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  \noindent
 September 6, 1999.   January 18, 2000. Kempered. August 6, 2003.  \\
   Eric Rasmusen, Erasmuse@indiana.edu.  Web:
Mypage.iu.edu/$\sim$erasmuse.
  
 
 
\section*{ *15 Entry  }   

 \bigskip
\noindent
 {\bf *15.1 Innovation and Patent Races} 

 
 
\noindent
   How do  firms come to enter particular industries?    Of the many
potential products that might be produced, firms choose a    small
number, and each product is only produced by a few firms.   Most
potential firms choose to remain potential, not actual.     Information
and strategic behavior are especially important in borderline industries
in which only  one or  two firms are active in production.
 
 This chapter begins with a discussion of  innovation with the
complications  of imitation by other firms and patent protection by the
government.  Section 15.2 looks at  a different way to enter a market:
by purchasing an existing firm, something that also provides help
against moral hazard on the part of company executives.  Section 15.3
analyzes a more traditional form of entry deterrence, predatory pricing,
using a Gang of Four model of a repeated game under incomplete
information. Section 15.4 returns to a simpler  model of predatory
pricing, but shows how the ability of the incumbent to credibly engage
in a price war can actually backfire by inducing entry for buyout.





\bigskip
\noindent
{\bf Market Power as a Precursor of Innovation}

\noindent
 Market power is not always inimical to social welfare.
Although restrictive monopoly output is inefficient,
the profits it generates encourage innovation, an
important source of both additional market power and
economic growth. The importance of innovation,
however, is diminished because of imitation, which can
so severely diminish its rewards as to entirely prevent 
it.  An innovator generally incurs some research cost,
but a discovery instantly imitated  can  yield  zero net
revenues. Table 15.1 shows how the payoffs look if the firm
that innovates incurs a cost of 1 but imitation is
costless and results in Bertrand competition.
Innovation is a dominated strategy. 

\begin{center}
{\bf Table 15.1} Imitation with Bertrand pricing 

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Brydox}\\
  &       &     &  {\it  Innovate}    &   &  $Imitate$      \\
  &   &  $ Innovate$       &    -1,-1 &    &  $-1,0$ \\
 & {\bf Apex} && & &   \\
&  &    {\it  Imitate}     &      $0,-1$  &    &{\bf 0,0} \\  
\multicolumn{6}{l}{\it Payoffs to: (Apex, Brydox)}
\end{tabular}                
\end{center}


   Under different assumptions, innovation   occurs even with
costless imitation.  The  key is whether duopoly profits are high enough
for one firm to recoup the entire costs of innovation.  If they are, the
payoffs are as shown in table 15.2, a version of
  Chicken. Although the firm that innovates pays the
entire cost and keeps only half the benefit, imitation is not   
dominant. Apex imitates if Brydox innovates, but not if Brydox
imitates.  If Apex could move first, it would bind itself not to
innovate, perhaps by disbanding  its research laboratory.

\begin{center}
{\bf Table 15.2} Imitation with profits in the product market 

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Brydox}\\
  &       &             &  {\it  Innovate}    &   &  $ Imitate$       \\
  &   &  $ Innovate$       &     1,1 &    & {\bf 1,2} \\
 & {\bf Apex} && & &   \\
&  &    {\it Imitate }      &  {\bf 2,1}   &    & 0,0 \\  
\multicolumn{6}{l}{\it Payoffs to: (Apex, Brydox)}
\end{tabular}                
\end{center}


   Without a first-mover advantage, the game has two pure strategy
 Nash  equilibria, $(Innovate, Imitate)$ and $(Imitate, Innovate)$, and
a symmetric   equilibrium in mixed strategies in which each firm
innovates with probability 0.5. The mixed-strategy equilibrium is
inefficient, since sometimes both firms innovate and sometimes
neither.

History might provide a focal point or explain why one player moves
first.  Japan was for many years incapable of doing basic scientific
research, and does relatively little even today. The United States
therefore had to innovate rather than imitate in the past, and today
continues to do much more basic research.

  Much of the literature on innovation compares the relative merits
of monopoly and competition.  One reason a monopoly might innovate
more is because it can capture more of the benefits, capturing the
entire benefit if perfect price discrimination is possible
(otherwise, some of the benefit goes to consumers).  In addition, the
monopoly avoids a second inefficiency: entrants innovating solely to
steal the old innovator's rents without much increasing consumer
surplus. The welfare aspects of innovation theory --  indeed, all
aspects -- are intricate, and the interested reader is referred to
the surveys by Kamien \& Schwartz (1982) and Reinganum (1989).

 \bigskip
  \noindent
  {\bf Patent Races}

\noindent
 One way that governments respond to imitation is by issuing patents:
exclusive rights to make, use, or sell an innovation.  If a firm
patents its discovery, other firms cannot imitate, or even  use the
discovery if the make it
 independently.  Research effort therefore has a
discontinuous payoff: if the researcher is the first to make a
discovery, he receives the patent; if he is second, nothing.  Patent
races are  examples of the tournaments discussed in section 8.2 except
that  if no player exerts any effort, none of them will get the
reward.  Patents are also special
because they lose their value if consumers find a substitute and stop
buying the patented product.  Moreover, the effort in tournaments is
usually exerted over a fixed time period, whereas research usually has
an endogenous time period, ending when the discovery is made.
Because of this endogeneity, we call the competition a {\bf patent
race.}

   We will consider two models of patents. On the technical side, the
first
model shows how to derive a continuous mixed strategies probability
distribution, instead of just the single number derived in chapter 3. On
the substantive side, it shows how patent races lead to
inefficiency.

\begin{center}
 {\bf    Patent Race for a New Market }
  \end{center}
 {\bf Players}\\
  Three identical firms, Apex, Brydox, and Central.

  
\noindent
 {\bf The Order of Play }\\
 Each firm simultaneously chooses research spending $x_i \geq 0$, $
(i = a,b,c)$.

 \noindent
 {\bf Payoffs}\\ 
 Firms are risk neutral and the discount rate is zero. Innovation
occurs at time $T(x_i)$ where $T' <0$.  The value of the patent is
$V$, and if several players innovate simultaneously they share its
value. 

 \begin{tabular}{ll}
  $ \pi_i =$& $\left\{
 \begin{tabular}{lll}
 $V-x_i$ & if $T(x_i) < T(x_j),\;\; (\forall j \neq i)$ & (Firm $i$
gets the patent)\\
 & & \\
 $\frac{V}{1+m} - x_i$ & if $T(x_i) = T(x_k),\;\; $ & (Firm $i$
shares the patent with\\
 & & $m=1$ or 2 other firms)\\
 & & \\
 $- x_i$ & if $T(x_i) > T(x_j)$ {\rm for some $j$} & (Firm $i$ does
not get the patent) \\
  \end{tabular}
\right.$ 
\end{tabular}\\

  The game does not have any pure strategy Nash equilibria, because
the payoff functions are discontinuous. A slight difference in
research by one player can make a big difference in the payoffs, as
shown in figure 15.1 on the next page for fixed values of $x_b$ and
$x_c$. The
research levels shown in figure 15.1 are not equilibrium values.  If
Apex chose any research level $x_a$ less than $V$, Brydox would
respond with $x_a + \varepsilon$ and win the patent. If Apex chose
$x_a = V$, then Brydox and Central would respond with $x_b = 0$ and
$x_c = 0$, which would make Apex want to switch to $x_a =
\varepsilon.$

\begin{center}
 {\bf Figure 15.1} The payoffs in Patent Race for a New Market
\end{center}


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   There does exist a symmetric mixed strategy
equilibrium.   We will derive    $M_i(x)$, the cumulative
density function for the equilibrium mixed strategy,
rather than the    density function  itself.     The
probability with which firm $i$ chooses a research level
less than or equal to $x$ will be   $M_i(x)$. In a mixed-strategy
equilibrium a player is indifferent between
any of the pure strategies among which he is mixing.  Since
we know that the pure strategies $x_a=0$ and $x_a= V$
yield zero payoffs,   if Apex mixes over the support
$[0,V]$ then the expected payoff for every strategy
mixed between must also equal zero.  The expected payoff
from the pure strategy $x_a$ is the expected value of
winning minus the cost of research. Letting $x$ stand for
nonrandom and $X$ for random variables, this is 
  \begin{equation} \label{e9}
 V \cdot Pr (x_a \geq X_b, x_a \geq X_c) - x_a = 0,
 \end{equation}
 which can be rewritten as
  \begin{equation} \label{e10}
 V \cdot Pr (X_b \leq x_a) Pr(X_c \leq x_a) - x_a = 0,
 \end{equation}
  or
 \begin{equation} \label{e14.11}
 V \cdot M_b(x_a)M_c(x_a) - x_a = 0.
 \end{equation}
 We can rearrange equation (15.\ref{e14.11}) to obtain
 \begin{equation} \label{e12}
 M_b(x_a)M_c(x_a) =\frac{ x_a}{V}.
 \end{equation}
  If all three firms choose the same mixing distribution $M$, then 
     \begin{equation} \label{e13}
 M(x) = \left( \frac{x}{V} \right)^{1/2} \;{\rm for}\; 0 \leq x \leq V.
  \end{equation}
    
  What is noteworthy about a patent race is not the nonexistence of a
pure strategy equilibrium  but the overexpenditure on research. All
three players have expected payoffs of zero, because the patent value
$V$ is completely dissipated in the race.  As in Brecht's {\it
Threepenny Opera}, ``When all race after happiness/Happiness comes in
last.''\footnote{  Act III, scene 7 of the {\it
Threepenny Opera}, translated by John Willett (Berthold Brecht, {\it
Collected Works}, London: Eyre Methuen (1987).}  To be sure, the
innovation is made earlier
than it would have been by a monopolist, but hurrying the innovation
is not worth the cost, from society's point of view, a result that
would persist even if the discount rate were positive.  The patent
race is an example of {\bf rent seeking} (see Posner [1975] and
Tullock [1967]), in which players dissipate the value of monopoly
rents in the struggle to acquire them. Indeed, Rogerson (1982) uses a
game very similar to ``Patent Race for a New
Market'' to analyze competition for a government monopoly franchise. 

 The second patent race we will analyze is asymmetric because one
player is an incumbent and the other an entrant.  The aim  is to
discover which firm spends more and to explain why firms acquire
valuable patents they do not use.  A typical story of a sleeping
innovation (though not in this case patented)  is  the story of
synthetic
caviar. In 1976, the American Romanoff Caviar Company said that it had
developed
synthetic caviar as a ``defensive marketing weapon'' which  it would
not introduce in the US unless the Soviet Union introduced  
synthetic caviar  first.  The new product
would sell for one quarter of the old price,  and {\it Business Week}
said that the reason Romanoff did not introduce it was to avoid
cannibalizing its old market ({\it Business Week}, June 28, 1976, p.
51). The game theoretic aspects of this situation put the claims of
all the  players in doubt, but  its dubiousness makes it all the
more typical of sleeping patent stories.

 The best-known   
model  of sleeping patents  is Gilbert \&
Newbery (1982). In that model,  the incumbent firm does research and
acquires
a sleeping patent, while the entrant does no research. We will look
at a slightly more complicated model which does not reach such an
extreme result.


 \begin{center}
{\bf   Patent Race for an Old Market  }
\end{center}
 {\bf Players}\\
  An incumbent and an entrant.

 

\noindent
 {\bf The Order of Play }\\
1 The firms simultaneously choose research spending $x_i$ and
$x_e$, which result in research achievements $f(x_i)$ and $f(x_e)$,
where $ f' > 0$ and $f''<0$.\\
2 Nature chooses which player wins the patent using a function $g$
that maps the difference in  research achievements to a probability
between zero and one.
 \begin{equation}\label{e14.14}
  Prob{\rm (incumbent \;wins\; patent)} = g[f(x_i) - f(x_e)],
\end{equation}
 where $ g' > 0,\; g(0) = 0.5$, and $0 \leq g \leq 1.$\\
3 The winner of the patent decides whether to spend $Z$ to
implement it.
 

\noindent
 {\bf Payoffs}\\
 The old patent yields revenue $y$ and the new patent yields $v$. The
payoffs are shown in table 15.3.

\begin{center}
{\bf Table 15.3} The payoffs in Patent Race for an Old Market 

 \begin{tabular}{ lll }
\hline
 {\bf Outcome } &      $\boldmath{ \pi_{incumbent}}$ &     $\boldmath{
\pi_{entrant}}$ \\
 \hline
 &  & \\
  The entrant wins and implements & $-x_i$ & $v-x_e - Z$\\
 &  & \\
The  incumbent wins and implements & $v-x_i - Z$ & $ -x_e  $\\
&  & \\
  Neither player implements & $y-x_i$ & $ -x_e  $\\
 & & \\
       \hline
\end{tabular}                
\end{center}

Equation (15.\ref{e14.14}) specifies the   function $g[f(x_i) - f(x_e)]$
to capture the three  ideas of (a) diminishing returns to inputs,  (b)
rivalry, and (c) winning a patent race as a probability. The $f(x)$
function  represents dimishing returns because $f$ increases at a
decreasing rate in the input $x$. Using the difference between   $f(x)$
for each firm  makes it relative effort which matters.  The $g(\cdot)$
function  turns this measure of relative effective input into a
probability between zero and one.
  


  The entrant will do no research unless he plans to implement, so we
will disregard the  strongly dominated strategy, ($x_e > 0$, $ no\;\;
implementation$).  The incumbent wins with probability $g$ and the
entrant with probability $1-g$, so from table 15.3 the expected
payoff functions are 
    \begin{equation}\label{e15}
 \pi_{incumbent} =(1- g[ f(x_i) - f(x_e)]) (- x_i) + g[ f(x_i) - f(x_e)]
Max
\{  v - x_i - Z, y - x_i\}
\end{equation}
 and
 \begin{equation}\label{e16}
 \pi_{entrant} =(1- g[ f(x_i) - f(x_e)])(v- x_e- Z) + g[ f(x_i) -
f(x_e)]
(-x_e).
 \end{equation}
  On differentiating and letting $f_i$ and $f_e$ denote $ f(x_i)$ and $
f(x_e)$ we obtain the
first order conditions 
 \begin{equation}\label{e17} 
 \frac{d \pi_i}{dx_i} = -(1- g[ f_i - f_e]) - g' f'_i ( - x_i) + g'
f'_i Max \{ v - x_i - Z, y - x_i\} - g[ f_i - f_e] = 0 
  \end{equation}
  and
 \begin{equation}\label{e18}
 \frac{ d\pi_e}{dx_e} =-(1 - g[ f_i - f_e]) + g' f'_e(v- x_e- Z) -
g[f_i - f_e] + g' f'_e x_e = 0.
 \end{equation}
 Equating (15.\ref{e17}) and (15.\ref{e18}), which both equal zero, we
obtain
 \begin{equation}\label{e19}
 -(1-g) - g'f_i'x_i + g'f_i' Max\{ v - x_i - Z, y - x_i \} -g 
 = -(1-g) + g'f_e'(v-x_e - Z) - g + g'f_e'x_e,
 \end{equation}
   which simplifies to
 \begin{equation}\label{e20}
  f_i'[ x_i +  Max\{ v - x_i - Z, y - x_i \}] = 
 f_e'[v-x_e - Z  + x_e],
\end{equation}
 or
 \begin{equation}\label{e21}
 \frac{ f_i'}{f_e'} = \frac{ v- Z }{ Max \{ v - Z, y \} }.
 \end{equation}
 We can use equation (15.\ref{e21}) to show that different parameters
generate two qualitatively different outcomes.

\noindent
  {\bf Outcome 1}.
  {\it The entrant and incumbent spend equal amounts, and each
implements if successful.}\\
   This happens if there is a big gain from
patent implementation, that is, if
 \begin{equation}\label{e22}
 v -Z \geq y,
  \end{equation}
 so that equation (15.\ref{e21}) becomes 
 \begin{equation}\label{e22a}
 \frac{ f_i'}{f_e'} = \frac{ v- Z }{  v- Z  } = 1, 
 \end{equation}
 which implies  that $x_i = x_e$.

\noindent 
 {\bf Outcome 2.}\
  {\it The incumbent spends more and does not implement if he is
successful (he acquires a sleeping patent).}\\
  This happens if the gain
from implementation is small, that is, if
 \begin{equation}\label{e24}
 v -Z < y, 
 \end{equation}
 so that equation (15.\ref{e21}) becomes
 \begin{equation}\label{e25}
 \frac{ f_i'}{f_e'} = \frac{ v- Z }{  y  } < 1, 
 \end{equation}
 which implies that $f_i' < f_e'$. Since we assumed that $f''< 0$,
$f'$ is decreasing in $x$, and it follows that $x_i > x_e$.

     This model shows that the presence of another player can
stimulate the incumbent to do research he otherwise would not, and
that he may or may not implement the discovery. The incumbent has at
least as much incentive for research as the entrant  because a large
part of a successful entrant's payoff comes at the incumbent's
expense. The benefit to the incumbent is the maximum of the benefit
from implementing and the benefit from stopping the entrant, but the
entrant's benefit can only come from implementing. Contrary to the
popular belief that sleeping patents are bad, here they can help
society by eliminating wasteful implementation.




 \vspace*{1in}
\noindent
 {\bf  *15.2 Takeovers and Greenmail} 

\noindent
{\bf The Free Rider Problem }
 
\noindent
  Game theory is well suited to modelling takeovers because the
takeover process depends crucially on information and includes a
number of sharply delineated actions and events.  Suppose that under
its current mismanagement, a firm has a value per share of $v$, but
no shareholder has enough shares to justify the expense of a proxy
fight to throw out the current managers, although doing so would
raise the value to $(v+x)$.  An outside bidder makes a tender offer
conditional upon obtaining a
majority.  Any bid $p$ between $v$ and $(v+x)$ can make both the
bidder and the shareholders better off. But do the shareholders
accept such an offer?

   We will see that they do not.  Quite simply, the only reason the
bidder makes a tender offer is that the value would rise higher than
his bid, so no shareholder should accept his bid.

\begin{center}
 {\bf   The Free Rider Problem in Takeovers }\\
  (Grossman \& Hart [1980])
  \end{center}
 {\bf Players}\\
  A bidder and a continuum of shareholders, with amount $m$ of
shares.

 
\noindent
 {\bf   The Order of Play } \\
 1 The bidder offers $p$ per share for the $m$ shares. \\
 2 Each shareholder decides whether to accept the bid (denote by
$\theta$ the fraction that accept).\\
 3 If $\theta \geq 0.5$, the bid price is paid out, and the value of
the firm rises from $v$ to $(v+x)$ per share.

\noindent
 {\bf Payoffs}  \\
 If $\theta < 0.5$, the takeover fails, the bidder's payoff is zero,
and the shareholder's payoff is $v$ per share. Otherwise,  

\noindent
 $$
  \pi_{bidder} =  \left\{  
\begin{tabular}{ll}
$ \theta m (v+x - p)$  & if  $\theta \geq 0.5$.\\ 
 0 &   {\rm  otherwise}\\
\end{tabular} \right. 
$$

\noindent
   $ \pi_{shareholder} =  \left\{ 
\begin{tabular}{ll}
 $ p$ & if the shareholder  accepts.\\
 $v+x$ & if the shareholder rejects.\\ \end{tabular}
  \right.$ \\

Bids above $(v+x)$ are dominated strategies, since the
bidder could not possibly profit from them.  But if the bid is any
lower, an individual shareholder should hold out for the new value of
$(v+x)$ rather than accepting $p$.  To be sure, when they all do
that, the offer fails and they end up with $v$, but no individual
wants to accept if he thinks the offer will succeed.  The only
equilibria are the many strategy combinations that lead to a failed
takeover, or a bid of $p = (v+x)$ accepted by a majority, which
succeeds but yields a payoff of zero to the bidder.  If organizing an
offer has even the slightest cost, the bidder would not do it.

  The free rider problem is clearest where there is a continuum of
shareholders, so that the decision of any individual does not affect
the success of the tender offer. If there were, instead, nine players
with one share each, then in one asymmetric equilibrium five of them
tender at a price just slightly above the old market price and four
hold out. Each of the five tenderers knows that if he held out, the
offer would fail and his payoff would be zero. This is an example of
the discontinuity problem of section 8.6.

  In practice, the free rider problem is not quite so severe even
with a continuum of shareholders.  If the bidder can quietly buy a
sizeable number of shares without driving up the price (something
severely restricted in the United States by the Williams Act), then
his capital gains on those shares can make a takeover profitable even
if he makes nothing from shares bought in the public offer.  Dilution
tactics such as freeze-out mergers also help the bidder (see Macey \&
McChesney [1985]).
  In a freeze-out, the bidder buys 51 percent of
the shares and merges the new acquisition with another firm he owns,
at a price below its full value. If dilution is strong enough, the
shareholders are willing to sell at a price less than $v+x$.

 Still another takeover tactic is the two-tier tender offer, a nice
application of  the  Prisoner's Dilemma. Suppose   the underlying
value of the firm is 30, which is the initial stock price.  A
monopolistic bidder offers a price of 10 for 51 percent of the stock
and 5 for the other 49 percent, conditional upon 51 percent
tendering.  It is then a dominant strategy to tender, even though all
the shareholders would be better off refusing to sell.


 
\bigskip
\noindent
{\bf Greenmail}

\noindent
     Greenmail occurs when managers buy out some 
shareholders at an inflated stock price to stop them from taking
over.  Opponents of greenmail explain this using the Corrupt
Managers model.  Suppose that a little dilution is possible, or the
bidder owns some shares to start with, so   he can take over the
firm but would lose most of the gains to the other
shareholders. The managers are willing to pay the bidder a large
amount of greenmail to keep their jobs, and both manager and bidder
prefer greenmail to an actual takeover, despite the fact that the
other shareholders are considerably worse off.  

   Managers often use what we might call the Noble Managers model
to justify greenmail.  In this model, current management knows the
true value of the firm, which is greater than both the current stock
price and the takeover bid. They pay greenmail to protect the
shareholders from selling their mistakenly undervalued shares.

 The Corrupt Managers model faces the objection that it fails to explain
why the corporate
charter does not prohibit greenmail. The Noble Managers model faces the
objection that it
 implies either that shareholders are irrational or that  stock
prices rise  after greenmail because shareholders know that the
greenmail signal (giving up the benefits of a takeover) is more
costly for a firm which really is not worth  more than the
takeover bid. 

  Shleifer \& Vishny (1986) have constructed a more sophisticated
model in which greenmail is in the interest of the shareholders.  The
idea is that greenmail encourages potential bidders to investigate
the firm, eventually leading to a takeover at a higher price than the
initial offer. Greenmail is costly, but for that very reason it is an
effective signal that the manager  thinks a better offer could come
along later. (Like  Noble Managers,  this assumes that the manager acts
in the interests of the shareholders.)  I will present a numerical
example in the spirit of Shleifer \& Vishny rather than following
them exactly, since their exposition is not directed towards the
behavior of the stock price.

  The story behind the model is that a manager has been approached by
a bidder, and he must decide whether to pay him greenmail in the
hopes that other bidders -- ``white knights'' -- will appear. The
manager has better information than the market as a whole about the
probability of other bidders appearing, and some other bidders can
only appear after they undertake costly investigation, which they
will not do if they think the takeover price will be bid up by
competition with the first bidder. The manager pays greenmail to
encourage new bidders by getting rid of their competition.

\bigskip

\begin{center}
{\bf  Greenmail to Attract White Knights }\\
 (Shleifer \& Vishny [1986])
  \end{center}
 {\bf Players}\\
  The manager, the market, and bidder Brydox. (Bidders Raider and
Apex do not make decisions.) 

 
\noindent
 {\bf The Order of Play } \\ 
  Figure 15.2 shows  the game tree.  After each time $t$, the
market picks a share price $p_t$.\\
 0 Unobserved by any player, Nature picks the state to be (A), (B),
(C), or (D), with probabilities 0.1, 0.3, 0.1, and 0.5, unobserved by
any player.\\
 1 Unless the state is (D), the Raider appears and offers a price
of 15.  The manager's information partition becomes \{(A), (B,C),
(D)\}; everyone else's becomes \{(A,B,C), (D)\}.  \\ 
 2 The manager decides whether to pay greenmail and extinguish the
Raider's offer at a cost of 5 per share.  \\
 3 If the state is (A),  Apex appears and offers a price of
25 if greenmail was paid, and 30 otherwise.\\
 4 If the state is (B),  Brydox decides whether to buy
information at a cost of 8 per share. If he does, then he can make an
offer of 20 if the Raider has been paid greenmail, or 27 
if he must compete with the Raider.\\
 5 Shareholders accept the best offer outstanding, which is the
final value of a share.  If no offer is outstanding, the final value
is 5 if greenmail was paid, 10 otherwise.

\noindent
 {\bf Payoffs}  \\
   The manager maximizes the final value.\\
 The market minimizes the absolute difference between $p_t$ and the
final value.\\
 If he buys information, Brydox receives 23 $(= 31-8)$ minus the
value of his offer; otherwise he receives zero.
  \bigskip


\begin{center}
 {\bf Figure 15.2} The game tree for Greenmail to Attract White Knights 
 \end{center}


 

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 The payoffs specify that the manager should maximize the final value
of the firm, rather than a weighted average of the prices $p_0$
through $p_5$. This assumption is reasonable because the only
shareholders to benefit from a high value of $p_t$ are those that
sell their stock at $t$.  The manager cannot say: ``The stock is
overvalued: Sell!'', because the market would learn the overvaluation
too, and refuse to buy.

  The prices 15, 20, 27, and 30 are assumed to be the results of
blackboxed bargaining games between the manager and the bidders.
Assuming that the value of the firm to Brydox is 31 ensures that he
will not buy information if he foresees that he would have to compete
with the Raider.  Since Brydox has a dominant strategy -- buy
information if the Raider has been paid greenmail and not
otherwise -- our focus will be on the market price and the decision
of whether to pay greenmail.  This model is also not designed to
answer the question of why the Raider appears. His behavior is
exogenous. As the model stands, his expected profit is positive since
he is sometimes paid greenmail, but if he actually had to buy the
firm he would regret it in states B and C, since the final value of
the firm would be 10.

 We will see that in equilibrium the manager pays greenmail in states
(B) and (C), but not in (A) or (D).  Table 15.4 shows the equilibrium
path of the market price.

\begin{center}
{\bf Table 15.4}  The equilibrium price in Greenmail to Attract White
Knights

 \begin{tabular}{ lllll llll }
\hline
 {\bf  State } &  {\bf Probability} & $p_0$ &  $p_1$ &$p_2$ &$p_3$
&$p_4$ &$p_5$ &     {\bf Final management} \\
 \hline
    &  &   &   &  &   &  &   &  \\
 (A) & 0.1 & 14.5 & 19 & 30 & 30 & 30 & 30 & Allied\\
  &  &   &   &  &   &  &   &  \\
(B) & 0.3 & 14.5 & 19 &  16.25 & 16.25 & 20 & 20 & Brydox\\
 &  &   &   &  &   &  &   &  \\
 (C) & 0.1 & 14.5 & 19 & 16.25 & 16.25 & 5 & 5 &  Old management\\
  &  &   &   &  &   &  &   &  \\
(D) & 0.5 & 14.5 & 10 & 10 & 10 & 10 & 10 & Old management\\
    &  &   &   &  &   &  &   &  \\
       \hline
\end{tabular}                
\end{center}


 The market's optimal strategy amounts to estimating the final value.
Before the market receives any information, its prior beliefs
estimate the final value to be 14.5 (= 0.1[30] + 0.3[20]+ 0.1[5] +
0.5[10]). If state (D) is ruled out by the arrival of the Raider, the
price rises to 19 (= 0.2[30] + 0.6[20]+ 0.2[5]).  If the Raider does
not appear, it becomes common knowledge that the state is (D), and
the price falls to 10.

 If the state is (A), the manager knows it and refuses to pay
greenmail in expectation of Apex's offer of 30.  Observing the lack
of greenmail, the market deduces that the state is (A), and the price
immediately rises to 30.

  If the state is (B) or (C) the manager does pay greenmail and the
market, ruling out (A), uses Bayes's Rule to assign probabilities of
0.75 to (B) and 0.25 to (C). The price falls from 19 to 16.25 $(=0.75
[20] + 0.25 [5])$. 

  It is clear that the manager should not pay greenmail in states (A)
or (D), when the manager knows that Brydox is not around to
investigate.  What if the manager deviates in the information set
(B,C) and refuses to pay greenmail?  The market would initially
believe that the state was (A), so the price would rise to $p_2 =
30$. But the price would fall again after Apex failed to make an
offer and the market realized that the manager had deviated. Brydox
would refuse to enter at time 3, and the Raider's offer of 15 would
be accepted. The payoff of 15 would be less than the expected payoff
of 16.25 from paying greenmail.

  The model does not say that greenmail is always good for the
shareholders, only that it  can be good {\it ex ante}. If the true state
turns out to be (C), then greenmail was a mistake, {\it ex post}, but
since state (B) is more likely, the manager is correct to pay
greenmail in information set (B,C).  What is  noteworthy is
that greenmail is optimal even though it drives down the stock price
from 19 to 16.25.  Greenmail communicates the
bad news that Apex is not around, but makes the best of that
misfortune by attracting Brydox.











  
\vspace{1in}
\noindent
 {\bf  *15.3 Predatory Pricing: The Kreps-Wilson Model}
 
 One traditional form  of  monopolization and  entry deterrence is
predatory pricing, in which the firm seeking to acquire the market
charges a low price  to drive out  its rival.   We have looked at
predation already in chapters 4, 5 and 6  in the ``Entry Deterrence''
games. The  major problem with  entry deterrence under complete
information is
  the chainstore paradox.
The heart of the paradox is the sequential rationality problem faced by
an
incumbent who wishes to threaten a prospective entrant with low
post-entry prices.  The incumbent can respond to entry in two ways.
He can collude with the entrant and share the profits, or he can
fight by lowering his price so that both firms make losses.  We have
seen that the incumbent would not fight in a perfect equilibrium if
the game has complete information. Foreseeing the incumbent's
accommodation, the potential entrant ignores the threats.  

  In Kreps \& Wilson (1982a),   an application of the gang of four model
of chapter 6, incomplete information allows the
threat of predatory pricing to successfully deter entry.  A
monopolist with outlets in $N$ towns   faces an entrant who can
enter each town. In our adaption of the model, we will start by
assuming that the order in which the towns can be entered is common
knowledge, and that if the entrant passes up his chance to enter a
town, he cannot enter it later.  The incomplete information takes the
form of a small
probability that the monopolist is ``strong'' and has nothing but
$Fight$ in his action set: he is an uncontrolled manager who
gratifies his passions in squelching entry instead of  maximizing
profits. 
 
   
\begin{center}
{\bf   Predatory Pricing  }\\
  (Kreps \& Wilson [1982a])
\end{center}
  {\bf Players}\\
  The entrant and the monopolist.

  

\noindent
 {\bf The Order of Play  } \\
 0 Nature chooses the monopolist to be $Strong$ with low probability
$\theta$  and $Weak$, with high probability $(1-\theta)$. Only the
monopolist observes Nature's move.\\
 1 The entrant chooses  $Enter$ or $Stay$ $Out$ for the first
town.  \\ 
 2 The monopolist chooses $Collude$ or $Fight$ if he is weak,
$Fight$ if he is strong.\\
 3 Steps (1) and (2) are repeated for towns 2 through $N$.

 \noindent
 {\bf Payoffs}\\
  The discount rate is zero. Table 15.5  gives the payoffs per period,
which are the same as in table 4.1.

 \begin{center}
{\bf Table 15.5} {  Predatory Pricing}  

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Weak incumbent}\\
  &       &             &  {\it   Collude}    &   &  $ Fight$      \\
  &   &  $  Enter$       &    {\bf 40,50} &    &  $-10,0$ \\
 & {\bf  Entrant} && & &   \\
&  &    {\it  Stay out }       &      $0,100$  &   & {\bf 0,100} \\  
\multicolumn{6}{l}{\it Payoffs to: (Entrant, Incumbent)}
\end{tabular}                
\end{center}


  In describing the equilibrium, we will denote towns by names such
as $i_{30}$ and $i_{5}$, where the numbers are to be taken purely
ordinally. The entrant has an opportunity to enter town $i_{30}$
before $i_{5}$, but there are not necessarily 25 towns between them.
The actual gap depends on $\theta$ but not $N$.


 \noindent
 \begin{center} 
 {\bf Part of the Equilibrium for {Predatory Pricing}   }
  \end{center}
\begin{description}
 \item[Entrant:]
 Enter first at town $i_{ -10}$. If entry has occurred
before $i_{10}$ and been answered with $Collude$, enter every town
after the first one entered. 
\item[{\bf Strong monopolist:}]
 Always fight  entry. 
\item[{\bf Weak monopolist:}] Fight any entry up to $i_{30}$.  Fight
the first entry after $i_{-30}$ with a probability $m(i)$ that
diminishes until it reaches zero at $i_{5}$. If $Collude$ is ever
chosen instead, always collude thereafter. If $Fight$ was chosen in
response to the first attempt at entry, increase the mixing
probability $m(i)$ in subsequent towns.
\end{description}

  This description, which is illustrated by figure 15.3, only covers
the equilibrium path and small deviations. Note that
out-of-equilibrium beliefs do not have to be specified (unlike in the
original model of Kreps and Wilson), since whenever a monopolist
colludes, in or out of equilibrium, Bayes's Rule says that the
entrant must believe him to be $Weak$.

\begin{center}
{\bf Figure 15.3 The equilibrium in {Predatory Pricing} }
\end{center}


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   The entrant will certainly stay out until $i_{30}$.  If no town
is entered until $i_{5}$ and the monopolist is $Weak$, then entry
at $i_{5}$ is undoubtedly profitable.  But entry is attempted at
$i_{10}$, because since $m(i)$ is diminishing in $i$, the weak
monopolist probably would not fight even there.

  Out of equilibrium, if an entrant were to enter at $i_{90}$, the
weak monopolist would be willing to fight, to maintain $i_{10}$
as the next town to be entered. If he did not, then the entrant,
realizing that he could not possibly be facing a strong monopolist,
would enter every subsequent town from $i_{89}$ to $i_{1}$. If no
town were entered until $i_{5}$, the weak monopolist would be
unwilling to fight in that town, because too few towns are left to
protect.  If a town between $i_{30}$ and $i_{5}$ has been entered
and fought over, the monopolist raises the mixing probability that he
fights in the next town entered,  because he has a more valuable
reputation to defend. By fighting in the first town he has increased
the belief that he is strong and increased the gap until the next town
is entered.

  What if the entrant deviated and entered town $i_{20}$? The
equilibrium calls for a mixed strategy response beginning with
$i_{30}$, so the weak monopolist must be indifferent between
fighting and not fighting. If he fights, he loses current revenue but
the entrant's posterior belief that he is strong rises, rising more if
the fight occurs late in the game.  The entrant knows that in
equilibrium the weak monopolist would fight with a probability
of, say, 0.9 in town $i_{20}$, so fighting there would not much
increase the belief that he was strong, but if he fought in town
$i_{13}$, where the mixing probability has fallen to 0.2, the belief
would rise much more. On the other hand, the gain from a given
reputation diminishes as fewer towns remain to be protected, so the
mixing probability falls over time. 

   The description of the equilibrium strategies is incomplete
because describing what happens after unsuccessful entry becomes
rather intricate.  Even in the simultaneous-move games of chapter 3,
we saw that games with mixed strategy equilibria have many different
possible realizations.  In repeated games like   Predatory Pricing, the
number of possible realizations makes an exact description very
complicated indeed.  If, for example, the entrant entered town
$i_{20}$ and the monopolist chose $Fight$, the entrant's belief that
he was strong would rise, pushing the next town entered to $i_{ -8}$
instead of $ i_{10}$. A complete description of the strategies would
say what would happen for every possible history of the game, which
is impractical at this book's level of detail.   

 Because of mixing,  even the equilibrium path becomes
nonunique after $i_{10}$, when the first town is entered.  When the
entrant enters at $i_{10}$, the weak monopolist chooses randomly
whether to fight, so the entrant's belief that the monopolist is
strong increases if he is fought. As a result, the next entry might be
not at $i_{9}$, but $i_{7}$.

 As a final note, let us return to the initial assumption that if the
entrant
decided not to enter town $i$, he could not change his mind later. We
have seen that no towns will be entered until near the last one,
because the incumbent wants to protect his reputation for strength.
But if the entrant can change his mind, the last town is never
approached. The entrant knows he would take losses in the first
$(N-30)$ towns, and it is not worth his while to reduce the number to
30  to make  the monopolist   choose $Collude$.
Paradoxically, allowing the entrant many chances to enter helps not him,
but the
incumbent.






 
  \bigskip
\noindent
 {\bf  15.4 *Entry for Buyout}

\noindent
  The previous section suggested that predatory pricing might actually
be  a credible threat if information were slightly incomplete, because
the incumbent might be willing to makes losses  fighting the first
entrant to deter future entry.  This is not the end of the story,
however, because
even if entry costs exceed  operating revenues,   entry might
still be profitable if the entrant  is bought out by the incumbent.

   To see this most simply, let us start
by thinking about how entry might be deterred under
complete information.  The incumbent needs some way to precommit
himself to unprofitable post-entry pricing.  Spence (1977) and Dixit
(1980) suggest that the incumbent could enlarge his initial capacity
to make    the post-entry price  naturally drop to below average
cost.  The post-entry price would still be above average variable
cost, so having already sunk the capacity cost the incumbent fights
entry without further expense. The entrant's capacity cost is not yet
sunk, so he refrains from entry.

      In the model   with the extensive form of figure 15.4, the
incumbent has the additional option of buying out the entrant. An
incumbent who fights entry bears two costs: the loss from selling at
a price below average total cost, and the opportunity cost of not
earning monopoly profits.  He can make the first a sunk cost, but not
the second. The entrant, foreseeing that the incumbent will buy him
out, enters despite knowing that the duopoly price will be less than
average total cost.  The incumbent faces a second perfectness
problem, for while he may try to deter entry by threatening not to
buy out the entrant, the threat is not credible.

\begin{center}
{\bf Figure 15.4}  Entry for Buyout
\end{center}

 

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\begin{center}
 {\bf  Entry for Buyout}\\
  (Rasmusen [1988a])
\end{center}
 {\bf Players}\\
  The incumbent and the entrant.

 
\noindent
 {\bf The Order of Play }\\
 1 The incumbent selects capacity $K_i$.\\
 2 The entrant decides whether to enter or stay out, choosing a
capacity $K_e \geq 0$.\\
 3 If the entrant picks a positive capacity, the incumbent
decides whether to buy him out at price $B$.\\
 4 If the entrant has been  bought out, the incumbent selects output  
$q_i \leq K_i + K_e$.\\
 5 If the entrant has not been bought out, each player decides
whether to stay in the market or exit.\\
 6 If a player has remained in the market, he selects the output
$q_i \leq K_i$ or $q_e \leq K_e$. 

\noindent
 {\bf Payoffs}\\
  Each unit of capacity costs $a$, the constant marginal cost is $c$,
a firm that stays in the market incurs fixed cost $F$, and there is
no discounting. There is only one period of production.\\
 If no entry occurs, $\pi_{inc} =[p(q_i)-c]q_i - aK_i - F$ and
$\pi_{ent} = 0$.\\
  If entry occurs and is bought out, $\pi_{inc} =[p(q_i)-c]q_i - aK_i
- B - F$ and $\pi_{ent} = B - aK_e$.\\
 Otherwise, \\ 
  \begin{tabular}{ll}
  $ \pi_{incumbent} =$& $\left\{
\begin{tabular}{ll} 
 $[p(q_i,  q_e)-c]q_i - aK_i- F$ & if the incumbent
stays.\\
 $ - aK_i $ &  if the incumbent exits.\\
 \end{tabular}
\right.$ \\
\end{tabular} 

  \begin{tabular}{ll}
$   \pi_{entrant} =$& $\left\{
\begin{tabular}{ll}
  $[p(q_i, q_e)-c]q_e - aK_e - F$ & if the entrant stays.\\
 $ - aK_e $ & if the entrant exits.\\
 \end{tabular}
\right.$ 
\end{tabular}
 

 

   Two things have yet to be specified: the buyout price $B$ and the
price function $p (q_i, q_e)$. To specify them requires particular
solution concepts for bargaining and duopoly, which chapters 12 and
14 have shown are not uncontroversial.  Here, they are subsidiary to
the main point and can be chosen according to the taste of the
modeller.  We have ``blackboxed'' the pricing and bargaining subgames
in order not to deflect attention to subsidiary parts of the model.
The numerical example below will name specific functions for those
subgames, but other numerical examples could use different functions
to illustrate the same points.

\bigskip
\noindent
{\bf A Numerical Example}
    
\noindent
 Assume that the market  demand curve is  
 \begin{equation} \label{e1}
 p = 100 - q_i - q_e.
\end{equation}
 Let the cost per unit of capacity be $a =10$, the marginal cost of
output be $c = 10$, and the fixed cost be $F = 601$.  Assume that
output follows Cournot behavior and the bargaining solution splits
the surplus equally, in accordance with the Nash bargaining solution
and Rubinstein (1982). 

    If the incumbent faced no threat of entry, he would behave as a
simple monopolist, choosing a capacity equal to the output which
solved
 \begin{equation} \label{e2}
\stackrel{Maximize}{q_i} ( 100 - q_i)q_i - 10q_i - 10q_i.
\end{equation}
 Problem (15.\ref{e2}) has the first-order condition 
 \begin{equation} \label{e3}
80 - 2q_i=0,
\end{equation}
 so the monopoly capacity and output would both equal 40, yielding a net
operating revenue of 1,399 ($ =[p - c]q_i - F$), well above the
capacity cost of 400.

  We will not go into details, but under these parameters   the
incumbent chooses the same output and capacity of 40 even if entry is
possible but buyout is not.  If the potential entrant were to enter, he
could do no better than to choose $K_e = 30$, which costs 300. With
capacities $K_i=40$ and $K_e = 30$, Cournot behavior leads the two firms
to solve
 \begin{equation} \label{e4}
 \stackrel{ Maximize}{q_i} ( 100 - q_i - q_e)q_i - 10q_i\;\; s.t.
\;\; q_i \leq 40
 \end{equation}
 and
\begin{equation} \label{e5}
\stackrel{ Maximize}{q_e} ( 100 - q_i - q_e)q_e - 10q_e 
\;\;s.t.\;\; q_e \leq 30,
\end{equation}
which have first order conditions
 \begin{equation} \label{e6}
90 - 2q_i -q_e=0
\end{equation}
and 
 \begin{equation} \label{e7}
90 - q_i - 2q_e =0.
\end{equation}
   The Cournot outputs both equal 30, yielding a price of 40 and net
revenues of $R^d_i = R^d_e = 299$ ($ = [p - c]q_i - F$).  The
entrant's profit net of capacity cost would be $-1$ ($= R^d_e -
30a$), less than the zero from not entering.

 What if both entry and buyout are possible, but the incumbent still
chooses $K_i = 40$? If the entrant chooses $K_e = 30$ again, then the
net revenues would be
$R^d_e = R^d_i = 299$, just as above.  If he buys out the entrant,
the incumbent, having increased his capacity to 70, produces a
monopoly output of 45. Half of the surplus from buyout is
 \begin{equation}\label{e8}
\begin{array}{lll}
 B& =& 1/2 \left[ \stackrel{Maximize}{q_i} \{ [p(q_i)-c]q_i|q_i \leq
70\}
- F - (R^d_e + R^d_i) \right]\\
 & & \\
& = &1/2 [(55-10)45 - 601 - (299 + 299)]
= 413.  
 \end{array}
 \end{equation}
  The entrant is bought out for his Cournot revenue of 299 plus the
413 which is his share of the buyout surplus, a total buyout price of
712.  Since 712 exceeds the entrant's capacity cost of 300, buyout
induces entry which would otherwise have been deterred.  Nor can the
incumbent deter entry by picking a different capacity.  Choosing any
$K_i$ greater than 30 leads to the same Cournot output of 60 and the
same buyout price of 712. Choosing $K_i$ less than 30 allows the
entrant to make a profit even without being bought out.

         Realizing that entry cannot be deterred, the incumbent would
choose a smaller initial capacity. A Cournot player whose capacity is
less than 30 would produce right up to  capacity.  Since
buyout will occur, if a firm starts with a capacity less than 30 and
adds one unit, the marginal cost of capacity is 10 and the marginal
benefit is the increase (for the entrant) or decrease (for the
incumbent) in the buyout price. If it is the entrant who adds a unit
of capacity, the net revenue $R^d_e$ rises by at least $(40-10)$, the
lowest possible
Cournot price minus the marginal cost of output. Moreover, $R^d_i$
falls because the entrant's extra output lowers the market price, so
under our bargaining solution the buyout price rises by more than 15
($=\frac{40-10}{2}$) and the entrant should add extra capacity up to
$K_e =
30$. A parallel argument shows why the incumbent should build a
capacity of at least 30. Increasing the capacities any further leaves
the buyout price unchanged, because the duopoly net revenues are
unaffected, so both firms choose exactly 30.

         The industry capacity equals 60 when buyout is allowed, but
after the buyout only 45 is used. Industry profits in the absence of
possible entry would have been 999 ($= 1,399-400$), but with buyout
they are 824 ($= 1,424 - 600$), so buyout has decreased industry
profits by 175.  Consumer surplus has risen from 800 ( = $0.5[100-
p(q|K=40)][q|K=40]$) to 1,012.5 ($=0.5[100-p(q|K=60)][q|K=60])$, a
gain of 212.5, so buyout raises total welfare in this example. The
increase in output outweighs the inefficiency of the entrant's
investment in capacity, an outcome that depends on the
particular parameters chosen.  

  The model is a tangle  of paradoxes.  The central paradox is that the
ability of the incumbent to destroy industry  profits after entry ends
up hurting  him  rather than helping  because it increases the buyout
price. This  has a similar flavor to the ``judo economics'' of Gelman \&
Salop (1983): the incumbent's very size and influence weighs against
him.
  In the numerical example,
allowing the incumbent to buy out the entrant raised total welfare,
even though it solidified monopoly power and resulted in wasteful
excess capacity. Under other parameters, the effect of excess
capacity dominates, and allowing buyout would lower welfare -- but
only because it encourages entry, of which we usually approve.
Adding more potential entrants would also have perverse effects. If
the incumbent's excess capacity can deter one entrant, it can deter
any number. We have seen that a single entrant might enter anyway,
for the sake of the buyout price.  But if there are many potential
entrants, it is easier to deter entry.  Buying out a single entrant
would not do the incumbent much good, so he would only be willing to
pay a small buyout price, and the small price would discourage any
entrant from being the first. The game becomes complicated, but
clearly the multiplicity of potential entrants makes entry more
difficult for any of them.

 


 \begin{small}

   

\bigskip
\noindent
{\bf  Notes}

\bigskip
\noindent
\noindent
{\bf N15.1} {\bf Innovation and patent races}
\begin{itemize}
 \item
 The idea of the patent race is described by Barzel (1968), although
his model showed the same effect of overhasty innovation even without
patents.



 \item
  Reinganum (1985) has shown that an important element of patent
races is whether increased research hastens the arrival of the patent
or just affects whether it is acquired. If more research hastens the
innovation, then the incumbent might spend less than the entrant
because the incumbent is enjoying a stream of profits from his
present position that the new innovation destroys.


 \item
 {\bf Uncertainty in innovation.} Patent Race for an Old Market, is only
one way to model innovation under uncertainty.  A more common
way is to use  continuous time with discrete discoveries and
specifies that discoveries arrive as a Poisson process with parameter
$\lambda (X)$, where $X$ is research expenditure, $\lambda'>0$, and
$\lambda''<0,$  as in  Loury (1979) and Dasgupta \&
Stiglitz (1980).   Then 
 \begin{equation}\label{e26z}
\begin{array}{ll}
  Prob{\rm (invention \; at\;} t) & = \lambda e^{-\lambda(X) t};\\
  Prob{\rm (invention \; before\;} t) & = 1-e^{-\lambda(X)t}.        \\
\end{array}
\end{equation}
   A little algebra gives us the current value of the firm, $R_0$, as
a function of the innovation rate, the interest rate, the
post-innovation value $V_1$, and the current revenue flow $R_0 $. The
return on the firm equals the current cash flow plus the probability
of a capital gain. 
 \begin{equation}\label{e27} 
rV_0 = R_0 - X + \lambda(V_1 - V_0),
\end{equation}
\noindent
 which implies
\begin{equation} \label{e28}
V_0 = \frac{\lambda V_1 + R_0 - X}{\lambda + r}.
\end{equation}
          Expression (15.\ref{e28} ) is frequently useful.

\item
 A common theme in entry models is   what has been called the
{\bf fat-cat effect} by Fudenberg \& Tirole (1986a, p.
23). Consider a two-stage game, in the first stage of
which an incumbent firm chooses its advertising level and in the second
stage  plays a Bertrand subgame with an
entrant.  If the advertising in the first stage gives the
incumbent a base of captive customers who have inelastic
demand, he will choose a higher price than the entrant.
The incumbent has become a ``fat cat.'' The effect is
present in many models.  In section 14.3's Hotelling
Pricing Game  a firm located so that it has a large ``safe''
market would choose a higher price. In  section 5.5's   
Customer Switching Costs a firm that has old customers
locked in would choose a higher price than a fresh entrant
in the last period of a finitely repeated game. 

 \end{itemize}



{\bf N15.2} {\bf Predatory Pricing: the Kreps-Wilson Model}
 \begin{itemize}
  \item
 For  other expositions of this model  see pages 77-82 of Martin (1993)
239-243  of Osborne \& Rubinstein (1994).

 \item
 Kreps \& Wilson (1982a) do not simply assume that one type of
monopolist always chooses $Fight$. They make the more elaborate but
primitive assumption that his payoff function makes fighting a
dominant strategy. Table 15.6 shows a set of payoffs for the strong
monopolist which generate this result. 

\begin{center}
{\bf Table 15.6} Predatory Pricing with a dominant strategy 

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf Strong Incumbent}\\
  &       &             &  {\it  Collude}    &   &  $ Fight$      \\
  &   &  $ Enter$        &    20,10 &    &  $-10,40$ \\
 & {\bf Entrant} && & &  \\
&  &    {\it Stay out }       &      $0, 100$  &    & {\bf 0,100} \\  
\multicolumn{6}{l}{\it Payoffs to: (Entrant, Incumbent)}
\end{tabular}                
\end{center}
 

$\;\;\;$ Under the Kreps-Wilson assumption, the strong monopolist
would actually choose to collude in the early periods of the game in
some perfect Bayesian equilibria.  Such an equilibrium could be
supported by out-of-equilibrium beliefs that the authors point out
are absurd: if the monopolist fights in the early periods, the
entrant believes he must be a weak monopolist. 

 \end{itemize}
 
  
  
\bigskip
\noindent
{\bf Problems}
 
  
\noindent
{\it 15.1: Crazy Predators} (adapted from Gintis [forthcoming], 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 {\it 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 doesn not learn from experience). Brydox initially
assesses the probability that Apex is crazy at $\theta$.

\begin{enumerate}
\item[15.1a] 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{e14.a1}
 -p_a +m < 2d 
\end{equation}

\item[15.1b] 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{e14.a2}
 \theta \geq \frac{d_b}{p_b+d_b}
\end{equation}

\item[15.1c] If neither of the two conditions above applies, 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
equilibrium values of $\alpha$ and $\beta$ are:
 \begin{equation} \label{e14.a3}
 \alpha =\frac{\theta d_b}{(1-\theta)p_b}
\end{equation}
\begin{equation} \label{e14.a4}
\beta =\frac{-p_a +m -2d_a}{m-d_a}
 \end{equation}

\item[15.1d]  Is this behavior related to any of the following
phenomenon: signalling, signal jamming, reputation, efficiency wages?
  \end{enumerate}

\bigskip

\noindent
{\it 15.2:  Rent Seeking}\\ I mentioned that  Rogerson (1982) uses a
game very similar to ``Patent Race for a New
Market'' to analyze competition for a government monopoly franchise.
See if you can do this too. What can you predict about the welfare
results of such competition?
\bigskip

\noindent
{\it 15.3:  A Patent Race}\\ See what happens in Patent Race for an Old
Market when specific functional forms and parameters are assumed. Set
$f(x) =  log(x)$, $g(y) = 0.5(1+ y/(1+y)$ if $ y \geq 0$, $g(y) = 0.5(1+
y/(1-y)$ if $ y \leq 0$,  $y=2$, and  $z = 1$. Figure out the  research
spending by each firm for the three cases of (a) $v=10$, (b) $v=4$, (c)
$v =2$ and (d) $v =1$.
  
\bigskip

\noindent
{\it 15.4: Entry for Buyout}\\ Find the equilibrium  in Entry for Buyout
if all the parameters of the numerical example are the same  except that
the marginal cost of output is $c=20$ instead of $c=10$.



\end{small}

\end{document}