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\noindent
     August 24, 1999. \\
   Eric Rasmusen, Erasmuse@indiana.edu.  Web:  Php.indiana.edu/$\sim$erasmuse. 

 This is a section from Chapter 13 (Pricing) of {\it Games and Information}.  that was dropped from the 3rd Edition.

\bigskip
\noindent
{\bf Conjectural Variation} 

\noindent
 Conjectural variation, an equilibrium concept different in flavor
from any that has yet appeared in this book, is a way to quantify
the degree of cooperation between oligopolists.  Let us continue to
specify the strategies as quantities.  In a  Nash
equilibrium, no player wants to deviate, and his beliefs
about how the other players would behave are confirmed whatever
nodes are reached.  Under conjectural variation, a player believes,
for  reasons outside the model, that if he deviated, the other players would
deviate in specified ways. This should seem quite an unnatural idea to anyone who has read this far in the book, since it violates  the basic assumptions of Bayesian games and it is rather hazy about what is happening in this simultaneous-move game. The idea  may be clearer in an example.
Returning to the two-player model, we can use equation (13.\ref{e13.3}) to
write Apex' self-perceived first order condition as 
 \begin{equation}\label{e13.15}
  \frac{d\pi_a}{dq_a} = 
 p + \left( \frac{dp}{dq} \right) \left( \frac{dq }{d q_a} \right)
q_a - \frac{dc}{dq_a} = 0.
  \end{equation}
 The difference between the first-order-conditions (13.\ref{e13.3}) and
(13.\ref{e13.15}) is that (13.\ref{e13.15}) contains
 \begin{equation}\label{e13.16}
 \frac{dq }{d q_a} = 1 + \frac{d q_b }{d q_a}.
  \end{equation}
  Equation (13.\ref{e13.16}) says that the expected effect on industry
output of an increase in $q_a$ by one unit has two components: a
direct increase of one unit, and an indirect increase from Brydox
increasing his output in response. The first-order condition (13.\ref{e13.15})
must be qualified by ``self-perceived'' because Apex might be
mistaken in his beliefs about Brydox' response.  The belief implicit
in Nash equilibrium, that Apex' deviation is not followed by a
response from Brydox, is the only belief that supports an
equilibrium in which one player or the other is not mistaken. But if
consistency of beliefs is not required, other beliefs are possible
that lead to different behavior.

 \noindent
{\it Firm $i$'s {\bf conjectural variation} is the rate $\frac{d
q_{-i} }{d q_i}$ at which he conjectures that the output of other
firms would change if $i$'s own output changed.}

\noindent
  {\it CV} = 0\\
  In a Cournot-Nash equilibrium, Apex believes that if he deviated
by producing more, Brydox would not deviate, so the conjectural
variation equals 0.

 \noindent
{\it CV} = $-1$\\
 If Apex believes that an increase in his output is matched by a
decrease in Brydox' output, so the total industry output is left
unchanged, the conjectural variation is $-1$.  If both firms use this
conjectural variation, the industry output is the competitive level;
firms ignore the effect of their output in depressing the price. Of
course, if both firms use a negative value, their beliefs are
inconsistent. 

 \noindent
  {\it CV} = 1\\
  If Apex believes that Brydox would exactly match his output
changes, the conjectural variation is 1.  With two firms, with identical cost curves, industry
output is at the cartel level, though an $n$-player game would need
$CV = n-1$ to achieve that level.

     In Stackelberg equilibrium (Section 3.5), the conjectural
variation of the Stackelberg follower is between 0 and 1, and takes
the value given by a reaction function like equation (13.\ref{e13.7}).

  In the world oil market, fringe producers like Britain face the
OPEC cartel.  If Britain's conjectural variation equals $-1$, Britain
believes that producing more would make OPEC cut back an equal
amount; if 0, that OPEC would ignore Britain; if 0.5, that OPEC would
follow with a smaller increase; if 1, that OPEC would match every
increase; and if 10, that OPEC would respond by flooding the market.
Setting up equations with the appropriate value for the conjectural
variations of all the players, we could solve for the equilibrium
output.  The idea is useful for organizing different models of
duopoly and it is simple enough to be empirically estimated. Even
without knowing the correct theory, an estimate could be made of how
much OPEC actually does respond to Britain.


 \noindent
{\bf Notes:

  The idea of conjectural variation is attributed to Bowley (1924)
and is discussed in Jacquemin (1985) and  Varian (1992, p. 302). 

 
   Do not confuse a conjectural variation of $-1$ with perfect
competition, even though both may lead to the efficient output. In
perfect competition, individuals do not believe that they affect the
rest of the market, but if $CV = -1$, a firm believes that other firms
will cut back when it produces more. Perfect competition is more like
a game with players so small relative to the market that even though
$CV = 0$, as in Nash equilibrium, each player correctly believes that
his actions have a trivial effect on the market price.



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