# Cautious Belief and Iterated Admissibility††thanks: A previous version of
this paper has been circulated under the title “Common Assumption of Cautious
Rationality and Iterated Admissibility.” We are indebted to Pierpaolo
Battigalli and Amanda Friedenberg for important inputs and suggestions about
our work. We also thank Gabriele Beneduci, Adam Brandenburger, Martin
Dufwenberg, Nicolò Generoso, Edward Green, Byung Soo Lee, Julien Manili,
Burkhard Schipper, Marciano Siniscalchi, Elias Tsakas and the attendants of
our talks at ESEM 2022 conference, LOFT 2014, AMES 2013 conference, Stern
School of Business, Bocconi University, Scuola Normale Superiore and
Politecnico di Milano for their valuable comments. Financial support from
European Research Council (STRATEMOTIONS–GA 324219) and from the Italian
Ministry of Education, PRIN 2017, Grant Number 2017K8ANN4, are gratefully
acknowledged.

Emiliano Catonini
New York University in Shanghai, Department of Economics
<EMAIL_ADDRESS>Nicodemo De Vito
Department of Decision Sciences - Bocconi University
<EMAIL_ADDRESS>
(Draft of May 2023)

###### Abstract

We define notions of cautiousness and cautious belief to provide epistemic
conditions for iterated admissibility in finite games. We show that iterated
admissibility characterizes the behavioral implications of “cautious
rationality and common cautious belief in cautious rationality” in a terminal
lexicographic type structure. For arbitrary type structures, the behavioral
implications of these epistemic assumptions are characterized by the solution
concept of self-admissible set (Brandenburger, Friedenberg and Keisler 2008).
We also show that analogous conclusions hold under alternative epistemic
assumptions, in particular if cautiousness is “transparent” to the players.

KEYWORDS: Epistemic game theory, iterated admissibility, weak dominance,
lexicographic probability systems.

JEL: C72.

##  1 Introduction

The iterated deletion of weakly dominated strategies, called iterated
admissibility (henceforth IA), is an important and widely applied solution
concept for games in strategic form. Shimoji (2004) shows that, in many
dynamic games with generic payoffs at terminal nodes, IA is outcome-equivalent
to Pearce’s (1984) extensive-form rationalizability, a prominent solution
concept whose foundations are well understood (Battigalli and Siniscalchi
2002). Applications of IA in games of interest are, for instance, voting
(Moulin 1984) and money-burning games (Ben-Porath and Dekel, 1992). Yet, while
IA has an independent intuitive appeal, its theoretical foundations have
proved to be elusive (see Samuelson 1992). Thus, the decision-theoretic
principles and the hypotheses about strategic reasoning that yield IA require
careful scrutiny.

A recent literature—starting with the seminal contribution of Brandenburger,
Friedenberg and Keisler (2008, henceforth BFK)—has tackled this issue building
on two key ideas. The decision-theoretic aspects of the problem have been
modeled through the _lexicographic expected utility_ theory of Blume et al.
(1991a). Lexicographic expected utility preferences are represented by
_lexicographic probability systems_(henceforth LPS’s), i.e., lists of
probabilistic conjectures in a priority order, each of which becomes relevant
when the previous ones fail to identify a unique best alternative. In games
with complete information, opponents’ strategies constitute the only payoff-
relevant uncertainty. In order to come up with an educated conjecture about
opponents’ strategies, a player naturally starts reasoning about opponents’
beliefs and choice criteria. BFK modeled this aspect with the tools of
_epistemic game theory_ , the formal, mathematical analysis of how players
reason about each other in games.111See Dekel and Siniscalchi (2015) for a
recent survey.

Inspired by BFK, in this paper we adopt lexicographic expected utility and
epistemic game theory for our foundation of IA in finite games. Specifically,
we use the formalism of lexicographic type structures to model players’
interactive beliefs. However, we start from partially different basic
principles. To motivate our analysis, we briefly mention BFK’s results.

BFK showed that, for every natural number $m$, the strategies that survive
$m+1$ rounds of iterated elimination of inadmissible (i.e., weakly dominated)
strategies are those consistent with the epistemic conditions of rationality
and $m$th-order assumption of rationality (henceforth R$m$AR). Such epistemic
conditions are represented as events in a type structure, and the result
requires a “richness” property—called completeness—of the type structure.
BFK’s notion of “rationality” incorporates a full-support requirement, which
reflects the idea that nothing is ruled out by the players. As we intuitively
explain below, the concept of assumption can be thought of as a strong form of
“virtually persistent belief” in an event. Rationality and common assumption
of rationality (henceforth RCAR) is the condition that R$m$AR holds for every
$m$. BFK considered the natural conjecture that, in complete type structures,
RCAR is an epistemic condition for IA—that is, the latter characterizes the
behavioral implications of the former. Yet, they show that RCAR is empty in
complete and continuous type structures, such as the canonical, universal type
structure (see Yang 2015, and Catonini and De Vito 2018), which represents all
hierarchies of lexicographic beliefs. Therefore, the event RCAR falls short of
providing a general justification of IA.

#### Our contribution

In this paper we propose an alternative approach to the foundations of IA.
Specifically, we provide notions of _rationality_ , _cautiousness_ and
_cautious belief_ that justify the choice of iteratively admissible strategies
in the following way: in “rich” type structures, IA characterizes the
behavioral implications of cautious rationality and common cautious belief in
cautious rationality (henceforth
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$). A prominent example of
“rich” type structure that works for this purpose is precisely the canonical,
universal type structure. Thus, our first result (Theorem 1) shows that
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$, unlike RCAR, is possible in
a complete and continuous type structure.

We explain in more detail the main ingredients of our analysis. First, we
define rationality as lexicographic expected utility maximization. The notion
of cautiousness requires that all payoff-relevant consequences be deemed
possible by the player; so, it describes a cautious attitude of the player
towards the opponents’ strategies. Cautious rationality is given by the
conjunction of rationality and cautiousness.

Cautious belief is a strengthening of the notion of weak belief (Catonini and
De Vito 2020), which is based on the preference-based concept of “infinitely
more likely” due to Lo (1999). Intuitively, a player deems an event $E$
infinitely more likely than $F$ if she strictly prefers to bet on $E$ rather
than on $F$ regardless of the size of the winning prizes for the two bets
(given the same losing outcome). With this, we say that a player cautiously
believes an event $E$ if she deems each payoff-relevant component of $E$
infinitely more likely than not-$E$. Put differently, cautious belief requires
that: (1) the player deems $E$ infinitely more likely than not-$E$, and (2)
before entertaining the possibility that $E$ does not occur, she takes into
account all the possible payoff-relevant consequences of $E$. Condition (1)
corresponds to weak belief; condition (2) says that the player is cautious
towards the (weakly) believed event. Thus cautious belief in $E$ is stronger
than weak belief as it captures cautiousness relative to $E$.

We further show how our approach allows to provide alternative epistemic
conditions for IA when there is common certainty that everybody is cautious.
Precisely, we prove that IA is also justified by rationality, transparency of
cautiousness, and common cautious belief thereof (Theorem 3). Here, we say
that an event $E$ is __ transparent if it is true and there is common certain
belief in $E$.222The precise, formal definition of “transparency” for an event
is given in Section 5.3. To gain intuition, the reader can temporarily think
of transparency as common knowledge, even if they are distinct notions. Hence,
our epistemic conditions motivate IA as a suitable solution concept for a
context in which there is common certainty that everyone is cautious. Such
result departs from previous justifications of IA—that is, those based on the
concept of assumption or weak assumption (BFK, Dekel et. al 2016, Yang 2015).
In particular, these justifications require that, for some games, players’
caution (however defined) _cannot_ be certainly believed, as we elaborate in
Section 6.

Theorems 1 and 3 identify terminality as the relevant richness property of
type structures for the justification of IA. Such property is satisfied by the
canonical type structure, which represents all hierarchies of beliefs
satisfying an intuitive coherency condition. Differently from most of the
results in epistemic game theory (see, e.g., Dekel and Siniscalchi 2015),
completeness plays no role in the statements and proofs of our results. As
shown by Friedenberg and Keisler (2021), completeness may not be the
appropriate notion of richness to provide epistemic foundations for iterated
strict dominance. Leveraging on the ideas and proofs in Friedenberg and
Keisler (2021), we show (Theorem 5) that, for any non-degenerate game, there
exists a complete type structure such that IA does not characterize the
behavioral implications of R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$.

Given this fact, we address the following question: What are the implications
of our epistemic assumptions across all type structures? Note that considering
small (“non-rich”) type structures is essentially equivalent to hypothesize
that belief-related, non-behavioral events are transparent to the players. In
other words, there is a “context” (previous history, social conventions etc.)
in which the game is played, and such context specifies what beliefs players
do vs. do not consider possible. For arbitrary type structures, we show
(Theorems 2 and 4) that the behavioral implications of the aforementioned
epistemic assumptions are characterized by the solution concept of self-
admissible set (henceforth SAS). This concept was introduced by BFK, who note
that a finite game may admit many SAS’s, and the IA set is one of them. BFK
show that, for a fixed type structure associated with a given game, the
behavioral implications of RCAR constitute an SAS; the first part of Theorem 2
shows that an analogous conclusion holds for
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$. It should be noted that,
since RCAR and R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ are distinct
epistemic conditions, they can yield different SAS’s in a given type
structure.333For instance, consider the canonical type structure. The
behavioral implications of R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$
and RCAR are characterized by, respectively, the IA set and the empty set. By
definition, the empty set is an SAS. Nonetheless, as BFK show for the case of
RCAR, the second part of Theorem 2 shows that every SAS represents the
behavioral implications of R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$
in some type structure. Theorem 4 shows analogous results when cautiousness is
transparent to the players.

#### Related literature

Cautious belief has many similarities with BFK’s notion of assumption. Yet,
the two concepts are distinct, and the main difference relies on how players
are cautious in “believing” an event $E$. To gain some intuition, also
assumption requires that $E$ be deemed infinitely more likely than not-$E$;
yet, as for cautious belief, this is not enough. Roughly speaking, assumption
requires that event $E$ be deemed infinitely more likely that not-$E$
conditional on every “virtual observation” consistent with $E$.444We use the
expression “virtual observation” to emphasize that there is no real
observation in this static setting. It is just a suggestive language. This is
in line with the full-support requirement in BFK’s notion of rationality,
which reflects the idea that “everything is possible.” By contrast, cautious
belief requires that event $E$ be deemed infinitely more likely than not-$E$
conditional on payoff-relevant “virtual observations” consistent with $E$.
Thus, it requires a weaker form of “virtual persistence of belief.”

The notion of _weak assumption_ , introduced by Catonini (2013) and Yang
(2015), solves the impossibility of RCAR by requiring only the payoff-relevant
parts of $E$ to be deemed infinitely more likely than not-$E$, in the same way
as cautious belief. However, weak assumption maintains the same preference-
based foundation of assumption, that is, the same notion of infinitely more
likely.

We chose Lo’s notion mainly for two reasons. First, as pointed out by Blume et
al. (1991a), their notion of infinitely more likely may fail to satisfy a
natural disjunction property. To clarify, consider pairwise disjoint events
$C$, $D$ and $E$, and suppose that both $C$ and $D$ are infinitely more likely
than $E$. Blume et al. (1991a, p. 70) show that $C\cup D$ (logically: $C$ or
$D$) need not be infinitely more likely than $E$; Section 6.2 provides a
concrete example.

Second, Lo’s (1999) notion of “infinitely more likely” is equivalent, in terms
of LPS’s, to the definition used by Stahl (1995) for the solution concept of
lexicographic rationalizability—a concept which coincides with IA.555Stahl
(1995) did not provide a preference-based foundation of infinitely more
likely. Lo’s (1999) definition applies to a wide class of preferences,
including the lexicographic expected utility model. Stahl introduced
lexicographic rationalizability as a refinement of permissibility
(Brandenburger 1992), an iterated elimination procedure for lexicographic
beliefs. Stahl’s analysis is “pre-epistemic” in the following sense: there is
no epistemic apparatus to formally express events such as rationality and some
forms of “belief” in rationality. In our view,
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ matches very closely the
logic of lexicographic rationalizability, exactly as the weaker condition of
“cautious rationality and common weak belief of cautious rationality” captures
the logic of permissibility, as shown in our previous work (Catonini and De
Vito 2020). In a sense, the contribution of this paper can be best seen as
providing foundations for lexicographic rationalizability (see Section 6.1).

Before moving on to the formal analysis, it is worth stressing other features
of our approach. First, unlike BFK, we do not restrict the analysis to type
structures where players’ beliefs are lexicographic conditional probability
systems (henceforth LCPS’s). Loosely speaking, LCPS’s are LPS’s such that the
supports of the component measures are pairwise disjoint. We instead allow for
arbitrary LPS-based type structures, as in Dekel et al. (2016). These authors
argue that, from the perspective of providing foundations for IA, it is
conceptually appropriate to consider unrestricted LPS’s. With this, they show
that all BFK’s results have analogues in the setting of LPS-based type
structures.

Finally, we define cautiousness as a full-support condition on the set of
strategies. This notion of caution can be found, with some minor differences,
in other works (Asheim and Dufwenberg 2003, Perea 2012, Heifetz et al. 2019,
Lee 2016, Catonini and De Vito 2020). An alternative to cautiousness is the
notion that requires the full-support condition on strategies and types.
Catonini and De Vito (2020) discuss the difference between them. Here, we just
mention that cautiousness is a condition expressed in terms of belief
hierarchies, and it is invariant to details of the type structure that are
unrelated to hierarchies (e.g., the topology of type spaces). By contrast, the
full-support condition is not invariant even to isomorphisms between type
structures. Therefore, we provide a justification of IA and SAS’s using
expressible epistemic assumptions about rationality and beliefs, that is,
assumptions which can be expressed in a language describing primitive terms
(strategies) and terms derived from the primitives (beliefs about strategies,
beliefs about strategies and beliefs of others, etc.)—cf. Battigalli et al.
(2021, Section 3.A).

We have mentioned some important contributions on the epistemic analysis of IA
in games. We provide detailed comments on the closest related literature in
Section 6.

#### Structure of the paper

Section 2 introduces IA and self-admissible sets. Section 3 provides formal
definitions of LPS’s and type structures. Section 4 analyzes cautious
rationality and cautious belief. Section 5 contains our epistemic
justifications of IA and self-admissible sets. Section 6 discusses certain
conceptual aspects of the analysis, and it compares our notion of cautious
belief to those derived from BFK’s approach. Appendix A provides decision-
theoretic foundations for cautiousness and cautious belief. Appendix B and
Appendix C collect the proofs omitted from the main text.666The Supplementary
Appendix contains elaborations and discussions on some results discussed
throughout the paper.

##  2 Iterated admissibility and self-admissible sets

Throughout, we consider finite games. A finite game is a structure
$G:=\left\langle I,(S_{i},\pi_{i})_{i\in I}\right\rangle$, where (a) $I$ is a
finite set of players with cardinality $\left|I\right|\geq 2$; (b) for each
player $i\in I$, $S_{i}$ is a finite, non-empty set of strategies; and (c)
$\pi_{i}:S\rightarrow\mathbb{R}$ is the payoff function.777Our notation is
standard. For any profile of sets $\left(X_{i}\right)_{i\in I}$, we let
$X:=\mathop{\textstyle\prod}_{i\in I}X_{i}$ and
$X_{-i}:=\mathop{\textstyle\prod}_{j\neq i}X_{j}$ with typical elements
$x:=\left(x_{i}\right)_{i\in I}\in X$ and $x_{-i}:=\left(x_{j}\right)_{j\neq
i}$ $\in X_{-i}$.

Each strategy set $S_{i}$ is given the obvious topology, i.e., the discrete
topology. We let $\mathcal{M}\left(X\right)$ denote the set of all Borel
probability measures on a topological space $X$. So, given a mixed strategy
profile $\sigma\in\mathop{\textstyle\prod}_{i\in I}\mathcal{M}(S_{i})$, we
will denote player $i$’s expected utility simply by
$\pi_{i}(\sigma_{i},\sigma_{-i})$, i.e.,

$\pi_{i}(\sigma_{i},\sigma_{-i}):=\mathop{\textstyle\sum}_{(s_{i},\left(s_{j}\right)_{j\neq
i})\in S_{i}\times
S_{-i}}\sigma_{i}(\left\\{s_{i}\right\\})\left(\mathop{\textstyle\prod}_{j\neq
i}\sigma_{j}(\left\\{s_{j}\right\\})\right)\pi_{i}(s_{i},\left(s_{j}\right)_{j\neq
i})\text{.}$

Similarly, given a pure strategy $s_{i}\in S_{i}$ and a probability measure
$\mu_{i}\in\mathcal{M}(S_{-i})$, we will denote player $i$’s expected utility
by $\pi_{i}(s_{i},\mu_{i}):=\mathop{\textstyle\sum}_{s_{-i}\in
S_{-i}}\pi_{i}(s_{i},s_{-i})\mu_{i}(\left\\{s_{-i}\right\\})$. With an abuse
of notation, we will also identify the pure strategy $s_{i}\in S_{i}$ with the
mixed strategy $\sigma_{i}\in\mathcal{M}(S_{i})$ such that
$\sigma_{i}(\left\\{s_{i}\right\\})=1$.

In the remainder of this section, we fix a finite game $G:=\left\langle
I,(S_{i},\pi_{i})_{i\in I}\right\rangle$. Let $\mathcal{Q}$ be the collection
of all subsets of $S$ with the cross-product form
$Q=\mathop{\textstyle\prod}_{i\in I}Q_{i}$, where $Q_{i}\subseteq S_{i}$ for
every $i$.

###### Definition 1

Fix a set $Q\in\mathcal{Q}$. A strategy $s_{i}\in Q_{i}$ is weakly dominated
with respect to $Q$ if there exists a mixed strategy
$\sigma_{i}\in\mathcal{M}(S_{i})$, with $\sigma_{i}\left(Q_{i}\right)=1$, such
that $\pi_{i}(\sigma_{i},s_{-i})\geq\pi_{i}(s_{i},s_{-i})$ for every
$s_{-i}\in Q_{-i}$ and
$\pi_{i}(\sigma_{i},s_{-i}^{\prime})>\pi_{i}(s_{i},s_{-i}^{\prime})$ for some
$s_{-i}^{\prime}\in Q_{-i}$. Otherwise, say $s_{i}$ is admissible with respect
to $Q$.

If $s_{i}\in S_{i}$ is weakly dominated (resp. admissible) with respect to
$S$, say $s_{i}$ is weakly dominated (resp. admissible).

###### Remark 1

Fix a set $Q\in\mathcal{Q}$. A standard result (Pearce 1984, Lemma 4) states
that a strategy $s_{i}\in Q_{i}$ is admissible with respect to $Q$ if and only
if there exists $\mu_{i}\in\mathcal{M}(S_{-i})$, with $\mu_{i}(Q_{-i})=1$,
such that $\mu_{i}(\left\\{s_{-i}\right\\})>0$ for every $s_{-i}\in Q_{-i}$,
and $\pi_{i}(s_{i},\mu_{i})\geq\pi_{i}(s_{i}^{\prime},\mu_{i})$ for every
$s_{i}^{\prime}\in Q_{i}$.

The set of iteratively admissible strategies (henceforth IA set) is defined
inductively.

###### Definition 2

For every $i\in I$, let $S_{i}^{0}:=S_{i}$, and for every $m\in\mathbb{N}$,
let $S_{i}^{m}$ be the set of all $s_{i}\in S_{i}^{m-1}$ that are admissible
with respect to $S^{m-1}:=\mathop{\textstyle\prod}_{j\in I}S_{j}^{m-1}$. A
strategy $s_{i}\in S_{i}^{m}$ is called $m$-admissible. A strategy $s_{i}\in
S_{i}^{\infty}:=\cap_{m=0}^{\infty}S_{i}^{m}$ is called iteratively
admissible.

By finiteness of the game, it follows that $S_{i}^{m}\not=\emptyset$ for all
$m\in\mathbb{N}$, and, since $S_{i}^{m}\supseteq S_{i}^{m+1}$ for all
$m\in\mathbb{N}$, there exists $M\in\mathbb{N}$ such that
$S_{i}^{\infty}=S_{i}^{M}$. Consequently, the IA set $S^{\infty}$ is non-
empty.

To formally introduce BFK’s notion of self-admissible set, we need an
additional definition. Say that a strategy $s_{i}^{\prime}\in S_{i}$ supports
$s_{i}\in S_{i}$, if there exists a mixed strategy
$\sigma_{i}\in\mathcal{M}(S_{i})$ such that
$\sigma_{i}(\left\\{s_{i}^{\prime}\right\\})>0$ and
$\pi_{i}(\sigma_{i},s_{-i})=\pi_{i}(s_{i},s_{-i})$ for all $s_{-i}\in S_{-i}$.

###### Definition 3

A set $Q\in\mathcal{Q}$ is a self-admissible set (SAS) if, for every player
$i$,

(a) every $s_{i}\in Q_{i}$ is admissible,

(b) every $s_{i}\in Q_{i}$ is admissible with respect to $S_{i}\times Q_{-i}$,

(c) for every $s_{i}\in Q_{i}$ and $s_{i}^{\prime}\in S_{i}$, if
$s_{i}^{\prime}$ supports $s_{i}$ then $s_{i}^{\prime}\in Q_{i}$.

Every finite game admits a non-empty SAS—in particular, the IA set is an SAS.
But, as shown by BFK, many games possess other SAS’s, which can be even
disjoint from the IA set. This is illustrated by the following example, which
is taken from Brandenburger et al. (2012, Example 5.9).

###### Example 1

Consider the following game between two players, Ann ($a$) and Bob ($b$):

$a\backslash b$ | $\ell$ | $r$
---|---|---
$u$ | $2,2$ | $2,2$
$m$ | $3,1$ | $0,0$
$d$ | $0,0$ | $1,3$

There are three non-empty SAS’s: $\left\\{u\right\\}\times\left\\{r\right\\}$,
$\left\\{u\right\\}\times\left\\{\ell,r\right\\}$ and
$\left\\{m\right\\}\times\left\\{\ell\right\\}$. The SAS
$\left\\{m\right\\}\times\left\\{\ell\right\\}$ is the IA set. $\blacklozenge$

A comprehensive analysis of the properties of SAS’s in a wide class of games
is given by Brandenburger and Friedenberg (2010).888Example 1 is the reduced
strategic form of the Battle of the Sexes with an Outside Option, which is
used by Battigalli and Friedenberg (2012a) to illustrate how “extensive-form
best response sets” (EFBR’s) are related to Pearce’s (1984) notion of
“extensive-form rationalizability.” In the example, the EFBR’s coincide with
the SAS’s, and Pearce’s extensive-form rationalizability coincides with IA.

##  3 Lexicographic beliefs and lexicographic type structures

###  3.1 Lexicographic probability systems

All the sets considered in this paper are assumed to be Polish spaces (that
is, topological spaces that are homeomorphic to complete, separable metric
spaces), and they are endowed with the Borel $\sigma$-field. We let
$\Sigma_{X}$ denote the Borel $\sigma$-field of a Polish space $X$, the
elements of which are called events. When it is clear from the context, we
suppress reference to $\Sigma_{X}$ and simply write $X$ to denote a measurable
space.

Given a sequence $\left(X_{n}\right)_{n\in\mathbb{N}}$ of pairwise disjoint
Polish spaces, the set $X:=\cup_{n\in\mathbb{N}}X_{n}$ is endowed with the
direct sum topology,999In this topology, a set $O\subseteq X$ is open if and
only if $O\cap X_{n}$ is open in $X_{n}$ for all $n\in\mathbb{N}$. The
assumption that the spaces $X_{n}$ are pairwise disjoint is without any loss
of generality, since they can be replaced by a homeomorphic copy, if needed
(see Engelking 1989, p. 75). so that $X$ is a Polish space. Moreover, we endow
each finite or countable product of Polish spaces with the product topology,
hence the product space is Polish as well.

Recall that $\mathcal{M}\left(X\right)$ denotes the set of Borel probability
measures on a topological space $X$. The set $\mathcal{M}\left(X\right)$ is
endowed with the weak*-topology. So, if $X$ is Polish, then
$\mathcal{M}\left(X\right)$ is also Polish. We let $\mathcal{N}\left(X\right)$
(resp. $\mathcal{N}_{n}\left(X\right)$) denote the set of all finite (resp.
length-$n$) sequences of Borel probability measures on $X$, that is,

$\mathcal{N}\left(X\right):=\mathop{\textstyle\bigcup}_{n\in\mathbb{N}}\mathcal{N}_{n}\left(X\right):=\mathop{\textstyle\bigcup}_{n\in\mathbb{N}}\left(\mathcal{M}\left(X\right)\right)^{n}\text{.}$

Each $\bar{\mu}:=\left(\mu^{1},...,\mu^{n}\right)\in\mathcal{N}\left(X\right)$
is called lexicographic probability system (LPS). In view of our assumptions,
the topological space $\mathcal{N}\left(X\right)$ is Polish.

For every Borel probability measure $\mu$ on $X$, the support of $\mu$,
denoted by $\mathrm{Supp}\mu$, is the smallest closed subset $C\subseteq X$
such that $\mu\left(C\right)=1$. The support of an LPS
$\bar{\mu}:=\left(\mu^{1},...,\mu^{n}\right)\in\mathcal{N}\left(X\right)$ is
defined as $\mathrm{Supp}\bar{\mu}:=\cup_{l\leq n}\mathrm{Supp}\mu^{l}$. So,
an LPS
$\bar{\mu}:=\left(\mu^{1},...,\mu^{n}\right)\in\mathcal{N}\left(X\right)$ is
of full-support if $\mathrm{Supp}\bar{\mu}=X$. We write
$\mathcal{N}^{+}\left(X\right)$ for the set of full-support LPS’s.

For future reference, we also record the following definition. An LPS
$\bar{\mu}:=\left(\mu^{1},...,\mu^{n}\right)\in\mathcal{N}\left(X\right)$ is
called lexicographic conditional probability system (LCPS) if there are events
$E_{1},...,E_{n}$ in $X$ such that, for every $l\leq n$,
$\mu^{l}\left(E_{l}\right)=1$ and $\mu^{l}\left(E_{m}\right)=0$ for $m\neq l$.
If $X$ is finite, then an LPS is an LCPS if and only if the supports of the
component measures are pairwise disjoint.

Fix Polish spaces $X$ and $Y$, and a Borel map $f:X\rightarrow Y$. The map
$\widetilde{f}:\mathcal{M}\left(X\right)\rightarrow\mathcal{M}\left(Y\right)$,
defined by

$\widetilde{f}\left(\mu\right)\left(E\right):=\mu\left(f^{-1}\left(E\right)\right),\text{
\ \ }E\in\Sigma_{Y}\text{, }\mu\in\mathcal{M}\left(X\right)\text{,}$

is called the image (or pushforward) measure map of $f$. For each
$n\in\mathbb{N}$, the map
$\widehat{f}_{\left(n\right)}:\mathcal{N}_{n}\left(X\right)\rightarrow\mathcal{N}_{n}\left(Y\right)$
is defined by

$\left(\mu^{1},...,\mu^{n}\right)\mapsto\widehat{f}_{\left(n\right)}\left(\left(\mu^{1},...,\mu^{n}\right)\right):=\left(\widetilde{f}\left(\mu^{k}\right)\right)_{k\leq
n}\text{.}$

With his, the map
$\widehat{f}:\mathcal{N}\left(X\right)\rightarrow\mathcal{N}\left(Y\right)$
defined by

$\widehat{f}\left(\bar{\mu}\right):=\widehat{f}_{\left(n\right)}\left(\bar{\mu}\right)\text{,
}\bar{\mu}\in\mathcal{N}_{n}\left(X\right)\text{,}$

is called the image LPS map of $f$. Alternatively put, the map $\widehat{f}$
is the union of the maps
$\left(\widehat{f}_{\left(n\right)}\right)_{n\in\mathbb{N}}$, and it is Borel
measurable.101010For details and proofs related to Borel measurability and
continuity of the involved maps, the reader can consult Catonini and De Vito
(2018).

Given Polish spaces $X$ and $Y$, we let $\mathrm{Proj}_{X}$ denote the
canonical projection from $X\times Y$ onto $X$; in view of our assumption, the
map $\mathrm{Proj}_{X}$ is continuous. The marginal measure of
$\mu\in\mathcal{M}\left(X\times Y\right)$ on $X$ is defined by
$\mathrm{marg}_{X}\mu:=\widetilde{\mathrm{Proj}}_{X}\left(\mu\right)$.
Consequently, the marginal of $\bar{\mu}\in\mathcal{N}\left(X\times Y\right)$
on $X$ is defined by
$\overline{\mathrm{marg}}_{X}\bar{\mu}:=\widehat{\mathrm{Proj}}_{X}\left(\bar{\mu}\right)$,
and the function $\widehat{\mathrm{Proj}}_{X}:\mathcal{N}\left(X\times
Y\right)\rightarrow\mathcal{N}\left(X\right)$ is continuous and surjective.

###  3.2 Lexicographic type structures

Fix a finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in I}\right\rangle$. A
_type structure_ (associated with $G$) formalizes an implicit approach to
model hierarchies of beliefs. The following is a generalization of the
standard definition of epistemic type structure with beliefs represented by
probability measures, i.e., length-$1$ LPS’s (cf. Heifetz and Samet 1998).

###### Definition 4

An $\left(S_{i}\right)_{i\in I}$-based lexicographic type structure is a
structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ where

  1. 1.

for each $i\in I$, $T_{i}$ is a Polish space;

  2. 2.

for each $i\in I$, the function
$\beta_{i}:T_{i}\rightarrow\mathcal{N}\left(S_{-i}\times T_{-i}\right)$ is
Borel measurable.

Each space $T_{i}$ is called type space and each $\beta_{i}$ is called belief
map.111111Some authors (e.g., Battigalli and Siniscalchi 1999, Heifetz and
Samet 1998) use the terminology “type space” for what is called “type
structure” here. Members of type spaces, viz. $t_{i}\in T_{i}$, are called
types. Each element $\left(s_{i},t_{i}\right)_{i\in
I}\in\mathop{\textstyle\prod}_{i\in I}\left(S_{i}\times T_{i}\right)$ is
called state (of the world).

In what follows, we will omit the qualifier “lexicographic,” and simply speak
of type structures when the underlying strategy sets $\left(S_{i}\right)_{i\in
I}$ are clear from the context. Furthermore, if every type in a type structure
$\mathcal{T}$ is associated with a probability measure, then we will say that
$\mathcal{T}$ is an ordinary type structure.

Type structures generate a collection of hierarchies of beliefs for each
player. For instance, type $t_{i}$’s first-order belief is an LPS on $S_{-i}$,
and is given by $\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$. A
standard inductive procedure (see Catonini and De Vito 2018, for details)
shows how to provide an explicit description of a hierarchy induced by a type.

We will be interested in type structures with one or more of the following
features, which do not make reference to hierarchies of beliefs or other type
structures.

###### Definition 5

A type structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$
is

  * •

finite if the cardinality of each type space $T_{i}$ is finite;

  * •

compact if each type space $T_{i}$ is compact;

  * •

belief-complete if each belief map $\beta_{i}$ is onto;

  * •

continuous if each belief map $\beta_{i}$ is continuous.

The idea of (belief-)completeness was introduced by Brandenburger (2003) and
adapted to the present context.121212In Dekel et al. (2016), a type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ is said to be
complete if the range of each belief map $\beta_{i}$ is a strict superset of
$\mathcal{N}^{+}\left(S_{-i}\times T_{-i}\right)$. A belief-complete type
structure is complete in the sense of Dekel et al. (2016); the converse does
not hold. Note that each type space in a belief-complete type structure has
the cardinality of the continuum. Finite type structures are compact and
continuous, but not belief-complete. No belief-complete lexicographic type
structure is also compact and continuous. To see this, observe that if the
type structure is compact and continuous, each $\beta_{i}(T_{i})$ is compact
but the space $\mathcal{N}\left(S_{-i}\times T_{-i}\right)$ is not
compact,131313The space $\mathcal{M}\left(X\right)$ is compact if and only if
$X$ is compact, and this in turn implies that the space
$\mathcal{N}_{n}\left(X\right)$ is also compact for every finite
$n\in\mathbb{N}$. But the same conclusion does not hold for the space
$\mathcal{N}\left(X\right)$. This is an instance of a well-known mathematical
fact (see Theorem 2.2.3 in Engelking 1989): If
$\left(X_{\theta}\right)_{\theta\in\Theta}$ is an indexed family of non-empty
compact spaces with $\left|X_{\theta}\right|>1$ for all $\theta\in\Theta$,
then the direct sum $\cup_{\theta\in\Theta}X_{\theta}$ is compact if and only
if the right-directed set $\Theta$ is finite. hence $\beta_{i}$ is not onto.

We next introduce the notion of type morphism, which captures the idea that a
type structure $\mathcal{T}$ is “contained in” another type structure
$\mathcal{T}^{\ast}$. In what follows, given a type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$, we let $T$
denote the Cartesian product of type spaces, that is,
$T:=\mathop{\textstyle\prod}_{i\in I}T_{i}$. Moreover, for any set $X$, we let
$\mathrm{Id}_{X}$ denote the identity map on $X$, that is,
$\mathrm{Id}_{X}\left(x\right):=x$ for all $x\in X$.

###### Definition 6

Fix type structures $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$ and $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$. For each $i\in I$, let
$\varphi_{i}:T_{i}\rightarrow T_{i}^{\ast}$ be a measurable map such that

$\beta_{i}^{\ast}\circ\varphi_{i}=\widehat{\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)}\circ\beta_{i}\text{.}$

where $\varphi_{-i}:=\left(\varphi_{j}\right)_{j\neq i}:T_{-i}\rightarrow
T_{-i}^{\ast}$. The function $\left(\varphi_{i}\right)_{i\in I}:T\rightarrow
T^{\ast}$ is called type morphism (from $\mathcal{T}$ to
$\mathcal{T}^{\ast}$).

The morphism is called bimeasurable if the map $\left(\varphi_{i}\right)_{i\in
I}$ is Borel bimeasurable.141414A Borel map $f:X\rightarrow Y$ between
separable metrizable spaces is bimeasurable if $f(E)$ is Borel in $Y$ provided
$E$ is Borel in $X$. The morphism is called type isomorphism if the map
$\left(\varphi_{i}\right)_{i\in I}$ is a Borel isomorphism. Say $\mathcal{T}$
and $\mathcal{T}^{\ast}$ are isomorphic if there is a type isomorphism between
them.

A type morphism requires consistency between the function
$\varphi_{i}:T_{i}\rightarrow T_{i}^{\ast}$ and the induced function
$\widehat{\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)}:\mathcal{N}\left(S_{-i}\times
T_{-i}\right)\rightarrow\mathcal{N}\left(S_{-i}\times T_{-i}^{\ast}\right)$.
That is, the following diagram commutes:

$\begin{CD}T_{i}@>{\beta_{i}}>{}>\mathcal{N}(S_{-i}\times\ T_{-i})\\\
@V{}V{\varphi_{i}}V@V{}V{\ \widehat{(\mathrm{Id}_{S_{-i}},\varphi_{-i})}}V\\\
T_{i}^{*}@>{\beta_{i}^{*}}>{}>\mathcal{N}(S_{-i}\times\
T_{-i}^{*})\end{CD}\text{.}$ (3.1)

Thus, a type morphism maps $\mathcal{T}$ into $\mathcal{T}^{\ast}$ in a way
that preserves the beliefs associated with types.

The notion of type morphism does not make reference to hierarchies of beliefs.
But the important property of type morphisms is that they preserve the
explicit description of lexicographic belief hierarchies: the
$\left(S_{i}\right)_{i\in I}$-based belief hierarchy generated by a type
$t_{i}\in T_{i}$ in $\mathcal{T}$ is also generated by its image
$\varphi_{i}(t_{i})\in T_{i}^{\ast}$ in $\mathcal{T}^{\ast}$. Heifetz and
Samet (1998, Proposition 5.1) show this result for the case of ordinary type
structures; the generalization to lexicographic type structures is
straightforward (see Catonini and De Vito 2018).

Next, we introduce the notion of terminality for a type structure.

###### Definition 7

Fix a class $\mathbb{T}$ of type structures. A type structure
$\mathcal{T}^{\ast}:=\langle S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in
I}$ is terminal with respect to $\mathbb{T}$ if for every type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ in $\mathbb{T}$,
there is a type morphism from $\mathcal{T}$ to $\mathcal{T}^{\ast}$.

Whenever $\mathcal{T}^{\ast}$ is terminal with respect to the class of all
type structures, we simply say, as customary, that $\mathcal{T}^{\ast}$ is
_terminal_. In Section 5 we will show that
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ justifies IA in every type
structure which is terminal with respect to the class of all finite type
structures, and that such a type structure exists.

##  4 Cautiousness and cautious belief

For this section, we fix a finite game $G:=\left\langle
I,(S_{i},\pi_{i})_{i\in I}\right\rangle$, and we append to $G$ a type
structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$.

###  4.1 Rationality and cautiousness

For any two vectors
$x:=\left(x_{l}\right)_{l=1}^{n},y:=\left(y_{l}\right)_{l=1}^{n}\in\mathbb{R}^{n}$,
we write $x\geq_{L}y$ if either (a) $x_{l}=y_{l}$ for every $l\leq n$, or (b)
there exists $m\leq n$ such that $x_{m}>y_{m}$ and $x_{l}=y_{l}$ for every
$l<m$; we write $x>_{L}y$ if condition (b) holds.

###### Definition 8

A strategy $s_{i}\in S_{i}$ is optimal under
$\beta_{i}(t_{i}):=\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\in\mathcal{N}(S_{-i}\times
T_{-i})$ if, for every $s_{i}^{\prime}\in S_{i}$,

$\left(\pi_{i}(s_{i},\mathrm{marg}_{S_{-i}}\mu_{i}^{l})\right)_{l=1}^{n}\geq_{L}\left(\pi_{i}(s_{i}^{\prime},\mathrm{marg}_{S_{-i}}\mu_{i}^{l})\right)_{l=1}^{n}\text{.}$

Say that $s_{i}$ is a lexicographic best reply to
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$ if it is optimal under
$\beta_{i}(t_{i})$.

This is the usual definition of optimality for a strategy, but this time
optimality is taken lexicographically. We next introduce the notion of
cautiousness.

###### Definition 9

A type $t_{i}\in T_{i}$ is cautious (in $\mathcal{T}$) if
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})\in\mathcal{N}^{+}\left(S_{-i}\right)$.

This notion of cautiousness requires that the first-order belief of a type be
a full-support LPS. That is, it requires that every payoff-relevant component,
viz. $\left\\{s_{-i}\right\\}\times T_{-i}$, be assigned strictly positive
probability by at least one of the measures of LPS $\beta_{i}(t_{i})$. For
each $i\in I$, we let $C_{i}$ denote the set of all pairs
$\left(s_{i},t_{i}\right)\in S_{i}\times T_{i}$ such that $t_{i}$ is cautious.

For strategy-type pairs we define the following notions.

###### Definition 10

Fix a strategy-type pair $\left(s_{i},t_{i}\right)\in S_{i}\times T_{i}$.

  1. 1.

Say $\left(s_{i},t_{i}\right)$ is rational (in $\mathcal{T}$) if $s_{i}$ is
optimal under $\beta_{i}\left(t_{i}\right)$.

  2. 2.

Say $\left(s_{i},t_{i}\right)$ is cautiously rational (in $\mathcal{T}$) if it
is rational and $t_{i}$ is cautious.

We let $R_{i}$ denote the set of all rational strategy-type pairs. As one
should expect, cautious rationality guarantees admissibility.

###### Proposition 1

If strategy-type pair $(s_{i},t_{i})\in S_{i}\times T_{i}$ is cautiously
rational, then $s_{i}$ is admissible.

The proof of Proposition 1 is in Appendix B.

###  4.2 Infinitely more likely and cautious belief

We say that player $i$ deems event $E$ infinitely more likely than event $F$
if she prefers to bet on $E$ rather than on $F$ no matter the prizes for the
two bets. This preference-based notion of “infinitely more likely” is due to
Lo (1999, Definition 1), and it is formalized in Appendix A, where we
introduce the appropriate language. Here, we provide the equivalent definition
of “infinitely more likely” in terms of the LPS that represents player $i$’s
preferences.

Given an LPS
$\bar{\mu}_{i}:=(\mu_{i}^{1},...,\mu_{i}^{n})\in\mathcal{N}(S_{-i}\times
T_{-i})$ and an event $E\subseteq S_{-i}\times T_{-i}$, let

$\mathcal{I}_{\bar{\mu}_{i}}\left(E\right):=\inf\left\\{l\in\left\\{1,...,n\right\\}:\mu_{i}^{l}\left(E\right)>0\right\\}\text{,}$

with the convention that $\inf\emptyset:=+\infty$. The following definition is
from Catonini and De Vito (2020); see also Stahl (1995).

###### Definition 11

Fix two disjoint events $E,F\subseteq S_{-i}\times T_{-i}$. Say that $E$ is
infinitely more likely than $F$ under $\bar{\mu}_{i}$ if
$\mathcal{I}_{\bar{\mu}_{i}}\left(E\right)<\mathcal{I}_{\bar{\mu}_{i}}\left(F\right)$.

It is straightforward to see that “infinitely more likely” is monotone. That
is, if $E$ is infinitely more likely than $F$ under $\bar{\mu}_{i}$ and $G$ is
an event such that $E\subseteq G$, then $G$ is infinitely more likely than $F$
under $\bar{\mu}_{i}$.

Consider now the following attitudes of player $i$ towards an event $E$.
First, player $i$ deems $E$ infinitely more likely than its complement.
Second, player $i$ has a cautious attitude towards the event: Before
considering its complement, she considers all the possible payoff-relevant
consequences of the event. The notion of cautious belief captures both
attitudes.

###### Definition 12

Fix a non-empty event $E\subseteq S_{-i}\times T_{-i}$ and a type $t_{i}\in
T_{i}$ with $\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$.
Event $E$ is cautiously believed under $\beta_{i}\left(t_{i}\right)$ at level
$m\leq n$ if the following conditions hold:

(i)

$\mu_{i}^{l}\left(E\right)=1$ for all $l\leq m$;

(ii)

for every elementary cylinder $\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times
T_{-i}$, if $E\cap\hat{C}_{s_{-i}}\not=\emptyset$ then
$\mu_{i}^{l}\left(E\cap\hat{C}_{s_{-i}}\right)>0$ for some $l\leq m$.

Event $E$ is cautiously believed under $\beta_{i}\left(t_{i}\right)$ if it is
cautiously believed under $\beta_{i}\left(t_{i}\right)$ at some level $m\leq
n$.

Type $t_{i}\in T_{i}$ cautiously believes $E$ if $E$ is cautiously believed
under $\beta_{i}\left(t_{i}\right)$.

Condition (i) captures the first attitude. Under condition (i), condition (ii)
is equivalent to saying that player $i$ deems all _payoff-relevant parts_ of
$E$ (i.e., the non-empty intersections of $E$ with each strategy-based
cylinder) infinitely more likely than not-$E$, so it captures the second
attitude.

The conceptual consistency between cautiousness and cautious belief is
highlighted by the following connection.

###### Remark 2

A type $t_{i}\in T_{i}$ is cautious if and only if $t_{i}$ cautiously believes
$S_{-i}\times T_{-i}$.

Appendix A provides a preference-based foundation for cautious belief in terms
of “infinitely more likely,” as well as a characterization in terms of
infinitesimal nonstandard numbers. Here, we just mention some properties of
cautious belief that will be useful for the proofs of our results.

###### Proposition 2

Fix a type $t_{i}\in T_{i}$ with
$\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$.

  1. 1.

Fix non-empty events $E_{1},E_{2},...$ in $S_{-i}\times T_{-i}$. If, for each
$k$, type $t_{i}$ cautiously believes $E_{k}$, then $t_{i}$ cautiously
believes $\cap_{k}E_{k}$ and $\cup_{k}E_{k}$.

  2. 2.

A non-empty event $E\subseteq S_{-i}\times T_{-i}$ is cautiously believed
under $\beta_{i}\left(t_{i}\right)$ if and only if there exists $m\leq n$ such
that $\beta_{i}\left(t_{i}\right)$ satisfies condition (i) of Definition 12
plus the following condition:

(ii’)

$\cup_{l\leq
m}\mathrm{Suppmarg}_{S_{-i}}\mu_{i}^{l}=\mathrm{Proj}_{S_{-i}}\left(E\right)$.

The proof of Proposition 2 is in Appendix B.

Proposition 2.1 states that cautious belief satisfies one direction of
conjunction as well as one direction of disjunction. Proposition 2.2 can be
viewed as a “marginalization” property of cautious belief: If $E$ is
cautiously believed under $\beta_{i}\left(t_{i}\right)$, then

(a) $\mathrm{Proj}_{S_{-i}}\left(E\right)$ is infinitely more likely than
$S_{-i}\backslash\mathrm{Proj}_{S_{-i}}\left(E\right)$ under
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$; and

(b) _every_ strategy in $\mathrm{Proj}_{S_{-i}}\left(E\right)$ is infinitely
more likely than (every strategy in)
$S_{-i}\backslash\mathrm{Proj}_{S_{-i}}\left(E\right)$ under
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$.

The failure of one direction of conjunction reveals that, although “infinitely
more likely” is monotone, cautious belief is not. That is, if $t_{i}$
cautiously believes $E$, then $t_{i}$ may not cautiously believe an event $F$
such that $E\subseteq F$. The reason why this can occur is that player $i$ may
not have towards $F$ the same cautious attitude that she has towards $E$. That
is, there may be some payoff-relevant components of $F\backslash E$ which are
not deemed infinitely more likely than not-$F$. This is illustrated by the
following example, which is from Catonini and De Vito (2020, Example 1).

###### Example 2

Consider a finite game with two players, Ann ($a$) and Bob ($b$), where the
strategy set of Bob is
$S_{b}:=\left\\{s_{b}^{1},s_{b}^{2},s_{b}^{3}\right\\}$. Append to this game a
type structure $\mathcal{T}$ such that $T_{b}:=\left\\{t_{b}^{\ast}\right\\}$.
Consider the LPS
$\bar{\mu}_{a}:=(\mu_{a}^{1},\mu_{a}^{2})\in\mathcal{N}(S_{b}\times T_{b})$
with
$\mu_{a}^{1}\left(\left\\{\left(s_{b}^{1},t_{b}^{\ast}\right)\right\\}\right)=1$
and
$\mu_{a}^{2}\left(\left\\{\left(s_{b}^{2},t_{b}^{\ast}\right)\right\\}\right)=\mu_{a}^{2}\left(\left\\{\left(s_{b}^{3},t_{b}^{\ast}\right)\right\\}\right)=\frac{1}{2}$.
Next, consider the events $E:=\left\\{s_{b}^{1}\right\\}\times T_{b}$ and
$F:=\left\\{s_{b}^{1},s_{b}^{2}\right\\}\times T_{b}$. Clearly, $E\subseteq
F$. Yet, $E$ is cautiously believed under $\bar{\mu}_{a}$ at level $1$, while
$F$ is not cautiously believed: indeed, $\mu_{a}^{1}\left(F\right)=1$ and
$\mu_{a}^{2}\left(F\right)=\frac{1}{2}$, and, with $l=1$, condition (ii) of
Definition 12 is not satisfied for
$\hat{C}_{s_{b}^{2}}:=\left\\{s_{b}^{2}\right\\}\times T_{b}$. $\blacklozenge$

However, it is easy to observe that cautious belief is monotone with respect
to events with the same behavioral implications.

###### Remark 3

Let $E_{-i},F_{-i}\subseteq S_{-i}\times T_{-i}\ $be events such that
$E_{-i}\subseteq F_{-i}$ and
$\mathrm{Proj}_{S_{-i}}\left(E_{-i}\right)=\mathrm{Proj}_{S_{-i}}\left(F_{-i}\right)$.
If a type $t_{i}$ cautiously believes $E_{-i}$, then $t_{i}$ cautiously
believes $F_{-i}$.

This “quasi-monotonicity” property will play a crucial role in the proof of
our main result.

For future reference, it is useful to mention the following notion of belief,
called certain belief (Halpern 2010). Fix a non-empty event $E\subseteq
S_{-i}\times T_{-i}$ and a type $t_{i}\in T_{i}$ with
$\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$. We say that $E$
is certainly believed under $\beta_{i}\left(t_{i}\right)$ if
$\mu_{i}^{l}\left(E\right)=1$ for all $l\leq n$. In other words, $E$ is
certainly believed under $\beta_{i}\left(t_{i}\right)$ if its complement is
deemed subjectively impossible by the player (see Appendix A for a preference-
based foundation).

Certain belief satisfies monotonicity: If $E$ is certainly believed under
$\beta_{i}\left(t_{i}\right)$ and $F$ is an event such that $E\subseteq F$,
then $F$ is certainly believed under $\beta_{i}\left(t_{i}\right)$.
Furthermore, certain belief is not a stronger concept than cautious belief. To
see this, consider a type $t_{i}\in T_{i}$ such that
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})\notin\mathcal{N}^{+}\left(S_{-i}\right)$.
Event $S_{-i}\times T_{-i}$ is certainly believed under
$\beta_{i}\left(t_{i}\right)$, but it is not cautiously believed because
$t_{i}$ is not cautious (see Remark 2). Yet, it is immediate to check that,
for cautious types, certain belief implies cautious belief.

##  5 Epistemic analysis

###  5.1 Epistemic analysis of IA

In what follows, we fix a finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in
I}\right\rangle$. Given an associated type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$, for each player $i\in I$, we let
$R_{i}^{1}:=R_{i}\cap C_{i}$ denote the set of cautiously rational strategy-
type pairs. Let $\mathbf{B}_{i}^{c}:\Sigma_{S_{-i}\times
T_{-i}}\rightarrow\Sigma_{S_{i}\times T_{i}}$ be the operator defined by

$\mathbf{B}_{i}^{c}\left(E_{-i}\right):=\left\\{\left(s_{i},t_{i}\right)\in
S_{i}\times T_{i}:t_{i}\text{ cautiously believes }E_{-i}\right\\}\text{,
}E_{-i}\in\Sigma_{S_{-i}\times T_{-i}}\text{.}$

Corollary C.1 in Appendix C shows that the set
$\mathbf{B}_{i}^{c}\left(E_{-i}\right)$ is Borel in $S_{i}\times T_{i}$ if
$E_{-i}\subseteq S_{-i}\times T_{-i}$ is an event; so the operator
$\mathbf{B}_{i}^{c}$ is well-defined.

For each $m\geq 1$, define $R_{i}^{m+1}$ recursively by

$R_{i}^{m+1}:=R_{i}^{m}\cap\mathbf{B}_{i}^{c}\left(R_{-i}^{m}\right)\text{,}$

where $R_{-i}^{m}:=\prod_{j\neq i}R_{j}^{m}$. Note that

$R_{i}^{m+1}=R_{i}^{1}\cap\left(\mathop{\textstyle\bigcap}_{l\leq
m}\mathbf{B}_{i}^{c}\left(R_{-i}^{l}\right)\right)\text{,}$

and each $R_{i}^{m}$ is Borel in $S_{i}\times T_{i}$ (see Lemma C.2 in
Appendix C).

We write $R_{i}^{\infty}:=\cap_{m\in\mathbb{N}}R_{i}^{m}$ for each $i\in I$.
If $\left(s_{i},t_{i}\right)_{i\in I}\in\mathop{\textstyle\prod}_{i\in
I}R_{i}^{m+1}$, we say that there is cautious rationality and $m$th-order
cautious belief in cautious rationality
(R${}^{\text{c}}m$B${}^{\text{c}}$R${}^{\text{c}}$) at this state. If
$\left(s_{i},t_{i}\right)_{i\in I}\in\mathop{\textstyle\prod}_{i\in
I}R_{i}^{\infty}$, we say that there is cautious rationality and common
cautious belief in cautious rationality
(R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$) at this state.

With this, we state the first main result of this paper.

###### Theorem 1

Fix a type structure $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$ which is terminal with
respect to the class of all finite type structures. Then:

(i)

for each $m\geq 1$, $\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,m}\right)=\prod_{i\in I}S_{i}^{m}$;

(ii)

$\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,\infty}\right)=\prod_{i\in
I}S_{i}^{\infty}$.

We point out that type structure $\mathcal{T}^{\ast}$ in Theorem 1 exists. In
particular, there exists a _universal_ type structure for LPS’s, that is, a
type structure which is terminal and for which the type morphism from every
other type structure is unique. Lee (2016b) shows the existence of a universal
type structure for a wide class of preferences, which includes those
represented by LPS’s. Yang (2015) and Catonini and De Vito (2018) construct
the canonical type structure for hierarchies of lexicographic beliefs;
Catonini and De Vito also show that this type structure is universal. Since
the canonical type structure is continuous and belief-complete, it follows
from Theorem 1 that R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ is
possible in a continuous, belief-complete type structure.

The idea of Theorem 1 stems from Theorem 3 in Friedenberg and Keisler (2020),
a result concerning epistemic foundations for iterated strict dominance in the
context of ordinary type structures. In the spirit of such result, Theorem 1
identifies a “richness” property of the type structure that depends on its
ability to capture sufficiently many hierarchies of beliefs (specifically, all
those induced by finite type structures).

The proof of Theorem 1, like the proof of Theorem 3 in Friedenberg and Keisler
(2020), is based on the following “embedding” argument. We first construct a
finite type structure $\mathcal{T}$ such that, for every $m\geq 1$, the
behavioral implications of R${}^{\text{c}}m$B${}^{\text{c}}$R${}^{\text{c}}$
are characterized by the set of $m$-admissible strategy profiles. Then, by the
terminality property of $\mathcal{T}^{\ast}$, we map $\mathcal{T}$ in
$\mathcal{T}^{\ast}$ via type morphism. While doing so, we show that cautious
rationality and all orders of belief in cautious rationality are preserved.
For this, we need the next three preparatory results. First, we need to show
the existence of $\mathcal{T}$.

###### Lemma 1

There exists a finite type structure $\mathcal{T}:=\left\langle
S_{i},T_{i},\beta_{i}\right\rangle_{i\in I}$ such that, for each $i\in I$ and
each $m\geq 1$, $\mathrm{Proj}_{S_{i}}\left(R_{i}^{m}\right)=S_{i}^{m}$.

The proof of Lemma 1 is in Appendix C. Here we just mention that in the finite
type structure we construct for Lemma 1 all types are cautious. This fact will
be used below (Section 5.3).

Second, we need to claim the invariance of cautious rationality under type
morphisms.

###### Lemma 2

Fix type structures $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$ and $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$. Suppose that there
exists a type morphism $\left(\varphi_{i}\right)_{i\in I}:T\rightarrow
T^{\ast}$ from $\mathcal{T}$ to $\mathcal{T}^{\ast}$, and fix a strategy-type
pair $\left(s_{i},t_{i}\right)\in S_{i}\times T_{i}$. Then:

(i) $t_{i}$ is cautious in $\mathcal{T}$ if and only if
$\varphi_{i}\left(t_{i}\right)$ is cautious in $\mathcal{T}^{\ast}$;

(ii) $\left(s_{i},t_{i}\right)$ is rational in $\mathcal{T}$ if and only if
$\left(s_{i},\varphi_{i}\left(t_{i}\right)\right)$ is rational in
$\mathcal{T}^{\ast}$.

The proof of Lemma 2 can be found in Catonini and De Vito (2020, Fact C.1).

Third, we need an analogous invariance property for cautious belief.

###### Lemma 3

Fix type structures $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$ and $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$. Suppose that there
exists a bimeasurable type morphism $\left(\varphi_{i}\right)_{i\in
I}:T\rightarrow T^{\ast}$ from $\mathcal{T}$ to $\mathcal{T}^{\ast}$. If a
type $t_{i}\in T_{i}$ cautiously believes event $E_{-i}\subseteq S_{-i}\times
T_{-i}$, then $\varphi_{i}(t_{i})$ cautiously believes
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)$.

The proof of Lemma 3 is in Appendix C. For our purpose, it is crucial to
observe that, by Remark 3, if $\varphi_{i}(t_{i})$ cautiously believes
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)$, then
$\varphi_{i}(t_{i})$ cautiously believes also every Borel superset
$E_{-i}^{\ast}$ such that
$\mathrm{Proj}_{S_{-i}}\left(E_{-i}^{\ast}\right)=\mathrm{Proj}_{S_{-i}}\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)$.

Finally, for the proof of Theorem 1, we find it convenient to single out the
following fact, whose proof is immediate.

###### Remark 4

Fix type structures $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$ and $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$. Suppose that there
exists a type morphism $\left(\varphi_{i}\right)_{i\in I}:T\rightarrow
T^{\ast}$ from $\mathcal{T}$ to $\mathcal{T}^{\ast}$. Then, for every
$E_{i}\subseteq S_{i}\times T_{i}$,

$\mathrm{Proj}_{S_{i}}\left(\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(E_{i})\right)=\mathrm{Proj}_{S_{i}}(E_{i})\text{.}$

With this, we are ready to prove Theorem 1.

Proof of Theorem 1. By Lemma 1, there is a finite type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ such that
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{m}\right)=S_{i}^{m}$ for each $m\geq 1$ and
for each $i\in I$.

Part (i): Fix a type morphism $(\varphi_{i})_{i\in I}:T\rightarrow T^{\ast}$
from $\mathcal{T}$ to $\mathcal{T}^{\ast}$. Structure $\mathcal{T}$ is finite,
so $(\varphi_{i})_{i\in I}$ is bimeasurable. We show by induction on $m\geq 1$
that $\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{m})\subseteq
R_{i}^{\ast,m}$ and
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,m}\right)=S_{i}^{m}$ for each $i\in
I$.

($m=1$) Fix $i\in I$. It is immediate from Lemma 2 that
$\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{1})\subseteq
R_{i}^{\ast,1}$. By Remark 4,
$\mathrm{Proj}_{S_{i}}\left(\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{1})\right)=\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)$,
and since $\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)=S_{i}^{1}$, we obtain
$S_{i}^{1}\subseteq\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,1}\right)$.
Conversely, Proposition 1 entails
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,1}\right)\subseteq S_{i}^{1}$.
Therefore, $\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,1}\right)=S_{i}^{1}$.

($m>1$) Fix $i\in I$ and $(s_{i},t_{i})\in R_{i}^{m}$. We want to show that
$(s_{i},\varphi_{i}(t_{i}))\in R_{i}^{\ast,m}$. Since $R_{i}^{m}\subseteq
R_{i}^{m-1}$, the induction hypothesis yields $(s_{i},\varphi_{i}(t_{i}))\in
R_{i}^{\ast,m-1}$. Hence, it suffices to show that $\varphi_{i}(t_{i})$
cautiously believes $R_{-i}^{\ast,m-1}$. Since $t_{i}$ cautiously believes
$R_{-i}^{m-1}$ and the type morphism $(\varphi_{i})_{i\in I}$ is bimeasurable,
it follows from Lemma 3 that $\varphi_{i}(t_{i})$ cautiously believes
$(\mathrm{Id}_{S_{-i}},\varphi_{-i})(R_{-i}^{m-1})$. Note that

$\displaystyle\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{\ast,m-1}\right)$
$\displaystyle=$ $\displaystyle S_{-i}^{m-1}$ $\displaystyle=$
$\displaystyle\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{m-1}\right)$
$\displaystyle=$
$\displaystyle\mathrm{Proj}_{S_{-i}}\left((\mathrm{Id}_{S_{-i}},\varphi_{-i})(R_{-i}^{m-1})\right)\text{,}$

where the first equality is the induction hypothesis, the second equality
follows from the property of $\mathcal{T}$, and the third equality follows
from Remark 4. We also know from the induction hypothesis that
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)(R_{-i}^{m-1})\subseteq
R_{-i}^{\ast,m-1}$; thus, Remark 3 allows us to conclude that
$\varphi_{i}(t_{i})$ cautiously believes $R_{-i}^{\ast,m-1}$.

So, we have shown that
$\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{m})\subseteq
R_{i}^{\ast,m}$. By the property of $\mathcal{T}$ and Remark 4, we obtain

$\displaystyle S_{i}^{m}$ $\displaystyle=$
$\displaystyle\mathrm{Proj}_{S_{i}}\left(R_{i}^{m}\right)$ $\displaystyle=$
$\displaystyle\mathrm{Proj}_{S_{i}}\left(\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{m})\right)$
$\displaystyle\subseteq$
$\displaystyle\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,m}\right)\text{.}$

To show the opposite inclusion, fix $(s_{i},t_{i}^{\ast})\in R_{i}^{\ast,m}$.
Since $R_{i}^{\ast,m}\subseteq R_{i}^{\ast,m-1}$, it follows from the
induction hypothesis that $s_{i}\in S_{i}^{m-1}$. Let
$\beta_{i}^{\ast}(t_{i}^{\ast}):=\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)$.
Since $t_{i}^{\ast}$ cautiously believes $R_{-i}^{\ast,m-1}$ at some level
$l$, it follows from Proposition 2.2 and the induction hypothesis that

$\mathop{\textstyle\bigcup}_{k\leq
l}\mathrm{Suppmarg}_{S_{-i}}\mu_{i}^{k}=S_{-i}^{m-1}\text{.}$

So, by Proposition 1 in Blume et al. (1991b), we can form a nested convex
combination of the measures $\mathrm{marg}_{S_{-i}}\mu_{i}^{k}$, for
$k=1,...,l$, to get a probability measure
$\nu_{i}\in\mathcal{M}\left(S_{-i}\right)$, with
$\mathrm{Supp}\nu_{i}=S_{-i}^{m-1}$, under which $s_{i}$ is optimal. Thus, by
Remark 1, $s_{i}$ is admissible with respect to $S_{i}\times S_{-i}^{m-1}$,
and a fortiori with respect to $S_{i}^{m-1}\times S_{-i}^{m-1}$. Hence,
$s_{i}\in S_{i}^{m}$.

Part (ii): Fix $i\in I$. Since $(R_{i}^{m})_{m\in\mathbb{N}}$ and
$(S_{i}^{m})_{m\in\mathbb{N}}$ are weakly decreasing sequences of finite, non-
empty sets, there exists $N\in\mathbb{N}$ such that $R_{i}^{N}=R_{i}^{\infty}$
and $S_{i}^{N}=S_{i}^{\infty}$. Lemma 1 implies
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=S_{i}^{\infty}$. Hence, for
every $s_{i}\in S_{i}^{\infty}$, there exists $t_{i}\in T_{i}$ such that
$(s_{i},t_{i})\in R_{i}^{m}$ for every $m\in\mathbb{N}$. We have shown in the
proof of Part (i) that, for every $m\in\mathbb{N}$,
$\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)(R_{i}^{m})\subseteq
R_{i}^{\ast,m}$. So
$\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)((s_{i},t_{i}))\in
R_{i}^{\ast,m}$ for every $m\in\mathbb{N}$, which implies that
$\left(\mathrm{Id}_{S_{i}},\varphi_{i}\right)((s_{i},t_{i}))\in
R_{i}^{\ast,\infty}$. Therefore,
$S_{i}^{\infty}\subseteq\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,\infty}\right)$.
Conversely, Part (i) entails
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,N}\right)=S_{i}^{N}=S_{i}^{\infty}$.
Hence $\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,\infty}\right)\subseteq
S_{i}^{\infty}$. We conclude that
$S_{i}^{\infty}=\mathrm{Proj}_{S_{i}}\left(R_{i}^{\ast,\infty}\right)$.
$\blacksquare$

###  5.2 Epistemic analysis of SAS’s

Fix a finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in I}\right\rangle$.
The following result states that, for every type structure associated with
game $G$, the behavioral implications of
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ constitute an SAS.
Conversely, every SAS corresponds to the behavioral implications of
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ in some type structure.

###### Theorem 2

(i)

Fix a type structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$. Then $\mathop{\textstyle\prod}_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)$ is an SAS.

(ii)

Fix an SAS $Q\in\mathcal{Q}$. There exists a finite type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ such that, for
each $i\in I$,

$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=Q_{i}\text{.}$

The idea of Theorem 2 stems from Theorem 8.1 in BFK, a result concerning the
epistemic justification of SAS’s with rationality and common assumption of
rationality. The proof of Theorem 2, which is similar to that in BFK, is in
Appendix C. Here we point out that in the finite type structure we construct
for Theorem 2.(ii) all types are cautious. We have already mentioned (Section
5.1) that the same property holds for the finite type structure we construct
for Lemma 1. This raises the question whether one could incorporate the
cautiousness assumption in the definition of type structures for the epistemic
analysis of IA and SAS’s. That is, could we restrict attention to the class
$\mathbb{T}$ of type structures where each type’s belief over the opponents’
strategies have full support? Are there analogues of Theorems 1 and 2 for such
a class? We explore this issue in the next section.

###  5.3 Alternative epistemic conditions for IA and SAS’s

Fix a finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in I}\right\rangle$
and an associated type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. Say that type $t_{i}$ certainly
believes a non-empty event $E_{-i}\subseteq S_{-i}\times T_{-i}$ if $E_{-i}$
is certainly believed under $\beta_{i}(t_{i})$. For each player $i\in I$, let
$\mathbf{B}_{i}:\Sigma_{S_{-i}\times T_{-i}}\rightarrow\Sigma_{S_{i}\times
T_{i}}$ be the operator defined by

$\mathbf{B}_{i}\left(E_{-i}\right):=\left\\{\left(s_{i},t_{i}\right)\in
S_{i}\times T_{i}:t_{i}\text{ certainly believes }E_{-i}\right\\}\text{,
}E_{-i}\in\Sigma_{S_{-i}\times T_{-i}}\text{.}$

As shown in Catonini and De Vito (2020), the set
$\mathbf{B}_{i}\left(E_{-i}\right)$ is Borel in $S_{i}\times T_{i}$ if
$E_{-i}\subseteq S_{-i}\times T_{-i}$ is an event; thus, the operator
$\mathbf{B}_{i}$ is well-defined.

Next, fix a non-empty event $E_{i}\subseteq S_{i}\times T_{i}$ for every $i\in
I$. Event $E:=\mathop{\textstyle\prod}_{i\in I}E_{i}$ is self-evident (in
$\mathcal{T}$) if it satisfies $E\subseteq\mathop{\textstyle\prod}_{i\in
I}\mathbf{B}_{i}\left(E_{-i}\right)$; standard results (e.g., Catonini and De
Vito 2018, Appendix 5.2) show that $E$ is self-evident if and only if at every
state $\left(s_{i},t_{i}\right)_{i\in I}\in E$ there is common certain belief
in $E$. With this, we say that there is transparency of $E$ at state
$\left(s_{i},t_{i}\right)_{i\in I}$ if $\left(s_{i},t_{i}\right)_{i\in I}\in
E$ and $E$ is self-evident.

We let $C^{\infty}\subseteq\mathop{\textstyle\prod}\nolimits_{i\in
I}S_{i}\times T_{i}$ denote the event corresponding to transparency of
cautiousness in $\mathcal{T}$, and we let $C_{i}^{\infty}$ denote the
corresponding projection on $S_{i}\times T_{i}$.151515Since the set $C_{i}$ is
a Borel subset of $S_{i}\times T_{i}$ (Catonini and De Vito 2020, Corollary
D.1), an argument by induction on the iteration of the operator
$\mathbf{B}_{i}$ shows that $C_{i}^{\infty}$ is also a Borel subset (event) of
$S_{i}\times T_{i}$. In particular, if $\mathcal{T}$ is the canonical type
structure, then it is possible to show that $C_{i}^{\infty}$ is a Polish
subset of $S_{i}\times T_{i}$. With this, for each $i\in I$, let
$\hat{R}_{i}^{1}:=R_{i}\cap C_{i}^{\infty}$. Then, for each $i\in I$ and
$m\geq 1$, define $\hat{R}_{i}^{m+1}$ recursively by

$\hat{R}_{i}^{m+1}:=\hat{R}_{i}^{m}\cap\mathbf{B}_{i}^{c}\left(\hat{R}_{-i}^{m}\right)\text{,}$

where $\hat{R}_{-i}^{m}:=\mathop{\textstyle\prod}_{j\neq i}\hat{R}_{j}^{m}$.
Write $\hat{R}_{i}^{\infty}:=\cap_{m\in\mathbb{N}}\hat{R}_{i}^{m}$ for each
$i\in I$. Therefore, if $\left(s_{i},t_{i}\right)_{i\in
I}\in\mathop{\textstyle\prod}_{i\in I}\hat{R}_{i}^{\infty}$, we say that at
state $\left(s_{i},t_{i}\right)_{i\in I}$ there is (a) rationality and
transparency of cautiousness, and (b) common cautious belief in (a).

Say that $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ is a
cautious type structure if $C_{i}=S_{i}\times T_{i}$ for every $i\in I$. Since
event $S\times T:=\mathop{\textstyle\prod}\nolimits_{i\in I}\left(S_{i}\times
T_{i}\right)$ is self-evident, it is easily seen that $C^{\infty}=S\times T$
in a cautious type structure. Therefore, if $\mathcal{T}$ is a cautious type
structure, then, for each $i\in I$,

$\hat{R}_{i}^{1}=R_{i}^{1}=R_{i}\text{;}$

that is, event $\mathop{\textstyle\prod}_{i\in I}\hat{R}_{i}^{1}$ is the set
of states where there is rationality. It follows by induction on $m\geq 1$
that $\hat{R}_{i}^{m}=R_{i}^{m}$ for each $i\in I$. So, if
$\left(s_{i},t_{i}\right)_{i\in I}\in\mathop{\textstyle\prod}_{i\in
I}\hat{R}_{i}^{m+1}$, we can say that there is rationality and $m$th-order
cautious belief in rationality at this state.

We are now in a position to state a characterization result, which is an
analogue of Theorem 1.

###### Theorem 3

(i)

Fix a type structure $\mathcal{T}^{\ast}:=\langle
S_{i},T_{i}^{\ast},\beta_{i}^{\ast}\rangle_{i\in I}$ which is terminal with
respect to the class of all finite type structures. Then:

(i.1)

for each $m\geq 1$, $\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\ast,m}\right)=\mathop{\textstyle\prod}_{i\in
I}S_{i}^{m}$;

(i.2)

$\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\ast,\infty}\right)=\mathop{\textstyle\prod}_{i\in
I}S_{i}^{\infty}$.

(ii)

The same conclusions as in (i.1) and (i.2) hold if $\mathcal{T}^{\ast}$ is a
cautious type structure which is terminal with respect to the class of all
finite, cautious type structures.

To see why Part (i) of Theorem 3 holds, let $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ be the finite type structure we
construct for the proof of Lemma 1. Since $\mathcal{T}$ is a cautious type
structure, $\hat{R}_{i}^{m}=R_{i}^{m}$ for each $i\in I$ and $m\geq 1$.
Finiteness of $\mathcal{T}$ guarantees that the type morphism $\varphi$ from
$\mathcal{T}$ to $\mathcal{T}^{\ast}$ is bimeasurable. Lemma 2.(i) entails
that $S\times\varphi(T)\subseteq C^{\ast}$; that is, the image of self-evident
event $C^{\infty}$ in $\mathcal{T}$ under type morphism $\varphi$ is a subset
of $C^{\ast}$. Proposition 12 in Catonini and De Vito (2018) shows that event
$S\times\varphi(T)$ is self-evident in $\mathcal{T}^{\ast}$.161616Lemma A3 in
Battigalli and Friedenberg (2012b) shows an analogous result for type
structures with beliefs represented by conditional probability systems. But
then, by the monotonicity property of certain belief, it is immediate to see
that at every state in $S\times\varphi(T)$ there is common certain belief in
$C^{\ast}$. We therefore conclude that $S\times\varphi(T)\subseteq
C^{\ast,\infty}$. With this, the proof of Theorem 3 is the same as that of
Theorem 1—just replace sets such as $R_{i}^{\ast,m}$ and $R_{-i}^{\ast,m}$
with the corresponding sets $\hat{R}_{i}^{\ast,m}$ and
$\hat{R}_{-i}^{\ast,m}$.

Theorem 3.(ii) is a characterization result for $m$-admissible (resp.
iteratively admissible) strategies in terms of rationality and $m$th-order
(resp. common) cautious belief in rationality. To see why Theorem 3.(ii)
holds, we first point out that a terminal, cautious type structure exists. Let
$\mathcal{T}^{U}$ be the canonical, universal type structure. By Proposition
12 in Catonini and De Vito (2018), the self-evident event $C^{U,\infty}$
identifies a “smaller,” cautious type structure $\mathcal{T}^{\ast}$. In such
a structure, $\hat{R}_{i}^{\ast,m}=R_{i}^{\ast,m}$ for each $i\in I$ and
$m\geq 1$. By the above argument, $\mathcal{T}^{\ast}$ is terminal with
respect to the class of finite, cautious type structures. With this, the proof
of Theorem 1 yields the result.

The following characterization result, pertaining to SAS’s, is an analogue of
Theorem 2.

###### Theorem 4

(i)

Fix a type structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in
I}$. Then $\mathop{\textstyle\prod}_{i\in
I}\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\infty}\right)$ is an SAS.

(ii)

Fix an SAS $Q\in\mathcal{Q}$. There exists a finite, cautious type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ such that, for
each $i\in I$,

$\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\infty}\right)=Q_{i}\text{.}$

The proof of Theorem 4.(i) is, with some minor modifications, identical to the
proof of Theorem 2.(i). Part (ii) of Theorem 4 is essentially Theorem 2.(ii).
As noted, in the proof of Theorem 2.(ii) we construct a finite, cautious type
structure $\mathcal{T}$ such that $\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=Q$. In such a type
structure, $\hat{R}_{i}^{\infty}=R_{i}^{\infty}$ for each $i\in I$.

Theorem 4 allows us to obtain an alternative characterization of SAS’s.
Specifically, if we restrict attention to the class of cautious type
structures, we can restate the results of Theorem 4 as follows: SAS’s
characterize the behavioral implications of rationality and common cautious
belief in rationality across all cautious type structures.

Some final remarks on the epistemic assumptions considered so far are in
order. First, events $\prod_{i\in I}R_{i}^{\infty}$ and $\prod_{i\in
I}\hat{R}_{i}^{\infty}$ are equivalent in cautious type structures, but they
are typically different in non-cautious type structures. In the Supplementary
Appendix we exhibit an example where the game in Example 1 is associated with
a finite, non-cautious type structure $\mathcal{T}$ such that $\prod_{i\in
I}R_{i}^{\infty}$ and $\prod_{i\in I}\hat{R}_{i}^{\infty}$ are disjoint. The
example shows that the behavioral implications of such epistemic assumptions
are characterized by two disjoint SAS’s—specifically, the sets
$\left\\{m\right\\}\times\left\\{\ell\right\\}$ and
$\left\\{u\right\\}\times\left\\{r\right\\}$, respectively.

To understand this point, fix an arbitrary type structure $\mathcal{T}$
associated with a finite game. Let us consider the first two steps in the
definitions of $\prod_{i\in I}R_{i}^{\infty}$ and $\prod_{i\in
I}\hat{R}_{i}^{\infty}$. At the first step, we have $\hat{R}_{i}^{1}\subseteq
R_{i}^{1}$ for each $i\in I$, because transparency of cautiousness implies
cautiousness. Does an analogous conclusion hold for the sets $\hat{R}_{i}^{2}$
and $R_{i}^{2}$ ($i\in I$)? The answer is no. Recall that, for each $i\in I$,

$\displaystyle\hat{R}_{i}^{2}$ $\displaystyle:$
$\displaystyle=\hat{R}_{i}^{1}\cap\mathbf{B}_{i}^{c}\left(\hat{R}_{-i}^{1}\right)\text{,}$
$\displaystyle R_{i}^{2}$ $\displaystyle:$
$\displaystyle=R_{i}^{1}\cap\mathbf{B}_{i}^{c}\left(R_{-i}^{1}\right)\text{.}$

Since cautious belief is not monotonic (see Section 4.2), for some player
$i\in I$ we can have
$\mathbf{B}_{i}^{c}\left(\hat{R}_{-i}^{1}\right)\nsubseteq\mathbf{B}_{i}^{c}\left(R_{-i}^{1}\right)$
even if $\hat{R}_{-i}^{1}\subseteq R_{-i}^{1}$; this is illustrated by the
example in the Supplementary Appendix. Yet, it should be noted that
$\prod_{i\in I}\hat{R}_{i}^{2}\subseteq\prod_{i\in I}R_{i}^{2}$ holds whenever
$\prod_{i\in I}\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{1}\right)=\prod_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)$. Indeed, under such condition,
the “quasi-monotonicity” property of cautious belief (Remark 3) entails that
$\mathbf{B}_{i}^{c}\left(\hat{R}_{-i}^{1}\right)\subseteq\mathbf{B}_{i}^{c}\left(R_{-i}^{1}\right)$
for each $i\in I$.

In the Supplementary Appendix we show that, if $\mathcal{T}$ is a “rich” type
structure, then

$\mathop{\textstyle\prod}_{i\in
I}\hat{R}_{i}^{\infty}\subseteq\mathop{\textstyle\prod}_{i\in
I}R_{i}^{\infty}\text{ \ \ and \ \ }\mathop{\textstyle\prod}_{i\in
I}\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\infty}\right)=\mathop{\textstyle\prod}_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)\text{.}$

In words, if structure $\mathcal{T}$ is “rich,” then the epistemic assumption
of “rationality, transparency of cautiousness, and common cautious belief in
both” is stronger than R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$;
nonetheless, they are equivalent in terms of behavioral implications.

As one should expect, examples of “rich” type structures satisfying the
aforementioned property are terminal structures such as $\mathcal{T}^{\ast}$
in the statement of Theorem 1. Other examples are belief-complete type
structures (Definition 5): this is formally shown in the Supplementary
Appendix, whose proof makes explicit use of the “quasi-monotonicity” property
of cautious belief. Thus, if $\mathcal{T}$ is belief-complete, we can
conclude, by Theorem 2, that the behavioral implications of both epistemic
assumptions are characterized by one specific SAS. But this SAS could be
different from the IA set, as the following section illustrates.

###  5.4 Belief-completeness vs terminality

Theorem 1 identifies a “richness” condition on type structures for the
epistemic justification of IA. A related “richness” condition is belief-
completeness (Definition 5): a belief-complete type structure induces all
possible beliefs about types. In light of this, one might conjecture that the
conclusions of Theorem 1 continue to hold for belief-complete type structures.
However, this is not the case.

Call a finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in I}\right\rangle$
non-degenerate if $\left|S_{i}\right|\geq 2$ for each $i\in I$. The following
result states that, for each non-degenerate, finite game, there exists a
continuous, belief-complete type structure where
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ is not possible at any
state. It follows that, in such a structure, the behavioral implications of
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ constitute the empty SAS
(cf. Theorem 2.(i)).

###### Theorem 5

Fix a non-degenerate finite game $G:=\left\langle I,(S_{i},\pi_{i})_{i\in
I}\right\rangle$. There exists a continuous, belief-complete type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ such that

$\mathop{\textstyle\prod}\nolimits_{i\in I}R_{i}^{\infty}=\emptyset\text{.}$

Theorem 5 is inspired by Friedenberg and Keisler (2021, Theorem 1).
Specifically, Friedenberg and Keisler consider finite games satisfying a non-
triviality condition that is stronger than non-degeneracy. With this, they
show the existence of a belief-complete, ordinary type structure in which
there is no state consistent with rationality and common belief in
rationality.

The proof of Theorem 5 adapts the arguments in Friedenberg and Keisler (2021)
to the lexicographic framework, and it can be found in Appendix C. Here, we
briefly explain why Theorem 5 holds.

Fix a belief-complete type structure $\mathcal{T}$. In such a structure,
$\prod_{i\in I}R_{i}^{m}\neq\emptyset$ for each $m\geq 1$. In particular, it
can be shown that, for each $m\geq 1$, the behavioral implications of
R${}^{\text{c}}m$B${}^{\text{c}}$R${}^{\text{c}}$ are characterized by the set
of $m$-admissible strategy profiles. The reason why the set of states
$\prod_{i\in I}R_{i}^{\infty}$ can be empty is—conceptually—the same as in
Friedenberg and Keisler (2021): While a belief-complete (lexicographic) type
structure induces all beliefs about types, it need not induce all possible
hierarchies of beliefs. Specifically, Theorems 1 and 5 imply that a belief-
complete type structure may not induce all hierarchies of beliefs that can
arise in finite type structures.

In the context of ordinary type structures, Friedenberg (2010, Theorem 3.1)
shows that a belief-complete type structure is terminal if each type space is
compact and each belief map is continuous. An analogue of Friedenberg’s result
does not exist in the lexicographic framework: as already remarked (see
Section 3.2), a belief-complete, lexicographic type structure cannot be
compact and continuous. This explains why the terminality property of the type
structure is made explicit in the statements (and proofs) of our results
(Theorems 1 and 3).

##  6 Discussion

Given the previous analysis, it is now possible to discuss in detail a set of
conceptual issues, some of which were informally addressed in the
Introduction.

###  6.1 IA and lexicographic rationalizability

In the Introduction, we have informally claimed that
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ matches very closely the
logic of lexicographic rationalizability (Stahl 1995), an iterated elimination
procedure for lexicographic beliefs. Here we make this informal claim precise.

Fix a player $i\in I$ and a non-empty set $Q_{-i}\subseteq S_{-i}$. We let
$r_{i}(\bar{\mu}_{i})$ denote the set of player $i$’s strategies which are
optimal under $\bar{\mu}_{i}\in\mathcal{N}(S_{-i})$, and

$\mathcal{B}_{c}^{+}(Q_{-i}):=\left\\{\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)\in\mathcal{N}^{+}(S_{-i}):\exists
m\leq
n,\mathop{\textstyle\bigcup}_{l=1}^{m}\mathrm{Supp}\mu_{i}^{l}=Q_{-i}\right\\}$

is the set of all full-support LPS’s $\bar{\mu}_{i}$ such that $Q_{-i}$ is
“cautiously believed” under $\bar{\mu}_{i}$ (cf. Proposition 2.2). In words,
$\mathcal{B}_{c}^{+}(Q_{-i})$ is the set of all full-support first-order
beliefs under which player $i$ deems _every_ co-players’ strategy profile in
$Q_{-i}$ infinitely more likely—in the sense of Lo (1999) or Stahl (1995)—than
every profile in $S_{-i}\backslash Q_{-i}$. Note: if $Q_{-i}^{\prime}\subseteq
S_{-i}$ is such that $Q_{-i}\subseteq Q_{-i}^{\prime}$, then it is not
necessarily true that
$\mathcal{B}_{c}^{+}(Q_{-i})\subseteq\mathcal{B}_{c}^{+}(Q_{-i}^{\prime})$.
Specifically,
$\mathcal{B}_{c}^{+}(Q_{-i})\nsubseteq\mathcal{B}_{c}^{+}(Q_{-i}^{\prime})$
whenever there exists $s_{-i}\in Q_{-i}^{\prime}\backslash Q_{-i}$ which is
not deemed—under some $\bar{\mu}_{i}\in\mathcal{B}_{c}^{+}(Q_{-i})$—infinitely
more likely than every profile in $S_{-i}\backslash Q_{-i}^{\prime}$ (cf.
Section 4.2 and Example 2). Furthermore, note that
$\mathcal{B}_{c}^{+}(S_{-i})=\mathcal{N}^{+}(S_{-i})$.

With this, we can define a sequence $(\hat{S}^{m})_{m\geq 0}$ of subsets of
$S$ as follows. For each $i\in I$, let $\hat{S}_{i}^{0}:=S_{i}$. Also, let
$\hat{S}^{0}:=\mathop{\textstyle\prod}_{i\in I}\hat{S}_{i}^{0}$ and
$\hat{S}_{-i}^{0}:=\mathop{\textstyle\prod}_{j\neq i}\hat{S}_{j}^{0}$.
Recursively define, for $m\geq 1$,

$\displaystyle\hat{S}_{i}^{m}$ $\displaystyle:$
$\displaystyle=\left\\{s_{i}\in
S_{i}:\exists\bar{\mu}_{i}\in\mathop{\textstyle\bigcap}_{l=0}^{m-1}\mathcal{B}_{c}^{+}(\hat{S}_{-i}^{l}),s_{i}\in
r_{i}(\bar{\mu}_{i})\right\\}\text{,}$ $\displaystyle\hat{S}^{m}$
$\displaystyle:$ $\displaystyle=\mathop{\textstyle\prod}_{i\in
I}\hat{S}_{i}^{m}\text{.}$

Profiles in $\hat{S}^{\infty}:=\cap_{m=0}^{\infty}\hat{S}^{m}$ are called
lexicographic rationalizable. Stahl (1995) shows that $S^{m}=\hat{S}^{m}$ for
every $m\in\mathbb{N}$, that is, the set of $m$-admissible strategies
coincides with the set of strategies surviving the first $m$ steps of the
procedure.

Similarly, SAS’s can be given a characterization in terms of justifiability
(“best reply to some belief”). As shown by De Vito (2023), a set
$Q\in\mathcal{Q}$ is an SAS if and only if, for each $i\in I$ and each
$s_{i}\in Q_{i}$, there exists $\bar{\mu}_{i}\in\mathcal{B}_{c}^{+}(Q_{-i})$
such that $s_{i}\in r_{i}(\bar{\mu}_{i})$ and $r_{i}(\bar{\mu}_{i})\subseteq
Q_{i}$. Such characterization of SAS is similar to the definition of
“extensive-form best response set” (EFBRS, Battigalli and Friedenberg 2012a).
Of course, SAS’s and EFBRS’s are distinct concepts. In particular, EFBRS’s are
defined in terms of conditional probability systems and strong belief
(Battigalli and Siniscalchi 2002).171717The recent paper by Brandenburger et
al. (2023) establishes a relationship between strong belief and BFK’s notion
of assumption. Examples of the difference between SAS’s and EFBRS’s can be
found in Battigalli and Friedenberg (2012b, Section 8.c).

###  6.2 Comparison to (weak) assumption

In Catonini (2013) and Yang (2015), an event $E\subseteq S_{-i}\times T_{-i}$
is “weakly assumed” under an LCPS $\bar{\mu}:=(\mu^{1},...,\mu^{n})$ if there
exists $m\leq n$ such that

(i)

$\mu^{l}\left(E\right)=1$ for all $l\leq m$,

(ii)

$\mu^{l}\left(E\right)=0$ for all $l>m$,

(iii)

for every elementary cylinder $\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times
T_{-i}$, if $E\cap\hat{C}_{s_{-i}}\not=\emptyset$ then
$\mu^{l}\left(E\cap\hat{C}_{s_{-i}}\right)>0$ for some $l\leq m$.

The difference between weak assumption and BFK’s assumption relies on
condition (iii): BFK require that, for every open set $O\subseteq S_{-i}\times
T_{-i}$, if $E\cap O\not=\emptyset$ then $\mu^{l}\left(E\cap O\right)>0$ for
some $l\leq m$. BFK’s assumption is stronger than weak assumption because,
technically, every elementary cylinder is an open set. The definition of weak
assumption can be extended to all LPS’s while preserving its preference-based
foundation—in the same way Dekel et al. (2016) extend BFK’s assumption—as
follows.181818Such extension can be formally shown by slightly adapting the
proofs in Dekel et al. (2016). The preference-based definition of BFK’s
assumption is based on two axioms: Strict Determination and Nontriviality.
Similarly, weak assumption requires Strict Determination and a weaker axiom
than Nontriviality. Say that event $E\subseteq S_{-i}\times T_{-i}$ is weakly
assumed under LPS $\bar{\mu}:=(\mu^{1},...,\mu^{n})$ if there exists $m\leq n$
such that conditions (i) and (iii) above hold, and

(ii)’

for each $l>m$, there exists
$\left(\alpha_{1}^{l},...,\alpha_{m}^{l}\right)\in\mathbb{R}^{m}$ such that
$\mu^{l}\left(F\right)=\mathop{\textstyle\sum}_{k=1}^{m}\alpha_{k}^{l}\mu^{k}\left(F\right)$
for each Borel set $F\subseteq E$.

Cautious belief requires only conditions (i) and (iii). Thus, weak assumption
implies cautious belief, but the converse does not hold. We show this by means
of an example, which is taken from BFK (cf. Dekel and Siniscalchi 2015,
Example 12.10).

###### Example 3

Consider the following game with two players, Ann ($a$) and Bob ($b$):

$a\backslash b$ | $\ell$ | $c$ | $r$
---|---|---|---
$u$ | $4,0$ | $4,1$ | $0,1$
$m$ | $0,0$ | $0,1$ | $4,1$
$d$ | $3,0$ | $2,1$ | $2,1$

The IA set is $S^{\infty}=\left\\{u,m,d\right\\}\times\left\\{c,r\right\\}$.
By Theorem 4.(ii), we can append to this game a _cautious_ , finite type
structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in\left\\{a,b\right\\}}$ such that
$S^{\infty}=\mathrm{Proj}_{S_{a}}\left(R_{a}^{\infty}\right)\times\mathrm{Proj}_{S_{b}}\left(R_{b}^{\infty}\right)$.
Consider any type $t_{a}$ of Ann such that

(1) strategy $d$ is optimal under
$\beta_{a}(t_{a}):=(\mu_{a}^{1},...,\mu_{a}^{n})$, and

(2) $t_{a}$ cautiously believes Bob’s (cautious) rationality.

We show that the event corresponding to Bob’s rationality, viz. $R_{b}^{1}$,
cannot be weakly assumed under $\beta_{a}(t_{a})$. To ease notation, let
$E_{s_{b}}:=\left\\{s_{b}\right\\}\times T_{b}$ for each
$s_{b}\in\left\\{\ell,c,r\right\\}$. There is no rational strategy-type pair
of Bob in $E_{\ell}$; hence, $R_{b}^{1}=E_{c}\cup E_{r}$. As $t_{a}$ is
cautious, there exists $k\leq n$ such that
$\mathrm{marg}_{S_{b}}\mu_{a}^{k}\left(\left\\{\ell\right\\}\right)=\mu_{a}^{k}(E_{\ell})>0$.
Let $k^{\ast}:=\inf\left\\{k\leq n:\mu_{a}^{k}(E_{\ell})>0\right\\}$. Since
$R_{b}^{1}$ is cautiously believed under $\beta_{a}(t_{a})$, it is the case
that $\mu_{a}^{1}\left(R_{b}^{1}\right)=1$, which implies
$\mu_{a}^{1}\left(E_{\ell}\right)=0$. This in turn yields $k^{\ast}\geq 2$.

Next note that, for every $\nu\in\mathcal{M}(S_{b})$ such that
$\mathrm{Supp}\nu\subseteq\left\\{c,r\right\\}$, strategy $d$ is optimal under
$\nu$ if and only if $\nu(\left\\{c\right\\})=\nu(\left\\{r\right\\})=1/2$;
this entails that also $u$ and $m$ are optimal under $\nu$. Hence we must have
$\mu_{a}^{l}(E_{c})=\mu_{a}^{l}(E_{r})=1/2$ for every $l=1,...,k^{\ast}-1$.
But $d$ must be optimal also under $\mu_{a}^{k^{\ast}}$, the first component
measure of $\beta_{a}(t_{a})$ which assigns strictly positive probability to
$E_{\ell}$. It follows that $\mu^{k^{\ast}}(E_{\ell})>0$ and
$\mu_{a}^{k^{\ast}}(E_{r})>\mu_{a}^{k^{\ast}}(E_{c})$.191919Let
$\nu\in\mathcal{M}(S_{b})$ be the marginal of $\mu_{a}^{k^{\ast}}$ on $S_{b}$,
so that $\nu(\left\\{\ell\right\\})>0$. We need
$\nu(\left\\{r\right\\})>\nu(\left\\{c\right\\})$ for strategy $d$ to be
optimal under $\nu$. For, if
$\nu(\left\\{r\right\\})\leq\nu(\left\\{c\right\\})$, then $u$ would be the
unique best reply to $\nu$. Furthermore,
$\mu_{a}^{k^{\ast}}\left(R_{b}^{1}\right)<1$ (for, if
$\mu_{a}^{k^{\ast}}\left(R_{b}^{1}\right)=1$, we would have
$\mu^{k^{\ast}}(E_{\ell})=0$). With this, we conclude that conditions (i) and
(iii) of weak assumption are satisfied for event $R_{b}^{1}$ at level
$m:=k^{\ast}-1$ of LPS $\beta_{a}(t_{a})$.

Yet, condition (ii)’ of weak assumption does not hold. To see this, note that
$E_{c}$ and $E_{r}$ are Borel subsets of $R_{b}^{1}$, and
$\mu_{a}^{m+1}\left(E_{r}\right)>\mu_{a}^{m+1}\left(E_{c}\right)$. Suppose,
per contra, that condition (ii)’ is satisfied. Then, there exists
$\left(\alpha_{1},...,\alpha_{m}\right)\in\mathbb{R}^{m}$ such that

$\mu_{a}^{m+1}\left(E_{r}\right)=\mathop{\textstyle\sum}_{l=1}^{m}\alpha_{l}\mu_{a}^{l}\left(E_{r}\right)=\tfrac{1}{2}\mathop{\textstyle\sum}_{l=1}^{m}\alpha_{l}=\mathop{\textstyle\sum}_{l=1}^{m}\alpha_{l}\mu_{a}^{l}\left(E_{c}\right)=\mu_{a}^{m+1}\left(E_{c}\right)\text{,}$

which contradicts
$\mu_{a}^{m+1}\left(E_{r}\right)>\mu_{a}^{m+1}\left(E_{c}\right)$. We
therefore conclude that every rational strategy-type pair
$\left(d,t_{a}\right)$ is not consistent with weak assumption of $R_{b}^{1}$.
$\blacklozenge$

Example 3 shows that weak assumption can be strictly stronger than cautious
belief. Furthermore, the example illustrates the difference between our
approach to IA and the one based on weak assumption. To clarify, suppose that
R${}^{\text{c}}$CB${}^{\text{c}}$R${}^{\text{c}}$ is replaced by the epistemic
notion of “cautious rationality and common weak assumption of cautious
rationality” (R${}^{\text{c}}$CA${}^{\text{w}}$R${}^{\text{c}}$). Are there
analogues of Theorems 1 and 2 under
R${}^{\text{c}}$CA${}^{\text{w}}$R${}^{\text{c}}$? Not surprisingly, the
answer is Yes.202020A proof is available upon request.

Yet, under weak assumption, it is not possible to provide epistemic
foundations for IA if we restrict attention to cautious type structures. Refer
back to the type structure of Example 3. In such a structure, events $E_{c}$
and $E_{r}$ are proper subsets of $R_{b}^{1}$, and condition (ii)’ of weak
assumption fails. This is not necessarily true _if the type structure contains
both cautious and non-cautious types_. Indeed, it is possible to construct a
non-cautious type structure whereby event $R_{b}^{1}:=R_{b}\cap C_{b}$ (“Bob’s
cautious rationality”) is such that $R_{b}^{1}\nsubseteq E_{c}\cup E_{r}$, and
any cautiously rational pair $\left(d,t_{a}\right)$ is consistent with weak
assumption of $R_{b}^{1}$.212121For instance, consider LPS
$\beta_{a}(t_{a}):=(\mu_{a}^{1},...,\mu_{a}^{n})$ such that
$\mu_{a}^{l}\left(R_{b}\right)=1$ for each $l\leq m$, and
$\mu_{a}^{l}\left(C_{b}\right)=0$ for each $l>m$. In this case, condition
(ii)’ of weak assumption is satisfied, because $\mu_{a}^{m+1},...,\mu_{a}^{n}$
assign zero probability to Bob’s cautious rationality (in particular, we have
$\mu_{a}^{m+1}\left(E_{r}\right)>0$ and
$\mu_{a}^{m+1}\left(R_{b}^{1}\right)=0$).

We spend a few words on the origin of this difference. Cautious belief is
based on the notion of “infinitely more likely” of Lo (1999), which can be
expressed in words as follows: event $E$ is deemed infinitely more likely than
event $F$ if the agent prefers to bet on $E$ rather than on $F$ no matter the
(different) winning prizes for the two bets. According to this notion of
“infinitely more likely,” $\beta_{a}(t_{a}^{d})$ deems every element of
$R_{b}^{1}$ infinitely more likely than the complement of $R_{b}^{1}$, that
is, $\left\\{(\ell,t_{b}^{\ast})\right\\}$, which is sufficient for cautious
belief (and implies that $R_{b}^{1}$ is deemed infinitely more likely than
$\left\\{(\ell,t_{b}^{\ast})\right\\}$—this notion of “infinitely more likely”
satisfies disjunction).

(Weak) Assumption is instead based on the notion of “infinitely more likely”
of Blume et al. (1991a). According to this notion, $\beta_{a}(t_{a}^{d})$
deems every element of $R_{b}^{1}$, i.e., $\left\\{(c,t_{b}^{\ast})\right\\}$
and $\left\\{(r,t_{b}^{\ast})\right\\}$, infinitely more likely than
$\left\\{(\ell,t_{b}^{\ast})\right\\}$.222222The notions of “infinitely more
likely” due to Lo (1999) and Blume et al. (1991a) coincide when the events
under consideration are singletons; see Blume et al. (1991a, Footnote 8).
However (and this is the key point!), such notion of “infinitely more likely”
does not satisfy disjunction: in Example 3,
$\left\\{(c,t_{b}^{\ast})\right\\}$ and $\left\\{(r,t_{b}^{\ast})\right\\}$
are both deemed under $\beta_{a}(t_{a}^{d})$ infinitely more likely than
$\left\\{(\ell,t_{b}^{\ast})\right\\}$, but their union ($R_{b}^{1}$) is not
deemed infinitely more likely than $\left\\{(\ell,t_{b}^{\ast})\right\\}$.
Blume et al. (1991a, p. 70) provide a similar example which shows how their
notion of “infinitely more likely” fails disjunction.

We conclude this section with an example of a game and an associated, “rich”
type structure such that: (1) the set of states consistent with
R${}^{\text{c}}$CA${}^{\text{w}}$R${}^{\text{c}}$ is distinct from the set of
states consistent with our epistemic assumptions, but (2) the behavioral
implications are the same, i.e., the IA set. Thus, the example shows that our
epistemic conditions for IA are different from those previously studied.

###### Example 4

Refer back the the game of Example 3. Let $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in\left\\{a,b\right\\}}$ be the canonical
structure associated with this game. First note that, in such a structure, the
set of states consistent with RCAR is empty. Next recall that, according to
our notation, $\hat{R}_{a}^{\infty}\times\hat{R}_{b}^{\infty}$ is the set of
states consistent with (a) rationality, (b) transparency of cautiousness, and
(c) common cautious belief in (a) and (b). Similarly, let
$\bar{R}_{a}^{\infty}\times\bar{R}_{b}^{\infty}$ be the set of states
consistent with R${}^{\text{c}}$CA${}^{\text{w}}$R${}^{\text{c}}$. As argued
above, and by Theorems 1 and 3, it is the case that, for every
$i\in\left\\{a,b\right\\}$,

$S_{i}^{\infty}=\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=\mathrm{Proj}_{S_{i}}\left(\hat{R}_{i}^{\infty}\right)=\mathrm{Proj}_{S_{i}}\left(\bar{R}_{i}^{\infty}\right)\text{.}$

Yet, we claim that

(1) $\hat{R}_{a}^{\infty}\times\hat{R}_{b}^{\infty}\subseteq
R_{a}^{\infty}\times R_{b}^{\infty}$, and

(2) any strategy-type pair $\left(d,t_{a}\right)\in\hat{R}_{a}^{\infty}$ of
Ann is such that $\left(d,t_{a}\right)\notin\bar{R}_{a}^{\infty}$.

With this, (1) and (2) entail that both
$\hat{R}_{a}^{\infty}\times\hat{R}_{b}^{\infty}$ and $R_{a}^{\infty}\times
R_{b}^{\infty}$ are distinct from
$\bar{R}_{a}^{\infty}\times\bar{R}_{b}^{\infty}$.

The set inclusion (which is strict) in (1) follows from belief-completeness of
$\mathcal{T}$ and from the “quasi-monotonicity” property of cautious
belief—see Remark 3 and the discussion below Theorem 4.

To prove claim (2), pick any $\left(d,t_{a}\right)\in\hat{R}_{a}^{\infty}$, so
that strategy $d$ is optimal under
$\beta_{a}(t_{a}):=(\mu_{a}^{1},...,\mu_{a}^{n})$. In view of claim (1), Ann’s
type $t_{a}$ cautiously believes the event corresponding to Bob’s cautious
rationality, viz. $R_{b}^{1}$. We now show that $R_{b}^{1}$ cannot be weakly
assumed under $\beta_{a}(t_{a})$. To this end, let
$E_{b}:=S_{b}\times\hat{T}_{b}$ denote the set of strategy-type pairs of Bob
consistent with transparency of cautiousness (i.e., according to the notation
in Section 5.3, $E_{b}$ is the projection onto $S_{b}\times T_{b}$ of the
event $C^{\infty}$ in $\mathcal{T}$), so that

$\mu_{a}^{1}\left(E_{b}\right)=...=\mu_{a}^{n}\left(E_{b}\right)=1\text{.}$

Proceeding as in Example 3, define
$E_{s_{b}}:=\left\\{s_{b}\right\\}\times\hat{T}_{b}$ for each
$s_{b}\in\left\\{\ell,C,R\right\\}$. There is no (cautiously) rational
strategy-type pair of Bob in $E_{\ell}$: we have $\hat{R}_{b}^{1}=E_{c}\cup
E_{r}\subseteq R_{b}^{1}$. Then, the same argument as in Example 3 shows that
$R_{b}^{1}$ cannot be weakly assumed under $\beta_{a}(t_{a})$. Hence, every
strategy-type pair $\left(D,t_{a}\right)\in\hat{R}_{a}^{\infty}$ is not
consistent with weak assumption of $R_{b}^{1}$. $\blacklozenge$

###  6.3 Further comments on the related literature

Other articles with epistemic conditions for IA include Keisler and Lee
(2023), Lee (2016a), Heifetz et al. (2019), Halpern and Pass (2019), and, in
non-lexicographic frameworks, Barelli and Galanis (2013) and Ziegler and
Zuazo-Garin (2020).

Keisler and Lee (2023) construct a _discontinuous_ and complete type structure
where RCAR is possible. Furthermore, they show that such type structure
generates the same set of belief hierarchies as a continuous one. An immediate
implication of this findings is that BFK’s results hinge on topological
details of the type structure that cannot be expressed in terms of belief
hierarchies. Keisler and Lee conclude that BFK’s negative result stems from
the fact that, in a continuous type structure, players are “too cautious”
towards the assumed events. Lee (2016a) relaxes the traditional coherency
condition on belief hierarchies while maintaining coherency of the represented
preferences. With this, he identifies hierarchies of lexicographic beliefs
without an upper bound on the length of the LPS’s that cannot be represented
by any type structure but capture “rationality” and common assumption of
“rationality”. The notion of “rationality” used by Lee is essentially
equivalent to cautious rationality.232323In a companion paper (Catonini and De
Vito 2017), we show that our results can be replicated in the hierarchical
space studied by Lee (2016a). Heifetz et al. (2019) put forward the solution
concept of comprehensive rationalizability, and they give it an epistemic
foundation in a universal type structure for LCPS’s. Comprehensive
rationalizability neither refines nor is refined by IA, but it coincides with
IA in many applications. Halpern and Pass (2019) use instead a modal-logic
framework to provide an epistemic characterization of IA. They put forward a
notion of “generalized belief” which is given semantics in terms of LPS’s.
Their characterization of IA relies on a corresponding operator (“All I know”)
which is taken with respect to an appropriate language.

Within a standard Bayesian decision model, Barelli and Galanis (2013) use the
idea that each player has a list of preferences which allows her to break
ties. With this, they provide an epistemic foundation for IA in an appropriate
framework for interactive beliefs. Ziegler and Zuazo-Garin (2020) use instead
a decision model of incomplete, but continuous preferences where each player’s
uncertainty is represented by a set of beliefs. They provide foundations for
IA and SAS’s in terms of interactive, ambiguous beliefs, rather than LPS’s.
Both Barelli and Galanis (2013) and Ziegler and Zuazo-Garin (2020) can be
regarded as complementary to the LPS-based approach.

## Appendix A. Preference basis

We develop preference foundations for cautiousness, certain belief and
cautious belief. In so doing, we adopt the following decision-theoretic setup.
A game form is a structure $\left\langle I,Z,\left(S_{i}\right)_{i\in
I},z\right\rangle$ where (a) $I$ is the finite set of players, (b) each
$S_{i}$ is the finite set of strategies, and (c) $z:S\rightarrow Z$ is a
surjective outcome function (hence, the set $Z$ is finite). Each player is
viewed as a Decision Maker (DM) facing a problem where his co-players’
strategies are part of the description of the states, and mixed strategies are
the feasible acts. So, fix an $\left(S_{i}\right)_{i\in I}$-based
lexicographic type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. We fix a player $i\in I$ (the DM),
and, to ease notation, we set $\Omega:=S_{-i}\times T_{-i}$. With this, the DM
is uncertain about what “state” (strategy-type profiles of the co-players) in
$\Omega$ will be realized, and he is endowed with a preference relation over
all (Borel) measurable functions that assign to each element of $\Omega$ an
objective randomization on $Z$. In Blume et al. (1991a), an act is a function
from an abstract, finite domain of uncertainty $\Omega$ to
$\mathcal{M}\left(Z\right)$.

A game is obtained by adding to the game form a profile of von Neumann-
Morgenstern utility functions $\left(v_{i}\right)_{i\in I}$, which represent
players’ preferences over lotteries of consequences, according to expected
utility calculations. In what follows, we assume that the codomain of any act
on $\Omega$ is in utils, i.e., randomizations on material consequences are
replaced by their von Neumann-Morgenstern utilities, which take value in the
interval $\left[0,1\right]$. Thus, an act on $\Omega$ is a Borel measurable
function $f:\Omega\rightarrow\left[0,1\right]$.242424We omit the formalism for
randomizations as all preferences considered below agree on constant acts on
$Z$, hence the utilities are uniquely defined. Moreover, the definition of act
used here is also used in BFK and Dekel et al. (2016). We let
ACT$\left(\Omega\right)$ denote the set of all acts on $\Omega$.

The DM has preferences over elements of ACT$\left(\Omega\right)$. For
$x\in\left[0,1\right]$, we write $\overrightarrow{x}$ for the constant act
associated with $x$, i.e., $\overrightarrow{x}(\omega):=x$ for all
$\omega\in\Omega$. Given a Borel set $E\subseteq\Omega$ and acts
$f,g\in$ACT$\left(\Omega\right)$, we define $(f_{E},g_{\Omega\backslash
E})\in$ACT$\left(\Omega\right)$ as follows:

$(f_{E},g_{\Omega\backslash
E})(\omega):=\left\\{\begin{tabular}[]{ll}$f(\omega)$,&if $\omega\in
E\text{,}$\\\ $g(\omega)$,&if $\omega\in\Omega\backslash
E\text{.}$\end{tabular}\right.$

Let $\succsim$ be a preference relation on ACT$\left(\Omega\right)$ and write
$\succ$ (resp. $\sim$) for strict preference (resp. indifference). We assume
that preference relation $\succsim$ satisfies the standard axioms of Order and
Independence (see BFK for a formal definition). Thus, we let $\succsim_{E}$
denote the _conditional preference_ given $E$, that is, $f\succsim_{E}g$ if
and only if $(f_{E},h_{\Omega\backslash E})\succsim(g_{E},h_{\Omega\backslash
E})$ for some $h\in$ACT$\left(\Omega\right)$. Standard results (see Blume et
al., 1991a, for a proof) show that, under the axioms of Order and
Independence, $(f_{E},h_{\Omega\backslash
E})\succsim(g_{E},h_{\Omega\backslash E})$ holds for all
$h\in$ACT$\left(\Omega\right)$ if it holds for some $h$.

An event $E\subseteq\Omega$ is Savage-null under $\succsim$ if $f\sim_{E}g$
for all $f,g\in$ACT$\left(\Omega\right)$. Say that $E$ is non-null under
$\succsim$ if it is not Savage-null under $\succsim$. With this, we can
introduce the notion of certain belief in terms of the preference relation
$\succsim$.

Definition A.1 Event $E\subseteq\Omega$ is certainly believed under $\succsim$
if $f\sim_{\Omega\backslash E}g$ for all $f,g\in$ACT$\left(\Omega\right)$.

Throughout, we maintain the assumption that
$\bar{\mu}\in\mathcal{N}\left(\Omega\right)$ is a lexicographic expected
utility representation of $\succsim$, i.e.,
$\succsim=\succsim^{\bar{\mu}}$.252525To ease notation, we drop player $i$’s
subscript from LPS $\bar{\mu}_{i}$ on $\Omega$. Savage-null events and certain
belief can be characterized in terms of LPS’s as follows.

Proposition A.1 Fix $\bar{\mu}:=(\mu^{1},...,\mu^{n})\in\mathcal{N}(\Omega)$
and event $E\subseteq\Omega$. Then:

(i) $E$ is Savage-null under $\succsim^{\bar{\mu}}$ if and only if
$\mu^{m}\left(E\right)=0$ for all $m\leq n$;

(ii) $E$ is certainly believed under $\succsim^{\bar{\mu}}$ if and only if it
is certainly believed under $\bar{\mu}$.

The proof of Part (i) of Proposition A.1 is quite immediate, and it can be
found in Dekel et al. (2016, Remark 2.1). Part (ii) follows from Part (i).

The following definition is due to Catonini and De Vito (2020).

Definition A.2 Fix $\bar{\mu}\in\mathcal{N}(\Omega)$ and a set of acts
ACT${}^{\ast}\left(\Omega\right)\subseteq$ACT$\left(\Omega\right)$. Say that
$\bar{\mu}$ exhibits cautiousness with respect to
ACT${}^{\ast}\left(\Omega\right)$ if, for all
$f,g\in$ACT${}^{\ast}\left(\Omega\right)$, the following condition holds:

(*) if $f(\omega)\geq g(\omega)$ for each $\omega\in\Omega$ and
$f(\omega^{\prime})>g(\omega^{\prime})$ for some $\omega^{\prime}\in\Omega$,
then $f\succ^{\bar{\mu}}g$.

Cautiousness is defined with respect to a set of acts that are conceivable
given the potential ability of the states to influence utilities. Since the DM
is a player $i$ in a game, and the domain of uncertainty is
$\Omega:=S_{-i}\times T_{-i}$, we find it appropriate to consider
ACT${}^{\ast}\left(\Omega\right)$ as the set of acts
$f\in$ACT$\left(\Omega\right)$ such that, for all $s_{-i}\in S_{-i}$ , the map
$f\left(s_{-i},\cdot\right):T_{-i}\rightarrow\left[0,1\right]$ is constant. We
let ACT${}^{S_{-i}}\left(\Omega\right)$ denote this set of acts. In words,
ACT${}^{S_{-i}}\left(\Omega\right)$ is the set of all acts which are
independent of payoff-irrelevant components of states
$(s_{-i},t_{-i})\in\Omega$, i.e., the types of $i$’s co-players. Note: every
mixed strategy $\sigma_{i}\in\mathcal{M}\left(S_{i}\right)$ in a game can be
identified with the (feasible) act $f_{\sigma_{i}}:S_{-i}\times
T_{-i}\rightarrow\left[0,1\right]$ such that
$f(s_{-i},t_{-i})=v_{i}\left(\sigma_{i},s_{-i}\right)$ for all
$(s_{-i},t_{-i})\in\Omega$; hence,
$f_{\sigma_{i}}\in$ACT${}^{S_{-i}}\left(\Omega\right)$.

The following result, which is proved in Catonini and De Vito (2020), provides
the preference-based foundation for the type-based definition of cautiousness
(Definition 9).

Proposition A.2 Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ and a type $t_{i}\in T_{i}$. Then
$t_{i}$ is cautious in $\mathcal{T}$ if and only if $\beta_{i}(t_{i})$
exhibits cautiousness with respect to ACT${}^{S_{-i}}\left(S_{-i}\times
T_{-i}\right)$.

A bet (or binary act) on $\Omega$ is an act of the form
$(\overrightarrow{x}_{E},\overrightarrow{y}_{\Omega\backslash E})$, where
$x,y\in\left[0,1\right]$ and $E\subseteq\Omega$ is an event.

Definition A.3 Fix events $E,F\subseteq\Omega$. Event $E$ is more likely than
$F$ under $\succsim^{\bar{\mu}}$ if for all $x,y\in\left[0,1\right]$ with
$x>y$,

$(\overrightarrow{x}_{E},\overrightarrow{y}_{\Omega\backslash
E})\succsim^{\bar{\mu}}(\overrightarrow{x}_{F},\overrightarrow{y}_{\Omega\backslash
F})\text{.}$

Event $E$ is deemed infinitely more likely than $F$ under
$\succsim^{\bar{\mu}}$, and write $E\gg^{\bar{\mu}}F$, if for all
$x,y,z\in\left[0,1\right]$ with $x>y$,

$(\overrightarrow{x}_{E},\overrightarrow{y}_{\Omega\backslash
E})\succ^{\bar{\mu}}(\overrightarrow{z}_{F},\overrightarrow{y}_{\Omega\backslash
F})\text{.}$

In words, $E$ is more likely than $F$ if the DM prefers to bet on $E$ rather
than on $F$ given the same prizes for the two bets. Event $E$ is infinitely
more likely than $F$ if betting on $E$ is strictly preferable to betting on
$F$, and strict preference persists no matter how bigger the prize $z$ for
winning the $F$ bet is. This notion of “infinitely more likely” is due to Lo
(1999, Definition 1). Note that, if $E\gg^{\bar{\mu}}F$, then $E$ is non-null
under $\succsim^{\bar{\mu}}$, while $F$ may, but need not, be Savage-null
under $\succsim^{\bar{\mu}}$. When $\succsim^{\bar{\mu}}$ has a subjective
expected utility representation, $E\gg^{\bar{\mu}}F$ implies that $F$ is
Savage-null.

As pointed out by Lo (1999), the likelihood relation $\gg^{\bar{\mu}}$
possesses some natural properties, such as irreflexivity, asymmetry and
transitivity. Furthermore, if $E\gg^{\bar{\mu}}F$, then

(P1)

$E$ is infinitely more likely than every Borel subset of $F$; and

(P2)

every Borel superset of $E$ is infinitely more likely than $F$.

The next step is to characterize the likelihood order $\gg^{\bar{\mu}}$
between pairwise disjoint events in terms of LPS’s representing
$\succsim^{\bar{\mu}}$ (see Definition 11 of Section 4.2). Recall that, given
$\bar{\mu}:=(\mu^{1},...,\mu^{n})\in\mathcal{N}(\Omega)$ and non-empty event
$E\subseteq\Omega$,

$\mathcal{I}_{\bar{\mu}}\left(E\right):=\inf\left\\{l\in\left\\{1,...,n\right\\}:\mu^{l}\left(E\right)>0\right\\}\text{,}$

with the convention that $\inf\emptyset:=+\infty$. The proof of the following
result can be found in Catonini and De Vito (2020).

Proposition A.3 Fix $\bar{\mu}:=(\mu^{1},...,\mu^{n})\in\mathcal{N}(\Omega)$
and disjoint events $E,F\subseteq\Omega$. Then, $E\gg^{\bar{\mu}}F$ if and
only if
$\mathcal{I}_{\bar{\mu}}\left(E\right)<\mathcal{I}_{\bar{\mu}}\left(F\right)$.

We now introduce the notion of cautious belief in terms of the likelihood
order $\gg^{\bar{\mu}}$. Recall that $\hat{C}_{s_{-i}}\subseteq\Omega$ is
called elementary cylinder if $\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times
T_{-i}$ for some $s_{-i}\in S_{-i}$. Given $s_{-i}$ and event $E$, we say that
$E_{s_{-i}}$ is a relevant part of the event $E$ if
$E_{s_{-i}}:=E\cap\hat{C}_{s_{-i}}\not=\emptyset$ for some $\hat{C}_{s_{-i}}$.
Clearly, every non-empty event $E$ can be written as a finite, disjoint union
of all its relevant parts.

Definition A.4 Fix $\bar{\mu}\in\mathcal{N}(\Omega)$. A non-empty event
$E\subseteq\Omega$ is cautiously believed under $\succsim^{\bar{\mu}}$ if it
satisfies the following condition:

(*) for every relevant part $E_{s_{-i}}$ of $E$,
$E_{s_{-i}}\gg^{\bar{\mu}}\Omega\backslash E$.

In words, event $E$ is cautiously believed under $\succsim^{\bar{\mu}}$ if
every relevant part of $E$ is deemed infinitely more likely than
$\Omega\backslash E$. Since $E$ can be written as a finite, disjoint union of
all its relevant parts, it follows from (P2) that $E$ is deemed infinitely
more likely than $\Omega\backslash E$, i.e., $E\gg^{\bar{\mu}}\Omega\backslash
E$.

However, the converse need not hold. That is, if
$E\gg^{\bar{\mu}}\Omega\backslash E$, then $E$ is non-null under
$\succsim^{\bar{\mu}}$, and there exists at least one relevant part
$E_{s_{-i}}$ of $E$ such that $E_{s_{-i}}\gg^{\bar{\mu}}\Omega\backslash E$.
But this does not rule out the existence of different relevant parts of $E$
that do not satisfy this property.

Example A.1 Refer back to Example 2 in Section 4.2. Event
$F:=\left\\{s_{b}^{1},s_{b}^{2}\right\\}\times T_{b}$ is infinitely more
likely under $\bar{\mu}_{a}$ than its complement
$\left\\{s_{b}^{3}\right\\}\times T_{b}$. Yet $F$ is not cautiously believed
under $\succsim^{\bar{\mu}_{a}}$: the relevant part
$F_{s_{b}^{2}}:=\left\\{s_{b}^{2}\right\\}\times T_{b}$ is more likely than
$\left\\{s_{b}^{3}\right\\}\times T_{b}$, but $F_{s_{b}^{2}}$ is not
infinitely more likely than $\left\\{s_{b}^{3}\right\\}\times T_{b}$.
$\blacklozenge$

We say that event $E\subseteq\Omega$ is weakly believed under
$\succsim^{\bar{\mu}}$ if $E\gg^{\bar{\mu}}\Omega\backslash E$. Event $F$ in
Example A.1 is weakly, but not cautiously believed under
$\succsim^{\bar{\mu}_{a}}$.

We next state and prove the characterization result for cautious belief. For
the reader’s convenience, we restate the LPS-based definition of cautious
belief given in the main text, but in terms of relevant parts.

Definition A.5 Fix $\bar{\mu}:=(\mu^{1},...,\mu^{n})\in\mathcal{N}(\Omega)$. A
non-empty event $E\subseteq\Omega$ is cautiously believed under $\bar{\mu}$ at
level $m\leq n$ if:

(i) $\mu^{l}\left(E\right)=1$ for all $l\leq m$;

(ii) for every relevant part $E_{s_{-i}}$ of $E$,
$\mu^{l}\left(E_{s_{-i}}\right)>0$ for some $l\leq m$.

Event $E$ is cautiously believed under $\bar{\mu}$ if it is cautiously
believed under $\bar{\mu}$ at some level $m\leq n$.

Theorem A.1 Fix $\bar{\mu}:=(\mu^{1},...,\mu^{n})\in\mathcal{N}(\Omega)$ and a
non-empty event $E\subseteq\Omega$. Then $E$ is cautiously believed under
$\succsim^{\bar{\mu}}$ if and only if $E$ is cautiously believed under
$\bar{\mu}$.

Proof. The proof is immediate if $\Omega\backslash E$ is Savage-null under
$\succsim^{\bar{\mu}}$, so, in what follows, let $\Omega\backslash E$ be non-
null under $\succsim^{\bar{\mu}}$.

Suppose first that $E$ is cautiously believed under $\succsim^{\bar{\mu}}$.
Since every relevant part $E_{s_{-i}}$ of $E$ satisfies
$E_{s_{-i}}\gg^{\bar{\mu}}\Omega\backslash E$, Proposition A.3 yields
$\mathcal{I}_{\bar{\mu}}\left(E_{s_{-i}}\right)<\mathcal{I}_{\bar{\mu}}\left(\Omega\backslash
E\right)$. Hence, $\mathcal{I}_{\bar{\mu}}\left(\Omega\backslash E\right)\geq
2$. Let $m:=\mathcal{I}_{\bar{\mu}}\left(\Omega\backslash E\right)-1$. Then
$\mathcal{I}_{\bar{\mu}}\left(E_{s_{-i}}\right)\leq m$. Moreover, for every
$k\leq m$, we have $\mu^{k}(\Omega\backslash E)=0$, hence $\mu^{k}(E)=1$.
Therefore conditions (i)-(ii) of Definition A.5 are satisfied.

Conversely, if $E$ is cautiously believed under $\bar{\mu}$ at level $m$, then
condition (i) of Definition A.5 implies
$\mathcal{I}_{\bar{\mu}}\left(\Omega\backslash E\right)>m$. With this,
condition (ii) yields that each $E_{s_{-i}}$ satisfies
$\mathcal{I}_{\bar{\mu}}\left(E_{s_{-i}}\right)$
$<\mathcal{I}_{\bar{\mu}}\left(\Omega\backslash E\right)$. Hence, by
Proposition A.3, $E_{s_{-i}}\gg^{\bar{\mu}}\Omega\backslash E$. $\blacksquare$

Cautious belief in $E$ can be given an alternative axiomatic treatment. The
Supplementary Appendix proposes two axioms: Relevance says that, conditional
on every relevant part of $E$, the DM can have strict preferences. Weak
Dominance Determination says that, for any pair of acts
$f,g\in$ACT${}^{S_{-i}}\left(\Omega\right)$, whenever $f$ “weakly dominates”
$g$ on $E$, the DM prefers $f$ to $g$ unconditionally. This notion of weak
dominance is preference-based, as it corresponds to the notion of P-weak
dominance of Dekel et al. (2016) for acts in
ACT${}^{S_{-i}}\left(\Omega\right)$.262626P-weak dominance is defined by Dekel
et al. (2016) for any pair of acts belonging to the set
ACT$\left(\Omega\right)$ (the set of all acts).

We conclude this section by providing a characterization of cautious belief in
terms of infinitesimal nonstandard numbers. A preference relation $\succsim$
on $\Omega$ that admits a lexicographic expected utility representation can be
equivalently described by an $\mathbb{F}$-valued probability measure on
$\Omega$. Here, $\mathbb{F}$ is a non-Archimedean ordered field which is a
strict extension of the set of real numbers $\mathbb{R}$ (see Blume et al.
1991a, Section 6). For instance, the LPS $\bar{\mu}:=(\mu^{1},\mu^{2})$ can be
represented by a nonstandard real valued probability
$\nu:=(1-\varepsilon)\mu^{1}+\varepsilon\mu^{2}$, where $\varepsilon>0$ is an
infinitesimal nonstandard real such that $x>n\varepsilon$ for each real number
$x>0$ and each $n\in\mathbb{N}$.

Given nonstandard reals $x$ and $y$, we say that $x$ is infinitely greater
than $y$ if $x>ny$ for each $n\in\mathbb{N}$. As discussed in Catonini and De
Vito (2020), the notion of infinitely more likely in Definition 11 corresponds
exactly to the “infinitely greater” relation between the nonstandard
probability values that provide an equivalent representation of preferences.
With this in mind, we show that cautious belief can be given an easy,
nonstandard characterization.

To this end, we first recall the notion of “standard part” of a nonstandard
real number. Fix a nonstandard real $x$ such that $-r<x<r$ for some real
number $r>0$. The standard part of $x$, which is denoted by
$\mathrm{st}\left(x\right)$, is the unique real number $y$ such that
$\left|y-x\right|$ is an infinitesimal. It is easy to check that, given
positive nonstandard numbers $x$ and $y$, if $x$ is infinitely greater than
$y$, then $\mathrm{st}\left(\frac{y}{x}\right)=0$; the reverse implication is
also true—see Halpern (2010, p. 159). Next, fix a non-empty event
$E\subseteq\Omega$ and an $\mathbb{F}$-valued probability measure $\nu$
representing $\succsim$. Event $E$ is cautiously believed under $\nu$ if, for
every relevant part $E_{s_{-i}}$ of $E$, it is the case that
$\mathrm{st}\left(\frac{\nu\left(\Omega\backslash
E\right)}{\nu\left(E_{s_{-i}}\right)}\right)=0$. Finally note that, as each
$\nu\left(E_{s_{-i}}\right)$ is infinitely greater than
$\nu\left(\Omega\backslash E\right)$, so is $\nu\left(E\right)$. This in turn
implies $\mathrm{st}\left(\nu\left(E\right)\right)=1$, i.e., event $E$ is
weakly believed under $\nu$ (see Halpern 2010, and Catonini and De Vito 2020).

## Appendix B. Proofs for Section 4

We begin with the proof of Proposition 1.

Proof of Proposition 1. By definition, if $(s_{i},t_{i})\in R_{i}\cap C_{i}$,
then $s_{i}$ is a lexicographic best reply to
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})\in\mathcal{N}^{+}(S_{-i})$.
Proposition 1 in Blume et al. (1991b) says that for every
$\bar{\mu}_{i}\in\mathcal{N}^{+}\left(S_{-i}\right)$ and for every
lexicographic best reply $s_{i}^{\prime}$ to $\bar{\mu}_{i}$, there exists a
probability measure $\nu_{i}\in\mathcal{M}\left(S_{-i}\right)$ such that
$\mathrm{Supp}\nu_{i}=S_{-i}$ and
$\pi_{i}(s_{i}^{\prime},\nu_{i})\geq\pi_{i}(s_{i}^{\prime\prime},\nu_{i})$ for
every $s_{i}^{\prime\prime}\in S_{i}$. Thus, by Remark 1, $s_{i}$ is
admissible. $\blacksquare$

Next, we prove Proposition 2. To this end, we find it convenient to state and
prove an auxiliary result, which is the analogue of Lemma B.1 in BFK.

Lemma B.1 Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. Fix also a type $t_{i}\in T_{i}$ with
$\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$ and a non-empty
event $E\subseteq S_{-i}\times T_{-i}$. Then, $E$ is cautiously believed under
$\beta_{i}\left(t_{i}\right)$ if and only if there exists $m\leq n$ such that
$\beta_{i}\left(t_{i}\right)$ satisfies condition (i) of Definition 12 plus
the following condition:

(ii”) $E\subseteq\left(\cup_{l\leq
m}\mathrm{Suppmarg}_{S_{-i}}\mu_{i}^{l}\right)\times T_{-i}$.

Proof. Suppose that $E$ is cautiously believed under
$\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$ at level $m$. We
show that $\beta_{i}\left(t_{i}\right)$ satisfies condition (ii”). For every
$s_{-i}\in$Proj${}_{S_{-i}}\left(E\right)$, we have

$\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)\cap E\neq\emptyset\text{.}$

By condition (ii) of Definition 12, there exists $k\leq m$ such that
$\mu_{i}^{k}\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)>0$. Thus,
$s_{-i}\in\mathrm{Suppmarg}_{S_{-i}}\mu_{i}^{k}$. Hence,

$E\subseteq\mathrm{Proj}_{S_{-i}}\left(E\right)\times
T_{-i}\subseteq\left(\mathop{\textstyle\bigcup}_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\right)\times T_{-i}\text{.}$

Conversely, suppose that conditions (i) and (ii”) hold. We show that condition
(ii) of Definition 12 holds. Fix $s_{-i}\in S_{-i}$ such that
$E_{s_{-i}}:=\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)\cap
E\neq\emptyset$. By condition (ii”), $E_{s_{-i}}\subseteq\left(\cup_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\right)\times T_{-i}$. Hence,
there exists $k\leq m$ such that
$s_{-i}\in\mathrm{Supp}$marg${}_{S_{-i}}\mu_{i}^{k}$. Thus,
$\mu_{i}^{k}\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)>0$. Moreover, by
condition (i), $\mu_{i}^{k}\left(E\right)=1$. Therefore,
$\mu_{i}^{k}\left(E_{s_{-i}}\right)>0$, as desired. $\blacksquare$

Proof of Proposition 2. Part 1: Let
$\bar{\mu}_{i}:=\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)$ and suppose that,
for each $k$, $E_{k}$ is cautiously believed under $\bar{\mu}_{i}$ at some
level $m_{k}$. Let $m_{K}:=\min\left\\{m_{k}:k=1,2,...\right\\}$. We show that
$E:=\cap_{k}E_{k}$ is cautiously believed at level $m_{K}$. For each $k$, it
holds that $\mu_{i}^{l}\left(E_{k}\right)=1$ for all $l\leq m_{K}$. By the
$\sigma$-additivity property of probability measures, it follows that
$\mu_{i}^{l}\left(E\right)=1$ for all $l\leq m_{K}$. Fix an elementary
cylinder $\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times T_{-i}$ such that
$E\cap\hat{C}_{s_{-i}}\not=\emptyset$. Let $E_{m_{K}}$ be an event in
$\left(E_{k}\right)_{k\geq 1}$ which is cautiously believed at level $m_{K}$.
Obviously, $E_{m_{K}}\cap\hat{C}_{s_{-i}}\not=\emptyset$. Since $E_{m_{K}}$ is
cautiously believed, by condition (ii) of Definition 12, we have
$\mu_{i}^{l}\left(E_{m_{K}}\cap\hat{C}_{s_{-i}}\right)>0$ for some $l\leq
m_{K}$. Since $\mu_{i}^{l}\left(E\right)=1$, we obtain

$0<\mu_{i}^{l}\left(E_{m_{K}}\cap\hat{C}_{s_{-i}}\right)=\mu_{i}^{l}\left(E_{m_{K}}\cap\hat{C}_{s_{-i}}\cap
E\right)\leq\mu_{i}^{l}\left(E\cap\hat{C}_{s_{-i}}\right)\text{.}$

Next, let $m_{K}:=\max\left\\{m_{k}:k=1,2,...\right\\}$. We show that
$E=\cup_{k}E_{k}$ is cautiously believed at level $m_{K}$. Let $E_{m_{K}}$ be
an event in $\left(E_{k}\right)_{k\geq 1}$ which is cautiously believed at
level $m_{K}$. For each $l\leq m_{K}$, we have
$1=\mu_{i}^{l}\left(E_{m_{K}}\right)\leq\mu_{i}^{l}\left(E\right)$. For each
elementary cylinder $\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times T_{-i}$
with $E\cap\hat{C}_{s_{-i}}\not=\emptyset$, there is $k$ such that
$E_{k}\cap\hat{C}_{s_{-i}}\not=\emptyset$. By condition (ii) of Definition 12,
it follows that
$0<\mu_{i}^{l}\left(E_{k}\cap\hat{C}_{s_{-i}}\right)\leq\mu_{i}^{l}\left(E\cap\hat{C}_{s_{-i}}\right)$
for some $l\leq m_{k}\leq m_{K}$.

Part 2: Suppose that condition (i) of Definition 12 and condition (ii’) are
satisfied. Then condition (ii’) implies

$E\subseteq\mathrm{Proj}_{S_{-i}}^{-1}\left(\text{{Proj}}_{S_{-i}}\left(E\right)\right)=\mathrm{Proj}_{S_{-i}}^{-1}\left(\left(\mathop{\textstyle\bigcup}_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\right)\right)=\left(\mathop{\textstyle\bigcup}_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\right)\times T_{-i}\text{,}$

i.e., condition (ii”) in Lemma B.1 holds. Hence $E$ is cautiously believed
under $\beta_{i}\left(t_{i}\right)$.

For the converse, suppose that $E$ is cautiously believed under
$\beta_{i}\left(t_{i}\right):=(\mu_{i}^{1},...,\mu_{i}^{n})$ at level $m$. By
Lemma B.1, it follows that

$\text{{Proj}}_{S_{-i}}\left(E\right)\subseteq\text{{Proj}}_{S_{-i}}\left(\left(\mathop{\textstyle\bigcup}_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\right)\times
T_{-i}\right)=\mathop{\textstyle\bigcup}_{l\leq
m}\mathrm{Supp}\text{{marg}}_{S_{-i}}\mu_{i}^{l}\text{.}$

To show that this set inclusion holds with equality, let
$s_{-i}\notin$Proj${}_{S_{-i}}\left(E\right)$. Then
$\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)\cap E=\emptyset$. By
condition (i) of Definition 12, $\mu_{i}^{l}\left(E\right)=1$ for each $l\leq
m$, so

$\mu_{i}^{l}\left(\left\\{s_{-i}\right\\}\times
T_{-i}\right)=\mathrm{marg}_{S_{-i}}\mu_{i}^{l}\left(\left\\{s_{-i}\right\\}\right)=0\text{.}$

This implies $s_{-i}\notin\mathrm{Supp}$marg${}_{S_{-i}}\mu_{i}^{l}$.
$\blacksquare$

## Appendix C. Proofs for Section 5

In this section we first show that, for a given type structure
$\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$, the sets
$R_{i}^{m}$, $m>1$, are Borel subsets of $S_{i}\times T_{i}$. Then we provide
the proofs of Lemmas 1-3, as well as the proofs of Theorem 2 and Theorem 5.

We begin by showing that, for a given type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$, the set
$\mathbf{B}_{i}^{c}(E)\subseteq S_{i}\times T_{i}$ is Borel for every event
$E\subseteq S_{-i}\times T_{-i}$.

Lemma C.1 Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ and non-empty event $E\subseteq
S_{-i}\times T_{-i}$. Then the set of all
$\bar{\mu}\in\mathcal{N}(S_{-i}\times T_{-i})$ under which $E$ is cautiously
believed is Borel in $\mathcal{N}(S_{-i}\times T_{-i})$.

Proof. Recall that, for any event $E\subseteq S_{-i}\times T_{-i}$, the set of
probability measures $\mu$ satisfying $\mu\left(E\right)=p$ for
$p\in\mathbb{Q}\cap\left[0,1\right]$ is measurable in
$\mathcal{M}(S_{-i}\times T_{-i})$. So the sets of all
$\mu\in\mathcal{M}(S_{-i}\times T_{-i})$ satisfying $\mu\left(E\right)=1$ or
$\mu\left(E\right)=0$ are Borel in $\mathcal{M}(S_{-i}\times T_{-i})$. Now,
fix $n$ and $m\leq n$. By the above argument and by definition of
$\mathcal{N}_{n}(S_{-i}\times T_{-i})$, it turns out that the set

$C_{n,m}^{1}:=\mathop{\textstyle\bigcap}_{l\leq
m}\left\\{\bar{\mu}\in\mathcal{N}_{n}(S_{-i}\times
T_{-i}):\mu^{l}\left(E\right)=1\right\\}$

is Borel in $\mathcal{N}_{n}(S_{-i}\times T_{-i})$. Note that $C_{n,m}^{1}$ is
the set of all $\bar{\mu}\in\mathcal{N}_{n}(S_{-i}\times T_{-i})$ for which
condition (i) of Definition 12 holds for level $m$.

By the same argument, it follows that, for every
$s_{-i}\in\mathrm{Proj}_{S_{-i}}\left(E\right)$, the set

$C_{n,m}^{s_{-i}}:=\mathop{\textstyle\bigcap}_{l\leq
m}\left\\{\bar{\mu}\in\mathcal{N}_{n}(S_{-i}\times
T_{-i}):\mu^{l}\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)=0\right\\}$

is Borel in $\mathcal{N}_{n}(S_{-i}\times T_{-i})$. Note that the set

$C_{n,m}^{2}:=\mathop{\textstyle\bigcap}_{s_{-i}\in\mathrm{Proj}_{S_{-i}}\left(E\right)}\left(\mathcal{N}_{n}(S_{-i}\times
T_{-i})\backslash C_{n,m}^{s_{-i}}\right)$

is the (measurable) set of all $\bar{\mu}\in\mathcal{N}_{n}(S_{-i}\times
T_{-i})$ satisfying condition (ii) of Definition 12 for level $m$. Define
$C_{n,m}:=C_{n,m}^{1}\cap C_{n,m}^{2}$; clearly, $C_{n,m}$ is a Borel subset
of $\mathcal{N}_{n}(S_{-i}\times T_{-i})$. Hence, the set of all
$\bar{\mu}\in\mathcal{N}(S_{-i}\times T_{-i})$ under which $E$ is cautiously
believed is given by $\cup_{n\in\mathbb{N}}\cup_{m\in\mathbb{N}}C_{n,m}$, so
it is Borel in $\mathcal{N}(S_{-i}\times T_{-i})$. $\blacksquare$

By measurability of each belief map in a lexicographic type structure, we
obtain the following result.

Corollary C.1 Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. For every $i\in I$, if $E\subseteq
S_{-i}\times T_{-i}$ is a non-empty event, then $\mathbf{B}_{i}^{c}(E)$ is a
Borel subset of $S_{i}\times T_{i}$.

We can state and prove the desired result.

Lemma C.2 Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. Then, for each $i\in I$ and each
$m\geq 1$,

$R_{i}^{m+1}=R_{i}^{1}\cap\left(\mathop{\textstyle\bigcap}_{l\leq
m}\mathbf{B}_{i}^{c}\left(R_{-i}^{l}\right)\right)\text{,}$

and $R_{i}^{m}$ is Borel in $S_{i}\times T_{i}$.

Proof. The equality $R_{i}^{m+1}=R_{i}^{1}\cap\left(\cap_{l\leq
m}\mathbf{B}_{i}^{c}\left(R_{-i}^{l}\right)\right)$ is obvious. By Corollary
D.2 in Catonini and De Vito (2020), it follows that, for each $i\in I$, the
set $R_{i}^{1}:=R_{i}\cap C_{i}$ is Borel in $S_{i}\times T_{i}$. By Corollary
C.1, the set $\mathbf{B}_{i}^{c}\left(R_{-i}^{1}\right)$ is Borel in
$S_{i}\times T_{i}$. The conclusion follows from an easy induction on $m$.
$\blacksquare$

Proof of Lemma 1. Let $M\geq 1$ be the smallest natural number such that
$\prod_{i\in I}S_{i}^{\infty}=\prod_{i\in I}S_{i}^{M}$.272727Note that, if
$S^{0}=S^{1}$, then $M$ is $1$ and not $0$. This will simplify the exposition.
By Lemma E.1 in BFK, for every $n\in\left\\{1,...,M+1\right\\}$ and $s_{i}\in
S_{i}^{n}$, there exists $\mu_{s_{i}}^{n}\in\mathcal{M}(S_{-i})$ such that
$\mathrm{Supp}\mu_{s_{i}}^{n}=S_{-i}^{n-1}$ and

$\pi_{i}(s_{i},\mu_{s_{i}}^{n})\geq\pi_{i}(s_{i}^{\prime},\mu_{s_{i}}^{n})\text{,
}\forall s_{i}^{\prime}\in S_{i}\text{.}$

We use this result to construct a finite type structure
$\mathcal{T}:=\left\langle S_{i},T_{i},\beta_{i}\right\rangle_{i\in I}$ as
follows.

For each $i\in I$, let $T_{i}:=S_{i}^{1}$, and define each belief map
$\beta_{i}:T_{i}\rightarrow\mathcal{N}(S_{-i}\times T_{-i})$ as follows. Pick
any $s_{i}\in T_{i}$. Fix also an arbitrary $\bar{s}_{-i}\in T_{-i}$, and
define $\nu_{s_{i}}^{1}\in\mathcal{M(}S_{-i}\times T_{-i})$ as

$\nu_{s_{i}}^{1}\left(\left\\{\left(s_{-i},\bar{s}_{-i}\right)\right\\}\right):=\mu_{s_{i}}^{1}\left(\left\\{s_{-i}\right\\}\right)\text{,
}\forall s_{-i}\in S_{-i}\text{.}$

Next, let $m:=\max\left\\{k\leq M+1:s_{i}\in S_{i}^{k}\right\\}$. (Note that
if $s_{i}\in S_{i}^{M}$, then $m=M+1$, because $S_{i}^{M}=S_{i}^{M+1}$.) So,
if $m=1$, let $\beta_{i}(s_{i}):=\left(\nu_{s_{i}}^{1}\right)$. Otherwise, for
each $k=2,...,m$, define $\nu_{s_{i}}^{k}\in\mathcal{M(}S_{-i}\times T_{-i})$
as

$\nu_{s_{i}}^{k}\left(\left\\{\left(s_{-i},s_{-i}\right)\right\\}\right):=\mu_{s_{i}}^{k}\left(\left\\{s_{-i}\right\\}\right)\text{,
}\forall s_{-i}\in S_{-i}^{k-1}\text{,}$

and let

$\beta_{i}(s_{i}):=\left(\nu_{s_{i}}^{m},...,\nu_{s_{i}}^{1}\right)\text{.}$

Finiteness of each type set guarantees that each belief map is Borel
measurable (in fact, continuous). This completes the definition of the type
structure $\mathcal{T}$.

We now show that $\mathcal{T}$ satisfies the required properties. To this end,
we find it convenient to define, for each $i\in I$ and $k=1,...,M$, the
following sets:

$\Delta_{S_{i}^{k}\times T_{i}}:=\left\\{(s_{i},s_{i}^{\prime})\in
S_{i}^{k}\times T_{i}:s_{i}=s_{i}^{\prime}\right\\}\text{.}$

That is, each set $\Delta_{S_{i}^{k}\times T_{i}}$ is homeomorphic to the
diagonal of $S_{i}^{k}\times S_{i}^{k}$.282828The diagonal of $S_{i}^{k}\times
S_{i}^{k}$ is the set $\left\\{\left(s_{i},s_{i}^{\prime}\right)\in
S_{i}^{k}\times S_{i}^{k}:s_{i}=s_{i}^{\prime}\right\\}$. Next note that, for
every $s_{i}\in S_{i}^{2}$, all the component measures of
$\beta_{i}(s_{i}):=\left(\nu_{s_{i}}^{m},...,\nu_{s_{i}}^{1}\right)$ except
for $\nu_{s_{i}}^{1}$ are concentrated on those “diagonal” sets, namely

$\mathrm{Supp}\nu_{s_{i}}^{k}=\Delta_{S_{-i}^{k-1}\times T_{-i}}\text{,
}k=2,...,m\text{,}$

which implies
$\mathrm{Supp}\nu_{s_{i}}^{k}\subseteq\mathrm{Supp}\nu_{s_{i}}^{k-1}$ for
$k\geq 3$.

The rest of the proof is by induction.

Induction Hypothesis ($n$): For each $i\in I$,
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{n}\right)=S_{i}^{n}$; moreover,
$\Delta_{S_{i}^{n}\times T_{i}}\subseteq R_{i}^{n}$ if $n\leq M$, and
$\Delta_{S_{i}^{M}\times T_{i}}\subseteq R_{i}^{n}$ if $n>M$.

Basis Step ($n=1$). Fix $i\in I$ and $s_{i}\in S_{i}^{1}$. Type $s_{i}$ is
cautious, because

$\mathrm{Suppmarg}_{S_{-i}}\nu_{s_{i}}^{1}=\mathrm{Supp}\mu_{s_{i}}^{1}=S_{-i}\text{,}$

and the strategy-type pair $(s_{i},s_{i})$ is rational, in that