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         {\bf  Choosing Among Signalling Equilibria in Lobbying Games
}\\
 (Published:  {\it Public Choice})\\
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\bigskip
July 17, 1996  \\
        \bigskip
      Eric Rasmusen  \\
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        {\it Abstract}
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         Randolph Sloof has written a comment on the lobbying-as-
signalling
model in Rasmusen (1993) in which he points out an equilibrium I
missed and
criticizes my emphasis on  a particular separating equilibrium.    In
this
response, I      discuss how to interpret multiple equilibria in games
and  how
to interpret mixed strategy equilibria in which two types of player
with
identical incentives must  pick different mixing probabilities.

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          \noindent
    Indiana University School of Business, Rm. 456,
  1309 E 10th Street,   Bloomington, Indiana, 47405-1701.
Secretary, 812-855-9219, direct 812-855-3356   Fax: 812-855-3354.
Email:
Erasmuse@indiana.edu.  For
the latest versions of this and other papers, go to my
Webpage, http://ezinfo.ucs.indiana.edu/$\sim$erasmuse.
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  Dr. Sloof's comments on Rasmusen (1993)  raise a number of
methodological
issues  that may be of interest to more readers of {\it Public Choice}
than just
those who study lobbying.   I will try, therefore, to  respond to them
in a way
intelligible to those who have not read the original article, using
the model
as an example of more general issues.
  I will discuss:
1.  How to react to   multiple   equilibria,  2. Mixed strategies when
different
types of player must choose different mixing probabilities, and
3.    Important and unimportant differences between equilibria.

 First, let me  recapitulate  Rasmusen (1993).   A  lobbyist wants a
politician
to choose a new policy.   The lobbyist knows whether the voters
will  later
reward
  the politician for the innovation, but the politician does not.   If
the
voters actually want the new policy, we will call the lobbyist  a
``truthful
lobbyist''; otherwise, we will call him a ``lying lobbyist.''   The
lobbyist
incurs a direct cost for lobbying, but not for  lying.  The politician
can, at
some cost,  try to verify  the lobbyist's assertions, and if the
lobbyist is
silent  the politician can pay for an independent investigation.

  The most interesting equilibrium, I asserted,  is  the  mixed
strategy
separating  equilibrium E3,  in which  lying lobbyists sometimes lobby
and
truthful lobbyists always lobby,  while  politicians sometimes verify
but  never
investigage independently.    Dr. Sloof   disputes my  emphasis on E3
and points
out  a new equilibrium, E5, that I   missed.

\bigskip
\noindent
 {\it  The General Problem of  Multiple  Equilibria.  }
    Games often have multiple Nash equilibria, and applying
uncontroversial
ideas like sequential rationality  often still leave multiple
equilibria.
Signalling games, in particular,  usually have  at least two
equilibria:  a
pooling one, where  signalling  has    no impact, and a separating
one, where it
does.    So what should the modeller's next step be?

  Approach 1  is to think of ways to rule out all but one equilibrium.
The
modeller could apply
  refinement  concepts such as divinity (Banks \& Sobel [1987])  or
the
intuitive criterion    (Cho \& Kreps [1987]),     but nobody has come
up with a
persuasive general refinement, and rather than just citing previous
work, the
modeller really has to argue for the refinement in his particular
case.   Sloof
shows how to use Approach 1 in his Lemma,   and  shows how even if  it
were
justified it would fail in this particular model.  In the text, he
uses what I
will call Approach 2, which is  to discuss  how a variety of
principles such as
insufficient reason and pareto optimality apply to the particular
model.    I
take that approach to some extent in Rasmusen (1993), but  my analysis
is really
based on yet a third approach.

 Approach 3  to the problem of multiple equilibria is  not to mind
them.
Remember that the point of  a model is to explain something in the
real world.
In the real world,
  it seems that uninformative lobbying  and political demonstrations
take
place.      How are we to explain it?   Must it be irrational, or just
fun for
the lobbyist  and demonstrator, or does even contentless political
activity have
some persuasive power?

 The explanation in Rasmusen (1993) is that lobbying is    part of  a
separating
equilibrium of a signalling model. When certain conditions on
information and
costs  are satisfied, and  people expect lobbying to work as a signal,
it will
work as  a signal.  This leaves unexplained why it is that people
expect
lobbying to work, when  another equilibrium is possible in which
people do not
expect lobbying to work, but I do not find that  very objectionable.
The
signalling equilibrium also   assumes    that lobbying costs are not
prohibitively high, without  explaining why that is so, and we make
such
assumptions all the time.    We can still argue about the plausibility
of
assumptions---  Approach 2, again--- but it should not be distressing
if we find
a number of  contradictory assumptions all realistic for different
real world
settings.


     It may well be that sometimes the real-world  expectations are
that
lobbying will not work as a signal.   That is fine:  we do not need or
want  a
model which  says that in every country in every era uninformative
lobbying
will take place.
     Suppose   two equilibria  are possible, depending on
expectations, and  we
observe uninformative lobbying taking place in country X.   We can
explain X as
the result of  rational signalling,  and the fact that  the pooling
equilibrium
is being played out in countries Y and Z is no  objection  to our
explanation
for X.






I view the  main criterion for a model as its explanatory power rather
than its
falsifiability or its ability to predict, but  note that  a model with
multiple
equilibria  has no problem with these other criteria either.   Suppose
that  in
the model  fact A implies equilibria B or C.   The model is falsified
if A and D
are observed, and it narrows the predictions down to B and C rather
than B,C, D,
E, F.   If I have a model which explains earthquakes  in such a way as
to imply
that there will be an earthquake on either  March 1 or March 5 if it
snows in
January, then not only is this model falsifiable, it is highly useful
if
correct, despite its ambiguous predictions.


 In the lobbying model, Sloof comments that   if the original regime
prohibits
lobbying, and then the prohibition is lifted, the  resulting
equilibrium might
be one in which lobbying never occur, rather than the lobbying
equilibrium E3.
If this is true, then  predictions   cannot be made without
qualification. To
take the simplest example, I cannot say  ``The amount of lobbying will
rise when
lobbying is made legal,''    but must say,  ``The amount of lobbying
will rise
or stay the same when lobbying is made legal.''      I think the
stronger
prediction is  nonetheless  valid.   If   legalization results in the
no-
lobbying equilibrium with 99 percent probability and the lobbying
equilibrium
with 1 percent probability, the   expected amount of lobbying
unambiguously
rises. If  a nation is trying to decide whether to legalize lobbying,
and
welfare will be higher if the lobbying equilibrium is played out and
unchanged
if  it is not,  the  policy implication is unambiguous.   Uncertain
predictions
can still be valuable predictions.


\bigskip
\noindent
 {\it   Players with the Same Incentives Choosing Different Mixing
Probabilities.}
  Although the preceding section suggests that  multiple equilibria
need not be
distressing, let us
 now return to Approach 2   I will try to expand on why I think
equilibrium E2
is implausible.


In E2, the story is something like this.  Verification and
investigation are
costly enough that the politician never does them.  If he does not see
lobbying,
he never adopts the new policy.  If he does, then he sometimes adopts
the new
policy, because   the  lying  lobbyist only lobbies with probability
.2, but
the truthful  lobbyist lobbies with probability .9.  The  lobbyist
types  are
willing to  pick .2 and  .9 not  because the lying lobbyist is more
afraid of
getting caught (the politician never tries to verify, remember), but
   because each type is indifferent between {\it any}    probability
between 0
and 1 in a mixed strategy equilibrium.

      It is always true in  a mixed strategy equilibrium   that the
mixing
player  is indifferent about his behavior, but must pick one
probability to
support the equilibrium.   Here, each truthful lobbyist  is
indifferent between
probability 0 and 1, but he picks .9.  The usual argument  is that
unobservable
characteristics of the lobbyist  or the time of choice determine this.
He
really pursues a pure strategy of 0 if  his mood  of optimism fails to
reach  a
certain good level, and 1 otherwise, and that certain mood  level is
reached on
90 percent of occasions.  (See Harsanyi   [1973]. )

Here, however,   it is more difficult to apply that argument.  The two
lobbyist
 types have identical incentives,  but must pick different mixing
probabilities
in the same equilibrium.  Since politicians never verify or
investigate,  true
and false lobbyists are faced with identical incentives, and their
problem is
one of pure signalling.    We cannot explain this using  unobservable
type
characteristics, so that actually in the population  each lobbyist
pursues a
pure strategy, because here  those unobservables have to split
lobbyists at
exactly .9 and exactly .2.  The lying  lobbyists require a different
critical
mood level than the truthful  lobbyists.  Why, when their payoff
functions are
identical?
One could argue that  the critical levels are for  costliness of
lobbying for
the truthful lobbyists and  degree of moral qualms for the lying
lobbyists,
but    whether something like this would work is unclear enough to
require
formal modelling.   (The  best place for
such   modelling would, of course, have been in  the original paper,
but I
encourage anyone interested in this topic, either as applied to
lobbying or more
generally, to follow it up.)

\bigskip

Let us now turn to equilibrium E5, which, as  Sloof correctly notes,
I simply
failed to detect.  This is an important omission in my paper, because
Equilibrium E5 has intuitive plausibility, though it shares  something
of the
delicacy of E2.

 In equilibrium E3,    the  lying lobbyist lobbies sometimes but the
truthful
lobbyist lobbies always.    The politician sometimes tries to verify
lobbying,
but never  investigates independently   if no lobbying occurs, since
he knows
that if change is desirable,  he will see lobbying.


 In equilibrium E5, both types of lobbyists lobby  sometimes, but  the
truthful
lobbyist   uses a higher probability.  The politician sometimes tries
to verify
lobbying, and sometimes investigates independently   if no lobbying
occurs.
This equilibrium only  exists, naturally,   if investigation is cheap
enough for
the politician.

   The    difference is   that in E5 the truthful lobbyist does not
always
lobby, and the politician  sometimes investigates independently.  As a
result,
it can happen in E5, but not E3, that
 innovation is desirable,  the lobbyist fails to  lobby,   and   the
new policy
is not adopted.

 The appealing intuition behind E5 is the idea that  when
investigation is cheap
enough, the lobbyist  will try to save on lobbying costs in the hope
that the
politician will investigate for himself.


As in  equilibrium E2, however,   player types with identical
incentives chose
different  mixing probabilities.     The expected benefit of  lobbying
is the
same for the truthful lobbyist as for the lying one, but it is not so
obvious,
because  the  components of the expected benefit are  different.
Suppose a
lobbyist is considering  switching from  not lobbying to lobbying.   A
cost
for either type is that he must pay the direct lobbying cost.  The
benefit for
the   lying  lobbyist is that
 maybe the politician will not verify, and will adopt the new policy.
The
benefit for the  truthful   lobbyist is  bigger in  that  the
politician will
certainly  adopt the new policy, whether he verifies or not,  but from
this
benefit must be subtracted the loss of the probability that the
politician would
have  investigated and changed his policy even if there were no
lobbying.  Since
the politician has identical probabilities of  investigation and
verification in
E5, the net benefit for the truthful lobbyist turns out to be exactly
the same
as the net benefit for the lying lobbyist.

E5 does  have more leverage than E2 as to why the    truthful lobbyist
would
behave differently from the lying lobbyist.   In E2, the only
difference was
that one  type might have more moral qualms to deal with.  In E5,
individual
differences between  politicians in either verification or
investigation costs
might  differentiate mixing situations.    In addition, with both
politician and
lobbyist mixing,   correlations between unobservables politician and
lobbyist
might differentiate truthful from lying lobbyists.
   Given this  greater complexity, I find E5 more plausible than E2.


\bigskip
\noindent
 {\it    Importand and Unimportant Differences in Equilibria.      }
   Let us suppose  E5  is as likely as E3.  Is that a problem?    ---
Yes, but
not in the most obvious way.

   In both E3 and E5, the lying  lobbyist lobbies some but not all of
the time,
and the truthful lobbyist lobbies more often, and the politician
checks up on
the lobbyist some but not all of the time.   The   difference is  that
the
honest lobbyist sometimes will not bother to lobby,  because he knows
that   the
politician will sometimes investigate for himself even if no lobbying
occurs.
As a result, sometimes desirable new policies will not be adopted.

 In terms of the most interesting implications--- that  lobbyists will
sometimes engage in non-informative lobbying and that politicians will
sometimes
refrain from   investigating issues because they rely on lobbyists---
E3 and E5
are the same.   The purpose of the lobbying model is to explain  those
behaviors,  and the presence of E5 does not  affect that.

 The model's prediction that  a desirable new policy will always be
adopted
flows out of the modelling, but  is, alas, not one of the  real-world
facts that
needs explaining, and was not one of the features of the model that I
emphasized.  Any model  requires assumptions, both positive ones and
the
negative  {\it  ceteris paribus} assumption, and we do not take our
models'
definitive predictions as  gospel.     Even without equilibrium E5,
the lobbying
model is not disproved if someone finds  many     new policies   which
the
electorate wants but for which no one lobbies or investigates.
Sometimes no
lobbyist is interested and the politician is too busy  or
miscalculates.
Thus, the qualitative  differences in behavior between E3 and E5 are
not serious
differences.


 What is more important  is   the difference in welfare results  that
arise from
the  inner workings of the two equilibria.  Dr. Sloof points out that
in E3,
welfare rises when  the politician's investigation cost  and the
politician's
certainty about the electorate's views rise,   but in E5 these changes
have
ambiguous effects, because they discourage the lobbying of the
truthful
lobbyist.   This is a genuine difference, and were I to rewrite the
paper I
would  want  to either rule out E5 on the lines discussed above, or,
more
likely,  weaken Proposition 2.   Here, I would just like to make the
point that
sometimes  apparently different  multiple equilibria   have   the same
properties as far as  a  model's purpose goes,   and one must look
carefully to
see  whether they have differences in the   features the model is
trying to
address.


  \bigskip

\noindent
 {\it References}


   Banks, J. and   Sobel, J.
(1987) ``Equilibrium selection in signaling games''   {\it
Econometrica.} May 1987.   55:       647-61.

 Cho, I.  and   Kreps, D.  (1987) ``Signaling games and stable
equilibria'' {\it Quarterly Journal of Economics.} May 1987.   102:
 179-221.


   Harsanyi, J. (1973) ``Games with randomly disturbed
payoffs: A new rationale for mixed strategy equilibrium points''
{\it
International Journal of Game Theory.} 1973.   2:       1-23.


Rasmusen, E.  (1993) ``Lobbying when the decisionmaker can acquire
independent
information,''  { \it Public Choice}, (1993) 77: 899-913.

Sloof, R.  (199x) ``Lobbying when the decisionmaker can acquire
independent
information: A comment''     {\it  Public Choice},  xxx.

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