Potters, J.J.M.

English

Research Papers in Economics

Economics  Letters  38  (1992)  43-47 
North-Holland 
43 
Fixed  cost  messages  * 
Jan  Potters 
Tinbergen  Institute  and  lJnir,ersity of  Amsterdam,  1011  NH  Amsterdam,  Netherlands 
Received  11  April  1991 
Accepted  3  October  1991 
The  occurrence  and  impact  of  fixed  cost  messages  are  analyzed  by  means  of  an  asymmetric  information  game  and  related  to 
the  sender’s  stake  in  persuading  the  receiver,  and  the  cost  of  a  message. 
1.  Introduction 
The  typical  base  for  the  occurrence  of  informative  signals  of  messages  in  asymmetric  information 
games  are  differential  exogenous  signalling  costs,  meaning  that  the  sender’s  signalling  costs  vary 
with  his  private  information  and/or  with  the  ‘content’  of  the  signal  [see,  for  example,  the  class  of 
games  in  Cho  and  Sobel  (1990)].  In  the  present  paper  it  is  shown  that  informative  messages  can  also 
occur  if  there  are  no  differential  signalling  costs,  but  if,  alternatively,  there  is  a  fixed  exogenous 
cost  of  sending  a message  (in  addition  to  the  endogenous  cost/benefit  of  a signal  due  to  the  impact 
on  the  receiver’s  action). 
Our  model  is  close  to  Crawford  and  Sobel’s  (1982),  in  which  it  is shown  that  information  transfer 
can  occur  in  equilibrium  even  if  messages  bear  izo  exogenous  cost  to  the  sender.  A  necessary 
condition  for  this  to  be  possible  is  that  the  sender’s  and  receiver’s  preference  orderings  are  ‘close’ 
(in  a  well-defined  sense).  Such  closeness  of  preferences  will  not  be  assumed  in  the  present  paper. 
Specifically,  it  will  be  assumed  that  the  sender’s  preference  ordering  over  actions,  contrary  to  the 
receiver’s,  is  independent  of  the  state  variable  which  is  private  information  of  the  sender. 
We  think  the  model  has  relevance  for  asymmetric  information  situations  in  which  signalling  is 
mainly  an  informational  but  not  costless  activity,  such  as  informational  lobbying  or  advertizing.  On 
the  one  hand,  there  may  be  a  substantial  difference  of  preferences  over  the  receiver’s  action  set, 
and  it  may  be  difficult  for  the  receiver  to  gather  the  relevant  information  herself.  On  the  other 
hand,  there  is  no  direct  link  between  the  cost  of  a  message  and  the  content  of  the  message  or  the 
private  information  of  the  sender.  In  order  words,  a  message  bears  a  cost  to  the  sender,  but  this 
cost  is  not  related  to  what  he  says  or  to  what  he  knows. 
2.  The  model 
There  are  two  players,  a  sender  S  and  a  receiver  R.  The  sender  has  private  information  on  the 
value  of  some  (state  of  the  world)  variable  t,  drawn  by  nature  from  the  compact  interval  T  =  [tl,  t2] 
Correspondence  to:  Jan  Potters,  University  of  Amsterdam,  Department  of  Economics,  Jodenbreestraat  8,  1011  NK 
Amsterdam,  the  Netherlands. 
*  Comments  by  Pierre  Pestieau,  Arthur  Schram  and  Frans  van  Winden  are  gratefully  acknowledged. 
0165-1765/92/$05.00  0  1992  -  Elsevier  Science  Publishers  B.V.  All  rights  reserved 44  J.  Potter.7  /  Fixed  cost  messages 
according  to  a  probability  distribution  F(t)  with  continuous  density  function  f(t).  After  learning 
the  value  of  t,  S decides  whether  or  not  to  send  a message  SEM  to  R, where  M  is  the  (measurable) 
set  of  feasible  messages.  [As  will  be  seen  in  section  3,  the  specification  of  M  is  immaterial  to  the 
equilibrium  outcomes  as  long  as  it  is  non-empty.]  A  sender  having  private  information  t  will  be 
referred  to  as  an  S  of  type  t.  If  S  decides  not  to  send  a message  this  will  be  denoted  by  s =  n.  After 
having  received  a  particular  ‘signal’  s  -  i.e.,  s EM  or  s =  n  -  R  chooses  an  action  x E IR  which 
affects  the  payoffs  of  both  R  and  S.  All  aspects  of  the  game  except  t,  which  is  private  information 
of  S,  are  common  knowledge. 
Let  U(X,  t)  denote  R’s  twice  continuously  differentiable  utility  function  over  the  consequences 
of  action-state  pairs.  It  is  assumed  that  U, ,(. > <  0,  u ,& .> >  0,  and  the  u,(x,  t)  = 0 for  some  x.  It 
follows  that  x(t)  := argmax,  t  ,R u(x,  t> is  increasing  in  t  and  that  R  cannot  be  induced  to  take  an 
action  outside  the  set  X:=  [x(t,),  x(t,)].  Let  4x,  t)  denote  S’s  utility  function,  with  u,(.)  >  0  for 
all  t E  T  and  x  EX.  Hence,  the  strategic  incentive  structure  of  the  game  is  such  that  every  type  of 
S has  an  interest  in  persuading  R  that  the  value  of  t  is  (likely  to  be)  ‘high’  since  that  will  induce  R 
to  take  ‘large’  action.  In  addition,  it  is  assumed  that  c,J  .)  >  0,  meaning  that  higher  types  have  a 
larger  stake  in  persuading  R.  It  is easy  to  show  that  no  information  transmission  can  take  place  if 
1:  ,2 <  0. 
It  is  assumed  that  sending  a message  bears  a  cost  to  S  but  not  to  R.  ’  The  central assumption  in 
our  model  is  that  this  cost  is  independent  of  both  the  ‘content’  of  the  message  (i.e.,  the  particular 
element  of  M)  and  the  private  information  f  of  S.  Sending  no  message  bears  no  exogenous  cost. 
Hence,  c(s)  =  0  if  s =  n  and  c(s)  =  c >  0  if  s EM. 
The  game  will  be  solved  by  means  of  the  Perfect  Bayesian  equilibrium  concept.  The  sender’s 
signalling  rule  is  denoted  by  a(s  I t>  and  gives  the  probability  that  S  type  t  sends  signal  s,  with 
s EMU  {n}.  The  receiver’s  action  rule  is  denoted  by  p(s)  and  gives  R’s  action  in  response  to  S’s 
signal.  [Note  that,  due  to  u,,  <  0,  R  will  never  play  a  mixed  strategy.]  Finally,  g(t  1  s)  denotes  R’s 
posterior  beliefs.  Formally,  p  and  u  constitute  a  Perfect  Bayesian  Equilibrium  (PBE)  if: 
(1)  for  each  t E  T, v(n  1  t)  +  ],a(m  It>  dm  =  1 and  if  (T(S I t> >  0 then  s  solves  max,,  ,.t,,[u(p(s), 
t> -  &)I, 
(2)  for  each  s,  p(s)  solves  max,,,/,u(x,  t)g(t  I s> dt, 
(3)  g(t  Is) = ds  I t)f(t)//Tv(s  I  t)f(t>  dt,  whenever  IT&  I t>f(t>  dt > 0. 
Condition  (1)  requires  that  the  sender’s  signalling  rule  is  a  best  reply  against  the  receiver’s  action 
rule.  Condition  (2)  says  that  the  receiver’s  action  rule  is  optimal  given  its  posterior  beliefs  about  t 
after  having  received  signal  s  and  (3)  requires  the  receiver’s  posterior  beliefs  to  be  Bayesian-con- 
sistent  with  its  prior  beliefs  f(t)  and  S’s  signalling  strategy. 
The  next  Lemma  shows  that  all  (sent)  messages  induce  the  same  action. 
Lemma  1.  All  messages  m E M  which are sent with positive probability  induce  the same  action. 
All  types  of  S  have  the  sames  preference  ordering  over  actions.  Hence,  the  messages  which  induce 
a  less  favourable  action  by  R  will  not  be  sent  in  equilibrium.  It  is  not  the  content  of  a  message  as 
such  that  discloses  information  but  merely  that  fact  that  a  message  is  being  received  (or  not). 
’  To  assume  that  a  message  imposes  an  exogenous  cost  on  R  as  well,  would  not  affect  the  equilibrium  strategies,  provided 
that  a  message  has  no  direct  impact  on  R’s  preferences  over  x  but  only  an  indirect  impact  via  the  posterior  beliefs.  Of 
course,  R’s  equilibrium  payoff  will  be  affected. .I. Potters  /  Fixed  cost  messuge.~  45 
Consequently,  nothing  is  lost  -  in  terms  of  R’s  equilibrium  actions  -  if we  henceforth  assume  that 
M  contains  only  one  element,  m  say. 
Before  characterizing  the  equilibria,  we  introduce  some  useful  notation.  Let  x(p,  4)  denote  R’s 
best  reply  given  that  p  2  t I  q,  that  is,  x(p,  q)  := argmax  X~X/;~(~,  z)f(t)  dt  if  p  <q,  and  X(P,  P) 
:=x(p).  Let  G(t)  denote  the  utility  gain  for  type  t  from  pooling  with  all  higher  types  instead  of  all 
lower  types,  that  is,  G(t)  :=  L’(x(~,  tz),  t)  -  u(x(t,,  t),  tX>  0). The  next  proposition  establishes  the 
possibility  of  an  informative  non-pooling  equilibrium  and  demonstrates  that  it  is  a  partition 
equilibrium  in  which  the  S  types  seperate  in  two  groups. 
Proposition.  If for  some  type p,  G(p)  = c,  then  u(n  I t)  =  1 for  all  t  <p,  u(m  I t)  =  1 for  all  t >p, 
p(n)  =x(t,,  p)  and  p(m)  =x(p,  tz)  is a  PBE. 
The  higher  types  (t  >p)  send  a  costly  message,  whereas  the  lower  types  (t  <p)  do  not.  Type  p  is 
indifferent  between  sending  a  message  or  not.  It  follows  that  p(n)  <x(t,,  t,)  <p(m);  both  a 
message  (m)  and  silence  (n>  make  a  difference  relative  to  the  prior  beliefs.  Moreover,  both  p(n) 
and  p(m)  increase  with  p,  that  is,  decrease  as  the  set  of  types  that  send  a  message  becomes  larger. 
The  proof  of  the  Proposition  follows  by  simple  verification  of  the  equilibrium  conditions. 
Moreover,  it  is  straightforward  to  proof  that  any  non-pooling  equilibrium  must  be  a  partition 
equilibrium  of  size  two.  However,  a  type  p  with  G(p)  = c  need  not  exist.  It  follows  from  the 
continuity  assumptions  that  G(t,)  <  c  <  G(t,)  is  a  sufficient  conditions  for  existence.  However,  it  is 
easily  checked  that  with  G(t,)  < c,  it  is  a  PBE  for  no  type  to  send  a  message,  a(n  ( t> =  1 for  all  t, 
and  for  R  to  respond  with  p(n)  =.x(tl,  t2)  and  p(m)  =x(t,).  In  this  case  the  cost  of  a  message  is 
prohibitive.  Even  the  highest  type  -  the  type  with  the  ‘good’  information  -  prefers  to  be  disguised 
rather  than  to  send  a  revealing  but  costly  message.  And  with  G(t,)  > c,  it  is  a  PBE  for  all  types  to 
send  a  message,  a(m  1  t> =  1 for  all  t,  and  for  R  to  respond  with  p(m)  =x(t,,  tz)  and  p(n)  =x(t  ,). 
In  this  case  even  the  lowest  type  prefers  to  send  a  costly  but  concealing  message  rather  than  to 
disclose  its  information  by  being  silent.  All  types  send  a message  which,  therefore,  has  no  impact  on 
R’s  beliefs  relative  to  the  prior  beliefs.  In  a  sense,  a  message  is  ‘too  cheap  talk’. 
Finally,  we  note  that  even  if  a non-pooling  PBE  exists  it  need  not  be  the  unique  PBE.  Of  course, 
the  shape  of  G(t)  is  crucial  in  this  respect,  and  without  any  additional  restrictions  on  the  utility 
function(s)  and  prior  beliefs,  nothing  can  be  said  about  it  in  general.  A  sufficient  condition  for 
uniqueness  is  that  G’(t)  >  0  for  all  t.  For  instance,  with  U(X,  t> =  u(x  -  t)  and  F(t)  uniform,  we 
always  have  uniqueness  [cf.  Crawford  and  Sobel  (198211.  We  will  come  back  to  this  in  the  next 
section. 
3.  Examples  and  comparative  statics 
Unambiguous  comparative  statics  results  cannot  be  derived  for  the  general  case.  As  the  next, 
somewhat  contrived,  example  shows,  it  may  even  be  the  case  the  more  types  send  a  message  if  the 
exogenous  cost  of  sending  a  message  increases! 
Example  I.  Let  ,R’s  and  S’s  utility  function  be  given  by  u(x,  t)  =  -(x  -  t)2  and  c(x,  t)  =  (t  +  2)x, 
respectively,  and  let  the  prior  beliefs  be  F(t)  =  t2,  with  t E  [0,  11. Note  that  u, r <  0,  u,~  >  0,  I:~ >  0, 
and  cl2  >  0,  are  required.  Simple  calculation  reveals  that  x(0,  t)  =  +t  and  x(t,  1)  =  $(t’  +  t  +  l)/(t 
+  1).  Hence,  x(t,  1) -x(0,  t> =  $(t  +  1))’  and  G(t)  =  :(t  +  2)(t  +  1))‘.  It  follows  that  there  is  a 
P  E  (0,  1)  with  G(p)  = c,  if  1 <c  <  $.  In  this  case,  a(n)  =  1,  for  t <p,  a(m)  =  1,  for  t >p, 
p(n)  =x(0,  p)  and  p(m)  =x(p,  l),  with  p  = ($  -  c)(c  -  $)-‘,  is  a  PBE.  Moreover,  it  follows  that 46  J. Pottm  /  Fixed  cost  messages 
dp/dc  <  0.  Hence,  more  types  will  send  a message  if the  cost  of  doing  so  increases.  However,  since 
G(t,)  =  G(1)  =  1 <  c,  it  follows  that  (~(n  ( t)  =  1 for  all  t,  and  p(n)  =x(0,  1) =*/3,  p(m)  =x(t,>  is 
also  a  PBE. 
It  is  not  trivial  to  specify  conditions  for  the  utility  function(s)  and  prior  beliefs  which  ensure  that 
the  occurence  of  a  message  is  less  likely  if  the  cost  of  a  message  increases.  At  a  non-pooling 
equilibrium  it  holds  that  G(p)  =  c,  and  hence,  that  dp/dc  =  [G’(p)]-‘.  Consequently,  dp/dc  >  0 
if  and  only  if  G’(p)  =  u,(x(p,  t2),  p)[dx(p,  t,)/dpl  -  u2(x(t,, P),  p)[d(x(t,,  p)/dpl  >  0.  Since 
u ,* =  LIZ,  >  0,  a  sufficient  condition  for  this  inequality  to  hold  -  and  for  uniqueness  of  the  PBE  -  is 
that  A’(t)  2  0,  for  all  t E  T,  where  A(t)  :=x(t,  t,)  --x(t,,  t).  In  turn,  a  sufficient  condition  for  the 
latter  inequality  to  hold  is  that  u(x,  t> = u(x  -  t>  and  F(t)  is  uniform  [cf.  Crawford  and  Sobel 
(1982)].  Moreover,  in  this  case  it  follows  that  (dp(m)/dp)  dp/dc  >  0,  and  A’(p)  dp/dc  >  0, 
meaning  that  the  impact  of  a  message  relative  to  prior  beliefs  and  relative  to  silence,  increases  as 
the  cost  of  a  message  increases. 
Example  2.  Let  F(t)  be  uniform  on  T =  [0,  11 and  let  R’s  and  S’s  utility  function  be  u(x,  t)  =  -(x 
-  t>*,  and  U(X,  t)  =  ptx,  with  p  >  0.  Now,  x(p,,  p2)  =  $(p,  +p2)  and  G(t)  =  $t.  The  definition 
of  p,  G(p)  =  c,  implies  that  p  =  2c/p.  Hence,  existence  and  uniqueness  of  a  non-pooling 
equilibrium  are  ensured  if  0 <c  <  ip.  Moreover,  we  see  that  more  types  will  send  a  message  if  c 
(the  cost  of  a  message)  decreases  or  if  p  (the  mariginal  utility  of  x  to  S)  increases,  but  at  the  same 
time,  since  p(m)  =x(p),  1) =  i(p  +  1) =  c/p  +  $,  such  changes  lower  the  impact  of  a  message. 
The  result  that  more  types  send  a  message  if  S’s  stake  in  persuading  R  increases  (dp/dp  <  0) 
can  easily  be  generalized.  Introduce  a  parameter  /3,  such  that  L’,,(x,  t;fi)  :=  a~,(.>/@3  > 0. By 
(aG/ap)  dp  +  (aG/$?)  dp  =  0  and  PG/ap  >  0,  it  follows  that  sign  (dp/dp)  =  -sign(aG/ap)  = 
-sign(u,(x(p,  t,>,  p;  /3) -  c,(x(t,,  P>,  p;  P)).  Since  1’  3, =  L’,~ >  0  and  x(p,  t2)  >x(t,,  p),  we  have 
dp/dp  <  0.  A  message  is  more  likely  to  be  sent  if  there  is  more  at  stake  for  S.  At  the  same  time, 
since  dp(m)/d/?  <  0,  R’s  response  to  message  is  negatively  related  to  S’s  stake  in  persuading.  This 
result  is  supported  by  the  experimental  observation  [see,  e.g.,  Tedeshi  et  al.  (1973,  Ch.  4)]  that  the 
persuasiveness  of  messages  decreases  with  the  ‘source’s’  stake  in  persuading  the  ‘target’.  Further- 
more,  it  is  consistent  with  the  last  part  of  Gross’s  (1971,  p.  269)  assertion  that  communications  may 
be  influential  ‘particularly  when  there  is  confidence  in  the  wisdom  or  disinterestedness  of  the 
proposers  or  advisers’. 
Moreover,  the  result  that  the  impact  of  a  message  is  increasing  in  c,  is  in  line  with  the 
observation  that  the  cost  or  trouble  of  writing  letters  is  positively  related  to  their  impact  on 
legislators  [Bauer  et  al.  (1963,  p.  439)l.  Further  tentative  support  for  this  result  can  be  found  in  Van 
der  Putten  (1980).  After  a  thorough  investigation  of  the  realization  of  some  important  policies  in 
the  Netherlands,  he  concluded  inter  alia  that  reports  from  official  advisory  councils  had  a negligible 
impact  on  the  policy  process,  whereas  messages  and  reports  from  unofficial  advizing  agents  did 
have  a  substantial  impact  on  the  policy  process,  especially  in  those  cases  where  they  were  not 
invited  by  policymakers  to  give  advice.  Since  official  advisory  councils  have  easy  access  to 
policymakers  and  are  often  even  invited  to  send  a  report,  they  bear  a  lower  cost  of  sending  a 
message  than  the  agents  (interest  groups)  which  have  to  take  own  initiative  in  getting  their 
uninvited  messages  accross  to  policymakers.  Therefore,  the  higher  costs  of  the  latter  messages 
could  be  (part  of)  an  explanation  for  their  greater  impact. 
Finally,  we  note  that  the  receiver’s  ex  ante  expected  utility,  E[ul  := l,C,,,,,u(p(s),  t)v(s  I t) 
f(t)  dt  is  a function  of  p,  and  consequently,  of  the  incentive  structure  of  the  game,  in  particular,  of 
c.  In  Example  2,  E[u]  =  F(p)Var[t  / t <p]  +  (1  -  F(p))Var[  t I t >p]  is  maximized  by  p  =  i,  that  is, 
if  c =  ip.  Hence,  if  R  could  influence  or  manipulate  c  then  R  will  generally  have  an  incentive  to J.  Potters  /  Fixed  cost messages  47 
do  so.  In  particular,  R  could  try  to  extract  an  ‘optimal  amount’  of  information  by  being  more 
accessible  or  hospitable  (a  low  c)  for  messages  from  less  interested  senders  (a  low  p). 
References 
Bauer,  R.,  1. De  Sola  Pool  and  L.  Dexter,  1963,  American  business  and  public  policy  (Aldine  Atherton,  Chicago,  IL). 
Cho,  1. and  J.  Sobel,  1990,  Strategic  stability  and  uniqueness  in  signaling  games,  Journal  of  Economic  Theory  50,  3X1-410. 
Crawford,  V.  and  J.  Sobel,  1982,  Stategic  information  transmission,  Econometrica  50,  1431~1451. 
Gross,  B.,  1972,  Political  process,  in:  The  international  encyclopedia  of  the  social  sciences,  Vol.  12  (Collier-Macmillan, 
London). 
Tedeshi,  J.,  B.  Schlenker  and  T.  Bonoma,  1973,  Conflict,  power  and  games  (Aldine,  Chicago,  IL). 
Van  der  Putten,  J..  1980,  Haagse  machten  (The  Hague’s  powers)  (Staatsuitgeverij,  The  Hague). 

Fixed cost messages.

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