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%From: Eric Rasmusen <erasmuse@rasmusen.bus.indiana.edu>
%Date: Wed, 24 Jul 96 16:38:04 -0600

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         {\bf  Choosing Among Signalling Equilibria in Lobbying Games   
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July 17, 1996  \\ 

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      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.  Web:  
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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