COMMON ASSUMPTION OF RATIONALITY ∗
H. Jerome Keisler†
Byung Soo Lee‡
October 6, 2015
Abstract
We build on the work of Brandenburger, Friedenberg, and Keisler (2008, BFK) by showing that
rationality and common assumption of rationality (RCAR) is possible in complete (lexicographic)
type structures and characterizes iterated admissibility (IA)—i.e., iterated elimination of weakly
dominated strategies. Our result is unexpected in light of BFK’s result proving the impossibility
of RCAR in complete continuous type structures. We reconcile the two by showing that complete
continuous type structures differ from complete type structures that admit RCAR only in how
much caution—in a sense that we formalize—is required to assume events.
K E Y W O R D S : Epistemic game theory, admissibility, iterated weak dominance, assumption, bestrationalization principle
J E L C L A S S I F I C AT I O N : C72, D80.
∗
We thank Adam Brandenburger and Amanda Friedenberg for many valuable discussions about this work. We
thank Ed Green, Ig Horstmann, Martin Osborne, Aviad Heifetz, and Geir Asheim for their helpful comments. We are
also grateful to the editor and three anonymous referees for their detailed and insightful comments.
†
E-mail: [email protected] / Address: Department of Mathematics, University of Wisconsin-Madison,
Madison, WI 53706.
‡
E-mail: [email protected] / Address: Rotman School of Management, University of Toronto,
Toronto, ON M5S 3E6.
1
INTRODUCTION
Brandenburger, Friedenberg, and Keisler (2008, henceforth BFK) solved a long-standing puzzle
in game theory that had been pointed out in Samuelson (1992): How does one model a player’s
belief that simultaneously includes every strategy of the opponent and excludes every weakly dominated strategy of the opponent? BFK cut this Gordian knot by modeling beliefs as lexicographic
probability systems (Blume et al., 1991), which generalize standard probability measures in ways
that eliminate this tension.
By taking advantage of this elegant solution, BFK obtained two results that came close to
providing the epistemic foundations of iterated admissibility (IA)1 using the epistemic conditions
rationality and common assumption of rationality (RCAR) and rationality and m-th order assumption
of rationality (RmAR). These can be worded as follows:
• For any given game, there exists some lexicographic type structure such that, the set of
strategies predicted by RCAR is exactly the IA set.
• For any given game and any complete lexicographic type structure, there is some m such that
the set of strategies predicted by RmAR is exactly the IA set.
These results fall short of the epistemic foundations that BFK sought to find, which they described
as follows:
One might hope to characterize the IA set as the projection of a set of states which is
constructed in a uniform way in all complete type structures. One would expect the
RCAR set to be a natural candidate for this set of states.
BFK proved instead a discouraging result, hereafter called the impossibility theorem, saying
that RCAR is impossible in any complete continuous lexicographic type structure.2 In light of this,
BFK posed the following question:
(i) For every finite game, does there exist a complete (and necessarily discontinuous) lexicographic type structures in which RCAR is possible?
In this paper, we answer this question affirmatively with a possibility theorem (Theorem 3.4).
Furthermore, we provide an epistemic foundation for IA by showing that the IA set is exactly the
set of strategies that players choose when RCAR holds in complete type structures (Theorem 3.5):
1
IA is iterated maximal elimination of inadmissible (i.e., weakly dominated) strategies. It is a procedure that
eliminates all weakly dominated strategies of all players in each round. It is well-known that other orders of elimination
may yield different results.
2
The impossibility theorem has motivated several recent papers that have sought foundations of IA outside of BFK’s
framework. See Barelli and Galanis (2013); Halpern and Pass (2009); Heifetz et al. (2010).
1
(ii) For every complete lexicographic type structure in which RCAR is possible, the IA set is
exactly the projection of the RCAR set.
An informal interpretation of these results is that the line of reasoning given below3 is possible
(by the affirmative answer to (i)) and that its predictions for the game are exactly the IA strategies
(by the result in (ii)).
a1: Ann is rational
b1: Bob is rational
i.e., she chooses optimally after consider-
i.e., he chooses optimally after considering
ing all possibilities about Bob
all possibilities about Ann
a2: a1 and Ann assumes b1
b2: b1 and Bob assumes a1
a3: a2 and Ann assumes b2
b3: b2 and Bob assumes a2
. . . and so on
. . . and so on
An early interpretation of BFK’s impossibility theorem was that RCAR is incompatible with
complete continuous type structures because they generate too many belief hierarchies. We dismiss
that interpretation by proving the following result (Corollary 5.12).
(iii) For any given complete type structure in which RCAR is possible, there is a complete continuous type structure that generates the same belief hierarchies.
As elucidated by Harsanyi (1967), a type structure is a model in which belief hierarchies are
represented implicitly as types. BFK therefore viewed complete continuous type structures as the
natural class of models in which statements of the form “Ann (Bob) considers all possibilities about
Bob (Ann)” could be expressed. The above interpretation of the impossibility theorem was based
on the idea that Ann considering all possibilities about Bob conflicts with her assuming too much
(i.e., all of b1, b2, b3, and so on. . . ) about him.
In Section 2, we will review the underlying framework from BFK. In Section 3, we review
some results from BFK and state our main results on an epistemic characterization of IA. The
proof of the affirmative answer to (i) is in Section 4. Section 5 deals with result (iii), which also
reconciles our possibility result with the impossibility theorem. Finally, Section 6 concludes with a
brief discussion of Yang (2015) and Battigalli and Siniscalchi (2002), which are the most closely
related papers.4 Further results characterizing RCAR sets in complete type structures may be found
in an earlier version of this paper (Keisler and Lee, 2011). New results which are not proved in
the text are proved in Appendix B.
3
This informal line of reasoning is from BFK
Similarities between Yang (2015), Battigalli and Siniscalchi (2002), and this paper are given further consideration
in Appendix A.
4
2
2
UNDERLYING FRAMEWORK
Throughout this paper, we fix a two-player game G = S a , S b , πa , π b with finite strategy sets S a , S b .
The indices a and b respectively stand for Ann and Bob. At times, we use c to denote a generic
player (either a or b), and d to denote the other. πc is c’s utility function on S a × S b . A player c
is said to be indifferent in the game G if πc (r c , s d ) = πc (s c , s d ) for all r c , s c , s d . We also let Ω be a
non-empty Polish space.5
2.1
Lexicographic probability system
M (Ω) denotes the Polish space of all Borel probability measures on Ω with the topology of weak* convergence. A lexicographic probability system (LPS) of length n on Ω is an n-tuple of
Qn
probability measures σ = (µ0 , . . . , µn−1 ) on Ω. Nn (Ω) ≡ k=1 M (Ω) denotes the space of lengthS
n LPS’s on Ω, with the product topology. N (Ω) ≡ n≥1 Nn (Ω) denotes the space of LPS’s on Ω,
with the union topology. Both of these spaces are Polish.
For any µ, ν ∈ M (Ω), µ and ν are mutually singular (written µ ⊥ ν) if there exist disjoint
Borel sets U, V ⊆ Ω such that µ(U) = 1 = ν(V ). A lexicographic conditional probability system
(LCPS) is an LPS (µ0 , . . . , µn−1 ) such that µ j ⊥ µk for all j 6= k.6 Ln (Ω) and L (Ω) respectively
denote the space of length-n LCPS’s on Ω and the space of LCPS’s on Ω. Both spaces are endowed
with the subspace topology relative to N (Ω).
The support of an LPS σ = (µ0 , . . . , µn−1 ) is denoted by supp σ ≡
S
j<n supp µ j .
The LPS σ has
+
full support on Ω if supp σ = Ω. N (Ω) denotes the space of full-support LPS’s on Ω. M + (Ω),
Nn+ (Ω), L + (Ω), and Ln+ (Ω) are defined analogously.
2.2
Assumption
BFK defined when a decision maker whose preferences are given by ¥ assumes an event E. Instead
of restating that definition, we rely on their precise characterization of the conditions under which
a belief σ represents the preferences of a decision maker who assumes E.
Proposition 2.1 (Proposition 5.1 in BFK). Fix an event E ⊆ Ω and a full-support LCPS σ =
(µ0 , . . . , µn−1 ) on Ω. An event E is assumed under σ at level j if and only if
(i) µi (E) = 1 for all i ≤ j;
(ii) µi (E) = 0 for all i > j;
(iii) if U is open with U ∩ E 6= ∅ then µi (U ∩ E) > 0 for some i.
5
6
A Polish space is a topological space that is separable and completely metrizable.
BFK used the term LPS for LCPS. We revert to the terminology of Blume et al. (1991) in this regard.
3
E is assumed under σ if and only if E is assumed under σ at some level j.
For brevity, we write “σ assumes E” to mean that E is assumed under σ. Assumption is not
monotonic, that is, σ does not necessarily assume F even if σ assumes E ⊆ F . However, it does
have the following properties. Note that, for each event E ⊆ Ω, E denotes its closure in Ω.
Proposition 2.2 (Properties 6.1–6.3 in BFK). For each full-support LCPS σ on Ω, we have:
(i) If σ assumes both E and F at level j, then σ assumes any event G such that E ∩ F ⊆ G ⊆ E ∪ F
at level j.
(ii) If σ assumes both E and F at level j, then E = F .
(iii) If σ assumes Em for each m ∈ N, then σ assumes both
2.3
T
m
Em and
S
m
Em .
Admissibility
We extend the notion of the (pure strategy) best-response (BR) set as follows. Given any σ =
(µ0 , . . . , µn−1 ) ∈ N (S d ), the set of c’s pure strategies that maximize lexicographic expected utility
(LEU) under σ in the game G is denoted by BRc (σ), which is equal to
§
€P
Šn−1
€P
Šn−1 ª
c
c
d
c c d
L
d
c
c d
s ∈ S ∀x c ∈ S c
µ
(s
)π
(s
,
s
)
≥
µ
(s
)π
(x
,
s
)
,
s d ∈S d j
s d ∈S d j
j=0
j=0
(1)
where ≥L is the usual lexicographic order, defined as follows. For any u = (u0 , . . . , uk−1 ), v =
(v0 , . . . , vk−1 ) ∈ Rn , we write u >L v if there is some k < n such that (u0 , . . . , uk−1 ) = (v0 , . . . , vk−1 )
and uk > vk . We write u ≥L v whenever u >L v or u = v.
Note that for every σ, the set BRc (σ) is non-empty. Given a set X ⊆ N (S d ), we define BRc (X ) ≡
{BRc (σ)|σ ∈ X }. A strategy s c is admissible—i.e., not weakly dominated—in the game G if s c is
c
a best response under some full-support belief on S d . Let S0c ≡ S c . For each m ≥ 1, Sm+1
denotes
a b a b
c’s admissible strategies in the reduced game Gm ≡ Sm , Sm , π , π . We slightly abuse notation by
letting the restriction of πc to Sma × Smb be denoted by πc . Strategy s c is m-admissible if it is in Smc .
T∞
c
It is iteratively admissible (IA) if it is in S∞
≡ m=0 Smc .
2.4
Type structures
An (S a , S b )-based lexicographic type structure for G is a tuple T = S a , S b , T a , T b , λa , λ b such
that, for each player c,
(i) c’s type space T c is non-empty and Polish;
(ii) c’s type-belief map λc : T c → L (S d × T d ) is Borel.
4
Elements of S a × T a × S b × T b are called states of the world. λc (t c ) is the lexicographic belief
of the type t c ∈ T c . We frequently omit the adjective “lexicographic” for the sake of brevity in
obvious contexts. T is continuous if λa and λ b are continuous. T is one-to-one if λa and λ b are
one-to-one maps.
T is complete if λc (T c ) = L (S d × T d ) for each player c. Our terminology differs slightly from
that in BFK, where T is called complete type structure if λc (T c ) ) L + (S d × T d ) for each player c.
Thus complete in our sense implies complete in the sense of BFK.
2.5
Rationality
Fix a lexicographic type structure T = 〈S a , S b , T a , T b , λa , λ b 〉. Player c assumes the event E ⊆ S d × T d
at the state of the world (s a , t a , s b , t b ) if E is assumed under λc (t c ).
Ac (E) ≡ {t c ∈ T c |E is assumed under λc (t c )}
(2)
Player c is rational at the pair (s c , t c ), and at the state (s a , t a , s b , t b ), if
(i) c’s type maps to a full-support belief—i.e., λc (t c ) ∈ L + (S d × T d );
(ii) c’s strategy is a best response under this belief—i.e., s c ∈ BRc (margS d λc (t c )).
The set Rcm is defined inductively for all m > 1 as follows. Let Rc1 denote the set of strategy-type
pairs (s c , t c ) ∈ S c × T c at which c is rational.
Rcm+1 ≡ Rcm ∩ S c × Ac (Rdm )
Rc∞ ≡
\
Rcm
(3)
m≥1
b
If (s a , t a , s b , t b ) ∈ Ram+1 × R m+1
, then we say that it satisfies rationality and m-th order assumption
b
, then we say that it satisfies rationality and
of rationality (RmAR). If (s a , t a , s b , t b ) ∈ Ra∞ × R∞
common assumption of rationality (RCAR). We say that a type structure admits RCAR if Ra∞ ×
b
R∞
6= ∅.
3
EPISTEMIC JUSTIFICATION FOR ITERATED ADMISSIBILITY
Consider the following line of reasoning (taken from BFK), which was also mentioned in the
Introduction.
5
a1: Ann is rational
b1: Bob is rational
i.e., she chooses optimally after consider-
i.e., he chooses optimally after considering
ing all possibilities about Bob
all possibilities about Ann
a2: a1 and Ann assumes b1
b2: b1 and Bob assumes a1
a3: a2 and Ann assumes b2
b3: b2 and Bob assumes a2
. . . and so on
. . . and so on
BFK showed two results that, when taken together, give an epistemic justification of m-admissible
strategies related to this line of reasoning. These results also hold for our notion of “completeness”,
which is slightly stronger than it is in BFK.7
Proposition 3.1 (Proposition 7.2 in BFK). There exists a complete continuous type structure.
Theorem 3.2 (Theorem 9.1 in BFK). Fix a complete type structure T. Then
∀m ∈ N
b
projS a ×S b (Ram × R m
) = Sma × Smb .
b
The fact that there is a complete continuous type structure such that Ram × R m
6= ∅ is informally
interpreted as saying that it is possible for the players to reason in ways described by statements
b
am and bm. Because the m-admissible set is exactly the projection of Ram × R m
in any such type
structure, the m-admissible set is exactly the set of predictions for the game when players reason
as described by am and bm. Given this interpretation, the following impossibility theorem of BFK
immediately cast serious doubt on whether the full line of reasoning described above (which
includes am and bm for all m) is possible.
Theorem 3.3 (Impossibility Theorem, 10.1 in BFK). Fix a complete continuous type structure T.
b
Unless both players are indifferent, we have Ra∞ × R∞
= ∅.8
We now give our main results. Theorem 3.4 below is surprising, because at first glance it appears
to intuitively contradict the impossibility theorem. We will reconcile the two in Section 5 through
a notion that we call “degree of caution”.
Theorem 3.4. There exists a complete type structure that admits RCAR.
7
Theorem 3.1 can be proved for our definition of completeness by modifying BFK’s proof so that Theorem 13.7
(Kechris, 1995) is used instead of Theorem 7.9 (Kechris, 1995). Theorem 3.2 is immediate because our notion of
completeness is stronger.
8
The statement in BFK had the hypothesis that some player is not indifferent, which is equivalent to saying “unless
both players are indifferent”. Every type structure that is complete in our sense is complete in the sense of BFK, so
the result as stated in BFK implies the result as stated here.
6
b
Theorem 3.5. Fix a complete type structure T such that Ra∞ × R∞
6= ∅. Then
b
a
b
projS a ×S b (Ra∞ × R∞
) = S∞
× S∞
.
These results say that the reasoning described by the full list of statements at the beginning
of this section is an epistemic justification of IA because the reasoning is possible (Theorem 3.4)
and its predictions are exactly the IA strategies (Theorem 3.5). Notice that, while the complete
type structures that admit RCAR (as in Theorem 3.4) may not be the same for every game, the
line of reasoning that is the epistemic justification does not depend on the game. Nevertheless, it
is natural to ask whether Theorem 3.4 can be strengthened in the following way.
Question 1. Given a finite pair of strategy sets S a , S b , does there exist a complete type structure that
admits RCAR for every game with these strategy sets?
We leave this as an open question. However, Theorem 3.4 can be strengthened in another way,
which shows that there are many different complete type structures admitting RCAR:
Theorem 3.6. For all uncountable Polish spaces T a , T b , there exists a complete type structure 〈S a , S b ,
T a , T b , λa , λ b 〉 that admits RCAR.
Finally, we remark that in Theorems 3.4 and 3.6, the type structures can be taken to be one-toone9 . As far as we know, the property that the type-belief maps are one-to-one has no economic
interpretation. However, the conceptual important notion of expressibility requires that different
types have different belief hierarchies, which implies one-to-one. Moreover, one-to-one is a very
useful technical property for maps between Polish spaces because it guarantees that images of
Borel sets are Borel sets.
4
PROOF OF THEOREM 3.4
Although Theorem 3.4 is an immediate corollary of Theorem 3.6, we offer a stand-alone proof of
Theorem 3.4 that does not depend on Theorem 3.6. Doing so allows us to include informative
diagrams that directly correspond to various steps of the proof. We postpone the proofs of two lemmas and Theorem 3.6 to the appendices. This section may be safely skipped if one is uninterested
in the details of proving Theorem 3.4.
4.1
A necessary condition
Before embarking on the proof of Theorem 3.4, we prove a condition on the RmAR sets that is
necessary for a complete type structure to admit RCAR. While this condition is not directly used
9
This is not obvious from the statements, but will follow from the proofs.
7
in the proof of Theorem 3.4, it is helpful to keep in mind as we proceed. Further discussion of this
result can be found in Appendix A.
Recall that in a topological space, a set is nowhere dense if its interior is empty, and is meager
it is the union of countably many nowhere dense sets. Intuitively, meager sets are small in a
topological sense. When X , Y are sets in a topological space and Y ⊆ X , we say that Y is nowhere
dense, or meager, in X if Y is nowhere dense, or meager, in the relative topology on X .
Proposition 4.1. Suppose T is a complete type structure that admits RCAR for a given game. Then
there is some M such that for each player c,
(i) For each m ≥ M , the set RcM \ Rcm is nowhere dense in RcM .
(ii) The set RcM \ Rc∞ is meager in RcM .
Proof of Proposition 4.1. We prove (i) first. Since T admits RCAR, there exists (s a , t a , s b , t b ) ∈
b
Ra∞ × R∞
. Then for each player c, λc (t c ) assumes Rdm for all (finite) m. λc (t c ) has finite length `.
By Proposition 2.1, there exists j ≤ ` such that λc (t c ) assumes Rdm at level j for infinitely many m.
Let M c be the least m such that λc (t c ) assumes Rdm at level j. The sets Rdm form a decreasing chain,
so by Proposition 2.2(i), λc (t c ) assumes Rdm at level j for all m ≥ M c . Then by Proposition 2.2(ii),
we have Rdm = RdM c for all m ≥ M c . Taking M to be the maximum of M a and M b , it follows that for
each player c, Rdm = RdM for all m ≥ M . Therefore RdM \ Rdm is nowhere dense in RdM for all m ≥ M .
S
(ii) follows from (i) because RdM \ Rd∞ = m≥M (RdM \ Rdm ).
4.2
Overview of proof of Theorem 3.4
To build the desired type structure T = S a , S b , T a , T b , λa , λ b , we first construct a sequence 〈Q am ×
b
Qm
〉m∈N of “candidate rationality sets” such that
b
• Q am × Q m
⊆ S a × T a × S b × T b for all m;
b
b
• Q am × Q m
⊇ Q am+1 × Q m+1
for all m; and
b
• Q a∞ × Q ∞
≡
T
a
m (Q m
b
× Qm
) 6= ∅.
b
In the type structure that we ultimately construct, Q am+1 × Q m+1
will be the set of states at which
there is RmAR. We will then construct the type-belief maps λa and λ b so that T is a complete
b
b
type structure with Q am = Ram and Q m
= Rm
for all m. In view of Proposition 4.1, the sequences
b
b
〈Q am × Q m
〉m∈N that we construct must have the property that for some M , Q aM \ Q am and Q bM \ Q m
are topologically small (nowhere dense) for all m ≥ M .
Our construction will have two stages. In the first stage, we will partition T c into a family
S
of (possibly empty) Borel sets Pmc (X ) with X ⊆ S d and m ≤ ∞. We will then let Q cm = X (X ×
8
S
k≥m
Pkc (X )). In the second stage, we will construct Borel functions λc that map each set Pmc (X ) onto
the set of LCPS’s σ such that X = BRc (margS d σ) and σ assumes Q d1 , . . . , Q dm−1 —i.e., σ represents
the beliefs of a player whose degree of “strategic sophistication” is m.
4.3
First stage: Partition by candidate sets
b
Figure 3 give a visual representation of the construction of the sequences 〈Q am 〉m∈N and 〈Q m
〉m∈N
when G is the game in Figure 1. The shaded areas show the parts of Q am in {U} × T a and {D} × T a ,
b
and the parts of Q m
in {L} × T b and {R} × T b .
Bob
Ann
U
D
L
1, 1
0, 2
R
0, 1
1, 0
Figure 1: Example from BFK
Step 1: Define T c , X c , and m(·). Let T c ≡ [0, 1]×[0, 1] be the unit square. Let X c ≡ BRc (N + (S d )),
the family of all best response sets of full-support LPS’s. For each m ∈ N let
d
Ymc ≡ BRc (M + (Smd ) × M + (Sm−1
) × · · · × M + (S0d )).
Ymc is the family of best response sets of (m + 1)-tuples (µ0 , . . . , µm ) where each µi is a full-support
T
d
c
measure on the strategy set Sm−i
. We note that ∅ 6= Ymc ⊆ X c . Let Y∞
≡ m Ymc . Define m: X c →
N ∪ {∞} by
m(X ) =

∞
c
if X ∈ Y∞
,
min mX ∈
/ Ymc
otherwise.
Ultimately, m(X ) will be the highest degree of “strategic sophistication” that can be attributed to
a player whose best-response set is X .
We will need the following lemma, whose proof will be given in Appendix B.
c
Lemma 4.2. For each m ∈ N \ {0} we have Ymc ⊆ Ym−1
.
c
Since Ymc is finite for each m ∈ N, it follows that there is an M ∈ N such that Y Mc = Y∞
.
c
Therefore Y∞
is non-empty.
¦
©
c
Step 2: Partition T c into vertical strips. Arrange the elements of X c in a list X c = X 1c , . . . X |X
c| .
Define VXc X ∈ X c ∪ {∅} , a partition of T c into the vertical strips.

˜
k−1 k
c
c
,
× [0, 1] for each X k ∈ X c
V∅ ≡ {0} × [0, 1],
VX k ≡
|X c | |X c |
V∅c will be the set of Player c’s non-full-support types. VXc will be the set of full-support types with
S
best response set X . In other words, we eventually want Rc1 = X ∈X c (X × VXc ).
9
Step 3: Partition each of the vertical strips VXc .
Figure 2 shows the vertical strips VXc when G
is the game in Figure 1.
For each m ∈ N \ {0}, define the horizontal line H mc ≡ [0, 1] × {1/2m }. We use these sets to define
the partition Pmc (X )1 ≤ m ≤ m(X ) of VXc for each X ∈ X c , where


V c ∩ Hc
if 1 ≤ m < m(X )

 X Sm Pmc (X ) ≡ VXc \
H kc 1 ≤ k < m(X )
if m = m(X )


∅
if m > m(X )
c
Note that Pmc (X ) is topologically small (i.e., meager) in VXc for m 6= m(X ). Meanwhile, Pm(X
(X ) has
)
c
the same topological size as VXc (i.e., their difference is meager). Pm(X
(X ) will ultimately be the
)
subset of T c that “best-rationalizes” X —i.e., the set of types with the highest degree of “strategic
sophistication” among the types whose best-response set is X .
Final step. All that remains is to define Q cm ≡
Q cm+1
c
X ∈X c X ×
k≥m Pk (X )
T
Q cm , Q c∞ = m Q cm , and
S
S
for m ∈ N ∪ {∞}. Note
that by construction, for m ∈ N we have
⊆
[
[
proj T c (Q cm \ Q cm+1 ) =
Pmc (X ),
projS c (Q cm \ Q cm+1 ) =
{X ∈ X c |m(X ) ≥ m} = Smc .
X ∈X c
4.4
Step 1.
Second stage: Gluing piecewise Borel isomorphisms
In the first stage, we constructed a partition {V∅c } ∪ {Pmc (X )|X ∈ X c , 1 ≤ m ≤ m(X )} of
T c . We want to define a partition of L (S d × T d ) that “mirrors” the above partition in a sense that
will become clear as we proceed.
Step 2. We define the partition {V̂Xc |X ∈ X c ∪ {∅}} of L (S d × T d ), where

L (S d × T d ) \ L + (S d × T d )
if X = ∅
c
V̂X ≡  σ ∈ L + (S d × T d )X = BRc (margS d σ)
if X ∈ X c .
Step 3. For each X ∈ X c , we define the partition { P̂mc (X )|1 ≤ m ≤ m(X )} of V̂Xc , where
∀m ∈ N P̂mc (X ) = V̂Xc ∩ σ ∈ L + (S d × T d )σ assumes Q dk for all k < m but not Q dm
c
and P̂∞
(X ) = V̂Xc ∩ σ ∈ L + (S d × T d )σ assumes Q dk for all k .
Then {V̂∅c } ∪ { P̂mc (X )|X ∈ X c , 1 ≤ m ≤ m(X )} is a partition of L (S d × T d ) that refines the partition
from Step 2 above and mirrors the partition of T c from Stage 1.
10
Step 1
X a = {{D}, {D, U}, {U}}
Ta
m({D}) = 1, m({D, U}) = 1, m({U}) = ∞
Step 2
a
V{D}
V∅a
a
V{D,U}
a
V{U}
Step 3
V∅a
P1a ({U})
P3a ({U})
P1a ({D})
P1a ({d, U})
a
P∞
({U})
Step 1
X b = {{L}}
Tb
m({L}) = ∞
Step 2
V∅b
b
V{L}
Step 3
V∅b
P1b ({L})
P3b ({L})
b
P∞
({L})
Figure 2: Partitions for the game in Figure 1.
These are building blocks for constructing the sets Rcm and Rc∞
11
P2b ({L})
P2a ({U})
Sa × T a
Q 1a
Q 2a
Q 3a
Q 1b
Q 2b
Q 3b
{U } × T a
{D} × T a
Sb × T b
{L} × T b
{R} × T b
b
Q∞
a
Q∞
{U }
{L}
Figure 3: Candidate RmAR, RCAR sets for the game in Figure 1.
A type structure will be constructed with Q cm = Rcm and Q c∞ = Rc∞ .
12
Step 4: Piecewise isomorphisms.
Now we show the existence of the following bijective Borel
maps for all (X , m) ∈ X × (N ∪ {∞}) and 1 ≤ m ≤ m(X ).
c
c
λ(∅,0)
: V∅c → V̂∅c
c
λ(X
: Pmc (X ) → P̂mc (X ).
,m)
The sets V∅c = {0} × [0, 1] and V̂∅c = L (S d × T d ) \ L + (S d × T d ) are both uncountable Borel subsets
of Polish spaces and therefore have the same cardinality as the real line. In Appendix B, we prove
the following technical lemma:
Lemma 4.3. Pmc (X ) and P̂mc (X ) are uncountable Borel sets whenever X ∈ X c and 1 ≤ m ≤ m(X ).
c
c
The existence of the maps λ(∅,0)
and λ(X
are directly implied by the Borel Isomorphism
,m)
Theorem below and Lemma 4.3
Theorem (Borel Isomorphism Theorem). Let A, B be Borel subspaces of Polish spaces. If A and B
have the same cardinality, then there is a one-to-one Borel mapping from A onto B.10
Now we can define λc : T c → L (S d × T d ) by “gluing” the Borel isomorphisms
Step 5: Gluing.
that we defined in the previous step,
λ (t ) ≡
c
c

λc (t c )
(∅,0)
if t c ∈ V∅c
λc
(t c ) if t c ∈ Pmc (X )
(X ,m)
Then λc is itself a Borel isomorphism because it is the union of countably many Borel isomorphisms
with pairwise disjoint domains and ranges.
Final step: Verification.
T = S a , S b , T a , T b , λa , λ b is a complete type structure because λa and
λ b are surjective Borel maps.
c
We show that Q c∞ is non-empty. By Lemma 4.2, there exists a set Y ∈ Y∞
, and hence m(Y ) =
S
c
c
∞. Y is non-empty, and by Lemma 4.3, P∞
(Y ) is non-empty. We have Q c∞ = X (X × P∞
(X )), so
Q c∞ is non-empty.
Lemma 4.4. Ra1 × R1b = Q a1 × Q 1b in the type structure T.
Proof.
Rc1 = (s c , t c ) ∈ S c × T c s c ∈ BRc (margS d λc (t c )) ∧ λc (t c ) ∈ N + (S d × T d )
[
= (s c , t c ) ∈ S c × T c s c ∈ BRc (margS d λc (t c )) ∧ λc (t c ) ∈
V̂Xc
c
X ∈X
[ c c
c
c c
c
c c
c c
=
(s , t ) ∈ S × T s ∈ BR (margS d λ (t )) ∧ λ (t ) ∈ V̂Xc
X ∈X c
10
This is Theorem 15.6 in (Kechris, 1995), where it is stated in terms of standard Borel spaces, which are the
measurable spaces associated with Borel subspaces of Polish spaces.
13
=
[ (s c , t c ) ∈ S c × T c s c ∈ X ∧ λc (t c ) ∈ V̂Xc
X ∈X c
=
[ [
X × VXc = Q c1
(s c , t c ) ∈ S c × T c s c ∈ X ∧ t c ∈ VXc =
X ∈X c
X ∈X c
b
b
Using Lemma 4.4 at the first step, it follows by induction on m that Ram × R m
= Q am × Q m
for all
b
b
m ∈ N. Therefore Ra∞ × R∞
= Q a∞ × Q ∞
6= ∅. This completes the proof of Theorem 3.4.
5
CAUTION, CONTINUITY, AND RCAR
As mentioned in the Introduction, an early interpretation of BFK’s impossibility theorem was that
a complete continuous type structures cannot admit RCAR because it generates too many belief
hierarchies. Intuitively, if the set of belief hierarchies generated by a type structure is small, then
the players can be said to know a lot of about each other since they rule out all hierarchies that it
does not generate. Similarly, if the set of belief hierarchies generated by a type structure is very
large, then the players can be said to know very little about each other. BFK’s interpretation was
therefore based on the idea that RCAR is possible only when players are familiar with each other
in the sense that we have just described. Indeed, BFK showed that it is easy to construct incomplete
type structures that admit RCAR.
Unlike B-S, BFK do not construct a complete type structure by first defining all hierarchies
of beliefs. However, the usual constructions of “universal” type structures satisfy analogs of both
continuity and completeness.11 Furthermore, Friedenberg (2010) had previously shown that—
when beliefs are standard probability measures—a complete12 type structure generates all belief
hierarchies if the type spaces are compact and the type-belief maps are continuous.
Given this preexisting association between continuity and richness, albeit in non-lexicographic
frameworks, the idea that continuous complete type structures were “too rich” had intuitive appeal.
In this section, we develop a notion of caution required to assume events and use it to dismiss BFK’s
earlier interpretation of the impossibility theorem.
5.1
Definition of hierarchies
Before we proceed, we need to be more precise about what we mean by “belief hierarchies”, so
that our statements about them are unambiguous.
Definition 5.1. Let A and B be non-empty Polish spaces and f : A → B be a Borel map. The pushforward map fb : N (A) → N (B) is defined as follows. For all σ = (µ0 , . . . , µn−1 ) ∈ N (A), fb(σ) =
b n−1 ), where µ
b j (E) = µ j ◦ f −1 (E) for all Borel E ⊆ B and 0 ≤ j ≤ n − 1.
(b
µ0 , . . . , µ
11
e.g., Mertens and Zamir (1985), Brandenburger and Dekel (1993), and B-S.
Friedenberg (2010)’s definition of complete type structures accordingly differs from ours in only one way: λc (T c ) =
M (S d × T d ) instead of λc (T c ) = L (S d × T d ).
12
14
Definition 5.2. Let Ξ0c ≡ S c and inductively define the sequence (Ξcm )∞
of Polish spaces by letting
m=0
Ξcm+1 ≡ Ξcm × N (Ξdm ) for all m ≥ 0. Player c’s (belief) hierarchy is a sequence hc = (hcm+1 )∞
∈
m=0
Q∞
N (Ξdm ) of LPS’s, where hcm is called c’s m-th order belief.
m=0
A type structure T = S a , S b , T a , T b , λa , λ b generates hierarchies by using the type-belief
maps. As an intermediate step we need to inductively define the Borel map ξcm : S c × T c → Ξcm for
m = 0, 1, . . . by letting, for all (s c , t c ) ∈ S c × T c ,
ξ0c (s c , t c ) ≡ s c
bd ◦ λc (t c )) for all m ≥ 0.
and ξcm+1 (s c , t c ) ≡ (ξcm (s c , t c ), ξ
m
bd ◦ λc (t c ))∞ .
Definition 5.3. The hierarchy generated by type t c ∈ T c is the sequence (ξ
m
m=0
5.2
Caution required to assume events
Definition 5.4. We say that two Polish spaces X and Y are Borel equivalent if they have the same
points and the same Borel sets. We say that two type structures
T = S a , S b , T a , T b , λa , λ b
and U = S a , S b , U a , U b , λa , λ b ,
are Borel equivalent if T and U have the same strategies, types, and type-belief maps, and T c , U c are
Borel equivalent for each player c.
The next remark follows easily from the definitions.
Remark 5.5. Suppose T and U are Borel equivalent. Then the following hold.
(i) T and U have the same LCPS’s.
(ii) U is complete if and only if T is complete.
(iii) T and U generate the same belief hierarchies.
Definition 5.6. A Borel refinement of a Polish space X is Polish space Y such that Y has the same
points as X , and every open set in X is open in Y . Given two type structures
T = S a , S b , T a , T b , λa , λ b
and U = S a , S b , U a , U b , λa , λ b ,
we say that U is a Borel refinement of T and write U ≥ref T if T and U have the same strategies,
types, and type-belief maps, and U c is a Borel refinement of T c for each player c.
The relation ≥ref is a partial order on complete type structures.
Whenever we speak of any type structures labeled U and T in the remainder, let Um = Uma × Umb
b
and R m = Ram × R m
denote their respective RmAR sets. In view of the next proposition, when
U ≥ref T, we will also say that U requires a greater degree of caution (in assuming events) than
T does.
Proposition 5.7. Suppose U ≥ref T. Then the following hold.
15
(i) T and U are Borel equivalent.
(ii) Every LCPS that assumes an event E in U assumes E in T.
(iii) For all m ≥ 0, if Um = R m , then Um+1 ⊆ R m+1 . In particular, U1 ⊆ R1 .
Proof. By hypothesis, T c and U c have the same points, and every open set in T c is open in U c .
Then by Exercise 15.4 in Kechris (1995), if T c and U c have exactly the same Borel sets, so T c is
Borel equivalent to U c . Therefore (i) holds. By Remark 5.5 (i), T and U have the same LCPS’s. (ii)
follows from this and and Proposition 2.1. Part (iii) follows from part (ii).
We wish to emphasize that all of the following may change when one Borel-refines a type
structure: the rationality sets, the set of LCPS’s that assume a given event, and the events that are
to be assumed by players if RCAR is to hold. Because U1 ⊆ R1 , one might guess that Um ⊆ R m for
all m, and hence that U has fewer RCAR states than T. This would be correct if assumption were
monotonic. But assumption is not monotonic, and the above guess turns out to be incorrect for
complete type structures.
Proposition 5.8. Suppose T and U are complete and Borel equivalent. Then either Um = R m for all
m, or there are only finitely many m such that Um ⊆ R m .
One might also expect that a Borel refinement of a continuous type structure is also continuous,
but this is not necessarily so because adding more open sets to the type space T c will also add open
sets to the space L (S d × T d ). Nevertheless, we show in Proposition 5.9 that RCAR is unattainable
in any type structure Borel-refines (and therefore requires a higher degree of caution than) a
complete continuous type structure.
Proposition 5.9. Suppose neither player is indifferent in G, T is a complete continuous type structure,
and U is a Borel refinement of T. Then U does not admit RCAR.
Proof. In this proof, for any subset E of S c × T c , E will always denote the closure of E in the sense
of T rather than of U. It suffices to show that in U, no type whose image under λc has length less
than m can belong to Umc .
Lemmas F.1 and F.2 in BFK show that for each m the sets Rcm \ Rcm+1 is uncountable. Their proof
c
also shows that Umc \ Um+1
is uncountable. Therefore, the sets Umc , m ∈ N all have different closures
in the sense of T. Every closed set in the sense of T is also closed in the sense of U. Therefore, by
Proposition 2.2(ii), no type in U can assume more than one of the sets Umc at the same level. The
result follows.
In light of the above discussion, it may be surprising that there is a complete continuous type
structure that is a refinement of a complete type structure that admits RCAR. This will be shown
in Corollary 5.11 below.
16
Theorem 5.10. Every lexicographic type structure T has a continuous Borel refinement U.13
Proof of Theorem 5.10. We will use Theorem 13.11 in Kechris (1995): Whenever f is a Borel
map from a Polish space X to a second countable space Z, X has a Borel refinement Y such that
c
f : Y → Z is continuous. Let T0c = T c . For each n, we inductively obtain a Borel refinement Tn+1
c
of Tnc such that λc is continuous from Tn+1
to L (S d × Tnd ) by directly applying Theorem 13.11
c
of Kechris (1995). Let Tnc denote the topology of Tnc and let T∞
denote the topology generated
S
c
c
by n Tn . Let U be the topological space that has the same points as T c and the topology T∞
.
Kechris (1995, 13.3) directly shows that U c is a Borel refinement of Tnc for all n. It follows that λc
is continuous from U c to L (S d × Tnd ) for each n.
Let uck → uc in U c . Then λc (uck ) → λc (uc ) in L (S d × Tnd ) for all n. Let M denote the length of
λc (uc )—i.e., λc (uc ) ∈ L M (S d × U d ). Then there is some K such that λc (uck ) ∈ L M (S d × U d ) for all
k ≥ K. Without loss of generality, we assume that K = 0 in the remainder of this proof.
Let λc (uck ) = (µ0k , . . . , µkM −1 ) and λc (uck ) = (µ0 , . . . , µ M −1 ). Therefore µkm → µm in M (S d × Tnd )
for all (m, n). By Kechris (1995, Theorem 17.20),14 lim infk µkm (O) ≥ µm (O) for each (m, n, O) such
that O is open in S d × Tnd . By the definition of U d , the open sets of S d × Tnd across all n jointly form
an open basis for S d × U d . It follows from (the other direction of) Kechris (1995, Theorem 17.20)
that µkm → µm in M (S d × U d ) for all m. Therefore λc (uck ) → λc (uc ) in L (S d × U d ), which shows
that λc is continuous from U c to L (S d × U d ).
The next two corollaries are immediate consequence of Theorems 3.4 and 5.10.
Corollary 5.11. There exists a complete continuous type structure that is a Borel refinement of a
complete type structure that admits RCAR.
Corollary 5.12. There exists a complete continuous type structure that generates the same hierarchies
as some complete type structure that admits RCAR.
These results shift our focus from the belief hierarchies generated by complete continuous type
structures to the degree of caution they require for the assumption of events. Corollary 5.12 shows
that there are structures T and U that generate the same hierarchies such that T admits RCAR
but U does not! With regard to beliefs, the only fundamental difference between U and T is that
players must be more cautious in assuming an event in U than they are in assuming the same
event in T. This suggests that the collection of open sets of T, at least in the role it plays in the
definition of assumption, should be viewed as a primitive of the model that has non-topological
conceptual functions—i.e., those unrelated to the closeness/convergence of types.
13
Note that Proposition 3.1 is an immediate consequence of Theorems 3.6 and 5.10.
Kechris (1995, Theorem 17.20): Let X be a Polish space and let O be an open basis for X . A sequence µk weakly
converges to µ in M (X ) if and only if lim infk µk (O) ≥ µ(O) for every O ∈ O . This result is stated for all open sets in
Kechris (1995), but the version stated here with an open basis follows from his proof.
14
17
The notions of Borel equivalence and Borel refinement allow the separation of the degree
of caution from continuity within BFK’s framework. An example is provided by Theorems 3.4
and 5.10, which show that one can start with a complete type structure T admitting RCAR, and then
take a Borel refinement U that is complete and continuous. In such an analysis, one is definining
rationality and assumption with respect to the coarse topology on T c , while defining continuity
will be with respect to the fine topology on U c . We leave open the following natural question about
whether we can go in the other direction:
Question 2. Is every complete continuous type structure a Borel refinement of some type structure
that admits RCAR?
6
RELATED LITERATURE
The two papers that are most closely related (other than BFK) are Battigalli and Siniscalchi (2002)
(henceforth B-S) and Yang (2015).
B-S used type structured that model beliefs as conditional probability systems instead of lexicographic probability systems. B-S showed that the extensive form rationalizable (EFR) strategies
are exactly the strategies played in states at which there is rationality and common strong belief
of rationality (RCSBR) in some canonical complete continuous type structure. A discussion of the
relationship between strong belief and assumption can be found in BFK.
Yang (2015) used lexicographic type structures as in BFK and it is therefore easier to identify
parallels between his approach and ours. As we mentioned in the Introduction, Yang (2015)
introduced a condition called RCWAR (rationality and common weak assumption of rationality)—
which is defined by replacing all instances of assumption in RCAR with weak assumption—and
showed that it characterizes IA in a canonical lexicographic type structure that is complete and
continuous. In light of our results, the approach taken by Yang (2015) can be viewed as one
example of defining the degree of caution required for assumption with respect to a topology
on types that does not coincide with the topology on types specified by the type structure. More
specifically, the weak assumption is the analogue of assumption where T c has the trivial topology
{∅, T c }.15 We therefore interpret Yang (2015)’s result as saying that a line of reasoning about the
game that involves caution with respect to the strategies of others but no caution with respect to
the types of others can characterize the IA set.16
BFK’s assumption is a natural generalization of Blume et al. (1991)’s notion of “infinitely more
likely” from finite spaces to infinite ones. It is a purely decision-theoretic notion that does not refer
to game-specific objects such as strategy sets and type spaces. Furthermore, through the notion of
15
Unless T c is a singleton, this topology is not Polish, and hence is not directly compatible with BFK’s framework.
Although Yang (2015) requires full-support beliefs for rationality, as shown by Catonini and Vito (2014), this is
can be weakened to full-support beliefs on the opponent’s strategies.
16
18
Borel refinements (and more generally, Borel equivalence) there is a possibility of analyses with
varying degrees of caution within BFK’s framework. On the other hand, the definition of weak
assumption refers to game-specific objects, and the degree of caution required for weak assumption
does not vary across Borel refinements of the same type structure.
The possibility results of B-S and Yang (2015) are analogous to our Theorem 3.4, but there
are two notable differences. First, the type structures in those cases are continuous, but as we
showed in Section 5, there is a very specific sense in which any type structure that admits RCAR
is equivalent to a continuous type structure. Second, the results in both B-S and Yang (2015) take
the strong form as in Question 1.
In addition to being non-monotonic, there are other similarities between the notions of assumption, weak assumption, and strong belief. We discuss these in further detail in Appendix A.
A
APPENDIX: OPEN SETS AS CONDITIONING EVENTS
In this section we explore more deeply the parallels between the BFK, B-S, and Yang (2015). All three
papers are motivated by the Best Rationalization Principle, which is stated in Battigalli (1996) as follows:
A player should always believe that her opponents are implementing one of the “most rational”
(or “least irrational”) strategy profiles which are consistent with her information.
In B-S, “most rational” is interpreted as “highest degree of strategic sophistication”, where degree of strategic
sophistication m means rationality and m-th order strong belief of rationality (RmSBR). The analogous notions
of strategic sophistication in the frameworks of Yang (2015) and BFK are summarized in the following table.
Strategic
Sophistication
≥ m+1
=∞
B-S
Yang
BFK
requires consistency with
RmSBR
RCSBR
RmWAR
RCWAR
RmAR
RCAR
Table 1: Strategic sophistication in B-S, Yang (2015), and BFK
Section 5 introduced the idea that the open sets of type spaces encode the degree of caution required to
assume events. In the online supplement to BFK, it is pointed out that the family of open sets in BFK plays
the same role as the family of conditioning events in B-S. In Yang (2015), the family of strategy cylinders
plays that role.
Keeping that in mind, we will expose striking parallels between assumption and open sets, weak assumption and strategy cylinders, and strong belief and conditioning events. This will also give another explanation
of the fact that RCAR is impossible in all complete continuous type structures, but possible in some complete
type structures.
In order to have a common treatment of all three settings, we first introduce some terminology in a
very general setting.
Definition A.1. A best rationalization system (BRS) is a triple
B = (Ω, C , 〈Rm , m ∈ N〉)
19
where Ω is a Polish space, C is a family of non-empty Borel subsets of Ω, and 〈Rm , m ∈ N〉 is a decreasing chain
of non-empty Borel subsets of Ω such that R0 = Ω (In the intended applications, Ω will be a type space, C will
be a set of conditioning events, and R m will be the set of strategy-type pairs that exhibit m-th degree strategic
sophistication).
We say that a set C ∈ C is best-rationalized at degree M in B if M is the greatest m ∈ N such that C ∩ R m
is non-empty. We say that an integer M is a finite bound for B if every C ∈ C that is best-rationalized at some
finite degree is best-rationalized at some degree ≤ M .
We will consider the property of having a finite bound in different settings. Note that if the set C is
finite, then the existence of a finite bound is trivial. For finite games, the set of strategy cylinders is finite,
so the following is immediate.
Proposition A.2. Suppose that T is a complete lexicographic type structure, c is a player, and B is a BRS where
Ω = S c × T c , and C is the set of strategy cylinders. Then B has a finite bound.
Given a Polish space Ω and a family C of non-empty Borel sets in Ω, Rényi (1955) introduced the notion
of a conditional probability system (CPS) on (Ω, C ). In the case that C is finite, B-S introduced the notion
of a CPS strongly believing an event. In B-S, C was generated from the information sets of dynamic games.
In the online supplement to BFK, the notion of strong belief was generalized to the case that the family C
is infinite, and the following result (Theorem S.1) was obtained.
Theorem A.3. For every BRS B, the following are equivalent:
T
(i) For each C ∈ C , either C meets the intersection m R m , or there is a greatest integer M such that C
meets R M .
(ii) There exists a CPS on (Ω, C ) that strongly believes each event R m .
This has a consequence concerning finite bounds in the strong belief setting, even in the case that the
family C is infinite.
Proposition A.4. Suppose B is a BRS such that C is closed under countable unions. If there exists a CPS on
(Ω, C ) that strongly believes each event R m , then B has a finite bound.
Proof. By hypothesis, Theorem A.3(ii) holds, and therefore Theorem A.3(i) holds. Suppose by way of
contradiction that B does not have a finite bound. Then there is an infinite subset
A ⊆ N such that for each
S
m ∈ A there exists Cm ∈ C that is best rationalized at degree
m.
Then
C
≡
C
m∈A m belongs to C , and C
T
meets R m for
each
m
∈
A,
so
by
Theorem
A.3(i),
C
meets
R
.
Therefore
there exists m ∈ A such that
n∈A n
T
Cm meets n∈A R n , so Cm meets R n for some n > m, contradicting the fact that Cm is best-rationalized at
degree m.
Note that the above proof did not use the definition of a CPS or strong belief at all. It only used
Theorem A.3. We now turn to assumption and open sets. The following result is the exact analogue of
Proposition A.4, and is also a generalization of Proposition 4.1.
Proposition A.5. Suppose B is a BRS such that C is the family of non-empty open subsets of Ω. If there exists
an LCPS on Ω that assumes each event R m , then B has a finite bound.
Proof. The proof of Proposition 4.1 shows that there exists M ∈ N such that for each m ≥ M , the set R M \R m
is nowhere dense in R M . This means that whenever m > M , each open set O ∈ C that meets R M meets R m .
Therefore no open set can be best-rationalized at a degree m > M , so M is a finite bound for B.
20
RCSBR requires that each player strongly believes a decreasing sequence of events—the RmSBR sets—
representing increasing strategic sophistication of the other players. Similarly, RCAR requires that each player
assumes a decreasing sequence of events—the RmAR sets—representing increasing strategic sophistication
of the other players. And RWCAR requires that each player weakly assumes a decreasing sequence of events—
the RmWAR sets—also representing increasing strategic sophistication of the other players. In light of these
requirements, Propositions A.4, A.5, and A.2 lead to the same principle in three different frameworks:
In order for the limit of strategic sophistication (RCSBR/RCAR/RCWAR) to be attainable, it is
necessary that the associated BRS has a finite bound.
Thus there is a finite M such that, for every conditioning event, if there is a highest finite degree m
of the opponent’s strategic sophistication that is consistent with that event, then m ≤ M . An increase in
strategic sophistication beyond M will have no additional implications for the players’ strategy choices.
In BFK, open sets play the role of conditioning events that must be best-rationalized. In a complete
continuous type structure, for every finite m, there is some conditioning event (i.e., open set) such that m is
the highest degree of the opponent’s strategic sophistication that is consistent with that event, so RCAR is
not attainable by the above principle.
The complete (but not continuous) type structure that we construct in Theorem 3.4 avoids the pitfall
because the construction ensures that there is an integer M such that the difference sets RcM \ Rcm are
topologically small for all m > M . There is no conditioning event (open set) that is best-rationalized at a
degree m > M , and RCAR is attained.
B
APPENDIX: PROOFS
b
Proof of Theorem 3.5. Fix some (s a , t a , s b , t b ) ∈ Ra∞ × R∞
. Then σ = λa (t a ) is a full-support LCPS that
b
b
assumes every event in the sequence (R1 , R2 , . . . ). Since σ has only finitely many levels, there exists a level
b
k at which σ assumes R m
for infinitely many m ∈ N. By Proposition 2.2(i), it follows that there exist some
b
k and smallest M such that σ assumes R m
at level k for all m ≥ M .
b
By Proposition 2.2(iii), σ assumes R∞ at level k. By Proposition 2.2(ii), whenever M ≤ m ∈ N we have
b
b . Since s b × T b is open for all s b ∈ S b , proj b R b = proj b R b . By Theorem 3.2, proj b R b = S b .
R∞
= Rm
S
S
S
∞
m
m
m
b
b
b
a
Therefore projS b R∞
= Sm
= S∞
for all m ≥ M . Analogously, projS a Ra∞ = S∞
.
Proof of Lemma 4.2. For an LPS µ ∈ N (S d ) and strategies r, s ∈ S c , write s µ r if the LEU for s under µ
is lexicographically greater that for r. For µ, ν ∈ N (S d ), write µ ∼ ν if the relations µ and ν on S c are
the same. Note that ∼ is an equivalence relation, that µ ∼ ν implies BRc (µ) = BRc (ν), and that µ ∼ µ0 and
ν ∼ ν0 implies µν ∼ µ0 ν0 .
c
Suppose m > 0 and X ∈ Ymc . We show that X ∈ Ym−1
. X = BRc (ν) for some ν = (ν0 , . . . , νm ) ∈ Ymc . For
d
each i ≤ m, we have νi ∈ M + (Sm−i
). By Proposition 1 in (Blume et al., 1991), for each i ≤ m there is a
+ d
µi ∈ M (Sm−i ) such that (µi ) ∼ (ν0 , . . . , νi ). Then for each i < m, (µi , νi+1 ) ∼ (ν0 , . . . , νi+1 ) ∼ (µi+1 ).
Now let i < m and consider any r, s ∈ S c . If s (µi ) r then s (µi ,µi+1 ) r, s (µi ,νi+1 ) r, and hence
s (µi+1 ) r hold, so s (µi ,µi+1 ) r iff s (µi+1 ) r. Similarly, if r (µi ) s then s (µi ,µi+1 ) r iff s (µi+1 ) r. If neither
s (µi ) r nor r (µi ) s, then again s (µi ,µi+1 ) r iff s (µi+1 ) r. Therefore (µi , µi+1 ) ∼ (µi+1 ), and hence
(µi , µi+1 , . . . , µm ) ∼ (µi+1 , . . . , µm ).
µm was chosen so that (µm ) ∼ ν. By the above, (µi+1 , . . . , µm ) ∼ ν implies (µi , µi+1 , . . . , µm ) ∼ ν. So by
c
induction we have (µ1 , . . . , µm ) ∼ ν. Therefore BRc ((µ1 , . . . , µm )) = BRc (ν) = X ∈ Ym−1
.
Lemma B.1. Let X = {x 1 , . . . , x n } and Y be non-empty Polish spaces and S ⊆ X × Y be a Borel set. Let
µ ∈ M (X × Y ) and ν ∈ M (X ) such that supp(margX µ) = supp ν. Then there exists some µ0 ∈ M (X × Y )
such that
21
(i) µ and µ0 are absolutely continuous with respect to each other; and
(ii) margX µ = margX µ0 .
P
Proof. Let µ0 (E) = x∈X ν(x)µ(E|{x} × Y ). It is easily verified that µ0 has the desired properties.
Lemma B.2 (Lemma B.1 in BFK). Let X be a Polish space, E be a Borel subset of X , σ = (µ0 , . . . , µn−1 ) be
a full-support LCPS on X , and k < n. Then σ assumes E at level k if and only if the following conditions are
met.17
(i) µi (E) = 1 for each i ≤ k;
(ii) µi (E) = 0 for each i > k; and
S
(iii) E ⊆ i≤k supp µi .
Lemma B.3 (Lemma C.3 in BFK). For each Polish space X and Borel set E in X , the set of σ ∈ L + (X ) that
assume E is Borel.
b ∈ L + (S d × T d )
Lemma B.4 (Lemma E.2 in BFK). For each σ ∈ L + (S d × T d ), there are continuum-many σ
b , and for each k ∈ N, σ and σ
b assume the same events at level k.
such that margS d σ = margS d σ
Proof of Lemma 4.3. By construction, Pmc (X ) is an uncountable Borel set for all (X , m) ∈ X c × (N ∪ {∞})
such that 1 ≤ m ≤ m(X ). By Lemma B.3, P̂mc (X ) is Borel for all such (X , m).
We only need to show that P̂mc (X ) is uncountable for all such (X , m). By Lemma B.4, showing P̂mc (X ) 6= ∅
is enough. Recall the definition of P̂mc (X ):
∀m ∈ N
b c ∩ σ ∈ L + (S d × T d )σ assumes Q d for all k < m but not Q d
P̂mc (X ) = V
X
m
k
c
b c ∩ σ ∈ L + (S d × T d )σ assumes Q d for all k
P̂∞
(X ) = V
X
k
Let M be the smallest M such that Q dM = Q d∞ . We can choose two sequences (b
µk ) and (e
µk ) of probability
d
d
d
measures in M (Q 0 ) = M (S × T ) so that
b k (Q dk ) = 1 and Q dk ⊆ supp µ
b k ; and
(i) µ
e k (Q dk \ Q dk+1 ) = 1 and Q dk \ Q dk+1 ⊆ supp µ
ek .
(ii) µ
d
)×
Case 1: m = m(X ) = ∞. The fact that m(X ) = ∞ implies that there is some (ν0 , . . . , ν M ) ∈ M + (S M
c
· · · × M + (S0d ) such that X = BRc ((ν0 , . . . , ν M )). We want show that there is some σ ∈ P̂∞
(X ). It is easily
verified (via Lemma B.2) that
0
e M −1 , µ
e M −2 , . . . , µ
e0 )
σ0 = (ρ00 , . . . , ρ M
) = (b
µ∞ , µ
0
d
is a full-support LCPS that assumes Q dk for all k. Note that margS d (ρ00 , . . . , ρ M
) ∈ M + (S M
) × · · · × M + (S0d ).
+ d
+ d
Because (ν0 , . . . , ν M ) is also in M (S M ) × · · · × M (S0 ), Lemma B.1 implies that there is some σ =
(ρ0 , . . . , ρ M ) such that
(i) margS d (ρ0 , . . . , ρ M ) = (ν0 , . . . , ν M ); and
(ii) ρk and ρk0 are absolutely continuous with respect to each other for all k ≤ M , which implies that σ
and σ0 assume exactly the same events.
c
Since X = BRc ((ν0 , . . . , ν M )) = BRc (margS d σ), we have shown that σ ∈ P̂∞
(X ).
17
The proof in BFK establishes this fact, but its statement was garbled in BFK.
22
Case 2: m ≤ m(X ) < ∞. The fact that m ≤ m(X ) < ∞ implies that there is some (ν0 , . . . , νm ) ∈
d
M + (Sm
) × · · · × M + (S0d ) such that X = BRc ((ν0 , . . . , νm )). We want show that there is some σ ∈ P̂mc (X ). It
is easily verified (via Lemma B.2) that
0
e m−1 , µ
e m−2 , . . . , µ
e0 )
σ0 = (ρ00 , . . . , ρm
) = ((0.5b
µm+1 + 0.5e
µm ), µ
0
d
is a full-support LCPS that assumes Q d1 , . . . , Q dm but not Q dm+1 . Note that margS d (ρ00 , . . . , ρm
) ∈ M + (Sm
)×
+ d
+ d
+ d
· · · × M (S0 ). Because (ν0 , . . . , νm ) is also in M (Sm ) × · · · × M (S0 ), Lemma B.1 implies that there is
some σ = (ρ0 , . . . , ρm ) such that
(i) margS d (ρ0 , . . . , ρm ) = (ν0 , . . . , νm ); and
(ii) ρk and ρk0 are absolutely continuous with respect to each other for all k ≤ m, which implies that σ
and σ0 assume exactly the same events.
Since X = BRc ((ν0 , . . . , νm )) = BRc (margS d σ), we have shown that σ ∈ P̂mc (X ).
Proof of Theorem 3.6. Define X c , Ymc , and m(·) as in the proof of Theorem 3.4. By Corollary 6.5 in Kechris
(1995), T c contains a homeomorphic copy of the Cantor space C = {0, 1}N , which is in turn homeomorphic
to C × C. We may assume that C × C is a subspace of T c . Pick a partition {Wkc |0 ≤ k ≤ |X c |} of C, where
Wkc is non-empty for each k, and Wkc is open for each S
k > 0. Form the partition {VX0c |X ∈ X c ∪ {∅}} of T c ,
0c
c
c
0c
c
where VX = Wk × C for each X k ∈ X , and V∅ = T \ {VX0c |X ∈ X c }. Pick countably many distinct points
k
0c
0c
x 1 , x 2 , . . . in C. For each m ≥ 1, let H m
be the “horizontal line” H m
= C × {x m }. Now argue exactly as in the
c
0c
c
0c
proof of Theorem 3.4, but replacing VX everywhere with VX , and replacing H m
everywhere with H m
.
Proof of Proposition 5.8. We show even more: If Um 6= R m , then Un and R n are incomparable for all n > m.
c
d
Suppose that Um
6= Rcm . It is obvious that m > 0. It suffices to show that Um+1
6⊆ Rdm+1 , because by
d
d
symmetry it would then follow that R m+1 6⊆ Um+1 . Since T and U are complete, it is enough to find an LCPS
σ that has full support in U c , and assumes Ukc for all k ≤ m but does not assume Rck for all k ≤ m.
c
By Lemma E.3 of BFK, for each k ∈ N, Rck+1 ( Rck and Uk+1
( Ukc . Let B be the finite Boolean algebra
c
c
generated by the sets R k and Uk for k ≤ m. Let C be a countable subset of S c × T c such that for every set
B ∈ B, C ∩ B is dense in B with respect to the topology of S c × U c . Let µ be a probability measure on
c
S c × T c such that µ(C) = 1 and µ({c}) > 0 for each c ∈ C. For each k < m, let µk (E) = µ(E | Ukc \ Uk+1
). Let
c
µm (E) = µ(E | Um ). Then σ = (µm , µm−1 , . . . , µ0 ) is an LCPS that has full support in U , and assumes Ukc at
level m − k for each k ≤ m.
Suppose σ assumes Rck for all k ≤ m. Since C meets Rck \ Rck+1 for each k < m, σ cannot assume more
c
than one of these sets at the same level, so σ assumes Rcm at level 0. Then µm (Rcm ) = 1, so Um
⊆ Rcm , and
c
c
c
c
c
µk (R m ) = 0 for all k < m, so R m ⊆ Um . This contradicts Um 6= R m , and completes the proof.
REFERENCES
B A R E L L I , P. A N D S. G A L A N I S (2013): “Admissibility and Event-Rationality,” Games and Economic Behavior,
77, 21–40.
B AT T I G A L L I , P. (1996): “Strategic Rationality Orderings and the Best Rationalization Principle,” Games
and Economic Behavior, 13, 178–200.
B AT T I G A L L I , P. A N D M. S I N I S C A L C H I (2002): “Strong Belief and Forward Induction Reasoning,” Journal
of Economic Theory, 106, 356–391.
23
B L U M E , L . , A . B R A N D E N B U R G E R , A N D E . D E K E L (1991): “Lexicographic Probabilities and Choice
Under Uncertainty,” Econometrica, 59, 61–79.
B R A N D E N B U R G E R , A. A N D E. D E K E L (1993): “Hierarchies of Beliefs and Common Knowledge,” Journal
of Economic Theory, 59, 189–198.
B R A N D E N B U R G E R , A., A. F R I E D E N B E R G ,
metrica, 76, 307–352.
AND
H. J. K E I S L E R (2008): “Admissibility in Games,” Econo-
C AT O N I N I , E . A N D N . D . V I T O (2014): “Common assumption of cautious rationality and iterated admissibility,” Working Paper.
F R I E D E N B E R G , A . (2010): “When do type structures contain all hierarchies of beliefs?” Games and Economic Behavior, 68, 108–129.
H A L P E R N , J. Y. A N D R. P A S S (2009): “A logical characterization of iterated admissibility,” in Proceedings
of the 12th Conference on the Theoretical Aspects of Rationality and Knowledge, 146–155.
H A R S A N Y I , J. C. (1967): “Games with Incomplete Information Played by “Bayesian” Players, I-III. Part I.
The Basic Model,” Management Science, 14, 159–182.
HE I F E T Z, A., M. ME I E R,
AND
B . C . S C H I P P E R (2010): “Comprehensive Rationalizability,” Mimeo.
K E C H R I S , A . S . (1995): Classical Descriptive Set Theory, New York: Springer-Verlag.
KE I S L E R, H. J.
AND
B. S.
L E E (2011):
Https://ideas.repec.org/p/pra/mprapa/34441.html.
“Common
Assumption
of
Rationality,”
M E R T E N S , J . - F. A N D S . Z A M I R (1985): “Formulation of Bayesian analysis for games with incomplete
information,” International Journal of Game Theory, 14, 1–29.
R É N Y I , A . (1955): “On a new axiomatic theory of probability,” Acta Mathematica Hungarica, 6, 285–335.
S A M U E L S O N , L. (1992): “Dominated strategies and common knowledge,” Games and Economic Behavior,
4, 284–313.
Y A N G , C . - C . (2015): “Weak assumption and iterated admissibility,” Journal of Economic Theory, 158,
87–101.
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