J. Watson

P. Battigalli

English

Research Papers in Economics

On "Reputation" Refinements with Heterogeneous Beliefs
Author(s): Pierpaolo Battigalli and Joel Watson
Source: Econometrica, Vol. 65, No. 2 (Mar., 1997), pp. 369-374
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/2171897
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http://www.jstor.orgEconometnica,  Vol. 65, No. 2 (March,  1997),  369-374 
NOTES  AND  COMMENTS 
ON "REPUTATION"  REFINEMENTS  WITH HETEROGENEOUS  BELIEFS 
BY  PIERPAOLO  BATIGALLI  AND JOEL WATSON1 
CONSIDER  A REPEATED  GAME with incomplete  information  in which a patient long run 
player,  whose type is unknown,  faces a sequence  of short  run opponents  (as in Fudenberg 
and Levine  (1989)).  The standard  "reputation"  result is that the patient long run player 
can obtain an average  long payoff almost equal to the Stackelberg  payoff of the stage 
game by consistently  playing  as a Stackelberg  leader.  In the analysis,  one generally  takes 
as fundamental  an assumption  that the players  have common  prior  beliefs on the states 
of  the  world and that behavior is  consistent with the  concept of  Bayesian Nash 
equilibrium.  However, these related suppositions  have been called into question as 
unrealistic  and too stringent  (cf. Gul (1991)).  In many  games  of incomplete  information  a 
player's  prior  probabilities  on types (or states of the world)  are best regarded  as purely 
subjective  psychological  parameters,  unknown  to the modeler  and to this player's  oppo- 
nents. Therefore,  it is important  to understand  whether  the standard  reputation  results 
(among  others)  are implied  by weaker  assumptions  on the knowledge  and behavior  of the 
players.  In fact, as Watson  (1993) demonstrates,  the reputation  result does not require 
equilibrium.  It is implied  by a weak notion of rationalizability  with some restrictions  on 
the beliefs of the players. 
Here we qualify  Watson's  (1993)  study  and extend  the line of inquiry  of Watson  (1993) 
and Battigalli  (1994) concerning  settings in which reputations  are effective.  As Watson 
shows,  two main conditions  on the beliefs of the players,  along  with weak rationalizabil- 
ity, imply  the reputation  result.  First,  there must be a strictly  positive  and uniform  lower 
bound  on the subjective  probability  that players  assign  to the "Stackelberg  type."  Second, 
the conditional  beliefs of the short  run players  must  not be too dispersed.  Watson  (1993) 
does not explicitly  indicate on what the updated beliefs of the short run players are 
conditioned.  We make this explicit and show that it is necessary  to assume that the 
conditional  beliefs of the short  run  players  satisfy  a stochastic  independence  property  (cf. 
Battigalli  (1996)).  We also comment  on the dispensability  of equilibrium  regarding  the 
reputation  result in games  with two long run players. 
1.  THE  PERTURBED  REPEATED  GAME  MODEL 
A finite two-player  stage game G = (A1,  A2; u1, u2} is infinitely  repeated. Let at = 
(at, at)  EA1 xA2  denote the pair of actions chosen in period t. A fixed individual  with 
objective  function  E,=  15  1ul(a')  plays  in the role of player  1. It is assumed  that player  1 
is patient  (i.e. her discount  factor 8 is arbitrarily  close to one). Player  1 faces a sequence 
of short  run opponents  2i, i = 1,  2,....  Player  2i's payoff  function  is U2(a9). ("2"  refers  to 
the set of short run opponents.) 
Watson  thanks  the National  Science  Foundation  for Support  (SBR-9422196). 
369 370  P.  BATTIGALLI AND  J.  WATSON 
Suppose for simplicity  that player 2  has a  single-valued  best response function 
BR: A1 -+A2. Player  l's Stackelberg  (stage  game)  payoff is 
u* =  max ul(al,  BR(al)). 
a1 EA1 
Player  l's Stackelberg  action is a fixed action a* which attains  the maximum  above;  i.e. 
u* = ui(a*,  BR(a*')). 
Let H, denote the set of histories through  period t. That is, Ht = (A1 xA2)t.  Let 
H=  U0  lHt, where Ho={+}  is  the  singleton containing the  empty initial history. 
Assuming  perfect  monitoring,  a strategy  for player  1 is a mapping  sl : H -+A1.  A strategy 
for player 2i is a mapping  A: Hi -*A2. Player l's strategy  space is denoted Sl, while 
player 2i's strategy  space is denoted S'. The set of strategy  profiles is thus S -  S, x 
n,= iS?  Let  2)-(A1  xA2)  be the set of infinite  histories.  Each  profile  s E S induces  a 
unique  history  h (s) E H,,.  For any  finite  history  ht E Ht, let S1(ht)  be the set of strategies 
for player  1 that are consistent  with history  h,. That is, s, E S1(ht)  if and only if there is a 
profile s-1  of strategies  for the players 2 such that h,  is the t-period truncation  of 
hoo(sl,  s_ 1). We focus on the set H:  = ({al*)  xA2)  of histories  in which  player  1 always 
plays  the Stackelberg  action.  Let Ht* =((a*) xA2)t and let H*  U,=o H 
At each information  set in the game (after all histories),  the players  entertain  beliefs 
about the strategies  of the others. For example,  conditional  on a given history,  each 
player  2i forms  a belief about  the strategies  of player  1 and the other  players  2. We focus 
on each player  2i's conjecture  concerning  the strategy  of player 1. Formally,  conditional 
on each history  h E H, player  2i's conjecture  is ,u4  E AS1,  a probability  distribution  over 
player l's strategies.2  Note that AO'  is this player's  conjecture  at the beginning  of the 
game. Player  2i also forms  conditional  beliefs about  the strategies  of the other players  2, 
but we do not model these directly.  Below we assume that the beliefs of the short run 
players obey Bayes' rule and have a  stochastic independence property.  (Our main 
contribution  is in making  the latter condition  explicit.) 
We consider  a perturbed  version of this repeated  game in which player 1 may be a 
"crazy"  type committed  to play a given strategy.  In particular,  player 1 may be the 
Stackelberg  type playing the strategy s*, where s4(h) =a  for all h e H. The set of 
possible types is otherwise  unrestricted.  Since the opponents'  payoffs are not directly 
affected by player l's  type, we do not need to model the type space explicitly.  We 
implicitly assume that player l's  opponents have some subjective  joint probability 
measure on player l's  types and strategies,  which assigns positive probability  to the 
Stackelberg  type and we consider  the marginal  on the strategy  space.3  Furthermore,  we 
assume that there is a common  lower  bound to the subjective  prior  probabilities  of the 
Stackelberg strategy (type): there is some real number E E (0,1),  known to player 1, such 
that 
(1)  +4({s*))  >  e  for each i. 
2 Endow  S1 with the o-algebra  induced  by finite  histories. 
3 For a discussion  of how the crazy  types  are interpreted,  see Watson  (1994). "REPUTATION"  REFINEMENTS  371 
2.  STOCHASTIC INDEPENDENCE 
Note that, although  the payoff  of a short run player  2i is not affected  by the behavior 
of other short run players,  in general player  2i's expectations  are given by probabilistic 
beliefs on the set of strategy profiles S1 x S2 X  *.  x S2- 1 x S2+ 1  ...  . These beliefs might 
exhibit correlation  across players. Even if they are ex ante uncorrelated,  conditional 
beliefs may  exhibit  correlation  after  a zero-probability  history.  This  means  that player  2i's 
conditional  probability  that player  1 is playing  a given  strategy  s, may  be directly  affected 
by previous  actions of some players  2j (j < i). We exclude this possibility  by assuming 
that expectations  satisfy  the following stochastic  independence  property:  for each player 
2i, each history  h E H, and each T1  c S1, 
(2)  .L(T1),4(S1(h))  =  14(T1 n Sl(h)). 
This assumption  states that, after histories  consistent  with his/her initial conjecture, 
player 2i's conditional  conjectures  about player l's  strategy  (a) do not depend upon 
expectations  about  the behavior  of the other short  run  players  and  (b) are consistent  with 
Bayes' rule. Obviously,  this condition must be satisfied for every positive probability 
history  if prior beliefs are uncorrelated.  But stochastic  independence  requires  that the 
condition  hold even if h contains  some unexpected  actions  by other short run players.4 
3.  REPUTATION 
In  this section we  clarify the  role  of  stochastic independence in  extending the 
reputation result of  Fudenberg and Levine (1989) to  a  nonequilibrium  framework 
without common  priors (Watson  (1993)). Given player 2i's conditional  beliefs and any 
history  h E H, let wih be the probability  that this player  assigns  to player  1 selecting  the 
Stackelberg  action al  after history  h. (Formally,  iih = p4'(Sl(h,  (al,a*))), where a2 eA2 
is arbitrary).  For any set X, let #X  denote the cardinality  of X. 
LEMMA  1 (cf. Fudenberg  and Levine  (1989)5):  Take  any  infinite  history  h,. E H* and  for 
each t let h, E Ht* be the t-period  truncation  of h.. If player  2i's beliefs  satisfy  conditions  (1) 
and (2), then  for all  E (0, 1), 
ln e 
(3)  #  <  {}  <  ln: 
PROOF:  Note  that  since  A4({s)  > 0  and  s* e 
Sl(ht+ 
1),  it  is  the  case  that 
,I(S1(h,+ 1))  > 0. Using this, along with the stochastic  independence  property,  we have 
that  Tiht = pit(S1(ht+ 1))  = ,4(S1(h,+ 1))/,41L(S1(ht)).  Note  that  S1(h0)  = S1  and  so 
,I.4(S1(h0)) = 1. Thus, using the equation  for  /iht, we have Hlk  0=rit  = /I4(Sl(hk+ )),  for 
each positive integer k. Since s* e Sl(hk+ 1)  for all k, our lower bound assumption 
implies  that /4(Sl(hk+  e))  2  E. Thus H= 0r>ht > e. Suppose  wiht <  ;  for K integers.  Since 
ihte  [0,1] for all t, it must be that  eK2e.  Taking logarithms  establishes  the result. 
Q.E.D. 
4 This formulation  is sufficient  for the purposes  of this note. A more complete formulation  in 
terms  of conditional  probability  systems  is put forward  in Battigalli  (1994, 1996). 
SFudenberg and Levine  (1989)  obtain equation  (3) for Stackelberg  histories  on the Bayes-Nash 
equilibrium  path,  while  we consider  arbitrary  Stackelberg  histories  which  may  have zero probability 
according  to the subjective  prior  ,u'. 372  P.  BATTIGALLI AND  J.  WATSON 
Stackelberg  Normal 
{E}  o  { 1-e} 
a*  a*  a 
T  3  TI  I  T1 
a*  ~a*  al  TI  1-TI 
22 
FIGURE 1.-Part  of the extensive  form  of the multistage  game.  Beliefs  are obtained  in the limit  as 
1-+  0. 
The proof makes  clear that under  the stated assumptions,  along any Stackelberg  path, 
the conditional probability  of  the Stackelberg  strategy (or type) is a  nondecreasing 
function  of time and  the conditional  probability  of the Stackelberg  action  is never  smaller 
than e. The example  depicted  in Figure 1 shows  that these properties  need not hold if 
the stochastic  independence  assumption  is violated. Consider  the conditional  probabili- 
ties obtained as the "correlated  nq-trembles"  depicted  in the figure  become negligible.6 
Under these conditional  probabilities,  player  22  sees player  21's choice of action 1 as a 
signal  that player  1 is the normal  type. Player  1 may  not be able to establish  a reputation 
when the players  2 hold such (unreasonable)  beliefs. Therefore,  Watson's  (1993) reputa- 
tion result does not hold if the stochastic  independence  assumption  is not satisfied.  The 
version  of Watson's  result stated  below makes  this assumption  explicit. 
Let v denote player l's prior expectation  regarding  the strategies  of the players  2. 
Then player  l's supremum  expected  payoff  is given by 
F001 
w1(V)=  Sup  (  [(1-)  at  lul(at(sl,s1))]v(dsi)J 
slES,  S-1  t=1 
where at(s) is the pair  of actions  induced  by profile  s in period t. The following  theorem 
shows  that if the short run opponents'  conditional  expectations  are not too diverse  and 
satisfy conditions (1)  and (2), then a  patient player 1 is  able to  get  (almost) the 
Stackelberg  payoff  in the long run. 
THEOREM 1 (cf. Watson (1993)): Fix e > 0. Let A be a set of systems  of conditional 
probabilities  /  =  ( /h)he  H  satisfying  conditions  (1) and (2). Furthermore,  assume  that A is 
compact  with  respect  to the  following  quasimetric:7 
d(  ,  )=  sup {17 h_  1) 
heH* 
6 Such conditional  probabilities  can be represented  as part of an extensive form assessment 
(behavior  strategies  and beliefs)  satisfying  Bayes'  rule wherever  possible. 
7The compactness  assumption  can be relaxed.  See the discussion  of a more general  assumption 
in Watson  (1993). "REPUTATION"  REFINEMENTS  373 
Assume that player 1 believes  that each short run opponent  maximizes  his conditional 
expected  payoff  given  some ,u  E A. Then  there  is a positive  integer  K,  which  depends  on A 
and e, such that 
(4)  wl(v)  2  5Ku* + (1  -  8K)  min  u(a,  a2). 
a2E A2 
PROOF: Watson's  (1993) proof invokes the lemma of Fudenberg  and Levine (1989) 
that we have generalized  here. It is enough to notice that stochastic  independence  is 
implicit  in Watson's  analysis.  One can then substitute  our lemma for that of Fudenberg 
and Levine  and  then follow  Watson's  proof.  Q.E.D. 
4.  REPUTATION  WITH  A  LONG  RUN  OPPONENT 
We wish to point out that the reputation  result also holds in a weak rationalizability 
setting  for games in which  both players  1 and 2 are long run players.  It is not difficult  to 
extend the result of Schmidt  (1993) on games of "conflicting  interests"  to the setting of 
weak rationalizability.  In such games,  player  l's Stackelberg  action holds player  2's stage 
game payoff to  the minmax level. Suppose player 2 believes with some probability 
14(s1)  2  e  that player 1 adopts  the Stackelberg  strategy.  Further  assume that player 1 
knows  this and believes that player  2 maximizes  his/her expected  payoff  given his/her 
belief. Then, letting 8  be the discount  factor  of player 1, we obtain  inequality  (4) again, 
where K  depends  on E and the discount  factor  of player  2.8 
We should note that this result can also be generalized  along the lines of Cripps, 
Schmidt,  and  Thomas  (1993),  who  find  bounds  on equilibrium  payoffs  in general  two-player 
repeated games. (Schmidt  (1993, p. 344) describes a version of this extension.) The 
bounds  are of the same form,  in that a player  can only  establish  a reputation  for playing  a 
particular  action each period,  but the bounds  are weaker  than the "Stackelberg"  variety. 
Watson  (1996) uses a different  methodology  to show that players  can establish  reputa- 
tions for using  more  complicated  strategies  in a nonequilibrium  context,  when  players  use 
"forgiving"  strategies. 
Dept. of Economics, Princeton University,  Princeton, NJ 08544-1021, U.S.A. 
and 
Dept. of Economics,  University  of Califomia, San Diego, 9500 Gilman Dr., La 
Jolla, CA 92093-0508, U.S.A. 
Manuscript  received  May,  1995;  final  revision  received  April,  1996. 
REFERENCES 
BATIiGALLI,  P. (1994): "Giochi ad Informazione Incompleta con Aspettative Soggettive Mutuamente 
Incoerenti," mimeo, Politecnico  di Milano. 
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Theory,  70, 201-234. 
CRIPPS, M.,  K  SCHMIDT, AND J.  THOMAS  (1993):  "Reputation  in  Perturbed  Repeated  Games," 
University of Bonn Discussion Paper A-410. 
8 An earlier version of this paper contains the details. 374  P.  BATTIGALLI AND  J.  WATSON 
FUDENBERG, D., AND D. K. LEVINE (1989):  "Reputation  and Equilibrium  Selection  in Games  with a 
Patient  Player,"  Econometrica,  57, 759-778. 
GUL,  F. (1991):  "On the Bayesian  View in Game Theory  and Economics,"  Research  Paper  #1991, 
Graduate  School  of Business,  Stanford  University. 
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Conflicting  Interests,"  Econometrica,  61, 325-351. 
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(1994):  "Strategy  Pertubations  in Repeated Games as Rules of Thumb,"  UCSD Working 
Paper  94-20. 
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On 'Reputation' Refinements with Heterogeneous Beliefs

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