  high probability  that the
market   was really large. Hence, the equilibrium must be in mixed
strategies, which is  equilibrium E3, or the incumbent must convince the rival 
to stay out by playing $R_0$, which  
is  equilibrium E4. 

  For the rival to mix, he   must
be indifferent between the second-period payoffs of 
 $ \pi_E (In|R_1) =   \frac{\theta}{\theta + (1-\theta)\alpha} (R_1-C) + 
 \frac{ (1-\theta)\alpha}{\theta + (1-\theta)\alpha} (R_2-C) $
  and 
 $ \pi_E (Out|R_1) = 0$.
 Equating these two payoffs and solving for $\alpha$ gives 
 $ \alpha  =\left( \frac{\theta}{1-\theta} \right) \left( \frac{C-R_1 }{R_2-C }
\right),  
 $  which is always non-negative, but   avoids exceeding one
only if 
\begin{equation} \label{e29a}
  \theta \leq \frac{R_2-C}{R_2-R_1}, 
   \end{equation}
 a necessary condition for equilibrium E3.   
For
the incumbent to mix when the market is large, he must be indifferent
between
$ \pi_I (R_2|Large) =  (M R_2-C) + (R_2-C)
$ and
$ \pi_I (R_1|Large) = (M R_1-C)  +  \beta (R_2-C)  + (1-\beta)(M R_2-C).
$ Equating these two payoffs and solving for $\beta$ gives 
 $ \beta  = \frac{MR_1-R_2  }{(M-1) R_2}, 
$ which is   strictly less than one,  and which is non-negative
 if $M R_1-R_2  \geq 0$,  condition  (\ref{e24}). 
 
 If the market
is small, the incumbent's alternative payoffs are the equilibrium payoff of 
 $ \pi_I (R_1|Small) = (M R_1-C)  +  \beta (R_1-C)  + (1-\beta)(M R_1-C)$ and 
the deviation payoff of
$ \pi_I (R_0|Small) = (M R_0-C) + (MR_1-C). 
$ The difference is 
  \begin{equation} \label{e35}
   \pi_I (R_1|Small)-\pi_I (R_0|Small) = [MR_1 + \beta R_1 + (1-\beta)MR_1 ]    
- [M R_0 +MR_1]
  \end{equation}
 This difference is 
 non-negative under either of two conditions. It is non-negative if $R_0$ is 
small enough; that is, if 
 \begin{equation} \label{e35a}
 R_0 \leq R_1\left(  1- \frac{R_1}{R_2}   \right). 
 \end{equation}
 Even if inequality (\ref{e35a}) is false, the difference is nonnegative if 
 $M$ is no greater than some amount $Z^{-1}$ defined as follows: 
  \begin{equation} \label{e36}
 M \leq  \left(   \frac{R_1}{R_2} - 1 + \frac{R_0 }{ R_1}  \right)^{-1} = Z^{-
1}.
 \end{equation}

 Note that  if condition (\ref{e35a}) is false, then $ Z^{-1}  > \frac{R_2}{R_1} 
$, because $ Z  < \frac{R_1}{R_2} $ and  $Z >0$.\footnote{xxx I need to check 
this carefully.} Thus, we can draw region E3 as it is shown in Figure 1.  
  
   \hspace*{.2in} (E4) SIGNALLING.  $R_0|Small$, $R_2|Large$,  $Out|R_0$, 
$In|R_1$, $In|R_2$.\\
     It follows from the discussion of E3 that if condition
(\ref{e29a}) is true but (\ref{e36}) is replaced by its converse, 
    then the unique equilibrium is for the incumbent to choose
$R_0|Small$.  Out-of-equilibrium beliefs that support this are that
if the rival observes $R_1$, he believes the market is large with
probability $\frac{(1-\theta)\alpha}{\theta + (1-\theta) \alpha}$, as
in equilibrium E3. Greater values of $Prob(Large|R_1)$ also support
the equilibrium, including the passive conjecture of
$Prob(Large|R_1)=1-\theta$.

 The signalling equilibrium is also an equilibrium for other
parameter regions.  Let the out-of-equilibrium belief be
$Prob(Large|R_1) = 1$.  The equilibrium payoff is 
 $ \pi_I (R_0|Small) = (M R_0-C) + (M R_1-C)$   
 and the deviation payoff is 
 $ \pi_I (R_1|Small) = (M R_1-C) + (R_1 -C)$.  
 The signalling equilibrium remains an equilibrium so long as 
 \begin{equation} \label{e37c}
  M \geq  \frac{R_1}{R_0}.  
 \end{equation}
   There exist multiple equilibria for those parts of E1,
E2, and E3 that overlap with the area defined by (xxx).
 $\Box$ 
   
       
In E1, limit pricing would not work and is not even attempted. In
E2, it is always used  successfully: the incumbent
sacrifices   profits in period one to avoid revealing the   market's size.   In 
E3,   limit
pricing is sometimes used and sometimes successful. In E4, limit
pricing is used, but to signal that the market is small rather than to
conceal that it is large.  
   These are, of course, the same equilibria that arose in the
predatory pricing model, but the size of the parameter regions have changed. One 
difference is that in limit pricing, for given values of $R_0$, $R_1$, and $R_2$ 
there may be no values of $\theta$ and $M$ that allow mixed signal jamming to be 
an equilibrium; region E3 may not exist. In addition, nonstrategic behavior is 
more attractive than in Model I, because the condition for its optimality is now  
$M < R_2/R_1$ instead of the improbable  $M < 2- R_1/R_2$, which    required 
$M<2$.  The size of $M$ that makes signalling more attractive than mixed signal 
jamming has also changed, but it remains true that   the  attractiveness of 
mixed signal jamming increases in $R_2$ and decreases in $R_0$.  



       
Proposition 1 says that there are four ways a rational incumbent
might behave towards the rival, each appropriate to its own
circumstances: (E1) to make no attempt to deter entry, (E2) to
use signal-jamming, (E3) indifferently to use
signal-jamming or accommodate, and (E4) to accommodate if the market
is large and signal if the market is small.
Equilibria E2 and E3 are similar to the signal jamming in Fudenberg
\& Tirole (1986) and Tirole (1988, p. 443), in which the incumbent
conceals the size of the market by his action. In the original
signal-jamming models, the incumbent himself does not know the size
of the market. Here, he does, but the outcome is the same: the
rival is forced to rely on data distorted by the incumbent. The
incumbent's knowledge, however, makes possible equilibrium E4,
equivalent to the separating equilibria in Milgrom \& Roberts (1982a)
and Roberts (1986), which does not exist under symmetric information.
In E4, the incumbent reduces his profits not to conceal that the
market is large, but to reveal that it is small. He is signalling
that is he is not signal jamming.

 The parameter regions in which these different strategies apply are
defined by the monopoly profit premium and the probability of a small
market.  If the monopoly profit is small enough (region E1),
strategic behavior is not worth the cost, whatever the probability of
a small market. Even if the market is almost surely small, it is not
worth pretending so, 
 and the difference in profits between a small market and a
large   is unimportant.  The  condition that defines  E1 is $ M     \leq 2 - 
\frac{R_1}{R_2}$, so  the region can exist only if $M <2$ and the product is 
differentiated enough that monopoly profits are less than twice duopoly profits.    
  If monopoly profits are higher, then strategic behavior of various
kinds becomes profitable.  If the market is very probably small
(region E2), then pure signal jamming is profitable, because it is
not very difficult to persuade the rival to stay out.  This is true
even if acquiring a monopoly is extremely profitable, and the rival
knows it is profitable, because he still views a low price as a
reliable sign of a small market. 

 If a small market is less probable, however (regions E3 and
E4),   behavior becomes complicated.  If monopoly profit is
moderate (region E3),  the equilibrium is in mixed strategies
because unless the incumbent sometimes chooses a high price in a large
market,   a low price is not a credible indicator of a small market.
True small markets are simply not common enough, so the rival is
dubious and will sometimes enter even on observing a low price. If,
however, monopoly profits are very high (region E4), then when the
market is small it is so important to the incumbent to prevent entry that
he is willing to take the extreme action $R_0$.     

  
 The signalling equilibrium is special  because it can coexist with any of the 
three other equilibria  and   it requires careful specification of   out-of-
equilibrium beliefs.  The incumbent chooses $R_1$ in neither a small nor a  
large market, and for this to be an equilibrium, the rival must believe that any 
incumbent who did choose $R_1$ was likely operating in  large market.  This is 
most plausible in parameter region E4, where the alternative to signalling is 
mixed signal jamming.

 In other regions, the necessary out-of-equilibrium beliefs are less plausible, 
as the following two arguments show. 
 
   
 First, the signalling equilibrium is not robust to a small
probability that the players are confused over which of the multiple
equilibria is being played out.  The incumbent moves first and
prefers any of the other three equilibria to signalling except in
region E4, and if such confusion were possible he could take
advantage of it. Suppose, for example, that the parameters were
located at point $P_1$ in Figure 1: a low monopoly premium and a
large probability that the market is small. This is in region E1, so
one equilibrium is nonstrategic, with $R_2|Large$ and $R_1|Small$,
and another is signalling with $R_2|Large$, $R_0|Small$, and the
out-of-equilibrium belief that $Prob(Large|R_1) = 1$. Such an
out-of-equilibrium belief does not seem reasonable, because although
$R_1$ is out-of-equilibrium behavior in the signalling equilibrium,
it is equilibrium behavior in the nonstrategic equilibrium. If we
relax slightly the standard assumption that the identity of the
equilibrium to be played out is common knowledge, then if the rival
thinks the equilibrium is signalling but observes $R_1$, he should
wonder whether he and the incumbent might have conflicting notions of
which equilibrium is being played out. It is equilibrium behavior for
the incumbent to choose $R_1$ if the incumbent thinks the equilibrium
is nonstrategic and the market is small, so, by this reasoning, the
rival should believe that the market is indeed small on observing
$R_1$. This breaks the signalling equilibrium. The same reasoning
eliminates the signalling equilibrium in every other region except
E4: if the rival interprets an action that is an equilibrium action
in equilibrium Y as indicating that the equilibrium being played out
by the other player is indeed Y, then the incumbent can effectively
choose whichever equilibrium he prefers, and he will not choose the
signalling equilibrium except in region E4.
 
 Second, except in region E4 the signalling equilibrium is not robust
to a small probability that the incumbent behaves nonstrategically.
Assume that with probability $\eta_1$ the rival is informed of the
market size and the incumbent receives a indicator to that effect, but
he receives the same indicator with an additional probability $\eta_2$
when the rival is actually uninformed. Assume that $\eta_2$ is
small enough that on receiving the indicator, the incumbent will find it
optimal to behave nonstrategically. 
 The incumbent will then sometimes play $R_1|Small$ in equilibrium,
which rules out the signalling equilibrium in its pure-strategy form
in every parameter region. In regions E1 and E2, the small-market
incumbent will deviate to $R_1$, breaking the signalling equilibrium.
In regions E3 and E4, both types of incumbents would deviate to $R_1$
to some extent, generating mixed-strategy equilibria. In region E3
this is simple enough, since the mixed-strategy equilibrium is close
to the mixed signal-jamming described in Proposition 1. In region E4,
the equilibrium involves a small amount of mixing but is essentially
the same as the original signalling equilibrium.  If the indicator is
received, the incumbent will play $\{R_2|Large$, $R_1|Small\}$.  If
the indicator is not received, the incumbent's equilibrium strategy
is
 $\{R_0|Small, R_1|Large $
with probability $\mu$, $R_2|Large$ with probability $(1-\mu)\}$. The
rival's equilibrium strategy is $\{In|Large$, $Out|Small\}$ if he is
informed, and 
$\{Out|R_0$, $In|R_1$ with probability $ \beta$, $Out|R_1$ with
probability $(1-\beta)$, $In|R_2  \}$ if he is uninformed.
The
values of $\beta$ are the same as in the mixed signal-jamming
equilibrium above.\footnote{xxx If the rival sees  $R_1$, that could be because 
the incubment has gotten a mistaken indicator from Nature, and the true market 
is Small. Or i, it could be that the incumbent is uninfomred and bluffing. }

 
The value of $\mu$  must make 
the rival
indifferent about entering when $R_1$ is observed. $R_1$ results from
a small market and the indicator with probability  $\theta \eta_2$,
and from a deceptive large-market incumbent with probability
$(1-\theta) \mu$. The rival's payoff is therefore
  \begin{equation} \label{e110}
 \pi(In|R_1)  = \frac{\theta \eta_2}{\theta \eta_2 + (1-\theta) \mu}
(R_1-C) +   \frac{(1-\theta) \mu}{\theta \eta_2 + (1-\theta) \mu}
(R_2-C).   
 \end{equation}
 Equating this to zero, the payoff from $Out|R_1$, and solving for
$\mu$ gives
  \begin{equation} \label{e18d}
   \mu = \frac{ \theta \eta_2 (C- R_1)}{(1-\theta)(R_2-C)}.
       \end{equation}
  For given $\theta$, as $\eta_2$ goes to zero the
equilibrium becomes arbitrarily close to the pure signalling
equilibrium, because $\mu$ is small if $\eta_2$ is small,
and if $\mu$ is small then $R_1$ is rarely chosen and the value of
$\beta$ rarely is relevant. Thus, a small probability that the
incumbent is behaving nonstrategically eliminates the signalling
equilibrium except in E4, and changes the equilibrium only slightly
in E4 itself. 
   

  

\bigskip
   \noindent
 {\bf  3. Interpretation}

    The first  involves the strategic activity that generates low revenue. In 
both models, this can be interpreted as a low price, which reduces the revenue 
observed by the rival  (his own in predatory pricing and the incumbent's in 
limit pricing).    In   limit pricing,  another interpretation is that the 
strategic activity is low quality or advertising, which would also reduce the 
incumbent's revenue. This interpretation does not carry over  comfortably to 
predatory pricing, because  when both firms are in the market such activities 
may increase the rival's revenue at the same time as they reduce the 
incumbent's.\footnote{ 
 The same difference in interpretation exists, {\it a fortiori} between cost-
based and demand-based limit pricing models. In a cost-based model, the 
incumbent  uses  low prices  to indicate that he has low costs. If his low 
prices might be due to his own  low quality, that just increases the rival's 
incentive to enter.   In a demand-based model, the rival is interested in 
discovering  market demand,  something facing both firms, so    low revenue can 
deter entry whether it is generated by an inappropriately low price or   
inappropriately low quality. }    
  
  

   Second, in limit pricing the incumbent may {\it expand}
output, capacity, and price   in  the second period, after entry. That is 
because the
incumbent's pre-entry   action may have been devoted to keeping
all of these variables small to make the market look unattractive to
the rival. If the rival  discovers the true state of the market
after entry,
 the incumbent will give up   concealment and maximize profits
with abandon.  This  story, in fact, might  explain the finding
of Lieberman (1987) that   entry into
concentrated markets in the chemical industry  was followed by incumbent
expansion, unlike entry into unconcentrated markets.

Third,   the limit pricing model provides an  explanation for why monopolies 
might seem  not to maximize profits. It explains apparently irrational actions,   
as an entry deterrence tactic: the monopoly deliberately reduces its profits to 
make the market unattractive. If entry occurs anyway, it will appear that 
competition has forced the monopoly to become more efficient, but what has 
happened is that it no longer worries about showing that high profits can be 
earned in this market.       


\bigskip
\noindent
 {\bf Predation}



 
 
   The rival will not deviate from equilibrium, because the
incumbent's choice fully reveals the type of market, and under the
assumptions that $R_2-C >0$ and $R_1-C <0$, remaining in the market
is only profitable if the market is large. 

   As in other models of predatory pricing, the incumbent is engaging
in activities that make the market unprofitable to both the rival
and himself, activities which the incumbent would cease if he thought
the rival would not exit the market. Here, however, the incumbent
is not threatening the rival, but confusing or warning him.  In
signal jamming, predatory pricing is profitable precisely because the
rival does not know it is predatory. The incumbent does not need to
make a threat of low prices credible, because he avoids making any
threat, blaming low profits on the small size of the market. In
signalling, on the other hand, the essence of the low price is that
the rival knows it is a strategic signal indicating a small market.
In both equilibria, the incumbent's present price is chosen to
communicate something about the exogenous parameters, not something
about the incumbent's future behavior.  


   
This model of predation, unlike others,  predicts that the incumbent will 
predate against rivals  but rivals will not predate against the incumbent. In 
Telser's deep-purse model, it might well be the rival that has the deep purse--- 
one thinks of chain stores predating against small grocery stores. In  the 
Kreps-Wilson (1982) and Milgrom-Roberts (1982b) repeated-game models, the rival 
might pretend to be irrational or have low costs just as easily as the 
incumbent---or more easily, since the rival is less well-known. In the present 
model, it is key that one firm knows the market better than the other, and this  
is what makes it possible for  the incumbent to predate. 


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

 \bigskip
\noindent
 {\bf Signalling Properties   of the Model}    
  
    Ordinarily, signalling models have three kinds of equilibria:
separating equilibria in which only the desirable type signals,
pooling equilibria in which neither type signals, and pooling
equilibria in which both types signal. The desirable type prefers
separation, the undesirable type prefers pooling with no signalling,
and pooling with signalling is preferred by neither type. The pooling
equilibria are vulnerable to elimination by various refinements of
out-of-equilibrium beliefs because the desirable type has a strong
incentive to separate out. In the present model, there is a
separating equilibrium in which only the desirable type signals (E4),
but also a pooling equilibrium in which only the undesirable type
signals (E2). Moreover, both types of incumbent prefer pooling, when
it exists as an equilibrium. The small-market incumbent has no
incentive to separate out, because, thanks to the discreteness of
entry, being pooled with the large-market incumbent has no ill
consequences. Hence, the pooling equilibrium at $ R_1$ is robust to
out-of-equilibrium beliefs---more robust, in fact, than the
separating equilibrium. This is why it is closer to signal jamming
than to pooling in a standard signalling model.  And this is why the
arguments from small amounts of uncertainty over the equilibrium
being played out and nonstrategic behavior make the signalling
equilibrium implausible except in region E4. 
         

  It is natural to wonder whether allowing a continuum of signal
levels instead of just three would matter.  Suppose that if the
market is large, the incumbent chooses $R \in (R_0, R_2]$, and if the
market is small, the incumbent chooses $R \in [-\infty, R_1]$.  If $R
> R_1$, the rival deduces that the market is large. If $R \in (R_0,
R_1]$, it is not clear what deduction should be drawn unless the
level of $R$ chosen is the level prescribed by the equilibrium,
because out-of-equilibrium beliefs must be imposed by the modeller.
There exists a continuum of pooling and separating equilibria, each
enforced by the belief that the incumbent's deviation from the
assigned $R$ is a sign of a large market.  The model with three
levels of $R$ strips this down to the revenue levels whose special
properties do not depend on out-of-equilibrium beliefs: $R_0$ and
$R_2$ definitely indicate the size of the market, and $R_1$ is the
profit-maximizing revenue in the small market.\footnote{The second
refining principle described above also reduces the number of
relevant revenue levels to these three, because $R_1$ could be used
to indicate a small market if the equilibrium specified a smaller
value for $R$.} 


   
   It is also interesting to ask what happens when the model is
modified so that no 
 revenue level $R_0$ exists that unmistakably indicates a small
market---effectively,   $R_0 = -\infty$. No signalling equilibrium
then exists, because even if the monopoly premium is large, the
small-market incumbent will not attempt to reduce revenues to
separate from the large-market incumbent because the large-market
incumbent would be equally willing to reduce his profits in
imitation. Region E3 increases to include E4, and even if the
monopoly premium is large, signal jamming will occur.  
 
%---------------------------------------------------------------

%---------------------------------------------------------------
\bigskip
 \noindent
 {\bf The Literature}
 
The most discussed kinds  of entry deterrence   are predatory pricing and 
 limit pricing.  Both practices involve  a  firm using a low price to keep  
competitors   out of the market, but they differ in whether the  competitors are 
initially in the market or not.
  In predatory pricing, a firm sets its price low in order to induce
exit of an existing competitor.  The predator's problem is to make
credible its threat to keep the price low until the competitor exits,
because the low price hurts itself as well as the victim.  The threat
might be credible because the victim has limited financial resources
and cannot continue operating even though it knows that the predator
will soon raise prices again--- the ``deep purse'' theory of Telser
(1967).  Or, the incumbent might have a reputation to make or
maintain, as in the reputation models of Kreps \& Wilson (1982) and
Milgrom \& Roberts (1982b). The incumbent is willing to take losses
because it can thereby successfully pretend to either be irrational
or have low costs. In both stories, predation works by making an
otherwise profitable market temporarily unprofitable.  
  A third story can be based on the ``signal jamming'' model of
Fudenberg \& Tirole (1986).  In this model of symmetric but imperfect
information, an rival does not know whether it can operate
profitably or not, because it is ignorant of its own fixed
cost.\footnote{The assumption that the rival can observe its
marginal but not its fixed cost is unrealistic, but it is useful for
simplifying the model. If it were marginal cost that was unknown,
then the rival's information would affect the output he chose; see
Riordan (1985) for analysis of this effect.  } It enters and tries to
use its profit to deduce the fixed cost, but profit also depends on
the toughness with which the incumbent competes, which is 
 unobserved. The incumbent may   act as  a tough
competitor to induce the rival to exit under the belief that it is  high fixed 
costs,  not tough competition, that is responsible for
low profits.  The signal jamming  model does not turn on the issue of 
credibility, because the victim does not know whether the incumbent is purposely 
reducing industry profits or not, and cannot predict an increase in profits 
after exit.     
 
  
  In limit pricing, a firm purposely reduces its profits---most
simply by not allowing its price to rise above a certain limit--- in
order to deter entry by firms not yet active in the market. 
 The  seminal modern limit-pricing model is   Milgrom \&
Roberts (1982a), which explains limit pricing as signalling. The
incumbent firm has high   or low costs, known only to itself, and the rival does 
not wish to 
 enter and compete with a low-cost incumbent.  In the absence of
possible entry, the low-cost incumbent would charge a lower price
than the high-cost incumbent. But if entry is possible, the high-cost
incumbent may wish to pretend that it is low-cost by charging less
than the high-cost monopoly price. Or, if customers believe that
high-cost firms might charge low prices the low-cost incumbent may
need to reveal its identity by charging so low a price that imitation
is unprofitable. Either way, some type of incumbent is using limit
pricing.
 
  The    model  of this paper bases the incumbent's behavior   on the      
motive  of trying to persuade  the competitor, truthfully or deceitfully, that 
market demand is too weak for two firms to   survive.  The
rival and incumbent are identical except that the incumbent
is permanently in the market and knows the market size, but the
rival must make its entry and exit decisions in ignorance.  There
are no entrance or exit fees, and no cost differences. The
incumbent's tactic is to purposely depress profits, either to
prevent profits from indicating the size of the market to the
rival, a form of signal jamming, or to signal that the incumbent is
not signal jamming.  As in previous models, there are multiple
equilibria, but arguments were made that for given parameter
values the predicted equilibria should be unique (except for weak
equilibria at boundaries), and that pooling equilibria, not just
separating equilibria, should survive refinement.  
   Besides showing that limit pricing and predatory pricing can have
a common motivation, the model showed  how the monopoly premium and
rival beliefs influence whether the rival can be deterred, and
how the incumbent may be driven to signalling that he is not signal
jamming. Under some parameter values, the possibility of strategic
behavior will hurt the incumbent, so that the idea that entry
deterrence might be desirable to encourage innovation into entirely
new markets will be invalid.  
  
  The model is similar in different ways to both Fudenberg \&
Tirole (1986) and Milgrom \& Roberts (1982a), although those models
are driven by uncertainty over costs rather than demand.  Cost
uncertainty is not unrealistic, but often what is most uncertain
about a market is not individual firms' costs, but the common demand
curve they face.\footnote{There may also be uncertainty over common
components of cost; see Harrington (1986) for a signalling model in
which this is true. Entry deterrence can then take the form of
charging a high price, not a low price, to indicate that the common
costs are high.} Demand may be even more important than costs to the
profitability of entry.  Strategic accommodation can permit a firm
with higher costs to survive, but only in a large market can the
rival enter at so small a scale as to make retaliation unprofitable
for the incumbent, as Gelman \& Salop (1983) suggest.  
  Extensions of the Milgrom-Roberts (1982a) model to the case where
the demand curve, not the cost curve, is private information have
been made by Matthews \& Mirman (1983), Roberts (1986), and Bagwell
\& Ramey (1990).  In Matthews \& Mirman (1983), the strength of
demand is a continuous variable known to the incumbent but not the
potential rival, who must estimate it by observing the market
price. The market price, in turn, is based on a choice by the
incumbent plus random noise. In equilibrium, incumbents in bigger
markets choose higher prices, and the rival enters if the observed
price is higher than a threshold level. Roberts (1986) is a predation
model  in which the predator has better information on demand than the victim
and can   choose output to induce the victim to
believe that the market is small and exit. Under suitable
assumptions on out-of-equilibrium beliefs, the equilibrium is a
separating one in which the predator chooses a price lower than the
monopoly level if demand is weak, and the victim exits.  Bagwell \&
Ramey (1990) is a limit pricing model in which the incumbent has
superior information on demand and can use both price and advertising
levels to try to comunicate this to the rival. Refinements of
equilibrium are explored, and the conclusion is that strategic
behavior exaggerates the effects of demand differences. In general,
these model predict separating equilibria, not pooling equilibria.
The Fudenberg-Tirole signal-jamming model can also be extended to
demand, and Tirole (1986, p. 443) shows how it might be based on
differences in the general profitability of different markets when
the incumbent can manipulate observed profitability even without
knowing the true size of the market.  
 
   The entry deterrence tactic at the heart of the present model is 
 signal jamming in the sense that one type of incumbent takes a
costly action to block information that would reach the rival in
the absence of strategic behavior.  Fudenberg \& Tirole use the term
``signal jamming'' because the incumbent is trying to prevent
information from reaching the rival, rather than to communicate
information, as in standard signalling.  One type of incumbent has
that same motivation in the pooling equilibrium of a signalling
model, but Fudenberg and Tirole use a model of symmetric information,
where the incumbent's action does not depend on type. 
  The present model has incomplete information, and signal jamming
will amount to one type taking a costly action to pool with another
type.  As will be seen, the properties of such pooling are closer to
symmetric-information signal-jamming than to the pooling equilibrium
of educational signalling, in which either zero or all types take a
costly action.\footnote{ 
 Some people prefer to use ``signal jamming,'' to describe
symmetric-information models or models in which the signal observed
by the rival is a noisy function of the incumbent's behavior,
instead of the deterministic function here. The use of the term in
this paper emphasizes the intentional and costly blocking of
information rather than symmetry of information or the presence of
noise.} Signalling will also occur in the model, however, because the
incumbent sometimes wishes to signal that it is not signal jamming.
This feature of the model will be closer to the results of Milgrom
and Roberts than to Fudenberg and Tirole. 



\bigskip
\noindent
{\bf 4. Model II : Innovative Entry and Limit Pricing } 

  We usually consider strategic entry deterrence a bad thing,
resulting in monopolies where there would otherwise be competitive
markets, but the choice might actually be between a monopolized
market and no provision of the good at all. It may be that the
incumbent has innovated by discovering a profitable market niche and
the rival is trying to seize some of the profits. The present model
focusses on small markets, which might be able to contain only one
firm, so it seems especially appropriate for looking at innovative
markets: 
  small monopolies that sell innovative products or sell in
geographically restricted markets. This is a model of a doctor in a
small town, not an automaker in a large country.\footnote{For an
analysis of prices in such markets, see Bresnahan \& Reiss (1991) on
concentration in small-town markets for services such as auto dealing
and veterinary medicine.} Moreover, small markets are a natural
setting for information-based models, because information acquisition
is subject to economies of scale and may be prohibitively costly for 
  a small market. Thus, the 
 implicit assumption
that   uninformed players will not simply buy  the  information they need is  
plausible. 

 It is a general feature of innovative markets that monopoly profits
may be a socially desirable spur to entry, since a monopoly is better
than no seller at all. This, of course, is the rationale behind
patents, and Hausman \& Mackie-Mason (1988) point out, for example,
that policy should encourage price discrimination in innovative
markets to encourage entry. Could limit pricing be useful in the same
way?  It prevents the rival from free-riding on the incumbent's
costly acquisition of the information that the market is not tiny,
acquisition which may be costly either because of a fixed cost of
research or because of the risk of failed entry. Model III will
investigate whether monopoly-facilitating practices do encourage
innovation in the present context.  It extends Model II to the
incumbent's original decision of whether to become the first firm in
the market, given that the market might be too small to generate
positive profits even for a monopoly.  The market will now be tiny
(profitable under no circumstances), small (profitable for one firm),
or large (profitable for two firms).  Either limit pricing or signal
jamming could be used; the model below assumes that the rival can
observe incumbent profits, so limit pricing is the relevant tactic.
Let   revenue in the tiny market be $R_{00}< R_0$, where $M R_{00}
-C <0$. 


\noindent
 {\bf The Order of Play in Model II }

 \begin{enumerate}
 \item
  Nature chooses the market to be $Tiny$ with
probability $\gamma$, $Small$ with probability $(1-\gamma)\theta$ and
$Large$ with probability $(1-\gamma)(1-\theta)$, observed by neither
player. 
  \item
 The incumbent decides whether to $Stay\; Out$, ending the game, or $Enter$.   
  \hspace*{.2in}   The incumbent observes the market size and chooses first-
period revenue to be $R_{00}$ if the market is tiny, $R_0$ or $R_1$ if it is 
small,  and $R_1$ or $R_2$ if it is large,  observed by
the rival.\\
  \item
The incumbent chooses   $Exit $, ending the game,  or $Stay \;In$. 
  \item
The rival decides whether to be  $In $ or $  Out$ for the second period.   \item
The incumbent chooses revenue to be $R_{00}$ if
the market is tiny, $R_0$ or $R_1$ if it is small, and $R_1$ or $R_2$
if it is large, observed by the rival. 
  \item
   The incumbent and rival collect their second-period profits,
which equal $R-C$ apiece if both are in the market, and $M R-C$ and 0
if the incumbent is alone. 
 \end{enumerate}

     Let us denote by $\pi $ the incumbent's equilibrium profits
in the ensuing subgame if the market is not tiny and he remains in
after entering. 
   The incumbent's payoff for the entire game is   either
 \begin{equation} \label{e38}
 \pi_{Stay \; Out} = 0
 \end{equation}
 or 
 \begin{equation} \label{e39}
 \pi{Enter} = \gamma (R_{00}-C) + (1-\gamma) \pi .
 \end{equation}
 Only if $\gamma$ falls below a certain critical level $\gamma^*$
will the incumbent be willing to enter.  $\gamma^*$ is found by
equating (\ref{e38}) and (\ref{e39}):
 \begin{equation} \label{e40}
 \gamma^* =\frac{ \pi }{  \pi  - (R_{00}-C) }.
 \end{equation}
If subgame profits increase, the critical level falls and the
incumbent is willing to enter   markets  that have  a higher probability of 
being tiny:  
 \begin{equation} \label{e41}
 \frac{\partial \gamma^*}{\partial \pi} = -\frac{ R_{00}-C }{  (\pi - (R_{00}-C) 
)^2}>0,
 \end{equation}
 where the inequality is true because $ R_{00}-C<0$. Hence, the question of 
whether strategic entry deterrence encourages innovation is the same as the 
question of how it affects $\pi$.  Proposition 3 compares the incentive of the 
incumbent to
enter when limit pricing is possible compared to when the rival can
discover the true state of the market regardless of the incumbent's
actions. 

\noindent
{\it  PROPOSITION 2:  Pure signal jamming encourages innovation, but mixed 
signal jamming or signalling discourages  it.}

  
\noindent
 PROOF:  
 If    the
subgame equilibrium is pure signal-jamming, the subgame profit is
 \begin{equation} \label{e42}
  \pi (LP) =  (M R_1- C)  +  \theta (M R_1-C) + (1-\theta) (M R_2-C),
       \end{equation}
 whereas if 
the rival  could observe the market's profitability directly, the
incumbent's subgame profit   would be 
 \begin{equation} \label{e43}
  \pi (no \;LP) =  \theta (M R_1-C)   + (1-\theta) (M R_2-C) +  \theta
 (M R_1-C) + (1-\theta)(R_2-C).
 \end{equation}
 The difference is 
 \begin{equation} \label{e44}
  \pi( LP) - \pi (no \;L P) =  (1-\theta)(M R_1 - R_2) \geq 0, 
 \end{equation}
  where the inequality follows from   condition (\ref{e24}), which holds 
whenever pure signal-jamming is an    equilibrium.
   Since $\frac{\partial
\gamma^*}{\partial \pi}>0$, there are thus values of $\gamma$
for which the difference between these two profits makes the
difference as to whether the incumbent enters, and limit pricing encourages
the incumbent's initial entry.

   If   the equilibrium is mixed signal-jamming,
the incumbent's two pure-strategy payoffs are equal, so we can use
either one to represent the limit-pricing subgame profit.
The overall subgame payoff   across both sizes of markets, using the payoff from 
$R_2|Large$, is 
 \begin{equation} \label{e45}
  \begin{array}{ll}
 \pi (LP) = & [\theta (M R_1-C ) + (1-\theta) (M R_2-C)] +[
\theta \beta (R_1-C)+ \\
  &
\theta (1-\beta) (M R_1-C) + (1-\theta) (R_2 -C ], 
 \end{array}
 \end{equation}
 whereas if the rival can observe the market's profitability
directly, the incumbent's subgame profits are 
as shown in equation (\ref{e43}).
 The difference is 
 \begin{equation} \label{e45a}
   \pi(LP) - \pi (no \;L P) = -\theta \beta (M-1)R_1,
    \end{equation} 
  which is negative. Limit pricing hurts the
incumbent's profits and deters his entry.  Similarly, if the equilibrium is 
signalling,  the
incumbent's profits   are
 \begin{equation} \label{e46}
 \pi (LP) = [\theta (M R_0-C ) + (1-\theta) (M R_2-C)]   +
 [\theta(M R_1-C) + (1-\theta) (R_2-C)]. 
 \end{equation}
 The difference between this and the profit under full information is
 \begin{equation} \label{e47}
   \pi( LP) - \pi (no \;L P) = -\theta M (R_1-R_0),
\end{equation}
 which is
negative. Under  signalling, the possibility of limit pricing
  hurts the incumbent's profits and deters his initial entry.$\Box$

 
  Thus, to the well-known idea that monopoly-facilitating tactics can
stimulate innovation by increasing profits is added a new idea: the
same tactics can discourage innovation by reducing profits, because the
rival is suspicious and makes mistakes.  Under mixed signal
jamming, he knows that the incumbent often is pretending that a large
market is small, so he enters randomly--- sometimes into a small
market, driving the incumbent's profit negative. Under signalling,
the underlying problem is still mistaken entry, but it has become so
costly that costly signalling is the preferred response.  If it were
common knowledge that the market were small or that limit pricing was
not being carried out, on the other hand, the worst the incumbent
could do would be a small positive profit. The incumbent
would like to be able to commit not to manipulate revenue, since
effective communication of the market size increases his profits on
average.

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

\bigskip
\noindent
{\bf 5.  Conclusion}



  An incumbent firm can use low prices to communicate information
about the size of the market in several different ways, ways that
apply whether its competitor is already in the market (predatory
pricing) or has not yet entered (limit pricing).  Milgrom \& Roberts
(1982a) and its successors suggest that the incumbent might use low
prices in separating equilibria to credibly indicate that the market
is unprofitable and deter entry that would hurt both firms, or use
moderate prices in a pooling equilibrium to cloud the market's
profitability.  Signal-jamming models in the tradition of Fudenberg
\& Tirole (1986) show that the incumbent might use low prices in pure
or mixed-strategy pooling equilibria to similarly obscure the
profitability of the market and perhaps deter entry, but without
out-of-equilibrium beliefs being relevant. The model here, based on
one firm's uncertainty over whether market demand is sufficiently
strong to accommodate two firms, combines the two ideas. If it is
required that the equilibrium be robust to uncertainty over which
equilibrium is being played out or to the possibility of nonstrategic
behavior, then the equilibrium is unique for given parameters, and it
may be a pooling equilibrium that survives. If the prior belief is
that the market is small, or if the premium from being a monopoly
takes a low value, the incumbent will use signal jamming to prevent
the rival from learning the true state of the market.
Mixed-strategy signal jamming is costly, however, since it sometimes
results in two firms mistakenly occupying a small market.  Therefore,
if the monopoly premium and the prior probability of a large market
are big enough, the incumbent will resort to true signalling
reminiscent of the separating equilibrium in Milgrom \& Roberts
(1982a): reducing profits to a level so low that it is clear the
market must be small. This is defensive signalling: signalling that
the incumbent is not signal jamming.  
 
 The model applies to small markets, where there is a strong
possibility that the minimum efficient scale will only allow one firm
to operate profitably. This suggests that the possibility of
strategic behavior would influence whether even one firm dares enter
the market.  Entry into a virgin market is a form of innovation, and
like other kinds of unpatentable discoveries, the discovery of a new
market is prone to free-riding by other firms. One might think that
the possibility of strategic behavior would act like a patent and
eliminate the free-riding problem at some small cost by allowing the
incumbent to monopolize the new market. When the prior probability
that the new market can contain only one firm profitably is high,
this is indeed the case, and the pure signal jamming that results
encourages innovation. When the prior is low, however, strategic
behavior is costly compared to honest disclosure of the market size,
and both signalling and mixed signal jamming reduce innovation.  
   
 
%---------------------------------------------------------------

\newpage
\noindent
 {\bf References}

 Bagwell, Kyle and Garey Ramey (1990) ``Advertising and Pricing to
Deter or Accommodate Entry when Demand is Unknown,''{\it
International Journal of Industrial Organization}. 8: 93-113.

 

Bresnahan, Timothy \& Peter Reiss (1991) ``Entry and Competition in Concentrated 
Markets,'' 
  {\it Journal of Political Economy.} 99: 977-1009.   


 Cho, In-Koo \& David Kreps (1987) ``Signalling Games and Stable
Equilibria,'' {\it Quarterly Journal of Economics.}    102, 2:
179-221.

Fudenberg, Drew and Jean Tirole (1986) ``A Signal-Jamming Theory of
Predation,'' {\it RAND Journal of Economics}. 17: 366-76.

 Gelman, J. and S. Salop (1983) ``Judo Economics: Capacity Limitation
and Coupon Competition,'' {\it Bell Journal of Economics}. 14: 315-25.



Harrington, Joseph (1986) ``Limit Pricing When the Potential rival
is Unsure of its Cost Function,'' {\it Econometrica}.  54:429-437.
 
 Hausman, J. \& J. Mackie-Mason (1988) ``Price Discrimination and
Patent Policy,'' {\it RAND Journal of Economics}  19:253-265.

Kreps, David \& Robert
Wilson  (1982) ``Reputation and Imperfect Information,''  {\it
Journal of Economic Theory.}  27, 2: 253-79.


Lieberman, Marvin (1987) ``Post-Entry Investment and Market Structure
in the Chemical Processing Industry,'' {\it RAND Journal of Economics}.
18: 533-549.

Matthews, S. and L. Mirman (1983) ``Equilibrium Limit Pricing: The
Effects of Private Information and Stochastic Demand,'' {\it
Econometrica}, 51: 981-95.

Milgrom, Paul and John Roberts (1982a) ``Limit Pricing and Entry Under
Incomplete Information: An Equilibrium Analysis,'' {\it
Econometrica}. 50:443-60.


 Milgrom, Paul  and
John Roberts (1982b) ``Predation, Reputation, and Entry Deterrence,''
{\it Journal of Economic Theory.}     27, 2: 280-312. 


  Riordan, Michael (1985) ``Imperfect Information and Dynamic
Conjectural Variations.'' {\it RAND Journal of Economics}, 15: 229-43.
 
Roberts, John (1986), ``A Signalling Model of Predatory Pricing,''
{\it Oxford Economic Papers} (supp). 38 (N.S). 75-93.

Telser, Lester (1966) ``Cutthroat Competition and the Long
Purse,'' {\it Journal of Law and Economics.}    9: 259-77.

Tirole, Jean (1988) {\it The Theory of Industrial Organization},
Cambridge, Mass: The MIT Press, 1988.


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