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working paper
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of economics
ON THE ROBUSTNESS OF EQUILIBRIUM REFINEMENTS
Drew Fuderiberg
David M. Kreps
David K. Levine
No. 454
July 1987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ON THE ROBUSTNESS OF EQUILIBRIUM REFINEMENTS
Drew Fuderiberg
David M. Kreps
David K. Levine
No. 454
July 1987
On
the Robustness of Equilibrium Refinements*
Drew Fudenberg, David M. Kreps, and David K. Levine
<
July 1987
Abstract. The philosophy of equilibrium refinements
knows things about the structure
The word "know"
unreasonable.
phasis. If in a fixed
game
game, can
of the
in the
that the analyst,
is
reject
some Nash
equilibria as
the analyst can reject a particular equilibrium outcome,
thoughts about rejecting the outcome.
between games, and we characterize
We
equilibrium, perfection, stability.
may have
second
consider several notions of distance
their implications for the robustness of equi-
librium refinements. (Keywords: noncooperative
JEL Numbers:
game
theory, refinements of
Nash
026, 213)
Introduction
1.
effort
he
preceding sentence deserves special em-
but he cannot do so for games arbitrarily "close by", then he
Much
if
has been devoted recently to refining the notion of a Nash equilibrium.
Beginning with Selten's [11,12] notions of perfection, concepts such as properness (Myerson
[8]),
sequentially (Kreps and Wilson
[7]),
often invoked or discussed in the literature.
and
stability
The guiding philosophy
that the analyst knows things about the structure of the
some of the Nash
know
in
equilibria as unreasonable.
The
game
[5])
are
of these refinements
is
that enable him to reject
point of this paper
is
that the word
the preceding sentence deserves emphasis. Specifically, the analyst creates a model
of the situation that
in the
(Kohlberg and Mertens
is
a simplification and (he hopes) an approximation. Suppose that,
model, the analyst can reject
particular equilibrium
a.
outcome using the various
refinements, but he cannot do so for models that are arbitrarily "close" to the one created.
Unless the analyst's faith in the model
about rejecting
To study
this
is
absolute, he
may
be wise to have second thoughts
outcome.
this "robustness" issue,
we carry out the
following program. Fix a space of
We
would like to thank Elon Kohlberg, Jean-Francois Mertens, Lloyd Shapley, two
and especially Eddie Dekel for helpful discussion and comments. Financial support
from National Science Foundation Grants SES84-08586, SES85-09484 and SES85-09697 and
from the Alfred P. Sloan Foundation is gratefully acknowledged.
*
referees
MIT Department of Economics, Stanford U. Graduate School of Business, and UCLA
Department of Economics and U. of Minnesota Department of Economics, respectively.
*
2009745
games and
Nash
a notion of "closeness" for the
equilibria, is strict
games
in
for each player, the strategy prescribed
if,
to the other players' strategies. This
is
equilibria of a given
game
such equilibria, near
strict.
equilibrium that
is
game
in
[5]
the space, a
a unique best response
is
we can think
as formidable a refinement criterion as
implying, for example, Kohlberg and Mertens'
of,
the space. For each
Now
hyperstability.
ask:
which Nash
are limit points of strict equilibria, for nearby games?
We
call
Following the discussion above, we would hesitate to reject any
near strict, insofar as the sense of closeness specified captures our doubts
about the exact specification of the game.
In section 2,
we take
for the space of
The
result
is
all
we measure
player and action) normal form, and
payoffs.
games
normal form games over
a.
fixed (finite
"closeness" with the Euclidean metric on
that every pure strategy Nash equilibrium
near
is
strict.
(We
also
give a weakening of the strictness requirement that accomodates every equilibrium, pure or
mixed.) This
motivated
is
hardly surprising, but then the various refinements of Nash equilibrium are
for the
most part by the analysis of extensive games.
turn to extensive games with the following philosophy:
may
in repeated
in
roughly, a (physical) extensive form
is,
is
information about earlier
is
But the analyst
given.
games with incomplete information. (See Kreps, Milgrom, Roberts and Wilson
and Fudenberg and Maskin
Our theory
[3].)
differs
from
one important way. The previous work considers the
sufficiently long,
are
certain (in his model)
is
that arise from such doubts are considered in the work on reputation
family of repeated games of varying lengths.
is
we
entertain doubts about the players' payoffs and/or their knowledge of the payoffs.
The perturbations
[6]
analyst
who moves when, with what
of the "physical" rules of the game:
moves, and so on. That
The
In the rest of the paper,
Nash (and even,
fixed,
The
this
effect of
work on reputation
a fixed perturbation on a
typical result there
see below, near strict) in the perturbed
and the perturbation
is
allowed to vary (and
game. In
made
players' payoffs himself, but
is,
that
game can be near
knows that these payoffs
an extensive form
the extensive form, and "closeness"
is
is
fixed, the space of
this
to vanish).
if
the horizon
paper the game
Hence only Nash
strict.
Section 3 concerns the program for cases in which the analyst
That
is
even outcomes that are not Nash equilibria of the unperturbed game
equilibrium outcomes of the original
players.
effects
are
may be unsure
of the
common knowledge among
games
is
the space of
measured by the Euclidean metric on
all
the
payoffs for
payoffs.
We
do
not get very clean results here. There do exist near strict equilibria that are not
strict,
but
the class of near strict equilibria has no apparent simple characterization.
Sections 4 and 5 are the heart of the paper.
Here we imagine that the analyst has
no doubts about the physical rules of the game, but
willing to
is
not quite certain of payoffs, and
is
admit the possibility that the players themselves are not quite certain of each
others' payoffs. Put another way, close to a given extensive
game
are
entertain slight doubts about each others' payoffs, and our analyst
an equilibrium that cannot be rejected
games that are nearby
in
games
is
in
which players
not prepared to reject
in this sense.
Section 4 deals with a definition of "closeness" under which every (pure) Nash equilib-
rium of the
original
game
4.1. Technical details
is
near
strict.
We
begin with
motivating example
and the definition of closeness are given
ness allows each player to be uncertain of his
may
a.
own
payoff,
as a small ex ante uncertainty can
In 4.3
we show
large
that, with this notion of closeness,
(Once again, mixed
strict.
become
equilibria,
all
it
subsection
This version of close-
and to believe that
(with small probability) have better information about
an unexpected deviation by one player can signal that
in 4.2.
in
his
than he does. In
opponents
this setting,
players should change their play,
upon observing an "unexpected"
all
action.
pure strategy Nash equilibria are near
can be accomodated with a minor weakening of the
definition.)
In section 5
we consider a second notion
of closeness, which requires that each player's
additional private information relate only to his
own
payoffs.
If
each player's additional
information must also be independent of the information of the others, then an unexpected
own
deviation by a player signals only that his
(In the literature
ipated.
on reputation
bations used satify this restriction.)
(pure)
Nash
in
With
equilibria are near strict.
payoffs are different than had been antic-
repeated games referenced above, the pertur-
this
However
more
all
restrictive notion of closeness, not all
pure strategy equilibria that are
trembling hand perfect in the normal form are near
subgame
perfect.
perfection.
tion
We
Finally,
may be
is
like
correlated.
including
some that are not
obtain a converse by weakening slightly the notion of trembling-hand
we consider
must be independently
equilibria that
strict,
strictly
relaxing the assumption that players' additional informa-
distributed.
Here we obtain a characterization of near
strict
trembling hand perfection, but where the "trembles" of various players
The idea that small
otherwise imperfect, equilibria
among
make
others,
we consider can render
uncertainties of the kind
this point.
is
Van
not original to us.
Our contribution
to define
is
Damme
some
perfect
and Myerson
[13]
and characterize more
[9],
precisely
the effects of various kinds of uncertainties corresponding to various measures of closeness.
2.
Fix a
finite player, finite action
games with
i
=
finitely
many
by S
where
The
does
E,-
set
j{i,s)
is
a
Two
strategies
ct_,-
=
utility
whether
?'
cr,-
is
's
of the other players,
i
a,
?'
this
throughout to
strategies.)
under strategy
s
We
i
take T
.
Let
Denote
= R IxS
,
.
natural topology, namely the Euclidean topology. So
and
...,ctj)
E
a\
(ai,...,aj)
i
,
and E = Yli=i ^«
are chosen by
use the
?
's
is,
opponents,
no matter what
all
players receive
l
for a given
mixing
Strict equilibria satisfy all the
is
wm
.
up to equivalent strategies
strict (unless the
(^?e
•
.)
are said to be equivalent
,
chooses a, or o\
equilibrium, taken as a singleton set,
many
normal form:
game 7
unique best response to the strategies
pure strategies can be
this definition.
,
(cj, ...,o",_i,<7i+i,
Nash equilibrium a =
player i,
i
possesses finitely
games over
profile to refer to elements of
strategies for player
restrict attention
£ 5, index the pure strategies of player
a,
the payoff to player
T comes endowed with
term strategy
A
,
whom
the space of mixed strategies for player
,
same
let
Let T be the space of
.
7 £ T
for
and
(We
normal form.
players, each of
1,...,/ index the players,
Yii=i ^i
the
Normal form games and payoff perturbations
for
is
2
i
cr_,-
is
=
called strict
if,
for each
(01, ...,ct,_i,ct, + i, ...,07)
Note that only an equilibrium
in
over equivalent strategies), according to
standard refinements;
in particular,
a
strict
hyperst&ble in the sense of Kohlberg and Mertens
3
[5].
1
Equivalent strategies arise naturally when the normal form game is constructed from
an extensive form game, and one player has multiple information sets in sequence. Then
if the player takes an action at an earlier information set that precludes reaching a later
information set, the choice of the player at that later information set is irrelevant. N.B.,
it is the player himself who precludes the later information set - equivalence of strategies
does not depend on the strategy choices a_,- of other players.
2
That
is,
any equally good best response a\ must be equivalent to C{ We could equally
normal form, in which all equivalent strategies are
.
well speak of strictness in the reduced
identified.
3
Recall that Kohlberg and Mertens
work with the reduced normal form. Conversely,
4
A
Definition.
is
£
strategy profile a G
a sequence of games 7™ (over the
{ct"}
such that
lim n a
n
= a
form
for
A
1.
if it is
The proof
is
7
7
same normal form
n
for each
(ii)
,
normal form
strict in the
un
,
is
as 7
a
for a
game 7
if
there
and a sequence of strategies
)
strict equilibrium of
7"
,
and
(iii)
.
Proposition
and only
=
lim n 7"
(i)
near
is
a pure strategy
To
quite simple.
is
a £ E
strategy combination
is
Nash equilibrium
near strict
for
7
in the
for
7
if
.
see that any strategy profile that
a Nash equilibrium of 7
normal form
is
near strict in the Normal
use the upper hemi-continuity of the Nash equilib-
,
rium correspondence. For the converse, consider the perturbation that adds a small amount
to the players' utilities at the
It is
outcome prescribed by the strategy
easy to deal with mixed strategies,
nition of near strictness. Recall
if
one permits a small extension to the
from Kohlberg and Mertens
form game, an equivalent normal form game
is
in question.
one
in
defi-
that for any given normal
[5]
which pure strategies that are con-
vex combinations of other pure strategies are added to or deleted from the original game.
Corresponding to every strategy
profile in
an original
game
are (possibly
many) strategy
a given equivalent game. Consider the following modification of the definition
profiles for
of near strictness in the normal form.
A
Definition.
normal forms
that
is
near
Proposition
only
if it is
strategy profile
for
game
if it
strict in the
2.
A
for a
normal form
strategy profile
is
depicted
easy.
A
o
for 2
near
strict in equivalent
strict in equivalent-
of an equivalent
normal forms
singleton set" instead of "strict".
in
7'
for
7
if
and
Consider
which player 2 randomizes
we have an equivalent game
added as a new strategy
restrict attention to
game
.
and the particular equilibrium
any pure strategy equilibrium that
we
1
near
for 7'
is
equally between L and R. In figure 2
as long as
is
simple example will illustrate the method of proof.
in figure 1,
equibbrium strategy
normal form game 7
corresponds to some strategy profile a
a Nash equilibrium of 7
Again the proof
the
7
a
for player 2.
7'
,
with the particular
In this
game,
this
added
is, as a singleton set, hyperstable, will be strict. Hence
pure strategy equilibria, we could use "hyperstable as a
M
strategy
corresponds to the mixed strategy of player 2
equilibrium in the original
game corresponds
game, and we can apply the
The
spirit of this
first
in
the original game; the Nash
to a pure strategy equilibrium in the equivalent
proposition.
extended definition
that players
is
may
derive a little extra utility
from playing a particular randomized strategy. This technique can be applied at any point
development to follow, to extend results from pure to mixed equilibria. However, as
in the
both referees emphatically observed, this
what
will, in
hence,
is
an
device of
artificial
follows, discuss only the near strictness of
3.
The
is
little
pure strategy
content.
Hence we
profiles.
Extensive games and payoff perturbations
result that every
pure Nash equilibria of 7
near hyperstable as a singleton set)
is
near strict in the normal form (and,
hardly surprising, since
is
sequence of (vanishing) payoff perturbations to the normal form game.
most normal form games, every Nash equilibrium
refinements were created initially to deal with
we might wish
form. Hence
4
satisfies
games that
we aUow
any
for
Recall that, for
the standard refinements; the
arise
from a specified extensive
game model,
to permit, as perturbations to an initial extensive
only perturbations that respect the structure of that extensive form.
We
E
form
still
can imagine that our outside analyst
,
concerned with a game over an extensive
is
and he entertains no doubts about the physical
our analyst
may have doubts about
the
full
rules of the
game
so specified.
specification of the situation.
But
He may,
for
example, be slightly unsure of the payoffs to the players or their probability assessments
concerning nature's moves.
5
assessments.
for nature's
7 6 T
4
,
Thus we can
moves
in
which strategy
The
We
E
.
will
ask:
suppose that there
is
no question about the probability
Fix an extensive form
Let T be the space of
all
E
and probability assessments
payoffs for players in
E
.
For a given
profiles are near strict in the following sense?
somewhat more satisfactory treatment for mixed strategy
which one considers as a perturbation of a given game the same game in
which a player is "replaced" by a continuum of identical copies, as per Harsanyi's [4] classic
treatment of purification. We will not follow that route here, except to suggest to the
interested reader that, when one comes to general elaborations in section 4, one will have
to be careful with this if one is not to get as near strict all correlated equilibria, precisely
referees suggested a
equilibria, in
in the spirit of
5
Myerson
[9].
The
usual accounting tricks make this without loss of generality,
about the support of those assessments.
6
if
there
is
no question
Fixing
Definition.
the extensive form
7 6 T, a (pure) strategy
if
We know
n -+
<r
from Section 2 that,
<7
strict
extensive form but that are not strict.
strict in
the extensive form
It is
is
taken from Kreps and Wilson
equilibrium (L,D)
will
be profiles that are near
(Our system
3.
[7].)
It
Indeed, for some extensive games, no equilibrium
The game
generally,
is
near
depicted in figure 4
But not every
diagramming extensive
for
should be easy to see that the Nash
is
near
strict in
is.
the extensive form.
an example: While (R,Uu) and (R,Ud) (and, more
any randomization between these two strategies) are very nice equilibria, neither
strict.
Normal form,
In fact, for an equilibrium of an extensive
player whose
game
necessary that any information set h that
it is
probability either
is
is
strict in the
near strict in the extensive
is
not near strict in the extensive form; only (R,U)
is
(i) is
move
it is
is
to be near strict in the
not reached with positive
precluded solely by the action at some earlier information set of the
at h or
a
(ii) is
irrelevant to everyone's payoff. This
"dummy"
is
information
set,
where the player's choice
a very simple manifestation of the type of problem
that leads Kohlberg and Mertens to define stable components of equilibria, and
similarly
strict
easy to show that every strategy profile that
Consider the extensive game in Figure
form games
in
themselves. Viewing a given normal form as
pure strategy Nash equilibrium over a given extensive form
form.
strict
are equilibria that are
pure strategy Nash equilibrium.
a
is
near
7 and (normal form)
>
some games, there
a simple extensive form, we see then that there
near
—
is
.
for at least
near strict in the normal form but not
is
a of the game 7
there exist a sequence of payoffs 7"
a" of 7" such that
equilibria
profile
attempt to define near
we could
components. However, we are more interested
strict
in the
implications of the less restrictive notions of "closeness" that correspond to the players not
being quite certain of each others' payoffs.
4.
4.1.
General Elaborations
An example
To motivate
found
in
Van
this
Damme
development, we provide a simple example.
[13].)
Consider
first
the
game
Imagine an outside analyst whose thought about
in figure 3
(A
similar
example
is
and the equilibrium (L,D)
this situation
run as follows:
know
"I
the particular physical rules of this game.
R
between L and
And
Player
I'm fairly sure that the payoffs are as in figure
certain of this.
am
I
U
and player 2 must choose between
,
willing to
admit that there
are quite different from those 6hown. Moreover,
between the players what those payoffs
information, in that players do not
is
That
are.
know the
D
and
first
R
if
is
choose
chosen.
But I'm not completely
3.
a small chance that the payoffs
might not be
it
must
1
is,
the
common knowledge
game may have incomplete
precise payoffs, although
all
the
players will have priors that give probability close to one for the payoffs in figure
3."
To model
we introduce a game that
Nature moves
follows.
is
this,
first,
more elaborate than that
is
selecting one of several "versions" of the
distinguished by having the
same extensive form
game. Each version
may have
as in figure 3 but
Nature chooses version having payoffs close to those
payoffs.
in figure 3, as
in figure 3
different
with probability
Each player has a partition over versions of the game, with each respective
close to one.
player being told in which
One such
of his partition the true version
cell
elaboration
is
given in figure
which occurs with probability
occurs with probability
picks, while player 2
is
from the original game
e
,
— €,
1
5.
Nature picks one of two versions. The
has the same payoffs as
has very different payoffs. Player
not told.
for small
Is this
f ?
elaboration of the
Our
lies.
in figure 3;
1
first,
the second, which
knows which version nature
game
in figure 3
very different
outside analyst, plagued by the sort of doubts
expressed above, might not be willing to rule out the possibility that the players perceive
the situation as being that in figure
The
(Zi-fto,
we
D
point of this example
D)
is
decrease
,
strict in the
normal form, as shown in figure
player l's choices at his
given the strategy of player
If
game
near
1,
player 2
is
given the
D
to measure distance so that, as
c
optimal the choice
then we would conclude, in the
strict.
This
is
see:
strictly
remains
if
strict as
will
nature picks version
2,
game approaches the
of earlier discussion, that (£,
(L,D)
play
optimal choices. And,
goes to zero, this
spirit
it
Given that player 2
move only
so even though, in figure 3,
8
figure 5, the equilibrium
Moreover,
6.
both
sets are
2.
in figure 3,
is
two information
by
strictly
we were
figure 3
now easy to make. For the game in
towards zero. This should not be bard to
e
which renders
is
5.
is
D)
in
subgame imperfect.
(Note that a nontrivial step
as
goes to zero?
e
With
We
is
implicit here: In
give a formal criterion below.)
we now develop
this as a prelude,
We
begin with a precise definition concerning when one
of another.
We
games
game
Fix an /-player extensive
(with nodes denoted by
various players,
initial
some
t )
which
W
;
The kind
game
And
E
game
will
w
/
that
W
£
H
game
tree
,
and terminal nodes
,
Z
z £
T
we have
in
an
;
with H{i) denoting the information
actions a £ A(h) at each information set;
;
assigning utilities to
E
This prescribes a
.
partitioned into sets of nodes T, that "belong" to the
node
all
and a
players.
mind
is
one
which one of TV possible
in
selected by nature at the outset, where each version has
is
game above and, except
for initial uncertainty as to nature's choice of
structure. Versions are distinguished by the players' payoffs.
be called elaborations of
N
this,
fold
at
(i,
n) for
copy of
utility to player
sets are
W
i
composed
,
t
.
all
E
.
N
fold
If
player
n
;
i.e.,
copy of
T,
i
T
moves
= T X
-
t
,
T
or
{1,
...,
.
r
;
For each player
i,
in
set of initial
a partition Pi
Actions at information sets are inherited
set in
in
scrutiny. Player
9
t
(Z,
E
the
,
n) to denote the
(if
t
£
then
T,- ),
nodes consists of the
node (w,n) given by p(w)fi(n)
and the information
The information structure bears some
node
use
E
.
The
denoted by u(i,z,n). Finally, information
as follows.
is
positive
on {l,...,A }. In
/j
N] we
...,
N} The
initial
node (z,n)
given,
X {1,
at (non-terminal)
at terminal
is
A
of perfect recall built up as follows.
with the probability of
denoted by Pi(n))
H(t) X Pi(n)
.
imagine a game
tree consists of an
moves
E
given, together with a probability distribution
is
n th copy of the node
cells
a small perturbation
players are unsure (in a general sense) as to which version prevails. Such perturbations
integer
N
R
same information
To formalize
i
—>
game above
tree of the
version, the
Z
I x
of perturbation of
"versions" of the
is
E
of perfect recall,
information sets h £
set containing the (nonterminal)
:
is
to be close.
of which are initial nodes
assessment p over
payoff function u
game
not develop a formal topology on the space of games, but instead we
will
give a simple sufficient condition for
of
a general treatment.
Elaboration perturbations
4.2.
the
what sense does {L\R.2,D) approach (L,D)
E
of {1,2,...,
A}
(with
with node (t,n) £ T{
is
the obvious fashion.
i 's
exogenously given informa-
tion concerning which version
That
is, if it is
H(t) and
his
construction.
player
i 's
n
turn to
move
knowledge about n
E
The game
chosen at the outset
is
is
node (t,n)
at
given by Pi(n)
in figure 5 consists of
.
P\
is
tions of
E
in
E
consider such elaborations
E
.
For
E
of a
E
game
to be a "small" perturbation,
t
is
.
given by
Figures 3 and 5 illustrate the basic
E
in figure 3. In the first
In the second (bottom)
in figure 3.
learns at the outset which version
1
the discrete partition. Player 2 learns nothing;
We
knowledge about
his
,
two versions of
(upper) version, the payoffs are identical with those
version, they are quite different. Player
described by the partition P;
is
Pi
i.e.,
as being
we take
it
is
is
chosen;
i.e.,
the trivial partition.
among
the possible perturba-
to be sufficient that the payoffs
should be, with high probability, approximately those in
E
.
Formally,
we pose the
following criterion.
Convergence Criterion. For a given game
E
To say that the sequence approaches
.
(i)
there
ration
(ii)
(iii)
,
{E k }
let
be a sequence of elaborations of
sufficient that:
it is
and a uniform bound on the absolute values of the u k
for each k,
i
and
hnu-^oo^l) =
Conditions
,
,
a uniform bound on the number of versions of the original
is
Ek
E
E
(ii)
and
z
,
lim/.—^, u
k
(i, z, 1)
=
u(i, z)
;
game
;
and
(iii)
state that, along the sequence, the probability that nature picks a
same
as in the original
game approaches
Note that a convergent sequence of extensive form payoff perturbations of
(trivial)
convergent sequence of elaborations.
6
The
first
it
cannot hurt. The second part of
the probability of other versions
of those versions, then
we can
is
going to zero,
if
(i) is
there
E
part of condition
unnecessary to support a notion of closeness, but since we are giving a
for convergence,
each elabo-
1.
version in which payoffs are asymptotically the
a
in
corresponds to
(i) is
probably
sufficient condition
quite important, however:
is
one.
Even
if
no uniform bound on the payoffs
inflate the payoffs in other versions so that, in expected utility
6
Strengthening condition (ii) to require that payoffs in the first version exactly equal
those in the original game is vacuous: Given a sequence of elaborations satisfying (ii), we
can replace version 1 by two versions, called 1' and 1" with payoffs in 1' exactly as in
the original game, and specify that no player can distinguish between 1' and 1"
With
appropriate care in picking payoffs in version 1" and the relative prior probabilities of the
two, the new elaborations would be strategically equivalent to the old ones.
,
.
10
terms, the players' expected payoffs over
With the uniform bound on
expected payoffs, only the
will
loom large
Is this
The
in
the versions are anything
all
payoffs, however,
(iii)
to be.
implies that in ex ante calculations of
version (which gives nearly the payoffs of the original game)
first
the limit.
games
a reasonable sufficient condition for closeness of
considerations put into the
would argue that
we wish them
it
mouth
of our outside analyst at the top of this section
may
although the reader
is,
form?
in the extensive
already see
how broad a
class of small
perturbations this allows. Insofar as the refinements of Nash equilibrium deal with out-ofequilibrium behavior, a small probability ex ante of very different payoffs
large ex post.
Near
As
strictness
follows,
we
will see
that this
E
do
we must
we wish
specify a
mode
given by
is
E
game. This
is
not completely
trivial,
in the original
because strategies
game.
Since
in
we take
only able to view actions in the physical
is
(and not those actions contingent on which version nature selected), one
to look at the (marginal) distribution induced by strategies on endpoints of the
"physical" game.
That
one could compute the distributions induced on
is,
Note that even
that they converge.
has a single version of
T
,
if all
this gives us
of entire strategies.
We
Definition.
An
(t,
,
and require
convergence of strategies only at information sets
is
weaker than the usual convergence
therefore use a slightly sharper convergence criterion:
convergence of behavior prescribed in an elaborated
contain the nodes
Z
elaborations are trivial, in the sense that each
that are reached with positive probability. Hence this
is
To
of convergence of a sequence of strategies for elaborations
the point of view that the outside analyst
there
this.
to identify (equilibrium) strategies for a given
an elaborated game are much richer than strategies
answer
and the reader should consider
that are near strict, in this case under the notion of closeness developed above.
to a strategy for the original
game
so,
is
quite
under general elaborations
the previous two sections,
in
game
so,
what
whether the uncertainty of an outside analyst may be so great as
carefully
4.3.
In
may loom
game
at
all
We
ask for
information sets h that
1)
strategy
a
for the
game
a sequence of elaborations of
E
E {E k }
,
near
is
,
under general elaborations
that converges to
convergence criterion given above, and a sequence of
11
strict
strict
E
in the sense of
(normal form) equilibria
if
the
{cr
k
}
for the respective elaborations, such that the behavior prescribed
games converges
in their respective
A
3.
elaborations
if
Proof. First
we show that any strategy
is
profile that
a Nash equilibrium. Suppose that some a
Then
there
some player
is
i
where u(i,a) denotes the
the strategy profile o
for all j
5",-
,
(<7,,<7_i)
is
>
e
1)
.
near strict under general
is
in
strict
but
E
.
not a Nash equilibrium.
is
such that u(i,
in the
i
(/.,
near strict under general elaborations
is
near
is
and
utility to player
and
,
strategy
,
E
nodes
at
prescribed by a
Nash equilibrium
a (pure strategy)
if it is
t
a for the game
pure strategy profile
Proposition
and only
node
to behavior at
by the a k
game
if
>
(<7,-,<7_j))
the outcome
+€
u(i,a)
determined by
is
the strategy profile composed of a, for
and Oj
i
7^ i
{E k } and
Fix a sequence of elaborations
converge to a
the sense
in
payoffs in version
we
{v k } which
strict equilibria
are using here. Payoffs are uniformly bounded in
the prior probability of version
approaches
1
all
versions,
k approach payoffs in the originally given game, and
of elaboration
1
corresponding
1
Therefore
.
overall expected utility in
i 's
elaboration k approaches u(i,a). Suppose that there were some &{ as above. Construct
d k so
that, in elaboration
prescription of d{
,
k, at information sets
and otherwise
(/i,P,-(l))
follows the prescription of
i
coverges to u(i, (ai,a_,)) so that for large enough k
in elaboration
k
,
Ek
strictly
node
z
€
dominant
T
,
ask
i
i 'S
i
,
fix
follows the
i
,
As k grows, u(i,(a k ))
b k would be better
for
i
than a\
a (pure strategy) Nash equilibrium a and construct an elab-
and pure strategy
for player
777,(2, Si)
payoffs at
i
=
.)
when
To give
— T7l;(2,S;)
Letting #5,- be the
771,(2, s^)
z lies along a
i
for
i
,
design payoffs for
player
i
i
that
make
this, at
had to deviate from
Si
strategy
each terminal
if
2
is
to be
path that passes through no information
the strict incentive to follow
sets
at every opportunity, set
s,-
.
number
are Yli=i(i^^i
+
1)
where each
is
drawn from the
r,
S{
throughout the course of play. To do
how many times
reached. (For example,
of
.
l
as follows.
For each player
S{
dk
H
a contradiction.
For the converse, we
oration
,
h £
for
of pure strategies for player
versions of the original game.
set
We
,
in
each elaboration there
label these versions by
5ilJ{0}. Inversion
12
i
(r :
,
....,r,)
(r l5
...,
of elaboration k
77)
,
the
payoffs to player
£ Si
(i) If r;
,
i
are as follows:
then player
?
assigned the payoffs described above that
is
make
a strictly
r,
dominant strategy.
(ii) If
=
r,-
and, for some j
^
i
rj
,
£ Sj
then player
,
make
assigned payoffs that
is
i
playing Oi a strictly dominant strategy.
If
(iii)
=
T{
to u(i,-)
—
and, moreover,
mi(-,<7i)/k
where u
,
=
r$
for all
j
,
then player
is
i
assigned utility equal
the originally specified utility function and
is
m,-
the
is
number-of-deviations function defined above.
The
oration
has prior probability
k
=
M«( r i)
prior distribution on these versions
(^
~
l)/'c
selected, at version
^
i
crazy as
Ijk
he
,
is
is
i
=
'
Mi( r !')
player
,
may seem
The
has pure strategies.
among
crazy or sane and,
if
where
/x;(r,)
the
?'
is
=
told the value of
quite complex, but
either "crazy" or "sane",
divided equally
?
Version
set as follows.
1/(A:
•
(rj, ...,?"/)
#5,)
if
r;
G
of elab5";
and
Finally, for his initial information concerning the version
•
(rj, ....,77)
This construction
player
r
[,-
I
is
many
crazy, the
it
has
7-,-
simple interpretation.
a.
and there are as many
overall chance that
different
i
.
different
A
his craziness takes.
selected independently. (Note that the payoffs of one player will
other, so the payoff perturbations are not independent.)
If
for
i
crazy in elaboration
is
forms of craziness.
form that
ways
player
Each
to be
k
is
told whether
is
Players' "types" are
depend on the type of the
a player
is
crazy according to
a certain pure strategy, then the player has payoffs that cause him to play that strategy
at every available opportunity.
'
If
a player
is
sane and some other player
type), then the sane player wants to follow the fixed
available opportunity.
It
in favor of
the fixed
crazy (of any
Nash equilibrium strategy
Finally, if all players are sane,
game, with a small "kick"
is
a,-
at every
then payoffs are as in the original
Nash equilibrium
strategies a;
.
should be evident that this sequence of elaborations converges to the originally given
game, according to the
criterion
ration, the following strategies d{
we have
given.
We
are strict: Player
7
claim, moreover, that in each elaboi
follows
a,-
if
his initial
information
This sort of construction, in which every strategy is played with positive probability,
used by Fudenberg and Maskin [3] to obtain a robust folk theorem for repeated games
with long but finite horizons and small levels of incomplete information.
is
13
is
=
r,-
and he follows
,
we
established,
will
7"j
if
information
his initial
{dj} of his opponents
cr;
perfect recall, the information sets of
h for
that
i
from
different
&i
is
b{
and
cr,-
earliest in
is
and
cr;
because this
information sets of
is
We
).
own
i 's
£',
.
Once
this claim
is
wish to establish that a
actions preclude.
terms of precedence among
in
which
best responses to the strategies
's
i
By
is
l
precisely
the assumption of
are ordered by precedence. Take any information
i
all
of
i 's
information sets in which
previous actions (which would be the same under
i 's
an earliest information set) do not themselves preclude h
we
satisfy these conditions,
i
one of
is
some elaboration k
(in
except possibly at information sets that
set
e
v,
have the proposition.
Suppose that some (pure) strategy
a,
some
is
.
If
no
are done. There are three possibilities to
consider:
The information
(i)
h
preclude
,
strategies in
h
set
and because every strategy
And
at that information set,
does strictly worse than
(ii)
<7;
,
information set that
equilibrium.
is
i
strict best
strictly better
is
latter
in
is
h has positive
?'
strictly better
.
i
a
with positive probability
which everyone
,
and
it
corresponds to an
in the original
in
game and
one of two ways: Either
sane, in which case foDowing d{
is
a
strict
best
(recall the "kicker" in defining utilities for this the all-sane version), or
someone
else
is
crazy and
i
is
than
a,
,
we
sane, in which case following
response for the rest of the game. In either case, there
is
a strategy for
ct;
i
and again we have a contradiction.
more than one
weakly best
hit
himself doesn't
possible under their
a strategy for
with positive probability under
(The kicker perturbation to
as there
a,
is
hit
it is
is
i
any action other than that prescribed by d{
belongs to the sane variety of
Then under a
Because
.
the information set
h
are in a version in which
a
i ),
i
opponents
(from then on). Hence there
CTj
set
in the version in
response for
is
is
which contradicts the assumed optimality of
The information
we are
profile of his
a (conditional on whatever type
prior probability.
than
belongs to a crazy variety of
utility in
the all-sane version would be unnecessary as long
player, since then
we would have
that following
a,-
is
at least
the all-sane version, strictly best in every someone-crazy version, and the
would have
strictly positive probability.
has zero probability.)
14
For one player games, however, the latter
(iii)
The information
set that
is
h belongs to the sane variety of
set
off the equilibrium
path
in
does not, by his actions, preclude h
strategy combination by
i
's
.
is
Since there
opponents, there
reached. Moreover, Bayes' rule forces
must be crazy
the original
i
is
i
to follow
a,-
,
corresponds to an information
it
game under
the strategy a
,
and player
h
positive prior probability that
to conclude that
at this point
i
positive prior probability of every other
is
at this point, at least insofar as they
strictly better for
i
all
some one or more of
his
a
follow their parts of
will
be
opponents
.
Hence
it
and henceforth, and we have the same
contradiction as in the previous two steps.
Q.E.D.
Proposition 3 shows that any Nash equilibrium can be "justified"
the right kinds of doubts about each other's payoffs.
this deviation.
However
all
his
way
in
of the Na,sh equilibria are "robust" in that sense that, given a
we can consider any "small enough"
same
conclusion.
justifies
a given equibbrium,
Loosely speaking, every (pure) Nash equibbrium
5.
games and get
further perturbations of the elaborated
"open cone" of elaborations whose vertex
5.1.
which a player responds to
opponents depends on the inferences that he draws from
sequence of elaborations that approaches the original game and
the
the players entertain
Naturally, different forms of payoff
uncertainty will justify different equibbria, because the
an unexpected deviation by
if
is
is
justified
by an
the original game.
Elaborations with Types
Independent types
In
terms of our outside analyst story, section 4 says that unless one
players wiU not
draw unmodelled inferences about
their
of their opponents, one should not reject any (pure)
that our outside analyst
is
own
is
certain that
payoffs from the deviations
Nash equibbrium. Suppose, though,
prepared to assert that the model as written captures ab of the
information that players have about each other; he entertains (only) the possibibty that
each player has unmodelled private information about his
own
payoffs.
from the bterature, the analyst entertains the possibibty that each player
types (specified by payoffs), and each player knows his
own
type.
If it is
Using language
is
one of several
also true that
no
player has information (beyond the information possessed by the analyst, which ab players
have) about the types of the others, then we will say that the types are independent.
15
In
this case the class of small perturbations that the analyst
and the
4.2,
set of near strict equilibria that result
(Note that
would consider
might be smaller as
is
smaller than
well.
considered in section 4, a player's payoffs were
in the general elaborations
not wholely determined by the player's "type". In particular, in the proof of proposition
the utility function of the "sane" type of player
player was "crazy." Hence
we
payoffs;
now
will
distributed; that
To
(L,D)
that the prior
we
1
lay with
D
.
on whether any other
crucially
game
was pertinent to player
(This, of course, supposes that
L
chooses
1
dominant
This
below)
is
will
2's best choice
the top version of the game.)
in
could not be given information about 2's payoffs superior
1
moving
is
not render near
In figure 8
informed about
sequential; a
8.
some
strict
know
8
that
In figure 7,
equilibria that are not themselves strict. Consider
we have an equilibrium (Lu,D) which
we have an elaboration
1
of this
only. (There are
game
in
which there
no information. In
his payoffs; player 2 gets
this
more involved elaboration with independent types
bottom version
of the
game, player
(Li^^i^, D)
1
plays
is
is
8
The
R% with
subgame
some uncer-
two possible types of player
renders (Lu,D) near strict under such elaborations.
native,
in the
not to say that "elaborations with independent types" (to be defined formally
tainty as to the payoffs to player
1 is
in
for him.
the games in figures 7 and
imperfect.
game
Hence,
2's payoffs.
information set that contains the node from the high probability version, would
is
a
in figure 5,
to the information that 2 receives, this would be impossible, as player 2,
U
own
and the Nash equi-
in figure 3
by choosing R, was communicating to 2 that
player
his
also require that the types be independently
near strict by formulating,
received information that
we supposed that
3,
have a product structure.)
will
this equilibrium
this elaboration, player 1,
if
f.i
depended
meet the requirement that each player knows
require this. Initially,
We made
.
which player
But
did not
i
see the force of this restriction, consider the
librium
in
is,
we
in
1.)
Player
game, (L^R^u-ido, D)
will
intuition
make
is
it
strict,
which
that in the alter-
positive probability
and
will
not strict because player l's choice between u^ and di
To make the choice matter, imagine that there
is independent uncertainty about 2's payoffs, where, with small prior probability, 2 has A
as a dominant strategy. Think of an elaboration with four versions, two (for each type of
player 1) by two (for each type of player 2), where each player knows his own type but not
the type of the other. If this is too sketchy, the general definition and construction to be
given in a moment will clarify matters.
is
In figure 8,
is
irrelevant given player 2's choice of
D
.
16
continue with d 2
R 2 d2
because in the bottom version,
,
dominant. Player
is
2,
then, given
the move, assesses probability one (at the L\ action equilibrium) that the bottom version
being played. As
is
With
must modify the
An
ration
elaboration
is
Ni
Yii
i
there
said to be an elaboration with independent types
is
an integer
is
for each
(iii)
there
is
and version
(iv)
for each
(j
and j G
i
u ...,'jj)
,
u(i,
TV,-
is
simply
i
/z,-
{1, ...,TV,}
;(ju
the information given to
prevails
we
such that the
number
if:
of versions in the elabo-
i
a probability distribution
i
First,
an allowable elaboration.
writing (ji,...,ji) as the index of one of the elaborations, where
(ii)
is
definition of
for each player
(i)
therefore continue with d 2 given the chance, 2 chooses D.
example to provide motivation, we turn to a formal development.
this
Definition.
1 will
if
...,j/))
=
on
{1,...,TV,}
with
a payoff function
6
ji
//((ji,
{1,...,TV,-}
ji))
....,
v(i, z;j)
,
there
— [} m(ji)
such that for each
;
i
v(i, --Ji)
the true version
is
(j\,.--,ji)
concerning which version
j,-
The same convergence
criterion as in section 4.2
is
we now
used, although
restrict to
elaborations with independent types for perturbations.
Definition.
A
strategy a for an extensive
independent types
{E
k
}
if
there
that converges to
,
k
strict equilibria {cr } for the
by the d k at nodes
To
give our result,
Definition.
A
each player
i
1)
if
converges to behavior at node
we need a
there
and index k
is
,
under elaborations with
t
prescribed by a
Damme
,
further definition.
game 7
a sequence of totally mixed strategies a k
o±
is
.
a
strict best
o
—
is
>
strictly perfect
a such that,
response (up to equivalent strategies for
k
[14].
17
for
i
)
9
.
The term "strictly perfect" is used by Okada [10] to mean something quite
same vein, we later use the term "quasi-perfect" in a sense different from
van
E
corresponding normal forms, such that the behavior prescribed
In the
in
strict
convergence criterion, and a sequence of
in the sense of the
to the other players' strategies prescribed by
9
near
(pure strategy) equilibrium a of a normal form
the normal form
in
(i,
is
a sequence of elaborations with independent types of
is
E
E
game
different.
its
usage
Remark. Figure
9
shows that (Lu D)
is
7
normal form. The num-
strictly perfect in the
bers in parentheses are "trembles" corresponding to a sequence of totally mixed strategies
converging to (Lx,D) such that
u.
Ru
,
are strict best responses for players
normal form. The key
respectively, in the reduced
often than onto
D
Lx and
even though, having played
is
that player
R
trembles onto
2,
Rd more
player one "should" continue with
,
.
1
and
1
10
Proposition
For a given extensive form
4.
normal form
in the
Proof.
E
,
every equilibrium o that
near strict under elaborations with independent types.
is
—
Fix a sequence of totally mixed strategies u k
o
best response to each
In the
.
of pure strategies of player
original
that type
Sj
i
if
s;
=
0{
for
i
is
chosen
game
strictly perfect
is
,
.
k th elaboration, N,
Payoffs for
and they make
is
>
i
sl
in versions
o such that each
is
set equal to
a,-
is
#5,-, the
corresponding to
a
strict
number
are as in the
S{
dominant otherwise. The marginal probability
exactly that assigned to
s;
in
a\
.
Hence these independent
elaborations do converge in the sense of the convergence criterion to the originally specified
game.
By
play a,
construction,
it
and to play
S{
,
is
a strict Nash equilibrium
when crazy
of type
s,
.
in
Ek
for each player
i
,
if
sane, to
This gives the result.
Q.E.D.
Given our previous
results, Proposition 3 should not
be surprising. Imperfect equilibria
that are perfect in the normal form correspond to players inferring that,
deviated from equilibrium play once, he
justified
if
is
likely to
do so again.
if
an opponent has
Such inferences can be
each player believes that with small probability his opponents' payoffs
may
be
quite different from payoffs that have high prior probability.
The converse
strict
to Proposition 4
is,
however,
false:
with respect to elaborations with independent types. The problem
equilibrium correspondence
is
L both
receive
1.
If either
is
that the perfect
poorly behaved with respect to extensive form payoff pertur-
bations. Consider the foDowing two person simultaneous
,
Imperfect equilibria can be near
chooses
R
,
move game.
If
both players choose
then both receive zero. Only (L,L)
10
is
perfect, but
Recall that Selten introduced perfectness in the agent-normal form precisely to rule
out such correlated trembles.
18
(R,R)
near
is
under payoff perturbations
strict
payoff to
(R,R) was
and
approach zero.
let
(
One way
e
in the extensive form:
for each player, holding the payoffs for
to obtain a partial converse to Proposition 4
is
Imagine thai the
{L,R) and (R, L)
at zero,
assume that each
players'
to
"main type" (the type corresponding to the main version of the game) has exactly the payoffs
given in the original game. That
is,
our outside analyst assesses small probability that he
has the payoffs off by (possibly) a large amount, but he assesses high prior probability that
he has precisely the correct payoffs, and each player knows
is
not particularly interesting. Instead,
taking
its
we modify the
E
an extensive form
m
for the extensive form, {u
}
,
is
with hmit u
normal forms corresponding to the u m
,
12
um
let
-
5.
.
a sequence of payoffs
if
there
such that a
is
strictly perfect in
is
u
quasi-perfect.
is
game E
and only
if
Suppose that o
be the payoff function
11
quasi-perfect.
Let
u
be the extensive form payoffs such that a
From Proposition
strategy profiles
dm
'
k
4,
.
Write
Em
for those elaborations are such that
& m,k
—
a
in
Em
'
k
are uniformly
take m;v large enough so that u mN
bounded
is
within
11
in
k
.
1/2N
E
for
,
and
normal
form game with payoffs
'
k
E m,k
of
Em
and
are strict equilibria in
Examination of the construction
number
m
am
is
quasi-perfect.
strictly perfect in the
is
for the extensive
Proposition 4 shows that we can assume that the
the elaborations
if it is
there are elaborations with independent types
the corresponding normal forms, and
a
a pure strategy equilibrium
,
under elaborations with independent types
is
each of the
.
For a given extensive form
forms corresponding to the u m
um
(pure strategy) equilibrium
quasi-perfect
Remark. Every perfect (pure strategy) equilibrium
Proposition
A
with payoff function u.
a of the corresponding normal form game
Proof.
definition of a perfect equibbrium by
closure with respect to payoff perturbations in the extensive form.
Definition. Fix
near-strict
This class of perturbation
this.
of versions and size of payoffs in
and k
of u
in
,
.
For each positive integer
and then take
k^j large
N
,
enough
Eddie Dekel has pointed out that this definition of quasi-perfection is equivalent to
a definition in which a need be only (weakly) perfect in each of the normal forms corresponding to the u m
12
We thank Eddie Dekel for finding an error in an earber version of this proof.
.
19
so that
the
(iii)
,
'
a mN
with
Hence a
To
is
'
is
N
a.
near
E
on strategies of
game
E
in
more
are within 1/27V of u mN
k»
in this elaboration. It is
;
and
immediate
N
(in
approaches a
strict equilibria for those elaborations that
E
game
a strategy for
is
ct,
1)/7V or
we begin with a
establish the converse,
If
.
—
>
)
.
strict.
vation. Fix an extensive
of
E mN
1
sequence of elaborations with independent types approaching
sequence of
a.
of
the payoffs in version
(ii)
;
version has probability (TV
first
{E mN k *}
that
E
N
>
kN
(i)
i
E
in
and
,
E
in
i
induced by
cr,-
let
E
,
we
.
That
and a preliminary obser-
piece of notation
be an elaboration (with independent types)
Ma,
will write
averaging over
is,
according to ^ (or averaging over
of
all
marginal distribution
for the
versions of the original
all
personal types according to
i 's
/x,
,
which gives the same result because types are independent) we compute the probability
with which
Now
strict
to
i
will
assume that M<7;
in
a
is
totally
and a pure strategy
E We
claim that
.
mixed, for each
i
E
in
This
.
is
,
we
Sj
i
,
has
can, in the
s;
as a
for
a,-
is
,
a
each player
and follow the dominant strategy
i
:
"Follow
for the
still
appended types,"
a
is
composite strategies for the others. (The constraint on the
a
where
to each
in the
strict best
extra
response
prior
strictly positive) so that the
from the
for types
a,-
is
loss of generality
augmented elaboration (with the extra types) choose the
probability of those extra types to be sufficiently small (but
composite strategy
E
in
i
a which
we can append
so because
dominant strategy. Because
profile
without
it is
of types a further type for each pure strategy of
list
type for strategy
i
E
suppose that we have an elaboration
equilibrium for the normal form of
player's
for
be "observed" to play various strategies
original elaboration,
strict best
response to the
size of the probability of
i 's
extra types depends on the extent to which the other players' strategies were strict best
responses.)
Suppose then that a
dependent types)
just above,
Ek
E
we can assume
For each player
tree into
—y
two
sets:
i
Let
is
near strict for
E
and corresponding
that
WLa k
is
,
and
fix
a sequence of elaborations (with
strict equilibria
ak
—
*
a
.
By
the observation
totally mixed.
(holding a fixed), partition the terminal nodes of the original
A,-
game
be those nodes which, for some combination of strategies by
opponents, can be reached when
i
uses
v (i,z; 1) to denote the payoffs to player
in-
a,-
i
,
and
in
20
let
B{ be the complementary
the main version of the
game
set.
i 's
Using
in elaboration
Ek
node
at
z
consider the payoffs
,
w k (i,z) =
supu'(?.
j>k
w '(i,z) =
We
assert that
Ma-
sequence
To
z
for
z; 1),
game with
and
£ A,
for
form given by
£ Bi.
z
wk
payoffs given by
against the test
,
7
.
we compare
see this,
strategy
against
Si
u J (-,
payoffs are
is
for the original extensive
inf u J'(i, z\ 1),
strictly perfect in the
is
<r
w k (i, z)
the difference to
wk
Mai,- when payoffs are given by
•;].), for
j
>
reached when strategies are
strategies are (s;,Mai,-)
a=
between playing
i
.
We
k
a;
and some other (pure)
with the difference when the
Write 11(2) for the probability that terminal node z
.
(a,-,
Mai,-)
and write
,
for this probability
II' (z)
when
the
wish, then, to sign
]T(n(z) - n'(z)K(»,
- £(n(*) - n'(z)y(i,
z)
z- 1).
Z
2
Partition the terminal nodes z into four classes: (1) nodes that are reached with positive
probability under both a; and
U(z)
up A,
11(2)
=
>
and
,
=
IL'(z)
;
(4)
s;
;
nodes with
classes (2)
and
II'(z) in class (1).
(4)
nodes such that
(2)
make up B
l
Now
.
-
zeClass (3)
But
a,-
ativity of
perfect
Thus a
is
A
when
is
a
II'(z)
z in class (3) are in A{
conclude, then, that
II'(z)
Note that
.
since
A
a,-
=
;
(3)
and
classes (1)
and
Sj
nodes with
(3)
make
are pure strategies,
for z in class (1) net to zero. For
k
]T
U'(z){w (i,z)-v^i,z-l)),
^eClass (4)
where we have used the facts that
class (4). Since
=
we can rewrite
n(z)(™%z)-v j (i,z;l))
J2
U(z)
Thus the terms making up
z in class (2), the terms are zero. Thus
A=
=
>
Il'(z)
Jl(z)
A
is
strict best
,
=
for
wk >
z
v^
in class (3)
there,
and
=
II(z)
and conversely
for
z in
We
for class (4).
nonnegative.
response to Mai,-
shows that the same
is
payoffs are given by
when
payoffs are given by v J
true
when
payoffs are given by
wk
And,
since
.
vi(-,-;l)
.
w k Thus
.
The nonnega
converges to u,
is
strictly
wk —
>
u.
quasi-perfect.
Q.E.D.
21
5.2.
Personal types
The
perturbations with independent types
effect of
and
tensive form payoff perturbations
different information sets of the
same
is
essentially
We
player.
can, and
to allow for ex-
between the "trembles" at
to allow correlation
(ii)
(i)
now
loosen the definition
will,
of elaborations with independent types to allow for the trembles to be correlated across
different players.
An
elaboration has personal types
if
each player's payoffs depend only on
his
own
infor-
mation about Nature's choice of version; the players' information need not be independent.
must
Formally, an elaboration with personal types
satisfy (i),
of an elaboration with independent types, but the overall
and
(iii)
(iv) of the definition
measure n on versions needn't
have the product form.
Consider the
R
game
and r, players
1,
and
2
in this
a
3 play
by
pair of independent strategies
least
13
in figure 10.
2x2x2
and
1
subgame where
In the
players
and
3 choose
3
have played
Against any
player 3 can guarantee himself a payoff of at
2,
Hence
any (normal form) perfect equilibrium of the game.
1
and
simultaneous move game.
B
subgame by playing a 50-50 mixture of A and
equilibrium of the game,
1
R
and
r
,
in
,
so
d
is
never part of
any normal form perfect
The subgame must
respectively.
along the equilibrium path in any perfect equilibrium, and in the subgame
1
and
2
lie
have
unique equilibrium strategies. (They play matching pennies no matter what 3 does.) Hence
3's best choice is
A
namely (R, r,A,
.577
this,
hence this
is
.
That
+
is,
-5T, .5h
game has a unique (Normal form)
this
+
.5t)
.
perfect equilibrium,
Perturbing the extensive form payoffs
the unique equilibrium that
is strict
will
not affect
under elaborations with independent
types.
But now consider elaborations with personal types. In particular, imagine that there
are two other types of players
(We
dominant strategy.
1,
RT
for
which
is
dominant.
t
is
13
[9]
This
game
For player
is a
all.
and
2.
refer to this as
dominant.
3 learns nothing at
1
2,
there
type
is
RE
is
.)
his
own
h
is
game suggested
making the same
point.
22
is
a
dominant and a type
type, so, in particular, player
crucial to our construction, the probability
variation on a similar
RE
In the second other type of player
a type for which
Each player learns only
And, what
gives another example,
In the first other type of player 1,
to us by J.-F. Mertens.
measure
Myerson
such that there
is
/i
is
high probability, say
RH
type
-
1
and 2
2<5
that
,
players are of the "standard"
all
types, probability 6 that
1
RT
and 2
the reader doesn't like this perfect correlation
and
2, it will suffice
type
is
/
(If
.
is
L by
14
equilibrium path.
1,
and
+
.5T, .5/i+ .5<) in the original
3 can
L
plays
This equilibrium
made near
is
The
been selected. Given the prior
H
with
and 2 with h
,
and
correlated continuation by
-.5
and so
,
in the
main
We
advance
may seem
.
nor
(Of course,
B
2 with
will
this choice
Nash
puts him off the
as follows: Since
some other
t
.
version has
(i) 1
When
will follow
3 assesses this
guarantee him a payoff above
by 3 validates
l's
choice of
D
definitions to follow in the following spirit.
to the reader to be too large a class of perturbation, because
know more about a second
player's payoffs than that second player
constraining ourselves to elaborations with the "type" structure,
suppose that each player has
all
the information about his
own
we
payoffs that any other player
further restriction to independent types implies that no player learns anything
about others' types from
in their past
learned
d
A
neither
example and the general
this
By
herself.
The
2,
and
1
Nash because
It is
as equally likely that
T
follow with
consider the
1
type
version.)
they allow one player to
has.
and
1
it
1 is
the types of
in
argument runs
signals to 3 that
posited, 3 sees
1 will
(ii)
3 optimally chooses
General elaborations
knows
we
\i
game.
that
/>
we adopt the personal type
strict if
intuitive
R
the main version, the choice of
in
Now
do anything he wishes, as L by
elaborations above, for 6 approaching zero.
1
and probability
,
to have high but less than perfect correlation.)
equilibrium outcome (L,d,a,.5H
d by 3 engenders
type h
is
his
own.
One might
instead imagine that players have "links"
- perhaps they went to the same school, or had similar teachers when they
game
theory, or whatever.
Knowing that these
links are possible, one suspects that
knowing something about the payoffs of one player might
of another. This
is
true, pari pasu, of players within a
tell
you something about payoffs
game. Which leads
to personal,
but
not independent, types.
Although
on
it
doesn't lead to quite the
this sort of elaboration.
Nash
equilibria,
one
is
By analyzing
same
the
thing,
we can
give another perspective
game noncooperatively and by
restricting to
supposing that players cannot access correlating devices, as defined in
14
Subsequent moves can be fiUed in any way the reader wishes, and we will still have a
Nash equilibrium. We have filled them in with the subgame perfect equilibrium strategies
for a reason that will become apparent momentarily.
23
Aumann
(See also Forges
[1].
Myerson
arid
[2]
of the outside analyst's state of knowledge
have access to correlating devices, he
perturbation of the
Suppose that a more accurate statement
[9].)
that while he
is
is
not completely certain of
is
game would be an
do not
fairly certain that players
Then an appropriate
this.
elaboration with personal types, where the utility
functions of the players in each version are close to (or perhaps even identical with) their
game modelled.
payoffs in the
had access to a correlating device, the
means that the
is
a
game
And
device will be of
his
information
set, afraid
that
have obtained access to the correlating device and
is
if
if
matching pennies
fact that they play
no use to them. But
even
players
if
one
2
subgame
in the
we change the subgame
and
1
so that
is
it
available.
unable to observe the outcome generated by the correlating device, then
is
he might choose d at
point
in figure 10,
of coordination, then they might well use a correlating device
player 3
if
example given
(In the
l's
will
R
choice of
indicates that
if
and 2
The
play a correlated equilibrium.)
that a low prior probability that players can access correlating devices
a high posterior probability,
1
may become
one observes an out of equilibrium action from one or more
of the players.
To
characterize the set of near strict equilibria under elaborations with personal types,
we introduce the
may
following definition, which
modify trembling hand perfection, to allow
Definitions. Fix a pure strategy profile
where
for each
i
and k
of players other than
pure strategy
profile
i
<f>\
,
,
is
a
is
a
be of independent interest.
.
The
equilibrium
for
a normal form game.
is
to
A
sequence {{<f>i}{-i}kLi
a totally mixed distribution over the pure strategy profiles
c- perfect test
sequence for a
a of a normal form game 7
is
idea
for "correlated trembles."
is
sequence {{^f}} for a such that for each player
4>i
The
strictly c-perfect
if
each a
i
-
t
for each
if,
c-perfect
if
a
,
$ (a)
—
>
1
.
A
there exists a c-perfect test
and index k
is
i
,
strict best
a
is
a best response to
response in the normal
form.
Remarks.
strategies
Note that the distributions
by the players.
<f>^
need not correspond to independently mixed
Also perfection and c-perfection are identical for the case of
two-player games.
Proposition
6.
If
an equilibrium o
is
strictly c-perfect,
to elaborations in personal types.
24
then
it is
near
strict
with respect
,
Proof. Let
k
be the c-perfect test sequence that makes a
<f>
=
elaborations (for k
game: There
will
where
runs over
i
for the other players.
And
all
In the
in
cr'_
i
version, player
(i, a'_i)
will
which players are
construct
of the original
sane, having the
all
game
there will be one version of the
the players and
game, and the other players
original
game
be a "main" version of the
payoffs given in the original game.
(i,a'_ { ),
composed of two kinds of versions
1,2,...) that are
We
strictly c-perfect.
for every pair
over the pure strategy combinations
will
i
have the payoffs given
in the
have payoffs that make the constituent, parts of
a'_
x
dominant
is
as in our previous constructions. In each elaboration, player
"sane" or not and,
for
an elaboration
To
version
of
is
We
1
is
not,
what are
k th elaboration,
his payoffs. This
it
— ne k
then meets the requirements
remains to specify the probabilities of the various
will assign probabilities so that, in
(for
some
versions in which
all
he
in private types.
specify the
versions.
if
told whether he
is
i
i
>
e^
is
elaboration k
,
and the marginal probability
to be determined in a bit),
sane and the others not will be
k
use p (i,c'_i)tk to denote the probability of the (i,a'_
1
u_
conditional probability of crazy types
{
,
{
)
the probability of the main
ek
More
.
version; that
conditional on
i
precisely,
is,
p
k
we
will
(i,a'_ i ) is the
being sane and the others
crazy.
Suppose that,
and according to
in elaboration
their
the probability that
i
bility
I
assesses that his
-
that everyone
k
(i, a'_i)
and, for
play according to a
when they
is
+ tkP k {i, di)
l-(n-l)£*
a'_ i
,
^
i
is
is
sane.
'
We will
want
=
€k
/(l-{n-l)e k )
25
is
the joint proba-
non-zero only
this to equal
,
p (t,a_i)
is
sane and the others play
yields
<r_,
being sane,
._,_.
sane and the others play a'_ { (which
i
i
are sane
will play a'_ i is
denotes the indicator function. The numerator in this expression
the marginal probability that
p
all
opponents
n€fc)V
plus the probability that only player
for
players
,
dominant strategies when not. Then conditional on
(1
where
k
a'_ i
;
if cr'_
i
—
cr_,-
the denominator
^H^'-i)
>
),
is
which, solving
To ensure that
;/"(?', CTj)
>
,
we must have that
1
That
is,
1
we choose
if
hand
k
.
>
e^
By
And
since
k
construction,
is
<f>
-#(*-,-)
n-(n-l)#(<7_0
to be less or equal to the
side of the previous inequality,
well defined.
is
fc
we must have
€fc<
So
- ne k
- (n - l)e
1
m°-i) >
and use
we
k
<f>
minimum
(over
to specify the
see immediately that
a c-perfect test sequence,
fc
:
(a_;)
—
»
1
i )
p
a
k
is
of the terms in the right
as above, the elaboration
strict in
each elaboration
implying that
,
ejt
—
,
so the
elaborations converge to the original game.
Q.E.D.
Finally,
we have a converse
to Proposition 6, exactly parallel to our converse to Propo-
sition 4.
An
Definition.
perfect
if
equilibrium
a of an extensive form game
k
there exists a sequence of payoffs u
—
E
with payoffs u
u such that a
is
is
quasi-c-
strictly c-perfect in the
corresponding normal form games.
Proposition
if it is
,
An
near strict
The proof
i
7.
is
equilibrium
in
a
of an extensive form
game
is
quasi-c-perfect
and only
elaborations with independent types.
just as in the proof of Proposition 5, except that
Mai
-
2
is
replaced, for each
by the conditional distribution of the actions of the others, conditional on
first
if
type.
26
i
being the
References
1.
Aumann,
R.
Subjectivity and correlation in randomized strategies,
J.
Math. Econ.
1
(1975), 67-96.
2.
An approach
F. Forges,
communication equilibrium, Econometrics 54 (1986), 1375-
to
1386.
3.
D. Fudenberg and E. Maskin, Folk theorems
for
repeated games with discounting or with
incomplete information, Econometrica 54 (1986), 533-554.
Harsanyi,
4. J.
Games with randomly
equilibrium points, Int.
5.
Kohlberg and
E.
J.
J.
disturbed payoffs:
Game Theory
F. Mertens,
On
A new rationale for mixed-strategy
2 (1973), 1-23.
the strategic stability of equilibria, Econometrica 54
(1986), 1003-1038.
6.
D. Kreps, P. Milgrom,
J.
repeated prisoners' dilemma,
Roberts and R. Wilson, Rational cooperation in the
J.
finitely-
Econ. Theory 27 (1982), 245-252.
7.
D. Kreps and R. Wilson, Sequential equilibria, Econometrica 50 (1982), 863-894.
8.
R. Myerson, Refinements of the nash equilibrium concept, Int.
J.
Game Theory 7
(1978),
73-80.
9.
10.
R. Myerson, Multistage games with communication, Econometrica 54 (1986), 323-358.
Okada,
On
the stability of perfect equilibrium points,
Int. J.
Game Theory 10
(1981),
67-73.
11.
R. Selten, Spieltheoretische behandlung eines oligopmodells mit nachfragetragheit, Z.
fur die
Gesamte Staatswissenschaft 121
12. R. Selten,
games,
13.
Int. J.
E. van
(1965), 301-324.
Re-examination of the perfectness concept
Game Theory 4
Damme,
for equilibrium points in extensive
(1975), 25-55.
"Refinements of the Nash Equilibrium Concept," Springer- Verlag,
Berlin/Heidelberg/New York, 1983.
14.
E. van
Damme, A
relation between perfect equilibria in extensive
proper equilibria in normal form games,
Int. J.
27
Game Theory 13
form games and
(1984), 1-13.
L
R
0,1
2,1
FIGURE
1
2
L
M
R
0,1
1,1
2,1
FIGURE 2
(2 0)
'
u
L
(1,1) -<
1
-
•
R
2
o
(-1,-1)
FIGURE
3
(1.1)
u
(0,0)
i
I
i
1
R
2
(0,0)
u
0,1)
FIGURE 4
u
U
(1.1)-*
(2,0)
R.
1
M
-*-•
(-1.-1)
u
(0,-1)
1
(-1,0)^
-o-
w
(0,0)
FIGURE
5
D
1
n-2
Lu = Ld
2,1
2,1
Ru
3,3
0,2
Rd
1,0
0,2
(*)
(£)
FIGURE
C
(tt)
(i)
(2,1) -*
o
•
•^
FIGURE
7
(3,3)
1
(2,1)-*
L
(-1.1)
R
1
•o
1
(3,3)
R.
*+
FIGURE S
D
u
L 1 L2
1-26
hR 2
1-6
1-6
,
,
1-26, 1-6
1-26
R1L2
2-36
R R2
2-26,-6
1-6
,
1-6
-1,-1+6
,0
-1+6 ,-1+6
1
FIGURE
9
361
I
C83
^(1,-1,-2.9)
--(-1,1,1.1)
(-V2.2)-*
(-2-3,-V2)
FIGURE
10
MIT LIBRARIES
3
TDflD
ODS 13
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