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RATIONAL BEHAVIOR WITH PAYOFF UNCERTAINTY
Eddie Dekel
and
Drew Fudenberg
Number 471
October 1987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
RATIONAL BEHAVIOR WITH PAYOFF UNCERTAINTY
Eddie Dekel
and
Drew Fudenberg
Number 471
October 1987
Rational Behavior with Payoff Uncertainty
Eddie Dekel and Drew Fudenberg*
October 1987
ABSTRACT
The
iterated deletion of
weakly dominated
However,
requirement for 'rational' play.
the players have
no doubts about
doubts are introduced,
deletion of
all that
weakly dominated
strongly dominated.
their
strategies has
this
been advanced as
necessary
a
requirement relies on the assumption that
opponents payoffs.
We
show
can be justified by an appeal to rationality
strategies, followed
by
once such
that
one round of
is
iterated deletion of those
which are
This extends the Fudenberg, Kreps and Levine (1987) study of the
robustness of Nash equilibrium refinements to the robustness of solution concepts based
only on
common knowledge
of rationality.
between various notions of what
*
Univerisity of California, Berkeley
work was begun while
the
means
for
and Massachusetts
second author was
port from the Miller Institute and the
it
NSF
is
at the
Our
results
also
payoff uncertainty
Institute of
clarify
to
2009749
relationship
be 'small'.
Technology, respectively.
University of California, Berkeley.
gratefully acknowledged.
the
This
Financial sup-
Introduction
Nash equilibrium and
tainty," in the sense that
regarding
how
the
game
refinements describe situations with
its
each player knows and
will
be played. While
is
little
or
no
"strategic uncer-
correct about the beliefs of the other players'
sometimes be the case,
this will
it
also interesting
is
to
understand what restrictions on predicted play can be obtained using only the assumption that
is
common knowledge
that the
players are rational.
Bemheim
(1984) and Pearce (1984) have
argued that these restrictions are captured by the concept of rationalizabilily.
notion
that
is
of iterated deletion of strongly dominated strategies, which
rationalizabilily.^
games
While (correlated)
has been argued that
it
extensive forms
it
rationalizabilily
does not capture
(Bemheim (1984, Section
6(b)),
all
be
that
is
the
be appropriate for generic normal form
the implications of 'rationality' in non-irivial
common knowledge
of rationality might seem
given by backwards induction.
Nash equilibrium refinements
are not
'robust'
in the
unreasonable.
Now
the
way
a player should
on how he expects the opponents
sistent with the players initial
for the deviation
is that
some out of equilibrium play
If ihe
observed play
lo date is not
understanding of the game, one plausible inference
the deviator's payoffs are different than
that
They then characterize
satisfy strong equilibrium refinements)
Extensive form
players
entertain
is
Werlang (1984).
is
discussed in
Aumann
small
their
ex-ante
which can
doubts
of rationality, and equilibri-
(1987), Brandenburger and Dekel (1987),' and
about
"justified"
for different classes of such doubts.
common knowledge
reason
FKL
Correlated rationalizability, in contrast to rationalizabilily, does not impose the restriction thai
each player believes the other players' strategy choices are independent. The relationship between
solution concepts
that ihe
con-
had originally been supposed.
the sets of equilibria
by allowing
these rationalizabilily concepts, formal definitions of
as
respond to a deviation by hisAier opponents depends
to play subsequently.
model these inferences by supposing
opponents' payoffs.
FKL) have argued
following sense.
refinements succeed in restricting the set of outcomes by rejecting
um
more general
equivalent to correlated
Recently, Fudenberg, Kreps and Levine (1987, henceforth referred to as
that
A
Pearce (1984, Section 4)) For example in games of
perfect information the only solution consistent with
to
may
it
Tan and
(made
to
-2-
The question of what players should
infer
from behavior they did not expect
restricted to equilibrium analysis: Rosenthal (1981),
Reny
(1985), Basu (1985) and
In this paper
alone.
we
adopt the
the payoffs are different than
FKL
not
of rationality
explanation that the reason for the imexpected play
had been supposed. Thus we are
is
Binmore (1987)
common knowledge
discuss this issue in the context of solution concepts based on
occur
to
is
that
led to characterize the implications
of small imcertainties about the payoffs for the predictions that can be based on the assumption of
We
'rational' play.
maintain that the assumption of payoff uncertainty
here than in the equilibrium context.
assume
that the payoffs are
This
is
if
anything,
because correlated rationalizability and
common knowledge,
its
more
apt
refinements
but allow the players to have imprecise and incon-
sistent beliefs (inconsistent in the sense that they
many
is,
may
Yet
disagree) about each others play.
situations with substantial strategic uncertainty, the
common knowledge
in
of payoffs assump-
tion is suspect as well.
There are two modeling issues which need
First, a
to
be considered
in
order to achieve our objectives.
sharp notion for the implications of rational behavior must be given for the games with
small doubts.
We
chose the notion of iterated deletion of weakly dominated
clearly incorporates the intuitive objectives of rationality postulates.^
strategies'^ since
The second modeling
related to the assumption of consistency. In the rationalizability approach to
modeling
uncertainty players are allowed to have inconsistent beliefs about each others' strategies.
seems natural
to
allow them to have inconsistent doubts about each others' payoffs, so
issue
it
is
strategic
Hence
it
we consider
both the case of inconsistent and consistent beliefs.
These modeling issues emphasize the
various solution concepts
given game.
a
sider.
Briefly,
is
fact that the
which sequence of games
in evaluating the
are to be considered
we
good approximations of
say that a sequence of games converges weakly to a limit
that the
if
we
each game
con-
in the
only difference between the games
This
is similar lo the use of strict equilibrium by FKL.
The relationship between backwards and forwards induction (two primary notions of
and weak dominance is discussed in Kohlbcrg and Menens (1986).
.
robustness of
Section 2 introduces our model and explains the notions of convergence
sequence has the same "physical extensive form," so
^
key question
rationality)
is
in the beliefs
the limit
about the payoffs, and moreover almost
game. The sequence converges strongly
all
types have almost the
almost
if
types have exactly the
all
weak dominance with
Section 3 proves our main result: The closure of iterated
strong convergence described above
the
weakly dominated
strategies.'*
The
strategies,
is
the set
we
call
same payoffs
\WIZ. This
set is
computed
same
payoffs.
respect to the
by deleting
first
and then continuing with iterated deletion of strongly dominated
intuition for this result is the following:
and so by our rationality postulate
will not
second round of deletion players must know
Each player knows his/her own payoffs,
In order to do a
choose a weakly dominated strategy.
that all the others will not
choose certain strategies.
small amount of payoff uncertainty caimot alter strong dominance relationships, but can break
ones, so that after the
round we can only proceed with the
iterated deletion of strongly
This result suggests reconsidering the intuition that since anything
inated strategies.
iterated deletion of
weakly dominated
anything might occm"
goes to
first
is
as in
strategies
is
appropriate.
The
point
is that if
may
A
weak
domoccur
the reason that
because of uncertainty about the payoffs, then iterated weak dominance
far.
Section 4 shows that
weak convergence
with respect to extensive form payoff perturbations.
also considers the closure of a slightly
more
IWIZ which
yields the set
To
facilitate
is
the closure of IVV/Z
comparisons with FKL, Section 4
restrictive version of iterated
weak dominance, namely
the iterated deletion of strategies that are never strict best replies. Section 5 discusses the altemative interpretation of the
how
unexpected, and
the
robusmess program
two interpretations
in terms
relate to
of
how
our two definitions of convergence.
more, using the notions of lexicographic beliefs derived
Dekel (1986)
it
is
* In
in
argued that the distinction between the two notions of convergence
IWIZ
two person games
lowing for correlation
—
cf.
Section 6 gives
this
coincides with Bernheim's (1984) extension of trembling hand perfec-
footnote
1.
For n person games
this differs
is
analogous
some examples
solution concept.
tion to the context of rationalizability.
Further-
Blume(1986) and Brandenburger and
between perfect and sequential equilibrium.
to the difference
help explain the
players interpret strategies which were
from Bernheim's notion by
al-
to
Perturbations, Elaborations, and Convergence
2.
Since this paper examines some implications of "small" amounts of payoff uncertainty, a crucial issue is to
consider what forms of uncertainty are small.
A
definitions for the convergence of sequences of games.
physical
the
is
common
those explicitly specified
cisely,
Y with
tree
e
we begin
U
=
{f\
formalized by using different
basic premise throughout the paper
with a
in
the given extensive form) are about each others' payoffs.
finite /
player
game of
perfect recall,
representative nodes y, terminal nodes
f:Z -^ R }
foT
each player
E.
information sets
z,
game E
This
t'
H, and
We
assume
of E,
for each possible^hoice
node y of Y then
just //'(}')
The
X
that
each player
move
is
/
utility
denoted by
at (j, r) for all
r
played. Each
summarized by
is
is
£
called
and
"personal
has one copy of
Y
move
at
Similarly ;'s information at node y
is
tree
?
e T = Yl^'-
e T.
t
is
utility function,
(This
function.
a utility function
£
informed of his/her
is
U. The game
identify T' with
by Nature, which
(s)he has a
of
If
pl^yer
/
has a
{;').
beliefs of each player
tional beliefs v'(-|r')
T'.
we can
game
have doubts about the
the players
that
which nature randomly chooses
in
receives no information regarding the other players'
types" in FKL.) Therefore
pre-
a utility function u'
and then an extensive form with the same structure as
e T'.
their
More
prescribes a
player's beliefs about the true payoffs, about his/her opponents' information, etc.
the players' type
that
/.
E
payoffs by considering "elaborations"
for each player,
is
knowledge, and the only doubts the payers entertain (other than
Following Harsanyi (1967-68), we model the idea
u'
is
form (who moves when and the players' information regarding
extensive
opponents' actions)
This
on the
set
/
arc derived
T~'
from
s FTr-' of
For technical reasons the measures p' and
pure strategies of player
/
in
game £
is
a prior p'
on the
set T,
which determines condi-
the other players types and marginal beliefs
v'
are
denoted S'.
assumed
to
have
finite
support.
The
^['
on
set
of
Player z"s mixed strategies are denoted by o'
€ Z' H A(5'), and beliefs over S~' are denoted by c~' e AiS~'\ where A(A')
is
the set of probabil-
ity
-
5
-
measures over X.
we
In general in this paper
denoted £„.
To
E
games
and sequences of elaborations of E,
distinguish between the strategy sets, utility functions, etc. in the elaborations
and the game £, we add
pure strategies of
citly as
will be considering
/
~ and
a
in £„.
To
£„
a subscript of n to the appropriate symbol, eg. S'^ denotes the
further
emphasize the distinction the game
an argument, eg. [W'{E„) denotes the
of weakly dominated strategies in the game £„.
set
of strategies of
When
may
be added
which survive
/
in expli-
iterated deletion
discussing a particular elaboration
£„
it
will
occasionally be necessary to refer to the utility functions or strategy choice of a player in a particular
version of Uie game, that
is
when each
player
is
This
of a particular type.
is
done by including
the type explicitly as an argument, eg. C^(f', /"') denotes the utility function for player
of type
f'.
Since this
from the notation.
utility
does not depend on the
version,
of the other players
Finally, the support of the limit of the beliefs
the support of the limit of the beliefs v^(-
Now we
tj^pes
|
r') will
ji'^
be denoted by m'(r')
which we
call
weak convergence, has
that the payoffs are "almost" as in the original
close utility functions, and the former
is
almost one to the payoffs being close
c
will
(The
latter
a probabilistic .statement
to those in the original
related notions of convergence are immediately apparent.
/
is
<z
T', and
T~'
be used.
the interpretation that each player
game.
when
will drop r~'
denoted by m'
is
can formalize the different forms of convergence which
we
:
is
The weakest
"almost" sure
"almost" requires a definition of
~
each player attaches probability
game).
One might
Two
suonger, and closely
require that the players are
"almost" sure that their payoffs are precisely as in the original game; or that they are absolutely
sure that the payoffs are "almost" as in the original game.
convergence and convergence
in payoffs respectively.
These two notions
will
In this section only strong
be examined, since the results are most intuitive and simplest
to
prove for
be called strong
convergence will
this case.
The other
notions, which will be discussed the next section, are important both for clarifying the relationship
of
this
paper with FKL, and
to help
understand certain issues related
to the re.sults in this paper.
In addition to the importance of distinguishing
between various notions of convergence,
it
is
important to consider the implications of assuming different restrictions on the information structure
For example,
of the games of incomplete information.
in the context
of consistent priors,
FKL
con-
sidered the implications of assuming that the players' beliefs over each others' types are indepen-
dent (whereas in the "personal types" model
p need
not be a product measure).
With independent
types player /'s observation of y's play can not effect /'s beliefs over k's type.
examine with care
the role of
assuming consistent priors
(p'
= p,
for
all
This
of our results hold with either consistent or inconsistent priors.
we
Interestingly, several
/).
is
In this paper
because the effects of
inconsistent priors over the payoffs can be duplicated by appropriately specified inconsistencies in
Thus, while the conceptual distinction between
the players' beliefs about each others' strategies.
strategic
uncertainty
(beliefs
about the strategies) and structural uncertainty (beliefs about the
payoffs and other parameters of the game)
tor'
in
Brandenburger and Dekel (1986) where
without loss of generality
is
assumptions about one of these kinds of imcer-
from assumptions about the other.
tainty cannot be separated
can be found
is clear,
whereas once consistency
is
when
it
is
Another discussion of similar issues
shown
that the existence
of a 'media-
beliefs over a state space are allowed to be inconsistent,
required this
is
In that paper a limited
no longer the case.
form of
consistency in the beliefs over the spaces of strategic uncertainty (namely the existence of a mediator) is
achieved by shifting the inconsistency to the beliefs over the state space.
the distinction
between Propositions
3.1
and 3.2 below
clarifies,
the
beliefs over the type spaces can be achieved (in Proposition 3.1) only
inconsistency over the beliefs over
ilie
strategy spaces.
So when
In this paper, as
consistency
the players
in
by incorporating
the latter
is
it
into an
ruled out (as in Pro-
position 3.2) the consistency of the beliefs over the type spaces can no longer be achieved.
In
order to state our main result strong convergence must be defined.
simpler for the case of consistent priors, so
the limit of
/j^
limit of the
common
(which
in
we
the consistent case
prior
/7„).
The
set
start
is
with that case.
f '.
is
definition
is
the support of
equal to the support of the marginal on T' of the
of possible types of
often be termed "sane" types iind denoted by
Recall that m'
The
/
according to the limit beliefs will
Clearly in the consistent case f'
= m'
DEFINITION
A
2.1:
sequence E„ of consistent elaborations of
E
converges strongly
to
E
(£„
—
£)if:
(a)
(i)
\T'(E„)\ <
<B
(b) \u'^\
(ii)For
all
7''
bounded
in n,
E
and
Note
the limit.
With
f~'.
where
all
this
u[(t'')
if:
(ii)
that
every "sane" type
for all
for all
e f',
Thus E„ -^
M
t'
(i)
/
/i,
and
=
n;
u'
the
number of types and
the set of types with payoffs different than those in
(with the obvious
E
has probability zero
in
because of the assumption of consistency the conditional beliefs v„(-|r') of
m
f' are that the other players are very likely
notion of convergence
versions in
the absolute value of the piiyoffs are uniformly
£
we
are treating as identical a
have the same payoffs as
mapping of
£.
in
be "sane," so
that m'{t')
c
game E and an
elaboration
E
to
So the two games
in
Figure
are identical
1
This way, each type plays a pure strategy, but
strategics of player 2).
a player can have a nondegenerate belief over the strategies of the other players (because beliefs
over their types
DEFINITION
lim
£
is
2.2:
nondegenerate) which
A
^i^,{t)^U[')
is
sequence of strategies
=
equivalent to them playing a mixed strategy.
d'„ will
be said
to
converge
to
a' (written
&'„
-^ a')
if
o'.
This notion of convergence requires that player
/'s
play converge to
cr'
at everj'
(even those which are not reached by g' regardless of the other players' strategies).
information
set
Iteraled
3.
A
u'(s'
is
strategy
a~') for
,
s'
all
is
weakly dominated
a~' e A{S~'), and
(i.e
Damme
there
is
the inequality
another strategy
for
is strict
such that u\s', a~') >
s'
some
<j~'
.
Any
strategy
which
s'
full support belief
the support of a~' is S~') over i's opponents strategies (Pearce (1984,
Appendix B), Van
is
said to be admissible, and
is
a best reply to
(1983), Gale and Sherman(1950)). Kohlberg and Mertens (1986) provide the following
argument
in support of iterated
weakly dominated)
their
if
if
some
not weakly dominated,
a~'
Weak Dominance and \WIZ
as a
weak dominance
minimal solution concept.
First note that if the players are imcertain
in fact that admissibility itself is a basic postulate
consequence of uncertainty about the environment.
under uncertainty which lead
to this postulate,
between strong dominance and expected
a
in
way which
is
analogous
to
Blume
provided
in
burger and Dekel (1986) and Luce and Raiffa (1957).) In any case
if
to
—
of decision theory
and not only a
Axiomatic characterizations of preferences
utility rationality, are
opponents payoffs, they should not expect them
of
(Kohlberg and Mertens
environment, they should never play a weakly dominated strategy.
(1986) argue
which are
(that is, ileratively deleting strategies
the relationship
(1986), Branden-
the players
know
their
play a weakly dominated strategy, and thus each
player should only play strategies which survive two rounds of deletion of weakly dominated strategies.
If the
denoted
W.
payoffs are
More
common knowledge
generally
kWlZ
is
used
this
argument leads
denote the
to
to iterated
set of strategies
remaining
of simultaneous deletion of weakly dominated strategies, followed by
strongly dominated strategies.
example IWIZ.
Each of
denotes the projection of
WTien k or
these sets
kWIZ on
is
/
is
infinity
there
would argue,
is
is
convenient
to
rounds
rounds of deletion of
use / (for iterated), for
kWlZ'
;'s strategy space.
sequence of ne:u-by games, and any strategy
if
is
/
after k
a Cartesian product of strategies for each player, so
Proposition 3.1 below says that any strategy' in
Thus
it
weak dominance,
in
IW
for
WVIZ
is
close to a strategy in
nearby games
is
IW
for
close to a strategy in
some
IWIZ.
"small" payoff uncertainty in the sense described by strong convergence (as,
typically the case) then ruling out
any strategy
in
IWIZ
is
questionable, even if
we
we
-9-
agrce to rule out
PROPOSITION
—>
E, and strategies
PROOF: Only
tI,
3.1: s'
=
[r',
f
'),
fW when
not in
all strategies
e IWIZ'(E)
rW{E„)
e
s'„
if
and only
such that
where «'(r) =
p
assigns probability
Thus,
ment
is
s'.
So
among
/
only
if
We
when
£„
in
claim
sane or
all
IW{EJ =
that:
Step J: is', s') € W{E„).
that
equivalent to a
£„
is
is
/
is
strategy choices.
The common
and for each player
r',
s;me and
hisAier play
crazy,
all
J'
full
is
1.
{{J'
,
s')
/
s'
J'
e 1U7Z'(£)
best reply to
d~'
support belief over S
.
~'.
when
Such
probability
c
So
crazy.
is
when
Since,
/
sane,
is
hisAier beliefs over S
~'
by ordered
this
ele-
first
(the
pairs
implies the
steps.
W'{E), there
(3"',^')
where the
eS'J Obviously
s'
the
all
that all the others are crazy
beliefs are denoted
€ 1U7Z'(£),
prior
the other players are crazy.
all
we can consider
Proving the claim involves two
Since
a
|
Let
constructed.
can either be a "sane" type (with payoffs as
written as an ordered pair {a'„, a'„)
is
sane, and the second
part of Proposiiion
A(5~') such
sequence of consistent elaborations E„
being of type
/
opponents strategies) as elements of A{S~') x A(S"').
a„"')
a
all his/lier
players) to the event that only
his/her opponents are either
(ct„"',
is
/(n-l)/(/(n-l)+l) and the conditional probability
is
play
I's
—>
0.
to all the players
a?
Player /'s strategy
l/(/(n-l)).
there
sane, Ihe conditional probability v'„{-\t') that (s)he assigns to the event that
is
/
players are sane
is
-
number of
the
when
1
=
and u'{t')
u'
E), or "crazy" and completely indifferent
is
si,
if
In this direction of the proof the sequence of elaborations
If:
in
l/nl (/
common knowledge.
payoffs are
exists a full support belief a"'
a best reply
For future reference
let
to
a be
(a~',a~') which
e
is
the smallest weight
assigned to any pure strategy s~' by a~'.
Step 2: If
need
to
(J', s')
show
superset of
e
nyiZ'(£) x S'
that (J', s')
IWIZ-'{E) x
is
then
(J'',
i'')
e 2W{E„).
a best reply to a full support belief
S"'.
Since j' e
lVyiZ'(£) there
is
This can be seen as follows.
over H''~'(£„), which by step
a a"'
e UV~'(£)
to
which
1
We
is
a
J' is a
10-
-
best reply. Specify that the sane types of the opponents play <y~' with probability
small and
is
specified below), and with
support distribution
ct"'
such
a strategy a'~'
over
that the
all
complementary probability P the sane types play any
the strategies in
IW'iE). The
opponents
/(/i-l)/(/(;j-l)+l)
(1-1/N)(1-/3)<t~'
is
(l-l/N){l-p)a~'
j3(A^-l)[CT~'
Step 2 can
now
if
will
be iterated
(l-l/A^);36-~'
to
full
crazy types of the opponents play
and a~')
show
This suggests that
strategy could be deleted
(For convenience set A'
.
The induced
+ {\/N)(r"' which we want
This
that if (I', s')
a~'
is
l/N.)
by
achieved
is
be a probability measure as long as
that in step 2 the fact that J'
J' is a best reply.
that
+
cr'~'
= l-\/N, and l/(/(«-l)) =
[l-(l-l/A^)(l_/3)]a"'.
- d~'] which
Remark: Note
which
+
is
weighted average (weighted by the probabilities of the crazy and sane
opponents, and of the sane opponents playing d~') of
s l{n-l). Then
1-^ {where p
j3
setting
a"'
equal
to
a~'
+
=
(s', s')
g rw'{E„).
fmding o~' e 1W"'(£)
to
not have found an elaboration to "justify"
s'
e liyiZ'(£) was used
by strong dominance.
payoffs should not be able to undo the iteration of
be
to
< a/{N-\).
e IWIZ'iE) x 5" then
we could
strategy for j's
strict
in
Intuitively,
"small" uncertainties about
dominance, so
should be necessary for a characterization of the "closure" of fH^.
This
that the
IWIZ
IZ step in
verified in the proof of
is
the "if direction below.
This direction also involves two steps.
If:
Step I:
is
e
si,
ni"(£J
a best reply to
2^v^(r~'
r
implies s\7') e
some
full
-»
1U"(£)
for
')cr~'(r~') w'hich is a full support belief
same
€
f.
This follows from the fact
support belief a~' over S~'. Hence x'„{r')
I
(s)he is of type t' is the
all 7'
a best reply to
is
over S~'. Since player /'s
as his/her utility function in
£,
clciirly 7'„ is
that J'„
utility
function
d~'
=
when
not weakly dominated in
£.
Step 2:
O'
'
=
s'„
€
nVlZ(£„)
implies s"„{7') g
2W(£).
y^5~'{r'')v^{r~' \t') for some 5~' which
is
We know
that J'„{7')
is
a best reply to
supported by strategies in U'~'(£„) since
some
J'„
e
11
-
As noted
2W'iE„).
earlier,
by condition
-
of the definition of convergence in types v„{t~'\t')
(ii)
converges to a measure supported by f "',
i.e.
same payoffs
for those types t~'
it
£. Further, by step
1W~'(£).
belief over
step)
as in
Taking
limits
has been shown that ?„(/')
now
be iterated to
The reason
now
is a
is
/
in the definition
best reply to lima"'
almost certain that the others have the
of a~'
which
we know
f~'
in
(in the
is
that after
In step
a~' was found, but
1
second sentence of
is
a
this
supported by IW~'{E), hence
show
that s'„it') is
an element of 1W/Z'(£).
^'
it
is
D
one round of deletion of weakly dominated strategics only strongly dom-
inated strategies could be deleted follows from the difference between steps
the proof.
that d„{t~')
dominated within UV"(£).
s'nit') is not strongly
Step 2 can
1,
player
the possibility of crazy types of j
full
^
and 2
a~' which has
full
support.
support within \W~'{E):
It's
support
a best reply to a strategy
does not have
1
and of course
i,
its
may have
limit
in the //part
of
In step 2 a similar
larger because of
is
smaller support than
IW-'(E).
Since the solution concept used here involves iterated deletion procedures
for inconsistencies in the su-atcgic beliefs of the players.
position 3.1 a players' beliefs in steps
2 and
1,
particular, in the first step the crazy types
in iterating the
else.
The
is
of step 2 need not be the same. In
to play
d~'
,
in the
second step o'~', and
second step the beliefs over the opponents would be different each time.
very similar to the hierarchies of beliefs
think that j
proof of the "only if part of Pro-
In the
in the iteration
were e.xpecied
inherently allows
it
in
Bemheim's
playing a certain strategy, but
;
definition of raiionalizability,
thinks j thinks
/
thinks that J
is
where
This
i
is
may
playing something
srraregic beliefs are not consistent. Tliis suggests that allowing in addition for incon-
sistent beliefs
over the rypes will not change the
result, as
Corollary
In order to formalize the inconsistent case the definition of
be extended accordingly.
Recall that in Definition 2.1 the
1
below confirms.
convergence of elaborations must
common
prior
p was used
in defining the
12
set
r' = m' of possible types of player
we
then required that for any player
game.
When
expanded definition of
type
r'
.
to ask
/,
whether
we
If so then
any player
in
t'
f' had the same payoffs as in the original
Of
we
will
define
f
to require that r'
as follows.
some permutation of players
/:,
/, ;,...,
converges strongly when conditions
extended definition of f'.
If
k,
(i)
t''
I,
and
7" (so
z's prior
should also be in
t'
/
thinks j tliinks
m'',
then
(ii)
i'
t'
e
e
m''{t''), t'
f. A
want
all
However we use an
r-'
in
m\
if
f '. Continuing
...
k thinks
/
e m'{t')
,...,
r-'
thinks a
iieratively,
may be
has payoffs as in the original giime E.
e
still
must asymptotically assign
Furthermore, for any
in £).
player h thinks
(in the limit)
want
c
we want m'
course
We
too weak.
is
have the same payoffs as in E.
has positive probability in the limit, then
we need
t
7"'.
in the limit to
/
having the same payoffs as
to
1
types
all
/
For a sequence of elaborations to converge
the player's priors differ this convergence requirement
the possible types of player
probability
in the limit.
/'
r*
of type
Formally, for
e m'(r') for
general sequence of elaborations then
of Definition 2.1 are satisfied with respect to the
Since the two definitions of f' coincide
when
p'
= p
for all;, the
extended definition of convergence agrees with the previous one when beliefs are consistent.
COROLLARY
strategies
s'„
PROOF: The
3.1: s'
s 1U7Z'(£)
e /U'"(£„) such
that J'„
proof of Proposition
above. The "only if direction
if
is
1
and only
-4
if there is a
sequence of elaborations E„ -^ E, and
s'
proves the corollary also,
exactly the same.
The
when 7"
is
of 7' in the inconsistent
iterative definition
case corresponds to the iteration applied in the proof of the "if direction.
redefined as discussed
D
4.
-
13
-
Payoff Perturbations and Strict Best Replies.
This section discusses the implications of using weak convergence, instead of strong conver-
The
gence, to characterize "small" doubts.
may have
payoffs
would expect,
strategies in
u'„
is that in
to the payoffs u' in
survive
FW
in
nearby games.
In fact,
llie
more sequences of elaborations converge
is
£, instead of
to
a given
resulting set
we denote
respect to extensive form payoff perturbations, which
IWIZ
weak convergence
1
never a
convergence and
u'
for all n.
strict
best reply, but the converse
strict best replies
we
is
in
As one
is that
the closure of
is
game £) we can show
the iterated deletion of strategies which are never strict best replies.
is
=
IWIZ
more
with
WJZ. Moreover (again because
close to a strategy which satisfies a stronger requirement than
dominated
u'„
the types in f'
consequence of allowing more convergent sequences of elaborations
the
E
which converge
difference
DV
A
in
any strategy
that
in
nearby games, namely
strategy
general false.
are also able to clarify the relationship
which
is
weakly
In considering
between our
weak
results
and those of FKL.
To understand
biliry.
In
the results of this section
Brandenburger and Dekel (1987)
as a posteriori equilibrium
is
it
is
it
helpful to review briefly a result
shown
(Aumann 1974) which
is
that correlated rationalizability is the
same
roughly the same as a Nash equilibrium with a
subjective correlating device (about which the players
may have
So, an alternative to I\V as a refinement of IZ
introduced.
on rationaliza-
inconsistent beliefs) explicitly
Nash equilibrium
look
at
strict
condition
(i)
of Definition 2.1 holds,
is
to
with subjective correlating devices.
DEFINITION
4.1:
E„ converges
£
in payoffs to
p
_
(£„ -? £)
if
and:
(ii)
For
all
t'
DEFINITION
eT\
4.2:
illit')
Two
-^ u'
strategies for player
/
arc cquivalcnr if Uicy lead to the
distribution over endpoints for all strategies of the opponents.
A Nash
same probability
equilibrium
(i'',...,
s')
is
-
each players' strategy
strict if
not equivalent to
LEMMA
than any other strategy
strictly better against s~'
~
If
best reply (up to equivalent strategies) to
i''
is
not weakly dominated then
each elaboration £„ so
in
knowledge. Let
u'„{z)
= u'{z)+\ln on
all
it
some a~' e
is
p
a best reply to
is
the
that
endpoints
utility
z
some
s~' with full support.
functions (defined next) are
reached by
E
where
A(5~').
s'
and a"', and
u'„{z)
4.1 provides the intuition for Proposition 4.1 below.
It
shows
that
Let
T
common
= u'{z)
by allowing
extensive form payoff perturbations, strategies which are not weakly dominated can be
best replies.
Proposition 4.1 below
weakly dominated"
for extensive
DEFINITION
is
strengthened to
is
oth-
for small
made
strict
an analog to Proposition 3.1, where the notion of "not
"is a strict best
reply" and convergence
is
weakened
to
allow
form payoff perturbations.
£„ converges weakly
4.3:
to
and:
£
(£„
-h^
£)
if
condition
(i)
of Definition 2.1 holds,
^
For
all t'
e f'
PROPOSITION
tegies
which
D
erwise.
(ii)
s'
not weakly dominated then there exists a consistent sequence E„ -^
is
be a singleton
Lemma
docs
-
s'
4.1: If s'
s' is a strict
PROOF:
s'
14
s'„
,
17^
4.1: If
-^ s\ such
Remark: Proposition
-^
i'
thai
Any
e l\VIZ'{E) then there
I,'
is
4.1 relics
equilibrium strategies thai
game £.
it'.
may
a sequence of elaborations
is
a strategy in a strict
£„
-=^
£, and
stra-
Nash equilibrium of £„.
on inconsistent elaborations
in
an essential
way
to obtain as a
Nash
not be played in any objective correlated equilibrium of the original
subjective correlated equilibrium
appropriate subjective correlating device
is
is
a
Nash equilibrium of
the
game where
explicitly incorporated into the strategy spaces.
the
The
-
point
is
that nature's
move
15
game, which determines the types of the players,
the beginning of the
at
serves also as a subjective correlating device.
-
(The difference between subjective and objective
correlating devices corresponds to the cases of consistent and inconsistent priors.)
PROOF: The
T'
possible types
is
£"
each elaboration each player's
set
of
partitioned into two sets, the "sane" types f' and the "crazy" types T'.
f'
is
elaborations
are constructed as follows.
isomorphic to \\VIZ'{E) and T'
these isomorphisms
we
isomorphic
is
will write t'
=
J',
and
In
of /'s pure strategics in E.
to the set S'
=
r'
s'
The
.)
priors
p'„
will be
types in T' are possible in the limit, which explains the abuse of notation.
7"',
we
to play
be
say that
was
"told" to play s[, and if /'s tj'pe
The payoffs and
J')..
is
=
tl
J',,
Note
f/.,
simply
that since these payoffs
have probability zero
To make
sequence e„ i
J;,
0.
in
set the payoffs
£,,
has
full
and thus m;ike
change the
IWIZ'
si.
is si
that
/
=
e
tl
was
told
and so that the elaborations converge
/,
uU^k) so
that
J^.
be
a
ver\' different than
strict
best
reply for type
e IWJZ' there exists a
we can
a strict best reply (up to
is
FKL
for an explicit construc-
those in £, the tj'pes in
T must
-»
Also there exists
support
si
may
in the limit.
Since
a best reply to a^'.
<j^'
type
e IWIZ' we say
equivalent strategies) to any belief a"' over the other players. (See
Since
If /'s
E.
to
For each crazy type
tion.)
chosen so ihat only
beliefs will be chosen so that in each elaboration playing as told will
best reply for each possible type of player
a strict
weakly
/
(Using
a
<7^'
:!.
we proceed
€ A(nS-') with
full
c^' e A(irTlU7Z-'). such that
increase the payoffs
a strict best reply against a;~'.
fact that Jl is a best reply against
CT;"'.
at all
is
follows.
First fix
a
support, such that si
is
is a
best reply to
cl^'.
endpoints reached under a^' and si by
Furthermore
This
J{.
as
this chcuige in payoffs will not
because no other pure strategy of
/
can
-
16
-
increase the probability of reaching the endpoints for which payoffs were increased.
Next we specify
tional
Let /"s beliefs over the others' types, condi-
the beliefs in an elaboration.
on his/her type be
For "sane" types
as follows.
/^
the beliefs
v'„(-|f^.):
(i)
assign probability
E„ to all the others being crazy, with the distribution of crazy types corresponding to ct^'; and
assign probability l-£„ to
For each
choose
/
a
all
sequence of marginals
which converges with probability one
the Proposition (say s\
=
For crazy types
the others being sane.
The
t{).
^'„
over 7" which has
to the sane type
priors
p'„
which
of player
/
arc generated
:[.,
full
are exactly the sane types f'.
Finally
told
is
brium.
we
Thus £„ converges weakly
of
are such that
iJ.'„
have positive probability
E.
/,
each type playing as
Nash
equili-
D
turn to the question of finding a converse to Proposition 4.1,
that the
converse
to Proposition 4.1
is
weakly
tion to
in the
incomplete information on the payoffs,
payoffs of £.
perturbations
bations.
gests that
ask which stra-
IWJZ{E). This
is
in
£
in a
which are
sequence of
because
sequence of elaborations converging
we
The problem
game.
to the original
to
£,
IWIZ
is
in addi-
allow for perturbations of the extensive form
Hence, roughly speaking, since the 'closure' of l\V allows for extensive form payoff
it
can only be equal
Since weak domin;mce
we
we
weakly dominated strategies
elaborations that converge to £, but which are not elements of
normal form solution concept, whereas
i.e.
There are strategies
not precisely correct.
the limit of strategies that survive iterated deletion of
to
...
7", and
in the hj'pothesis
by the v^ and
observe that by construction, for each n and each player
tegies can be justified using elaborations that converge
a
u
support on f'
a strict best reply to the others playing as told, hence playing as told is a strict
Now we
is
to
the beliefs are arbitrary.
which was
the sets of types which, in the limit, players think that others think that
(ii)
to a solution
is
concept which
is
closed with respect to such pertur-
not closed in this sense, neither
could achieve a generic converse to Proposition 2.1,
we
is
IWJZ.
believe
Although
it
is
more
this
sug-
interesting
provide a complete ch;u-acterizalion. In order to chuify the nature of the converse direction
we
begin with a partial converse that
with respect
e 5"
identical to
£„.
E
if there exists a
except for the payoffs which are
no such sequence, then
If there is
s'
round of deletion
first
not weakly* dominated
is
PROOF: Step
s'„
which
s'
is
instead of
not weakly dominated in E„
full
J'^
^
£, j;
Consider the versions of E„ where
I'„
is
i
.v'
is
s'
-^
s' is
not
e 2H"'(£J, J^Cr')
best reply to
is a
replaced by
e S'{E), then
measure which
is
supported by types
£).
Hence by
1,
for those types r~' in the support of lim v„(-]
A{lW*~'{Ey).
step
Taking
limits in the
lirst
l\V '(£„).
t~'
;')).
e 1W*/Z'(£).
that
satisfies u'„{t')
= ^a-~'(r~')v(r"'
with payoffs
which u~'{t~')
limcf„"'(^~') is
—
>
|
t') for
u'„{t').
Hence
we
—
t
'
|
;') for
from £„ -^
u~' (this follows
{')
we know
then have that
s'
is
i.e. s'
e
some
converges
|f')
that
a~Xr~') e
a best reply to
supported by 1W*~'(£) (and where the
Hence
s'
is
a best reply to the strategy lim d~'
which
is
supported
n
by lW*-'(£),
«'.
n
n
v'„(-|
s'
sequence of beliefs v„ii
sentence of step 2
some weighted average of lima~'(f~') where
weights are given by
for
Tlie
an analog
is
weak convergence.
Y cf~'(r~')\'„(f"'
some &~' =
to a
in
say that
e 1W*(£).
i.e. s'
which
supported by strategies
f'
£
<j„
is
Proposition 4.2
a best reply to d"'
weak* dominated,
we
Formally,
denote strategies which sur-
IW.
of any type
is
players agree about
not weakly dominated in
is
We
Let £„ be an extensive fomi identical to
support.
e S' such that
Step 2: Since
^
e !W'{E„), £„
4.2: If J^
I:
some d~' with
any
W*
1
'closure'
sequence of extensive form games E„
weakly* dominated.
is
in that the
tlie
is,
taken.
is
such that
u'
>
"if part of Proposition 3.1 when strong convergence
PROPOSITION
Any
—
u'„
vive deletion of weakly* dominated strategies by
to the
is partial
taken in a consistent manner (that
is
the perturbation) and only the closure of the
a strategy s'
The converse
easier to prove.
is
payoff perturbations
to
-
17
-
IW* 1Z'(£).
Step 2 can be iterated
to
show
that s'
e nV'*/Z'(£).
D
Proposition 4.2
not a converse to Proposition 4.1, so
is
we have
weak convergence of
sure' of the set of itcratively admissible strategics with respect to
do so both Propositions
In order to
tions.
and \\V*IZ' must be replaced by the same
respect to convergence in payoffs.
DEFINITION
lions
This set
IWIZ'{E)
4.4: s' e
and
set,
to
The
be strengthened.
elabora-
IWIZ'
sets
be the 'closure' of \WIZ' with
this set will
denoted by IWIZ.
is
-^
if 7'„
and 4.2 need
4.1
not yet characterized the 'clo-
and
i'
e
s'„
IWIZ E„
for
some sequence of elabora-
_ P
E„ —^ E.
PROPOSITION
PROOF:
€ IW{E„), £„
4..3: s'„
^
E,sl
s'
^ E,
if
and only
if s'
e IWIZ\E).
This follows from a simple diagonal argument and Proposition 3.1.
//.
then there exists a sequence
-^ E„ and
exists Ei,„
^
T^. „
~
£„
->
p
—
>
J'„
with
with
s'„
—
>
iind s'„
s'
€ IW\e;,).
f^. „
By
e IWIZ'CE„).
Clearly
"^ „ -^
If s'
g \WIZ'{E)
Proposition 3.1 there
^
and E'„_„
s'
£
as
required.
Only
is
We
if:
are given a sequence
R'^= limsup^^ where
k;,
=
£„ -^ £. Let R' denote
{s'
e S'
|
for
some
ii,
the strategies played
e IW'{E„)and some
7'
Construct the following elaborations £„ which will converge in payoffs to £.
types for each player
will be denoted
all tj'pes
lows.
4
and
by
is
/
s',.{m)
isomorphic
where
.s[
to
M'
copies of ^', where
€ R' for k =
1
K, and
M' =
m =
by sane types,
that
=
s'J.
€ f',
The
If!^''!-
1,...,
M'. For
7'„{t')
set
of possible
The t>pes
a given
;
and k
slim) have the same payoffs (independent of m), and these payoffs are determined as
Since
s'„
4
e
/?',
then taking a subsequence
G I\V'{E„).
with payoffs as
in
£„.
Hence
there exists a~' €
That me;ms
in
if
necessary, there exists
A(AV"'(£„)) such
particular
thai
s"„{t')
=
I'^
and
r'
is
a
best
fol-
with 7l,{t')
that s'„ is a best reply to
s'f.
£'^
in
reply
lo
=
g~'
a~'
=
-19-
^a„"'(r"')v(f"'|7'), with payoffs
lima"',
it
u',X'')-
Although
a best reply to &~' if the payoffs at
is
s'„{t') is not necessarily a best reply to
by
the endpoints reached
all
=
a~'
the strategies s'„{t')
n
and a"' are increased by a sufficiently large 'bonus' of
verges to zero since \imJ'„it')
Furthermore, the bonus required con-
£„.
_
slim) be equal
-
i
plays
si-
is
E„ such
IWJZ{E„). Recall
is in
Nash equilibrium
a
that /'s
beliefs be such that if (s)he
type R~'
X
(k),
s'j.{m)
playing
all
full
support
when
this strategy /-tuple
opponents will be an /-I tuple of types
in /?"'
x
Ij'pc
undominated
a x^'
the payoffs are
is
determined by a~' (see above).
which
to
Now we show
e A(5~') such
w^(r').
The
their type
that J'„{t')
strategy
a{.
is
payoffs are changed to include the bonus £„ described above.
best reply to the full support strategy of the opponents
because of the second stage
gated
all
the types
equilibrium.
bility
is
in the
into
be expanded
proof above.
one type
.?|.(1)
to
And
is
it
is
in
a best reply to t"' with
s'i,{m)
the payoffs as described
is a
when
playing
R~' x {kj plays
then each type sUl) playing si
for each k the tjpe ^^-(1)
a
sj^
t^T'.
the
is
a
D
rather than using only /?',
/?',
This
is
not weakly dominated.
So each type
copies of
Then s[{m)
best reply to z~'
a
where each type
M'
With
4
=
particular full support strategy of the opponents to
a best reply.
Let /'s
matched. This shows that
is
that
still
But these strategies are not necess;uily admissible.
we need one
ing si"
s^m)
to
K].
{I
slim), then (s)he believes that the opponents can only be of
Nash equilibrium.
is
Remark: The type spaces need
where each type 4('") of
that the strategy /-tuple
playing the strategy
Since J^(7') e I\V'{E^), there
claim that there
hence
of
s[ is a
We now
E.
>•
strategies,
in
and the distribution over R~'
best reply to the opponents
each type
is
P
—
Since £„ -» 0, £„
to u'„(r') with the £„ bonus.
exist beliefs v„(-|-) for the elaboration
each player
Let the payoffs of type
a best reply to a~' with payoffs limu'„{t').
is
is
above
would
because
to
if
still
we
aggre-
be a Nash
show admissi-
which "each type sl(l) play-
best reply against a different full support
-20
strategy t^
Hence we need
'.
The above
5.2).
Any
to
allow different types for
/
's
results suggest an additional interpretation of the following result in
£
quasi-c-perfect equilibrium in
is
FKL
equal to the the limit of a sequence of
perfect equilibrium.
A
c-perfect equilibrium
is
may be
correlated and inconsistent (that
trembles).
A
c-perfect equilibrium
strict
E. The
of
A
best replies to the test sequence.
strictly c-perfect equilibria in a
observation
first
is that
strictly c-perfect equilibria
Nash
Tliis
Lemma
convergence
closure with respect to (consistent)
in
payoffs of
undominated)
is
/
and j
may
test
strict
the
same
where the
The obser-
sequences for
is
limit strategies are
the limit of a sequence of
is
the
elaborations, the set of limits
Nash
c-perfect equilibria are
equilibria
the set of strict
where we showed
same
is that
So, the theorem cited above says that
payoffs of undominated
weak convergence of
1H7Z (which
in payoffs to
by the small payoff perturbations allowed by weak
weakly dominated.
in
of a quasi-c-
as the set of limits of c-perfect equilibria:
The second observation
4.1.
Nash
assign a third player k different
a c-perfccl equilibrium
is
result is closest in spirit to Proposition 4.3,
vergence
is
when considering weakly convergent
equilibria in strategies that are not
the closure with respect lo
where the
quasi-c-perfect equilibrium
of games E„
follows from
recall the defmiiion
sequence of consistent elaborations E„ which converge
any c-perfect equilibrium can be made
convergence.
is
we
a perfect equilibrium
each player
strict
First
(Section
strict
equilibrium of a sequence of consistent elaborations E„ which converges weakly to E.
vations below provide a simpler statement of this result.
D
opponents as a function of k.
Nash
is
equal to the
equilibria.
This
that the closure with respect to con-
as correlated rationalizable strategies
which
equal to the closure with respect to (inconsistent) weak convergence of the set
are
FW
-
5.
We
when
way
sketches a different
is Q.'
/
Alternative Interpretation
motivated the consideration of payoff uncertainty by asking what players should infer
they observe play that
player
An
the set of
all
is
not consistent with their understanding of the game.
specify i's beliefs over
possible utility functions for
surprised
if
is
is
questioned in this paper
u' if marg^-i^'
same way
=
0.
is
that
This approach
is
7-'.
is to
show how our
believes
at
node a
L; player 2 believes
expect to play.
1
that
1
for
Supp mmgjiq'i-
that player 2 will
L and
natural form of backwards
\
c
H~'
|
S~')
strategy choices.
This
3.1.
payoffs are as
in
node since
induction rationality
is
it
when
L
as
1
in this
In Figure 2,
will play
1
to assigns probability
expected
players'
lo the
£, and so player 2 does not
already clear that
the
is
best seen
is
play'L and that the payoffs are as in £, so
that the
H~')
Our purpose
proof of Proposition
in the
<?'(•
convergence used are related
At node b 2 has been surprised and updates his beliefs
to specify the beliefs at the third
litera-
determined by a sequence
in the elaborations.
when observing unexpected
playing Rl and the payoffs being as in £", so 2 will play
need
has partitions on
/
Here a conditional probability
results and the different notions of
will play
only update their beliefs
related to the formalization in this paper in essentially the
simple example which mimics the construction
1
s U,
(Recall that T^
Q.'.
Each player
.
that beliefs at all information sets in sequential equilibrium are
idea of updating beliefs on payoffs
player
A
that players
Here we allow
determined by a sequence of elaborations and the strategies
in a
e
even when observing an unexpected strategy choice
of beliefs (generated by completely mixed strategies).
section
q'
assumption (implicit in the refinements
traditional
the player docs not update his/her beliefs on
9t
by
Q.'
The assumption
/.)
This section
begin, note that the state space for each
formalized by Supp margj^^' = u'
Q' determined by the extensive form. The
which
To
to formalize those inferences.
= Yl^S^ x T^) and
about the payoffs
ture)
-
21
at
1
beliefs
node
a.
playing
one
There
L
is
to
no
satisfies a
over payoffs can be
updated.
The above argument shows how strong convergence corresponds precisely
updating beliefs. That
is, it
satisfies
Supp Tn^gj,q'{-
\
H~') =
u' if
H' was
to
the
ideas of
assigned positive prior
-
probability by q'
.
when
is
the player
22-
This interpretation does not allow for the payoffs to be "almost" equal
to
u-'
not surprised, which points out an interesting distinction between strong and
weak convergence, analogous
between sequential and perfect equilibrium.
the difference
to
In
sequential equilibrium each player's beliefs at information sets along the equilibrium path are pre-
Similarly the definition of strong conver-
cisely that the equilibrium strategies are being played.
gence requires that
game) any types which receive
(in the limit
have precisely the payoffs of the limit game.
beliefs are that the
game
= (L
Therefore IVV/Z
x
)
3b (so
as in Figure
is
{/
).
On
the other
So
1
hand
at
positive probability are believed to
player 2's information set in Figure 3a, 2's
R
can not play
because
in perfect equilibriinn,
it
is
even
weakly dominated).
information sets
at
The
along the equilibrium path the players allow for "trembles" in the opponents strategies.
that "trembles" are
allowed for even along the equilibriimi path can be formally understood using
the approach of lexicographic beliefs in
papers suggest that the limit
(where £
is
{^,^1 X
game which corresponds
R
is
to
weak convergence
is
(1986). These
below
as in Figure 3c
no longer weakly dominated, hence
1H7Z =
in fact
(r,/).
this
comparison between weak and strong convergence, we note
seems more appropriate
opponent's payoffs
modeling
if
/
modeling the idea
for
closure
that a player
/
obser\'es an unexpected strategy choice
the question of robustness of a refinement
wider class of perturbed games which
the
Blume (1986) and Brandenburger and Dekel
an infinitessimal). In this case
To conclude
for
fact
of iteratively
is
admissible
it
may
by
that the latter
update his/her beliefs about an
the opponent.
On
seems more natural
to
the other
hand
allow for the
formalized by weak convergence. Tlie similarity between
strategies
with
respect
to
either
notion
of convergence
emphasizes the close relationship between these two objectives.
One more
point regarding this interpretation of the
model
is
worth clarifying.
Our approach
allows a player to update his/her beliefs about the opponents' payoffs whenever surprised
there
is
a "rational" explanation
interesting extension of this
by an opponent's strategy
which does not require changing
model involves imposing
first
tries to
—
even
beliefs about the payoffs.
the restriction that a player
who
is
if
One
surprised
explain the observation without violating the assumptions
-23
that payoffs
and rationality are
common knowledge.
-
Instead the player assumes that his/her beliefs
about the opponents strategy choice (or the opponents beliefs about other players' strategies,
were wTong.
Only
if the
"deviation" can not be explained by questioning the players' beliefs over
the elements of strategic uncertainty
payoffs doubted.
etc.)
is
the
more
basic assumption regarding
common knowledge
of
-24-
6.
We
tion of
The
conclude with two examples.
weakly dominated strategies
between our
tegies.
Examples and Conclusion
results
In the
first is
in the first stage
meant both
motivate the simultaneous dele-
to
of \WfZ, and to further clarify the relationship
and the idea of updating beliefs about payoffs
example of Figure 4 the order
in
which
observing unexpected
after
first
x (L). The argument
round follows the intuition of backwards induction:
payoffs are certain to be as specified, player
both
L and R
tegies.
1.
reasonable.
However
1
will
M, and knowing
never play
This argument yields (U, D) x (L, R) as the
the only reason for 2 to be willing to play
But precisely these doubts are needed
Deleting both
strategies are deleted matters.
players' dominated strategies simultaneously, and then iterating, yields |U)
against simultaneous deletion in the
R
set
this
in the intuition for ruling out a
If the
2 should find
of reasonable stra-
(s)he entertains
is if
stra-
no doubts about
weakly dominated strategy
for either player.
Our second example helps explain why we do not
IW.
In this
example
should be sure that (s)he
what
is
will
node
is at
So 2
b.
if
2 plays
L which
is strictly
In conclusion
we would
like to
We
or
might argue
-
M
what explanation
that
and then both
U
and
C
is
should play
review the main points of
weak dominance (Section
3).
being
1
this paper.
when
is
true that 2
U
because
at c?
at c
is
play
to
Perhaps
1
we argued
rationality and payoffs.
a
in
surprised," are particularly
Including
sharp and intuitive characteri-
Also, the distinction between
the two objectives of this line of research,
the updating of beliefs on null events.
C
First, as
helpful in understanding the relationships between
weak dominance; and between
It
can be justified.
payoff imcertainty in the model and using weak convergence yields
and strong convergence
1
there for
is
assume only common knowledge of
zation of the "closure" of iterated
will play U.
M. But confronted with playing
the Introduction, the questions of robustness and "what to believe
relevant in models which
1
dominated by M, and hence 2 shouldn't be expected
But since 2 also shouldn't have played
doubt his beliefs about 2's payoffs
L
will not play
comfortable with a prediction based on
whether
like to ask is
the appropriate thought process for 1?
only best
L.
we would
the question
feel
weak
strict best replies
and
namely robustness and
25
-
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—
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Nature
Figure
1
E'
Figure
Figure
2:
3:
Figure 4:
U
2,1
1,1
M
2,1
0,0
D
1,1
2.1
L
M
R
5,-9 -2,-2
0,1
2,-9
0,1
Figure 5:
C
D
1,-1
-3,-3 -5, 4 -5,3
36(1
102
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