with Infinite Sets of Types and Actions
Sequential Equilibria of Multi-Stage Games
by
Roger B. Myerson and Philip J. Reny*
Draft notes October 2011
http://home.uchicago.edu/~preny/papers/bigseqm.pdf
essential sequential equilibrium while maintaining general existence. Abstract: We formulate a definition
of basic sequential equilibrium for multi-stage games with infinite difficulties of this basic concept and, in
light of them, propose the more restrictive definition of type sets and infinite action sets, and we prove its
general existence. We then explore several
* Department of Economics, University of Chicago
1
type sets and infinite action sets, and prove general existence. Tirole 1991) have been used
for infinite games. No general existence. but rigorously defined extensions to infinite games
have been lacking. Sequential equilibria were defined for finite games by Kreps-Wilson
1982, Harris-Stinchcombe-Zame 2000 explored definitions with nonstandard analysis.
Various formulations of "perfect bayesian eqm" (defined for finite games in
Fudenberg-Goal: formulate a definition of sequential equilibrium for multi-stage games
with infinite
Others should explore alternative definitions. include reasonable equilibria and exclude
unreasonable equilibria. Here we present the best definitions that we have been able to
find, game (in some sense) but have limits of equilibria that seem wrong. The problem is
to define what finite games are good approximations. taking limits of sequential
equilibria of finite games that approximate it. but of course this depends on intuitive
judgments about what is "reasonable". It is well understood that sequential equilibria of
an infinite game can be defined by We must try to define a class of finite approximations
that yield limit-equilibria which It is easy to define sequences of finite games that seem
to be converging to the infinite
2
T
= {possible informatio
actions for player i at pe
under finite  and com
ik
Ak  {1,...,K} periods of the game. Dynamic multi-stage games
=(,N,K,A,T,p,,v)  = {initial state space}, i  N = {players}, finite
set. Let L = {(i,k)| iN, k{1,...,K}} = {dated players}. We write ik for
(i,k).
, including all one-point sets as measurable. for  and T
There is a finitely additive probabilit
are measurable.
ik
. ikIf  is a finite set, then all subs
A =  A = {possible sequences of actions in the whole game}.
hK iN ih

The subscript, <k, denotes the projection onto periods before k. For example,
A =
= {}), = {possible action
(A

 A
<
k
h<k
i
N
i
h
<
1
=

and for aA,
a
<
k
h<k
i
N

a
i
h
Any player i's information at any period k is specified by a type function

such that, aA, (, ) is a measurable function of .
a

i
k
:
A
 is the parti
T
ik < i
k k
<
k
Assume perfect recall: ikL, m<k,

ikm
i
k
i
m
( (, )) =
(,
a
a
(
{ |
( ) {
t  t R
a
ik
<k
ikm ik

A
such that
i
m

ikm
ik
:T 
T
i
m
im
im
),a ), , aA,
and
<m
}} is measurable in
T
i
m

T
i
k
Each player i has a bounded utility function :A such that im
v
vi(,a) is a measurable function of , aA. Boundi(,a)|  ,
|v
(i,,a).
i
.  , ,  m
A a R
i i i
mmm
3
a[Prob
lems of
finite
suppor
t
mixtur
es in
appro
ximati
ng
games]
Examp
le.
(Kuhn)
Consid
er a
zero-su
m
game
in
which
player
1
choose
sa
numbe
r
Play
er 1
can
guar
ante
e 1/2
by
choo
sing
agua
rante
e 1/2
by
choo
sing
f(x)
=
1/2
for
all x.
Lebe
sgue
integ
ral
must
be
1/2.
Play
er
1’s
payo
ff is
f(aea
ch of
play
er
1’s
finit
ely
man
y
avail
able
actio
ns,
play
er
1’s
equil
ibriu
m
payo
ff is
0.
But
in
any
finit
e
appr
oxi
mati
on in
whic
h
play
er 2
can
choo
se a
func
tion
that
is
zero
at
One
soluti
on is
to
replac
e
ordin
ary
action
sets
with
mixed
action
sets.
Algeb
ras
(close
d
under
finite

and
compl
ement
s) of
meas
urabl
e
subset
s are
specif
ied
r each A
a
l
s
o
f
o
1
, including all one-point sets as
measurable.
1
1
). in [0,1], and player 2 chooses a
uniformly
continuous function, f, from [0,1] into itself
from [0,1] and
whose
player 2 can
Each  (, ) and each vi(,a) is assumed jointly measurable in (,a).
ik
a <k
Let ~ =
[0,1]). In the state ~ =
), nature draws  from the given p, and,
ikL
(
(,(~
ik
for
each
ik,
draws
~

[
0
,
1
]
i
n
d
e
p
e
n
d
e
n
t
l
y
f
r
o
m
L
e
b
e
s
g
u
e
m
e
a
s
u
r
e
.
ik
~This
define
sa
new
distri
butio
n p on
~.
No
player
obser
ves
any of
the
new
~
i
k
i
k
i
k
L
e
t
Ã
=
:
.
ik
{
[
} be
m player ik’s set of mixed
0 actions.
e
,
a
1
s
]
u
r

a

b
A
l
e
m
a
p
s
ã
)
ik
ik <k
(~,ã) =
v
Then ṽ
i
) = (
<k),
ik
ã(,ã(
meas
urable
in ~
for
(,ã(~)
) and
~
each
profil
e of
mixed
action
s ã.
A
l
l
o
u
r
i
,
<
k
(

~
,
ã
i
a
n
a
l
y
s
i
s
c
o
u
l
d
b
e
d
o
n
e
f
o
r
t
h
i
s
m
o
d
e
l
w
i
t
h
Ã
,


~
,
p
~
i
n
s
t
e
a
d
o
f
A
,


,
p
.
4
Problems of spurious signaling in approximating games Example. =  and chooses
Nature chooses {-1,1}, prob. ½ each. Player 1 observes t
1
[-1,1]. Player 2 observes t = a and chooses a
a
1
{-1,1}.
Payoffs are as follows.
1
= -1 a
(0,2) 2(1,0) 2
a
 = 1 (0,0) (1,1)
No observation t
=1
2
2
 = -1
2
2
= 1 since player 1 always
[-1,1] should lead player 2 to choose a
2
prefers a
=1
= 1 and the set of signals player 1 can send does not depend on . But any
approximation where 1’s finitely many actions are, e.g., all rational when t
To avoid such spurious signaling, the finite set of actions available to a player should not
1
= 1 when t
is any rational
number.
and are all irrational when t= -1 reveals , so a1
2
2
depend on his informational type.
depend on his type and in which the action aBut now consider any finite approximation = x > 0 is
in which player 1’s finite action set does not
a
1
= -x is not. Then t
2
= x implies  = 1, leading player 2 to choose a = 1.
1
2
, an
ik
type-space partitions become arbitrarily fine.we will expand the players’ik
5
finite action sets to fill in the action spaces ATo avoid also the latter
spurious signaling, we should limit player 2’s ability to coarsen the
players’ information by finitely partitioning each of their type spaces,
Tdistinguish her types and allow player 1 any finite subset of his actions.
So, we will
faster than th
Probl
ems
of
spuri
ous
knowl
edge
in
appro
ximat
ing
game
s
players
inform
ation
that
they
do not
have
in the
origina
l
game.
Some
approx
imatio
ns of
the
state
space
 can
change
the
inform
ation
structu
re and
give
the
first
(1's
cost to
sell
some
object
) is
obser
ved
only
by
player
1,
Exam
ple.

includ
es two
indep
enden
t
unifor
m
[0,1]
rando
m
variab
les:
the
secon
d (2's
value
of
buyin
g this
object
) is
obser
ved
only
by
player
2. For
any
intege
rm>
0, we
might
finitel
y
appro
ximat
e this
game
by
one in
which
the
state
2m
m
equally likely pairs of numbers:
[0,1];
m-digit decimals in
m-digit decimals but the last m digits repeat the first number in the pair.
but they represent games in which player 2 knows 1's type. As m,
these pairs uniformly fill the unit square in the state space ,
separately, we avoid giving spurious knowledge to any player. we can keep the
original state space  and probability distribution p in all finite approximations,
avoiding the above information distortion.
space inclu
the first nu
decimal in
With finite
By partitio
ik
or the ability to In some games, the
ik
A
convergence of a net of approximations (not just a sequence) that are
and all finite partitions of T subsets of A
ik
distinguish some special subsets of
i
kT
ik
finite partition of types are available when considering the properties of the limits. . So,
any finite collection of actions and any
6
Observable events and action approximations
For any aA and any measurable R  , let P(R |a) = p({|
T

i
i
i
i < i
) }). (,
k
k
k
kk k
R
a
(For k=1, we could write
|a), ignoring the trivial a =.) i1
P(R
The set of observable events for i at k that can have positive probability is
Q = {R  | R is measurable and a  A such that |a) > 0}.
ik
ik
ik ik
Tik
P(R
Let Q = 
ikikL Q (a disjoint union) denote the set of all events that can be observed with positive
probability by some dated player.
<
1
.
ik
An action approximation is any C =when players are restricted to actions in some
subsets of the AWe extend this notation to describe observable events that can have
positive probability
A
,
a
n
subset of
is a nonem
d so CA.ikL
ik
ik
i
k
For any action approximation C  A, let
Q
Let Q(C) =  ik
ikL
7
Q (C) = {R T
| is measurable and cC such that P(R |c) > 0}.
ik
ik
ik ik
R ik
Q is the union of all Q(C) over all finite approximations C. probability by some dated player when a
oAction
approximations are partially ordered by
If C C inclusion.

o
o
then Q(C) 
).
Fact. If C C
Q(C

o
then C
to the true action sets A. C
Finite approximations of the game An
information approximation is any  =
. (So elements of

ik
L
measurable subsets of Tik
Information approximations are partially ordered by the fineness of their partitions. Say
ikL
that =
i
k
o
(If  is finer than

ikL
i
k
is a finite
i
k

L.
i
k
o
i
k
i
k
to the true type sets T.) , then  is a

approximation (,C) of the game G . An information approximation  and an action ) in the app
of
approximation C together define a finite
After any history (,c
<k
) and is also informed of his past actions
 that contains 
ik
the
c<k
ik(,c
ik

. The la
ensures that action-recall is not violated in (,C) when we expand C to fill in the
i(<k)
action
i(<k)
=
,c i(<k
)
C
)
ik
{(

Let S
ik
ik
space A for any fixed information approximation .
i(<
the
finite
set of
possib
le
types s
of dated player ik in the fi
approximation (,C).
Fix . The approximation (,C) will satisfy perfect recall C if: ikL,

Let F
denot
e the
set of
i
 m
s.t.
{(,a)A|
finite
appro
ximat
ions
with
perfec
t
recall.
o
Fact. For any information approximation , there is a finer approximation  such that

(,C
)
satisfi
es
perfec
t
recall
for
every
C,
and
so
(,C
)F
C.
8
S 

i
k

C
i
k
is the finite set of types of player ik.
i(<k
)
)
Strategy profiles for finite approximations
Here let (,C)F be a given finite approximation of the game and so
A strategy profile for the finite approximation (,C) is any  = (
i ik i
(C
, 
(|s )=1, s
k L k
s.t. each
 ik ik
ik
C_ik
:Sikik
Let [t ] denote the element of
containing t T .
ik
ik
ik
ik
ik
Z
(c
)],c
(
|[
(,
))p(d).

L
i(<k)
c
mble Z and cC, let P(Z,c|) =

<k
ik ik
ik

i
i
i
i
c
For any measurable R
(,c
k
k
k
k
C
ik

T
<k
, let
P(R
|) =

P({|

)
R
},c|)
.
A totally mixed strategy profile  for (,C) has
ik

Any totally mixed  for (,C) yields
|) > 0,
P(R
R

C
(cik |s
ik
i
k
Sik. )>0
sik
c
i
,
k
(C), ikL.
ik
Q
ik
For all such Y and c C
, define
,cik
,( , )).
,) = P(Y|Rik c ik -ik
ik
P(Y|R
ik
P(Y(c),c|R ,) for any YA such that,
ik
P(Y| ,) = 
cC
ik
R
With
|) > 0, let
P(R
P(Z,c|R
We
can
exten
d this
proba
bility
functi
on to
define
for
each
cC,
ik
ik
ik
,) =
P(Z{|ik
(, )R
c <k
i
k },c|)/
P(R
|).
ik
ik
Y(c) =
{|
(,c)
Y}
is a
measu
rable
subset
of .
Note.
Chang
ing
the
action
of i at
k does
not
chang
e the
proba
bility
of i's
types
at k,
so
P(R |c , ) = P(R |) >
ik
ik ik ik
0.
9
ik
)=
Approximate sequential equilibria
Here let (,C)F be a given finite approximation of the game, 
and let  be a totally mixed strategy profile for (,C). With
the probability function P defined above, we can define
i
(,c) P(d,c|R
v
V
For any ikL and any cik
only in that i chooses action Changing the action of i at k does not change the probability of i's t
c
P(R |c , ) = P(R |) > 0 ,
 Q (C).
ik
ik ik ik
ik
Rik
i
i(cSo we can similarly define
,
k
ik
V
period k, given the observable event
R
at ) to be i's pay
ik
ik
-approximate sequential equilibrium for the finite approximation
 (,C)
For >0,  is an
(c ,
iff  is a totally mixed strategy for (,C) and
)  V(|
V
s
i
ik
.
i
Facts. Any finite approximation has an -approximate sequential equilibrium. In finite games
with perfect recall, a strategy profile is part of a sequential equilibrium if of a sequence of
-approximate sequential equilibria.
and only if it is the limit as → 0
10
Assessments Let Y denote the set of all outcome events Y 
A such that
{| (,a)Y} is a measurable subset of , aA.
For any action approximation, C, let W(C) = Q(C)  (
ikL i
Q A
k
) (the union of all such W(C)).
Let W = Q 
i
ik
k
(ikL
An
asse
ssme
nt is
a
vect
or 
of
cond
ition
al
prob
). (C) Q
C
i
k
abilit
ies
(Y|
W)
[0,1]
Y
Y,
W
W.
YW
So  is in the compact (product topology) set
,
[0,1]
, ik)[0,1],
i.e., (Y|Q )[0,1] and
ik
a ikL,YY,Q
(Y|Qik
. A ,a Q
ik
Basic sequential equilibria
An assessment  is a basic sequential equilibrium iff: for every  > 0, for every
finite subset F of YW, for every information approximation 
information approximation  such that for every action approximation o, there
C
is an action approximation C  o such that F  YW(C), (,C)F and (,C)
C
ik
ik
i
k
o
, there is a fin
The
proo
f is
base
d on
Tych
onof
f's
Theo
rem
on
com
pact
ness
of
prod
ucts
of
com
pact
sets.
has some -approximate sequential equilibrium  such that
|(Y|W)P(Y|W,)|  , (Y,W)F.

in the
Theorem. The set of basic sequential equilibria is nonempty and is a closed subset of
Y W
[0,1]
Kelley, 1955, p143.
11
Defining basic sequential equilibria with nets A partially ordered set is directed if for d}
any two members there is a member at least as large as both. A net of real numbers is a
collection of real numbers indexed by a directed
set (D,). A net of real numbers {x
is an index s.t. is within e of whenever o
d
x
x
e
d
d  d
e. Write lim
are
directe
d sets.
The set
of
action
d


D
x
. Also, define
lim
x
d
D
d
=
lim
x ). (infdDo = x
d’d d’
d
approxi
mation
s,
partiall
y
ordered
by set
inclusi
on, and
the set
of
inform
ation
approxi
mation
s,
partiall
y
ordered
by the
finenes
s of the
type-sp
ace
partitio
ns,
Then
 is a
basic
seque
ntial
equili
brium
iff for
every
finite
subset
F of
YW
, there
is a
[]
mappi
ng
such
that
lim lim
F
= 0,
|(Y|W) –
C:(,C)
lim→ 0+
P(Y|W,[,,C])|
where each [,,C] is some -approximate sequential equilibrium of the finite
ma
x
(Y,W)F
a
pproximation (,C).
probability by [,,C] for all large enough C. Note. All of the finitely many
conditioning events determined by F are given positive
12
Elementary properties of basic sequential equilibria , let Q ik
For any dated player ik and any observable event Q
I(Q
(,a )Q }.
ik
) = {(,a)A| ikik <k
Let 
be a
basic
seque
ntial
equili
brium.
Then
 has
the
follow
ing
gener
ik
al
proper
ties,
for
any
outco
me
events
Y and
Z
and any observable events
Qik
if YZ= then
(YZ|Q
(Y|Q )[0,1],
ik
(A|Qik
)|
(Y| ) =
Q
(YI(Q Q
jm
and
:
Q
)+
)=
(Z|Q (Y|Q
)=0
(probabilities);
i
i
)
(conditional
k
k
support);
) = 1,
(|Q
) (finite
additivity);
i
k
ik
ik
(YI(
Q
)
(I(Q
ik
) = (Y|T
jm
ik
jm
)=
(YI(Q
)
(I(Q
), for all ik and jm in L.
)|Q
ik
) (Bayes
consistency).
j
)|Q jm )|Q)|Qjm
m
ik
ik
Bayes
consist
ency
implies
that
(Y|T
So the
uncond
itional
distribu
tion on
outcom
es
A
for a
basic
sequent
ial
equilibr
ium
can be
define
d by
(Y)
=
(Y|
T
The
uncon
dition
al
margi
nal
distrib
ution
of 
on 
is the
given
prior
p:
(Z
A) =
p(Z),
for
any
measu
rable
Z.
Fact
(seque
ntial
ration
ality).
If 
is a
basic
seque
ntial
equili
brium
then,
ik
L,
Q Q , a’ A
ik
ik
ik
ik
, i(,a)
v (d,da|Q
,a’
ik
i
).
k
(,a)
(d,da|Q
The left-hand side integral is player i’s equilibrium payoff conditional on
Q
’
side integral is i’s payoff from deviating at date k to a
ik
. The right-hand
.
ik
)  v
i
ik
13
ik
conditio
Finite
additi
vity
of
soluti
ons
strictl
y
positi
ve
numb
er that
is
availa
ble to
him.
Exam
ple:
Consi
der a
game
where
the
winne
r is
the
player
who
choos
es the
smalle
st
strictl
y
positi
ve
numb
er.
In any
finite
appro
ximati
on, let
each
player
choos
e the
smalle
st
x) =
1,
1
x>0,
but
(a
in any =0) = 0.
1
1
1
.
2
after observing this a
1
2
1
x>0,
(a
2
.
1 1
< |a
a
x) =
1.
) in
{0,1}.
=x) = 1 and
(0<a1
But multiple equilibria can have any prior P(2 wins) =
(a2
By
<
a
1
<aIn the limit,
additivity, a
infinitesima
additive, onl
, 2 would alw
before choosing a {1,2}, and p
the limit, we
consid
ering
a
subnet
of
finite
appro
ximati
ons in
which
1
alway
s has
smalle
r
action
s
<athan 2, we can get an equilibrium with
(a12
) = 0 even though
(a
=x) = 1 x>0 <a |a
1
1
2
and
(a
>0)=1.
1
1
In this su
outcome cannot be derived by backward induction from 2's limiting strategy.
Strategic actions avoid such problems, as would considering subnets where later players'
1 4
a rlier players' (fast inner limits for later C
c
t
i
o
n
s
g
r
o
w
f
a
s
t
e
r
t
h
a
n
e
a
).
k
[Games with perfect information] |>1 for at most one player iN, and }, then a dynamic multi-st
If ={0k{1,...,K}, |A
ik
L,
(No
two
player
s
make
choice
s at
the
same
date,
and
all
player
s
obser
ve the
past
histor
y of
play.)
: {
ik 0
ik
}A
 T is one to one.
ik
<k
}, has a basic
Theorem. Every dynamic multi-stage game of perfect information with continuous 0
payoffs, compact Hausdorff action sets and one state of nature, i.e., ={
sequential equilibrium, , in which all prior probabilities are 0 or 1. That is,
({ }B){0,1},
0
BA.
o
Proof: Fix d (0,1). For every >0 and every information approximation compactness , by
and continuity, there is a finer information approximation such that the last
may repeat the argument for the second last player after possibly further refining his
player has, for each element of his partition, a single action that is -optimal against all
histories leading to that partition element. With this strategy fixed for the last player we
information partition, etc. Hence, for each >0 there is an information approximation 
-appro
ximate
sequent
ial
equilibr
ium in
which
some
outcom
e
receive
s
probabi
lity at
least
res
ult
be
ca
use
the
set
of
bas
ic
seq
ue
nti
al
eq
uili
bri
a is
clo
sed
in
the
pro
du
ct
top
olo
gy.
1d,
so
tha
t
the
pro
ba
bili
ty
of
an
y
B

A
is
in
[0,
d]

[1d
,1].
Let
tin
gd

0
yie
lds
the
15
that is finer than o
and is such that C large enough the game (,C) admits an
Strat
egic
entan
gleme
nt
(Milg
romWebe
r
1985)
Exam
ple: 
is
unifor
m
[0,1],
obser
ved
by
both
player
s1
and 2,
who
then
simult
aneou
sly
play a battle-of-sexes game where
= ={1,2},
A
A
1
2
v
=i, = 1 if a 1 2i, and = 0 if
. a2i
) = 2 if a,a12 v
i
1
a
v
i
Consider finite-approximation equilibria such that, for a large integer m, they do
a
1
2
1
=a
=a
(,
1
2
a
if the m'th digit of  is odd, = =2 if the m'th digit of  is even.
a
a
As
m
,
these
equili
=1 =
a
2
bria
conve
rge to
a
limit
where
the
player
s
rando
mize
betwe
en
action
s
(1,1)
and
(2,2),
each
with
proba
bility
1/2,
indep
enden
tly of
.
In the
limit,
the
player
s'
action
s are
not
indep
enden
t
given
 in
any
positi
ve
interv
al.
They
are
correl
ated
by
comm
only
obser
ved
infinit
esima
l
detail
s of .
Given
 in
any
positiv
e
interva
l,
player
1's
expect
ed
payoff
is 1.5
in eqm,
and
1's
condit
ional
payof
f from
deviat
ing to
a1
but 1's conditional payoff from deviating to a1=1 is 0.5(2)+0.5(0) = 1,
=2 is 0.5(0)+0.5(1) = 0.5.

A
Thus, the sequential-rationality inequalities can be strict for all actions. equilibrium, we
could consider also deviations of the form: "deviate to action cTo tighten the
sequential-rationality lower bounds for conditional expected payoffs in
B
the equilibrium strategy would select an action in the set B
i
A .
k
ik
i
k
i
k
ik
sequential equilibrium. In the next example, strategic entanglement is unavoidable,
occurring in the only basic
16
whe
ik
and any
1
fr
o
m
[1,
1]
,
pl
a
y
er
2
c
h
o
o
se
s
fr
o
m
{
L
,
R
}.
E
x
a
m
pl
e:
D
at
e
1:
Pl
a
y
er
1
c
h
o
o
se
s
a[
(
U
n
a
v
oi
d
a
bl
e)
S
tr
at
e
gi
c
e
n
ta
n
gl
e
m
e
n
t]
(
H
ar
ri
sR
e
n
yR
o
b
s
o
n
1
9
9
5)
D
at
e
2:
Pl
a
y
er
s
3
a
n
d
4
o
b
se
rv
e
th
e
d
at
e
1
c
h
oi
c
es
a
n
d
e
a
c
h
c
h
o
o
se
fr
o
m
{
L
,
R
}.
For i=3,4, player i’s payoff is -a
if i chooses L and a if i chooses R.
1
1
|, if they mismatch he gets |a |;
1
(First term) If 2 and 3 match he gets -|a
1
Pla
yer
2’s
pa
yof
f
de
pe
nd
s
on
wh
eth
er
she
ma
tch
es
3’s
ch
oic
e.
If
2
ch
oo
ses
R
the
n
she
get
s2
if
pla
yer
3
ch
oo
ses
R
but
-2
if
3
ch
oo
ses
L.
If
2
ch
oo
ses
L
the
n
she
get
s1
if
pla
yer
3
ch
oo
ses
L
but
-1
if
3
ch
oo
ses
R;
an
d
Pl
a
y
er
1’
s
p
a
y
of
f
is
th
e
s
u
m
of
th
re
e
te
r
m
s:
pl
u
s
(s
e
c
o
n
d
te
r
m
)
if
3
a
n
d
4
m
at
c
h
h
e
g
et
s
0,
if
th
e
y
m
is
m
at
c
h
h
e
g
et
s
-1
0;
plus (third term) he gets
-|a
su
12
|.
bg
am
e
per
fec
t
(he
nc
e
seq
ue
nti
al)
eq
uili
bri
um
in
wh
ich
pla
yer
1
ch
oo
ses
±1/
m
wit
h
pro
ba
bili
ty
½
eac
h,
pla
yer
2
ch
oo
ses
L
an
d
R
eac
h
wit
h
pro
ba
bili
ty
½,
an
d
pla
yer
s
i=3
,4
Ap
pro
xi
ma
tio
ns
in
wh
ich
1’s
act
ion
set
is
{-1
,…
,-2
/m,
-1/
m,
1/
m,
2/
m,
…,
1}
ha
ve
a
uni
qu
e
Pl
a
y
er
3’
s
a
n
d
pl
a
y
er
4’
s
st
ra
te
gi
es
ar
e
e
nt
a
n
gl
e
d
in
th
e
li
m
it.
T
h
e
li
m
it
of
e
v
er
y
a
p
pr
o
xi
m
at
io
n
pr
o
d
u
c
es
(t
h
e
sa
m
e)
st
ra
te
gi
c
e
nt
a
n
gl
e
m
e
nt
.
both choose L if a=-1/m and both choose R if a =1/m.
1
1
17
then simultaneously play the following 2x2 game. [(Spurious) Strategic entanglement]
(Harris, Stinchcombe, Zame 2000) Example:  is uniform [0,1], payoff irrelevant,
observed by both players 1 and 2, who
L R
L1,13,2
R2,30,0
Consider finite approximations in which, for large integers m, each player observes the
whether the m’th digit is 2 or not. first m-1 digits of the ternary expansion of .
Additionally, player 1 observes whether the m’th digit is 1 or not and player 2 observes
Consider equilibria in which each player i chooses R if the m’th digit of the ternary
equilibrium of the 2x2 game where each cell but (R,R) obtains with probability 1/3,
expansion of  is i and chooses L otherwise. As m, these converge to a correlated
independently of . Not in the convex hull of Nash equilibria!
two examples. The type of strategic entanglement generated here is impossible to generate
in some approximations—it depends on the fine details—unlike the entanglement in the
previous
18
type tProblems of spurious concealment in approximating games Example. The
state  is a 0-or-1 random variable, which is observed by player 1 as his
, and then c
1
19
players 2 and 3.
player 1, receiving 1 if they do but 0 otherwise. Example: Three players choose
simultaneously from [0,1]. [Problems of spuriously exclusive coordination in
approximating games] Player 1 wishes to choose the smallest positive number. Players
2 and 3 wish to match
It should not be possible for 2, but not 3, to match 1.
player 1 is available to player 2, but not to player 3. But consider finite approximations
in which the smallest positive action available to
matches 1 but 3 does not. choosing any (other) action. Hence, there are basic
sequential equilibria in which 2 The only equilibria have 1 and 2 choosing that special
common action and player 3
approximation. The exclusive-coordination equilibrium here depends on the fine
details of the
the same finite approximations. (How to recognize sameness?) Possible solution: Require
that type sets and action sets which are "the same" must have
20
Limitations of step-strategy approximations Example: [0,1],
 is uniform [0,1], player 1 observes , chooses a
1
1
=, v(,a )=0 if a.
1
v )=1 if a(,a11
1
1
In any
finite
appro
ximat
ion,
player
1's
finite
action
set
canno
t
allow
any
positi
ve
probability of cIf we gave player 1 an action that simply applied this strategy, he
would use it! utility function is discontinuous. Step functions close to this strategy
yield very different expected payoffs because the approximated by step functions
when 1 has finitely many feasible actions. 1Player 1 would like to use the strategy
"choose a
(It may be hard to choose any real number exactly, but easy to say "I choose .")
=," but th
1
game can significantly change our sequential equilibria. actions separately, which is
needed to prevent spurious signaling. This problem follows from our principle of
finitely approximating information and Thus, adding a strategic action that implements
a strategy which is feasible in the limit
Example (Akerlof):  uniform[0,1], 1 observes  =, 1 chooses [0, 1.5],
a
i
=a
(,a
2 observes 
1
1
2
= 0. Given any finite set of strategies for 1, we could always add some strategy such
b
() < 1.5 , and () has probability 1 of being in a range that has 1
that  < bb
1
1
probability 0 under the other strategies in the given set.
21
This must be done with care to avoid the problem of spurious signaling. Using strategic
actions to solve problems of step-strategy approximations Algebras (closed under finite 
and complements) of measurable subsets are specified One way to avoid the problems of
step-strategies is to permit the use of strategic actions.
, including all one-point sets as measurable. can be assum
also for each AikikIf  is a finite set,
then all subsets of A
For any msble g: → A, all (,g( ) and vi(,g()) are assumed msble in .
<k
)
ik
. → ik *:T ik
A strategic action for ikL is a measurable function,
A ik
a
Let A * denote ik’s set of strategic actions.
ik
The set of intrinsic actions A* because for each a* contains the
constant strategic action taking the value a
, A
ik
.
ik
p
r
ikFor every a*A* and every , let (,a*) denote the action profile aA that
o
is
fi
determined when the state is  and each player plays according to the strategic action le
aps:
A
T * = T (

i
k
i
k
A
defined by
h<k
any a * are measurable, and

*). ), *)=(*(, ih
*), with measu
a
a
<k
(,a*)= (,(,a*)). (,a) from A to 
v
v
i
i
i*) and
v
Note that for every ikL and every a*A*,  *(,a
(,a*) are measurable ik
<k
i
functions of .
Extend each
v
22
employ all of the notation and definitions we previously developed. This defines a
dynamic multi-stage game, G *, with perfect recall and we may therefore
Q
* is the set o
i ik
m
probability in *. Q* is the (disjoint) union of these Q
Approximations of G * are defined, as before, by (,C), where C is any finite set of
(strategic) action profiles in A* and  is any information approximation of all the T
* over
ik
intrinsic actions. Thus, there is perfect recall with respect to strategic actions but not with
respect to recall the action in Adate and the strategic action chosen there. Perfect recall
in finite approximations of * actually taken at the previous date. action there is not
constant across the projection of those types onto TIf the partition element contains more
than one of his information types and his strategic In a finite approximation of G*, perfect
recall means that a player ik remembers, looking back to a previous date m<k, the partition
element containing his information-type at that
23
intrinsic action, then the action too is recalled. Of course, if the strategic action chosen
is a constant function, i.e., equivalent to an
Robustness to approximations choose the smallest strictly positive number). An alternative is
to seek “as much robustness as possible” subject to existence. details of the approximation.
One might then insist that equilibria be robust to all exclusive coordination arise when limits of
approximate equilibria depend on the fine The problems of spurious strategic entanglement,
spurious concealment, and spuriously approximations. But this is not compatible with
existence (e.g., two players each wish to
probabilities agree with P(Y|W,) when (Y,W)YW(C), and are zero otherwise.
Essential sequential equilibria
containing every closed set of assessments M that is minimal with respect to the The set
of essential sequential equilibria is the smallest closed set of assessments
following property: for every open set U containing M, there is a [] mapping such that
where I
sequential equilibrium
() is the indicator function for U, and each [,,C] is some
-approximate
Theorem. The set of
the set of basic seque
U
(By Zorn’s lemma and Tychonoff’s Theorem  such a minimal set M that is nonempty.)
F I
lim lim
(m([,,C])= 1,
C:(,C)
lim→ 0+
)
U
For  completely mixed in (,C), let
m()[0,1]
YW
be the assessment whose
References: Journal of Economic Theory 53:236-260 (1991).
http://www.laits.utexas.edu/~maxwell/finsee4.pdf Theory of Infinite Games," UTexas.edu
working paper. J. L. Kelley, General Topology (Springer-Verlag, 1955). Information,"
Mathematics of Operations Research, 10, 619-32. David Kreps and Robert Wilson,
"Sequential Equilibria," 50:863-894 (1982). C. Harris, P. Reny, A. Robson, "The existence
of subgame-perfect equilibrium in Drew Fudenberg and Jean Tirole, "Perfect Bayesian and
Sequential Equilibrium,", Christopher J. Harris, Maxwell B. Stinchcombe, and William R.
Zame, "The Finitistic Milgrom, P. and R. Weber (1985): "Distributional Strategies for
Games with Incomplete continuous games with almost perfect information," Econometrica
63(3):507-544 (1995).
25