%Paper: ewp-game/9703005
%From: blipman@sulawesi.gsia.cmu.edu
%Date: Fri, 21 Mar 97 15:44:59 CST

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\begin{document} 
 
\title{Approximately Common Priors\thanks{I wish to thank Jim
Bergin, Eddie Dekel, and Steve Morris for helpful
discussions, SSHRCC for financial support, and the University of
Pennsylvania for a very enjoyable visit 
during which some of this research
was carried out.}}
 
\author{Barton L.~Lipman\\
~\\ 
Department of Economics\\
Social Science Centre\\
University of Western Ontario\\
London, Ontario N6A 5C2\\
~\\
email:  blipman@julian.uwo.ca}
 
\date{October 1995 \\ 
~\\
First Draft}
 
\maketitle
  
\begin{abstract}
I show that if one is only interested in  local results --- that is, theorems 
about what is true 
at a state --- which only depend on finitely many orders of beliefs
about beliefs, then the common prior assumption is essentially 
irrelevant.  More precisely, given 
any model where priors have the same support and any finite $N$, 
there is another model with common priors
which has the same $n^{\rm th}$ order beliefs for all $n\le N$.  In
this sense, priors with the same support are (locally) approximately common.
\end{abstract}
  
\section{Introduction}\label{sec-intro}
 
The common prior assumption --- the assumption that all agents have the
same prior over the relevant state space --- is used in virtually all of 
game theory and information economics.  It is a crucial assumption
for many important results such as Aumann's agreeing--to--disagree
result (\cite{Aumann1976}), the no--trade theorem (see, {\it e.g.},
Milgrom and Stokey~\cite{MilgromStokey}),
and Aumann's~\cite{Aumann1987} characterization of
correlated equilibria.  There has been a great deal of debate regarding
the assumption --- see, for example,
Morris~\cite{MorrisEcPhil} for a summary of some of the arguments.
 
I provide a new result clarifying what the common prior assumption
means as a statement about the agents' hierarchies of beliefs.  My
result, very loosely, is that the common prior assumption has only very
weak implications for beliefs of finite order --- that is, beliefs about
the beliefs of others about the beliefs of $\ldots$ about the beliefs of 
others, where ``beliefs
about'' is repeated only a finite number of times.   Phrasing the result 
differently, most priors
are locally approximately common.
One implication of
this is that dropping the common prior assumption is
(almost) the same thing as replacing common knowledge with a high but finite
level of mutual knowledge. 

A precise statement of the result requires some background.  Imagine
we are given a complete description of the world by an outside
observer.  This description includes all true statements about some
parameter which is unknown to the agents,
the beliefs of the agents (their {\it first--order beliefs}) about
this parameter, their beliefs about the beliefs of others ({\it
second--order beliefs}), their beliefs about the others' beliefs
about the others ({\it third--order beliefs}), etc.  I will refer to 
a specification of 
the true parameter together with an infinite hierarchy of beliefs for
each player as a {\it world}, reserving the term {\it state} for a related
but different notion.  As
Mertens and Zamir~\cite{MertensZamir} showed in their classic paper,
such a
description of the {\em actual\/} world generates a
collection of {\em possible} worlds, one of which is the actual
world.  These other worlds 
can then be seen as fictitious constructs, used to clarify
our understanding of the actual world.\footnote{ See
Dekel and Gul~\cite{DekelGul1995} for a similar perspective.}
 
The actual world also identifies a prior over the set of
possible worlds for each agent.  The prior for a given agent is not
unique since any prior yielding the same conditional probabilities on
the appropriate events would serve the same purpose.  However, the
point is that the actual world identifies whether or not there is a
single prior we could attribute to all the agents which generates the
beliefs of each agent.
 
In other words, the actual world --- the infinite
hierarchy of beliefs about beliefs for each player 
--- completely determines whether
or not the beliefs are consistent with the common prior assumption.
Hence even though we typically think of the common prior assumption
as an {\it ex ante} statement --- a statement regarding the agent's
beliefs before getting information --- we can view whether or not
this assumption holds as something determined at the interim level
--- that is, in the actual world or
after receipt of information.
 
To state the result more precisely, then, suppose the agents' beliefs
at a world $\omega$ are weakly consistent in the sense that it is common
knowledge that all agents' beliefs at all levels give positive
probability to the truth.  Then for any finite $N$, there is
another world, $\omega^N$, which is consistent with the
common prior assumption and where for all $n\le N$, 
$n^{\rm th}$ order beliefs
at $\omega^N$ are the same as those at world $\omega$.  In this sense, we
can always approximate weakly consistent priors by common priors, at
least for local results.
 
This result has several implications which are discussed in detail in
Section~\ref{sec-impl}.  One is the implication
mentioned above: dropping the common prior assumption is almost the same
thing as replacing common knowledge with a high but finite degree of
mutual knowledge.  That is, given a world where a particular
set of facts are common knowledge but priors are different, we can
find a model with common priors in which nothing changes except that
these facts are arbitrarily high mutual knowledge but {\em not}
common knowledge.  This has
important methodological implications which are discussed in more detail
in Section~\ref{impl-methodology}.
 
The theorem also tells us where the real ``bite'' of the common prior
assumption lies.  The theorem relies on an assumption of weak 
consistency --- and the result is not true without this assumption.
However, the theorem shows that weak consistency is the only
implication the common prior assumption has if we are only concerned 
with finitely many orders of beliefs.
In this sense, the common prior
assumption is primarily an ``infinite order'' assumption since we know
that the common prior assumption has a very significant effect at least
as measured by the kinds of theorems that are true with it and false
without it.
 
Finally, the result enables me to show that the set of worlds
consistent with common priors is dense in the set of all worlds ---
that is, it is dense in the Mertens--Zamir
universal belief space.  In this sense, {\em all} belief hierarchies are
approximately consistent with common priors.  This implication is explained in
Section~\ref{sec-topology}.  This is a very surprising result in 
light of the apparent
generic failure of common priors when viewed differently.  Suppose
we fix a state set and choose priors randomly for the agents.
Clearly, the probability that we choose the same priors for all
agents is very small, suggesting that the common prior assumption, in
some sense, fails generically.\footnote{ This description overlooks the
fact that it is only conditional beliefs that are relevant.  However, as
Nyarko~\cite{Nyarko1991} shows, this statement can be formalized and
demonstrated even taking this fact into account.} My result shows that when
we take the set of belief hierarchies as the relevant space, rather
than the set of probability distributions on a fixed state space, we
get a startlingly different view.
 
It is also worth mentioning that the result shows that the line
between differences in information and differences in beliefs is more
blurred than one might suppose.  As advocates of the common prior
assumption have long noted, the assumption is a formal statement of
the intuitive idea that differences in beliefs should all stem from
differences in information.  My result indicates that if
we only care about finitely many orders of belief, a ``pure''
difference of beliefs (noncommon priors) can always be translated
into a difference in beliefs stemming from a difference in
information.  Section~\ref{sec-ex} gives an example showing how this
translation works.
 
\section{The Model and an Illustrative Example}
 
\subsection{Model}
 
Tan and Werlang~\cite{TanWerlang1988} and Brandenburger and
Dekel~\cite{BrandenburgerDekel1993} have shown that 
we can generate any point in the universal
beliefs space by use of what I will term a {\it partitions model}.
Given a {\it parameter space} $S$, and a finite 
set of players ${\cal I} = \{1,\ldots, I\}$, a partitions
model consists of
 
\begin{enumerate}
 
\item a set of {\em states}, $\Omega$
 
\item a function $f:\Omega \rightarrow S$ telling what the unknown
parameter is in each state
 
\item for each player $i\in {\cal I}$, a partition $\Pi_i$ of
$\Omega$
 
\item for each player $i\in {\cal I}$, a prior $\mu_i$ over $\Omega$.
 
\end{enumerate}
 
I will use ${\cal M}$ to denote a typical partitions model
$(\Omega,f,\{(\Pi_i,\mu_i)\}_{i\in {\cal I}})$.  Given ${\cal M}$
and a state $\omega\in \Omega$, we can define a point in the
Mertens--Zamir universal beliefs space by ``unravelling.''  That is,
player $\imath$'s first--order beliefs at $\omega$ are those 
induced by updating
$\imath$'s prior given the event $\Pi_i(\omega)$ and using the
function $f$ to convert this to a belief on $S$.  Formally, the 
first--order beliefs of $i$ at state
$\omega$, say $\delta^1_i[\omega]$, are defined by
 \[ \delta^1_i[\omega](B) = \mu_i\left(\{\omega'\in \Omega\mid
    f(\omega') \in B\} \mid \Pi_i(\omega) \right) \]
for each measurable $B\subseteq S$.  The second--order
beliefs are calculated by performing the same updating but then using
the partitions for players other than $i$ to calculate $i$'s beliefs
about these players' beliefs, etc.  That is, if we have two players, 
then player 1's second--order beliefs at $\omega$, say $\delta^2_1[\omega]$,
are a probability distribution over $S\times \Delta(S)$, where $\Delta(S)$ is
the set of first--order beliefs for player 2.\footnote{We could equivalently
define second--order beliefs to be a distribution over $S\times \Delta(S)\times 
\Delta(S)$ and then impose the condition that 1 knows his own beliefs.  The
approach in the text follows Brandenburger and 
Dekel~\cite{BrandenburgerDekel1993}, while the approach
described in this footnote follows the original Mertens and Zamir formulation.}
Formally, then,
 \[ \delta^2_1[\omega](B) = \mu_i\left(\left\{\omega'\in \Omega\mid (f(\omega),
     \delta^1_2(\omega))\in B\right\}\mid \Pi_1(\omega)\right) \]
for each measurable $B\subseteq S\times \Delta(S)$.  Etc.
 
This equivalence works in reverse as well: any point in the
universal beliefs space can be constructed in this fashion from some
partitions model.\footnote{ To be more precise, we may need 
$\sigma$--fields instead of partitions.  See 
Brandenburger and Dekel~\cite{BrandenburgerDekel1993}.}  
 
\subsection{An Example}\label{sec-ex}
 
Suppose we have two players and two possible values of the unknown parameter, 
$S=\{s_1,
s_2\}$.  We also have two states, $\Omega=\{\omega_1,
\omega_2\}$.  The function $f$ is given by $f(\omega_j) = s_j$,
$j=1,2$.  Both players' partitions are trivial: $\Pi_i = \{\Omega\}$
for $i=1,2$.  Player 1's prior is $\mu_1(\omega_1)=2/3$, while player
2's prior has $\mu_2(\omega_1)=1/3$.  Hence we have a ``pure''
difference in beliefs, not stemming from any difference in
information.  Let ${\cal M}$ denote this partitions model.
 
I now show that we can match beliefs at each state to an arbitrarily
high order.  I demonstrate this by constructing a sequence of new
partitions models where each model has a state $\omega^n$ which matches the
beliefs at $\omega$ in ${\cal M}$ up to order $N$.  In each of
these new models, the difference in beliefs is replaced by a
difference in information.  However, as we will see, this replacement
is not without cost: we {\em cannot} avoid changing the beliefs at
some very high orders.
 
So fix an integer $N$ and construct the new model as follows.  
The new set of states, $\bar \Omega^N$, is taken
to be
 \[ \bar \Omega^N = \Omega\times \{1,\ldots, 2^N\}. \]
Intuitively, think of state $(\omega, k)$ as the $k^{\rm th}$
copy of state $\omega$ from our original model.  In line with this
intuition, the new function relating states and the value of the 
unknown parameter, 
$\bar f^N$, is given by
 \[ \bar f^N(\omega,k) = f(\omega). \]
The common prior of both players is that all states in $\bar
\Omega^N$ are equally likely.
 
To generate the appropriate beliefs for player 1, then, we will have
to have his information reflect a greater likelihood of state
$\omega_1$ than $\omega_2$.  Hence his partition, $\bar \Pi_1^N$,
includes all events of the form $\{(\omega_1,2k-1),(\omega_1, 2k),
(\omega_2,k)\}$ for $k=1,\ldots,2^{N-1}$.  Since player 1 ``uses up''
his $\omega_1$ copies faster than his $\omega_2$ copies, we will also
end up with an event $\{(\omega_2, 2^{N-1}+1),\ldots,(\omega_2,
2^N)\}$.  Note that at {\em every} event except this last one, player
1's beliefs over $S$ give probability 2/3 to $s_1$ and 1/3 to $s_2$.
 
The partition for player 2, $\bar \Pi_2^N$, is analogous.  Since
player 2 thinks $\omega_2$ is more likely, we include every event of
the form $\{(\omega_1, k), (\omega_2, 2k-1), (\omega_2, 2k)\}$ for
$k=1, \ldots, 2^{N-1}$.  Analogously to the case of player 1, this
leaves us with ``extra'' $\omega_1$ copies, so player 2's partition
also includes the event $\{(\omega_1, 2^{N-1}+1),\ldots, (\omega_1,
2^N)\}$.  Again, at all events except this last one, player 2's
beliefs over $S$ match his beliefs from the original model.  Let
${\cal M}^N$ denote this partitions model.
 
It is not hard to show
that $n^{\rm th}$ order beliefs at $(\omega, 1)$ 
in ${\cal M}^N$ are the same as $n^{\rm th}$ order beliefs
at $\omega$ in ${\cal M}$ for all $n\le N$ for both 
$\omega=\omega_1$ and $\omega =\omega_2$.  To see this, first note that
we have obviously matched first--order beliefs.  In state $(\omega, 1)$,
player 1's first--order beliefs put 
probability 2/3 on $s_1$ and 1/3 on $s_2$ or, in
the obvious vector notation, $(2/3, 1/3)$.  This is precisely the
same as player 1's first--order beliefs at $\omega_1$ in the original
partitions model.  Similarly for player 2.
 
To get second--order beliefs for player 1, we must identify the
events of 2's partition he considers possible.  These events are 
 \[ \{(\omega_1,1), (\omega_2,1), (\omega_2,2)\} \]
and
 \[ \{(\omega_1,2), (\omega_2,3), (\omega_2, 4)\}. \]
The set of first--order 
beliefs for 2 at these events form the support of player
1's second--order beliefs.  Note that 2's beliefs over $S$ at each of
these events are $(1/3, 2/3)$, so that player 1 puts probability 1 on
these being player 2's first--order beliefs.  This exactly matches
player 1's second--order beliefs at $\omega_1$ in the original partition
model.  Again, the analogous argument for player 2 works.
 
More generally, we see that player 1's $n^{\rm th}$ order beliefs are
completely determined by the events that are $n-1$ ``steps'' from
$(\omega,1)$.  More precisely, an event is $n-1$ steps from $(\omega,1)$ if
we can construct a sequence of $n$
events starting from $\{(\omega_1,1), (\omega_1,2), (\omega_2,1)\}$ such
that as we move from one event to the next, we move from an event in one
player's partition to an intersecting event in the other player's
partition.  
It is not hard to see that for $n\le N$,
all events reachable this way have the
same first--order beliefs as in the original model.  Hence $(\mbox{player 
1 attaches probability
1 to player 2 attaching probability 1 to})^n$ to player 1's beliefs 
being $(2/3, 1/3)$ and player 2's beliefs being $(1/3,2/3)$ as long as 
$n \le N$.
 
Note that the new partitions model is ``like'' the old one in the
sense that beliefs at certain states are the same as beliefs in the
original model up to a high but finite order.  On the other hand, the
{\it ex ante} models are quite different.  In the original model,
there is zero probability that, say, player 1 puts probability 1 on
$s_2$.  Yet in the new model, there is a positive probability of this
occurrence.  Put differently, the notion of ``closeness'' here is an
interim notion --- focusing on the closeness of actual worlds, rather
than the closeness of {\it ex ante} models.
 
A related point is that while 
the new model matches statements about mutual
knowledge to a very high degree, the fact that it does so only up to
a {\em finite} degree means that statements about common knowledge
are not matched.  In particular, in the original model, beliefs over
$S$ are common knowledge.  In the new model, both players know that
both know that $\ldots$ that both know the beliefs over $S$ for a
very large number of ``know that''$\,$'s.  But the beliefs are {\em
not} common knowledge.

A natural question to ask is whether this approach can be continued
for all orders instead of only up to some finite upper bound.  One
answer is simply that extending the construction above does not work.  If
we replace $K$ with $\infty$, we end up with a prior which is uniform
on a countable set --- an impossibility.
 
More generally, we know from Mertens--Zamir that no such construction
is possible.  As discussed in Section~\ref{sec-intro}, Mertens--Zamir
showed that if we specify beliefs about beliefs for all finite
orders, this completely determines whether or not the beliefs of the
players are consistent with the common prior assumption.  Hence if
they are not, there is no way to change the model to one with a
common prior and simultaneously match {\em all} orders of
beliefs.\footnote{ There is one way that this might be possible.
Heifetz~\cite{Heifetz} 
has shown that Mertens--Zamir rely on an assumption that we
might term common knowledge of countable additivity.  More
specifically, he shows that while there is a unique countably
additive belief which extends all finite orders of belief, there may
be many finitely additive extensions.  In light of this, it
is conceivable that there are finitely additive extensions which are
consistent with common priors even when the unique countably additive
extension is not.  While my construction does not seem to converge in any
meaningful sense to such finitely additive
extensions, it is possible that some other construction would
yield such a ``continuity'' result.}
 
\section{The Result}
 
I will say that a partitions model $(S, \Omega, f, \{(\Pi_i,
\mu_i)\}_{i\in {\cal I}})$ is {\em finite} if $\Omega$ is finite and
{\em countable} if $\Omega$ is finite or countably infinite.
Given any probability distribution $\mu$ on
$\Omega$, let $\supp(\mu)$ denote the support of $\mu$.
 
\begin{definition}
A partitions model is {\em weakly consistent} if
$\supp(\mu_i) = \supp(\mu_j)$ for all $i,j\in {\cal I}$.
\end{definition}
 
In the introduction, I defined weak consistency as
common knowledge that all players' beliefs at all levels give positive
probability to the truth.  Certainly, a common support seems to capture
this idea to some extent.  A more formal link is given in 
Section~\ref{sec-topology} where I show that weak consistency as defined
here for the partitions model is equivalent to weak consistency as
defined in the introduction for the equivalent belief hierarchies.
 
It is easy to see that weak consistency means that I may as well
assume that every agent puts positive probability on every
state.\footnote{ This overlooks the issue of how we would define beliefs
on events with prior probability zero.  If we simply require all events
to have positive prior probability (and if we only care about
conditional probabilities, the fact that $\Omega$ will be assumed to be
countable means that we may as well assume this), then states all agents
give zero prior probability to can be ignored.}
Otherwise, I can restrict attention to the subset of $\Omega$ which
is the common support for all agents' beliefs --- and the fact that
$\Omega$ is countable will mean that this support is
countable also.
 
\begin{theorem}\label{main}
Let ${\cal M}$ denote any weakly consistent countable partitions
model.  For any finite $N$, there exists a countable partitions model
${\cal M}^N$ in which 
 
\begin{enumerate}
 
\item the common prior assumption is satisfied and
 
\item for every $\omega\in {\cal M}$, there is an $\omega^N\in {\cal
M}^N$ such that $n^{\rm th}$ order beliefs at $\omega$ in ${\cal M}$ 
are the same as $n^{\rm th}$ order beliefs at $\omega^N$ in ${\cal M}^N$
for all $n\le N$.
 
\end{enumerate}
\end{theorem}
 
The proof, which is contained in the Appendix, 
is tedious but not difficult.  It provides a construction
which generalizes the construction in the example of
Section~\ref{sec-ex}.
 
\section{Implications}\label{sec-impl}
 
The result has a variety of implications which I now discuss in turn.
 
\subsection{Implications for the Structure of the Universal Beliefs 
Space}\label{sec-topology}
 
The main result, as stated in the introduction, referred to infinite
hierarchies of beliefs and used the assumption that it is common knowledge that
all players give
positive probability to the true world.  But
Theorem~\ref{main} was a result about partitions models and assumed that
the players' beliefs have the same support.  In this section, I first
show that the statement in the introduction is, in
fact, equivalent to Theorem~\ref{main}.  Once the theorem is restated
in terms of belief hierarchies, it is not difficult to use it to show
that every
world in the Mertens--Zamir universal beliefs space is
arbitrarily close to a world which satisfies in the common prior
assumption.  That is, the set of worlds satisfying the common prior
assumption is dense in the universal beliefs space.
 
I will try to minimize the extra notation needed to make these statements 
precise and so will refer to Mertens--Zamir~\cite{MertensZamir} 
for numerous technical details which are not
directly relevant for my purposes.  
As before, we have a set of players $\{1,\ldots, I\}$ and 
a parameter space $S$.  I assume $S$ is compact.  For any
compact set $X$, let $\Delta(X)$ denote the set of probability measures
on $X$ endowed with the weak$^*$ topology.  This set is compact as well.
Then we can define an infinite sequence of sets recursively by
 \[ \begin{array}{l}
   X_0 = S \\
   T_{k+1} = \Delta(X_k)\\
   X_{k+1} = X_k\times [T_{k+1}]^I
   \end{array} \]
To understand this definition, note that $T_1 = \Delta(X_0) = \Delta(S)$ 
is the set of 
first--order beliefs for an agent.  
Hence $X_1$ is just $S$ times the set of first--order
beliefs for each agent.  
Hence this is the set over which second--order beliefs are
defined, so $T_2=\Delta(X_1)$ is the set of second--order beliefs, etc.
Let $X=S\times\prod_{k=1}^{\infty} [T_k]^I$.  This is a well--defined, 
compact space. 
 
The universal beliefs space, which I denote ${\cal U}$, is the largest
subset of $X$ which, in the terminology of Brandenburger and 
Dekel~\cite{BrandenburgerDekel1993}, satisfies common knowledge of
coherence.  The exact definition of this term is irrelevant for my
purposes.  I will refer to a point in ${\cal U}$ as a {\em world}.
 
The main result of Mertens--Zamir is that there exists a set
of {\em types}, ${\cal T}$, such that ${\cal U}$ is homeomorphic to
$S\times {\cal T}^I$ and ${\cal T}$ is homeomorphic to $\Delta(S\times
{\cal T}^{I-1})$.  In other words, we can think of a world as specifying
the true value of the unknown parameter and the type of each player
where a type is a probability distribution on $S$ and the types of the
other players.  Equivalently, a type for player $i$ 
is a probability distribution on
the set of worlds with the property that player $i$ puts probability 1
on his own true type.  
 
If some player $i$ at world $u$ has a belief which
contains $u'$ in its support, I will say that $u'$ is {\em believed
possible} by $i$ at $u$.  A set of worlds $U\subseteq {\cal U}$ is 
{\em belief--closed} if for every $u\in U$ and every $u'$ believed
possible by some player at $u$, we have $u'\in U$.  In other words, every world
believed possible by someone at a world in $U$ is also in $U$.
 
Given a world, $u\in {\cal U}$, the {\em belief--closed subspace generated by
$u$} is the smallest belief--closed set containing all worlds believed
possible by any player at $u$.

My analysis makes use of four facts.  First, 
as discussed in Section 2.1, any partitions model
together with any state in that model uniquely identifies a
particular world in the universal beliefs space by the unravelling
procedure described earlier.  
 
Second, as alluded to in Section 2.1, any
belief--closed subspace $U$ of ${\cal U}$ generates a partitions
model.  More specifically, if $U$ is countable, we can find a
partitions model with the property that the state set in the partitions
model is one--to--one with $U$ and where the infinite
hierarchy of beliefs for any given player at a given state (defined by unravelling) 
is precisely the
infinite hierarchy of that player at the corresponding world.  (When $U$
is uncountable, we need $\sigma$--fields in place of partitions, but this
issue is irrelevant for my purposes.)  When a partitions model ${\cal
M}$ has this relationship to a belief--closed set $U$, I will say that
${\cal M}$ and $U$ are {\em equivalent}.  Similarly, I will say that a
state $\omega$ in ${\cal M}$ is {\em equivalent} to a world $u\in U$ if
the unraveling of $\omega$ generates $u$.
 
Third, as noted, ${\cal U}$ is a subset
of $X$.  In other words, it is a subset of an infinite product space.
For this reason, it is natural to follow Mertens--Zamir in using for a topology on
${\cal U}$ the 
(relativized) product topology.  
 
The final fact I require is a trivial implication of Mertens--Zamir's
Theorem 3.1.  The result is that for every $u\in {\cal U}$, there
exists a sequence $\{u^N\}$ in ${\cal U}$ converging to $u$ such that
every $u^N$ generates a finite belief--closed set.  Less formally, any
world $u\in {\cal U}$ is arbitrarily close to a world which generates a
finite belief--closed subspace.
 
I now give a more precise definition of a weakly consistent world in
preparation for restating Theorem~\ref{main} for belief hierarchies.
 
\begin{definition}
Let $u\in {\cal U}$ generate belief--closed subspace $U$.  Then $u$ is
weakly consistent if for all $u'\in U~\cup~\{u\}$, every player believes
$u'$ possible at $u'$.
\end{definition}
 
Intuitively, if every player believes the true world possible, this
means that each player's first--order beliefs
contain the true parameter in its support, each player's 
second--order beliefs contain the pair giving the true parameter value and 
true first--order beliefs in its support, etc.  If this holds at every
world in a belief--closed set, it means that every player knows that
this holds, every player knows that every player knows it, etc.  In
other words, as stated in the introduction, weak consistency simply
means that it is common knowledge that every player puts positive
probability on the truth at every level of belief.
  
The following lemma uses the first two facts above and 
enables me to relate Theorem~\ref{main} to the version of the
result stated in the introduction.  The proof is in the Appendix.

\begin{lemma}\label{wc=wc}
If $u\in {\cal U}$ is weakly consistent and generates a countable 
belief--closed subspace $U$, then $U$ is equivalent to a weakly consistent
countable partitions model.
\end{lemma}
 
One more definition is required.
 
\begin{definition}
A world $u\in {\cal U}$ is {\em consistent with the common prior 
assumption} if the belief--closed subspace it generates 
is equivalent to a partitions model with common priors.
\end{definition}
 
\begin{theorem}\label{equiv}
If $u\in {\cal U}$ is weakly consistent and generates a
countable belief--closed set $U$, then there is a sequence of worlds
$\{u^N\}$ converging to $u$ 
such that each $u^N$ is consistent with the common prior
assumption.
\end{theorem}
 
\nipar {\it Proof.}  Fix such a $u$ and let $U$ be the belief--closed 
set it generates.  
By Lemma~\ref{wc=wc}, it has an equivalent partitions model, say ${\cal M}$, 
which is weakly
consistent.  Obviously, ${\cal M}$ is countable since $\Omega$ and $U$ 
must be one--to--one.  Let $\omega$ be the state in ${\cal M}$ which 
is equivalent to $u$.
By Theorem~\ref{main}, for any $N$, we can find a
partitions model satisfying common priors and a state $\omega^N$ in that
model such that the $n^{\rm th}$ order 
beliefs at $\omega^N$ are the same as those at
$\omega$ for all $n\le N$.  Let $u^N$ be the world in ${\cal U}$ which is
generated from $\omega^N$ by unraveling.  By definition, $u^N$ is
consistent with
the common prior assumption.  Because $u^N$ is the world generated by
$\omega^N$, $u^N$ has the same parameter value as $u$ and has the same
$n^{\rm th}$ order beliefs for each player as $u$ for all $n\le N$.  
Hence
$u^N$ converges to $u$ pointwise as $N\rightarrow \infty$.  That is,
$u^N \rightarrow u$.~\qed
 
In other words, every weakly consistent world which generates a
countable belief--closed set is arbitrarily close to a world
with common priors.  
 
In fact, 
every world is arbitrarily close to one which is weakly consistent and
generates a finite belief--closed set.  Hence {\em every} world is
arbitrarily close to one consistent with common priors.  
The proof of the following lemma is in the Appendix.
 
\begin{lemma}\label{near-consis}
For any world $u\in {\cal U}$, there exists a sequence of worlds
$\{u^N\}$ converging to $u$ such that each $u^N$ is weakly consistent
and generates a finite belief--closed set.
\end{lemma}
 
Putting these results together gives
 
\begin{theorem}
For any world $u\in {\cal U}$, there exists a sequence of worlds $\{u^N\}$ 
converging to $u$ such that each $u^N$ is consistent with the common
prior assumption.  Hence the closure of the set of worlds consistent
with common priors is ${\cal U}$.
\end{theorem}
 
\nipar {\it Proof.}  Fix any world $u\in {\cal U}$.  By 
Lemma~\ref{near-consis}, we know that this world is arbitrarily close 
to a weakly consistent
world
generating a
finite belief--closed subspace.  By Theorem~\ref{equiv}, each such world 
is arbitrarily close to a world which is consistent with common 
priors.~\qed
 
\subsection{Methodological Implications}\label{impl-methodology}
 
In a sense, this result says that 
dropping the common prior assumption adds (weakly) fewer
possibilities than dropping common knowledge assumptions.  In other words, any
prediction we could obtain from a model with common knowledge and noncommon
priors could also be generated without common knowledge but with common
priors, while the reverse may not be true.  In this sense, if we wish to
generalize from the usual common knowledge and common prior assumptions but to
maintain as much predictive power as possible, it is better to drop the common
prior assumption than the common knowledge assumptions.  To see the
point,
suppose we knew that a particular outcome was possible in some state
where certain facts were common knowledge but priors were not the same.
If these priors satisfy weak consistency, then it would necessarily be
true that this same outcome would be possible in a model with common
priors but where the common knowledge facts are now only mutual
knowledge of some high but finite order.
 
Another methodological implication:  If we are willing to
assume weak consistency and only care about beliefs up to some finite
order, then there is no reason not to assume common priors.  Put
differently, any (local) result which only depends on finitely
many orders of belief and which is proved with the common prior
assumption is also true if we replace the common prior assumption by
weak consistency.  
 
By ``local,'' I mean a result which is true about
a state, not about the model as a whole.  For example,
Aumann's~\cite{Aumann1976} agreeing to disagree result is a local
result.
The theorem
hypothesizes that posterior beliefs are common knowledge at a state and
asks what else must be true at that state.  Similarly, no--trade
theorems begin with the hypothesis that it is common knowledge at a
state that a particular trade is mutually beneficial and characterize
other things which must be true at that state.  By contrast,
Aumann's~\cite{Aumann1987} result characterizing correlated equilibrium
is a global result since it says what must be true in expectation over
the entire set of states.  My result says nothing about
global results since the translation I do requires changing global
properties of the model in order to preserve local properties.
 
Rephrasing this last methodological implication, we see that 
any result which requires common priors must depend on infinitely many
orders of beliefs and hence on common knowledge
assumptions.  For example, 
the fact that the agreeing to disagree result and no--trade theorems
don't hold without common priors\footnote{ See Morris~\cite{Morris1994} for 
a characterization how relaxations of the common prior assumption affect 
no--trade results.} indicates that these results
must also rely on their common knowledge assumptions, a fact which is
well--known.
 
\subsection{Interpretation of the Common Prior Assumption}\label{sec-disc}
 
Given the controversial nature of the common prior
assumption, it is natural
to ask whether my results are a defense or a criticism of the
assumption.  I believe one can find ammunition for either position.
 
The points which support the common prior assumption are obvious.
First, if we are only interested in finitely many orders of beliefs
about beliefs, then the common prior assumption is really no stronger
than weak consistency.  Second, there is a sense in which every
world is arbitrarily close to one
in which the common prior assumption holds.
 
On the other hand, we typically are {\em not} only interested in
finitely many orders of beliefs about beliefs.  If we restrict
attention to finitely many orders, we cannot discuss common
knowledge, making it obvious that this is quite a severe restriction.
Similarly, the ``closeness'' of a world with common priors is
misleading: the topology in which we obtain this result is not a
topology in which our results are continuous.  In other words, even
though all possible beliefs hierarchies are ``close'' to
satisfying the common prior assumption, this does not say that all
behavior based on belief hierarchies is ``close'' to the behavior
predicted by common priors.
 
One implication of the result can be taken as a criticism of the
common prior assumption.  The result clearly shows that the common
prior assumption is, primarily, an ``infinite order'' restriction.
That is, since it only imposes a minimal restriction on beliefs up to
any finite order, the primary bite of the assumption is on the full
infinite hierarchy of beliefs.  Since we know that the common prior 
assumption has quite a lot of bite to it, the infinite order
restrictions must be strong.  This fact makes clear that a
(nontrivial) characterization of what the common prior assumption
means in terms of the hierarchies of 
beliefs at the actual world is likely to be
extremely difficult and perhaps uninterpretable.
 
\vfill\eject
 
\appendix
 
\section{Proof of Theorem~\ref{main}}  

Fix an arbitrary integer $N$.  As noted in the text, 
weak consistency means that we
may as well assume $\mu_i(\omega) > 0$ for all $\omega\in \Omega$ and
all $i\in {\cal I}$.  For each $\omega$, define a permutation
$\iota(\omega) = (\iota_1(\omega), \ldots, \iota_I(\omega))$ of
$\{1,\ldots, I\}$ 
such that 
 \[ \mu_{\iota_j(\omega)}(\omega) \le
  \mu_{\iota_{j+1}(\omega)}(\omega), ~~j=1,\ldots, I-1. \]
In other words, $\iota(\omega)$ renumbers the agents so that lower
numbered agents give less prior probability to $\omega$.
 
Define
 \[ \ell_1(\omega) = \mu_{\iota_1(\omega)}(\omega) \]
and for $j=2, \ldots, I$,
 \[ \ell_j(\omega) = \mu_{\iota_j(\omega)}(\omega) -
  \mu_{\iota_{j-1}(\omega)}(\omega). \]
Let
 \[ \max(\omega) = \sum_{j=1}^I \ell_j(\omega) = \max\{\mu_1(\omega),
  \ldots, \mu_I(\omega)\}. \]
It is useful to define a function
$j(i, \omega)$ giving the ``rank'' $\iota$ assigns to $i$ for
$\omega$ --- that is, $j(i, \omega)$ is defined by
 \[ \iota_{j(i,\omega)}(\omega) = i. \]
Note for future use that
 \[ \sum_{j=1}^{j(i,\omega)} \ell_j(\omega) = \mu_i(\omega). \]
Also, let 
 \[ \beta = 1 + \max_{\omega\in \Omega}{\max(\omega) - \ell_1(\omega)
   \over \ell_1(\omega)}. \]  
Let
 \[ \bar \Omega^N = \Omega\times \{1,\ldots, NI\}. \]
Naturally, let $\bar f^N(\omega,k) = f(\omega)$.  The common prior
$\bar \mu^N$ is given by 
 \[ \bar\mu^N(\omega,k) = \alpha z_k(\omega) \] 
where $\alpha > 0$ is a constant to be determined and $z_k(\omega)$
is defined as follows.  For $k=1,\ldots, I$,
 \[ z_k(\omega) = \ell_k(\omega) \]
Also, 
 \[ z_{I+1}(\omega) = (\beta + 1) \ell_1(\omega) - \max(\omega). \] 
Finally, for $k\ge I+2$,
 \[ z_k(\omega) = (\beta + 1)z_{k-I}(\omega). \]
If all the $z_k$'s are positive, then we can choose $\alpha$ so that
the $\bar \mu^N$'s sum to 1 and ensure that this is a probability
distribution.  By construction, $\ell_k(\omega) \ge 0$ for all $k$,
strictly so for $k=1$.  Hence this is a legitimate
probability distribution as long as
 \[ \beta\ell_1(\omega) \ge \max(\omega) - \ell_1(\omega) \]
which is guaranteed for all $\omega$ from the definition of $\beta$.
 
The partitions are as follows.  First,
 \[ \bar \Pi^N_i(\omega', 1) =
  \left\{(\omega,k)\mid \omega\in  \Pi_i(\omega') ~~\mbox{and}~~
   k\le j(\omega,i)\right\}. \]
For $n=1,\ldots, N-1$,
 \[ \bar \Pi^N_i(\omega', (n-1)I + j(i,\omega') + 1) = \left\{(\omega,k)
   \left| 
 \begin{array}{l}  
    \omega\in  \Pi_i(\omega') ~~\mbox{and}~~\\
    k\in \{(n-1)I+j(i, \omega) + 1,\ldots, nI + j(i,\omega)\}
 \end{array}
  \right. \right\}. \]
Finally, for $\omega'$ such that $j(i,\omega') < I$,
 \[ \bar \Pi^N_i(\omega', (N-1)I + j(i, \omega') + 1) =
  \left\{(\omega,k)\mid j(i, \omega) < I~ \mbox{and}~ k\in\{ (N-1)I +
  j(i, \omega)+1, \ldots, NI\} \right\}. \]
It is not difficult to verify that the events so defined form a
partition of $\bar \Omega^N$.  Let ${\cal M}^N$ denote the 
partitions model so constructed.  
 
The key
property of this construction is that certain conditional 
beliefs are preserved.  The following definition specifies the events
for which these conditional beliefs are appropriately preserved.
 
\begin{definition}
An event $E\subseteq \Omega^N$ is {\em $\Omega$--measurable} 
if there exists
$B\subseteq \Omega$ such that
 \[ E = \{(\omega,k)\in \bar \Omega^N\mid \omega\in B\} . \]
In this case, call $B$ the {\em $\Omega$ projection} of $E$.
An event $E\subseteq \Omega^N$ is {\em conditionally 
$\Omega$--measurable for $i$ at $(\omega,k)$} if there exists an
$\Omega$--measurable event $F$ such that
 \[ E~\cap~\bar \Pi_i^N(\omega,k) = F~\cap~\bar \Pi^N_i(\omega,k). \]
In this case, define the $\Omega$ projection of $E$ to be the 
$\Omega$ projection of $F$. 
\end{definition}
 
Intuitively, if an event $E$ in $\bar \Omega^N$ is of the form
$\{(\omega,k)\mid \omega\in B\}$, then it naturally corresponds to the
event $B$ in $\Omega$ --- hence the name $\Omega$--measurable.  
Even if $E$ is not $\Omega$--measurable, it may be 
true that conditional on $\imath$'s information, $E$ may as well be
in the sense that the conditional probability $\imath$
attaches to $E$ is the same as what he would give to some event which is
$\Omega$--measurable.
 
The construction
given above guarantees that conditional 
probabilities attached to the conditionally
$\Omega$--measurable events 
correspond appropriately to the conditional 
probabilities given the analogous events in
$\Omega$.  The following lemma states this more precisely.
 
\begin{lemma}\label{meas-match}
For all $\omega, 
\omega'\in \Omega$, all $i$, and all $k'\le (N-1)I + j(i,\omega')+1$,
 \[ \sum_{k} \bar \mu^N[(\omega,k)\mid \bar \Pi_i^N(\omega', k')] 
  = \mu_i[\omega\mid \Pi_i(\omega')]. \]
Hence for any $i$, any $\omega'$, any $k'\le (N-1)I + j(i,\omega')+1$,
and any event $E$ which is conditionally $\Omega$--measurable by $i$ at 
$(\omega', k')$, the probability $i$ gives to $E$ at $(\omega',k')$ in 
${\cal M}^N$ 
is the same as the probability $i$ gives to the $\Omega$ projection of
$E$ at $\omega'$ at ${\cal M}$.
\end{lemma}
 
\nipar{\em Proof of Lemma.}  First, 
note that if $\omega\notin \Pi_i(\omega')$, then the statement of the
lemma holds trivially since both sides of the 
equation are zero.  So suppose $\omega \in
\Pi_i(\omega')$.  Then we wish to show that
\begin{equation}\label{eq}
 \sum_{k\in R_i(\omega',k')} {\bar \mu^N(\omega,k)\over \sum_{(\omega'',k'')\in
 \bar \Pi_i^N(\omega',k')} \bar \mu^N(\omega'',k'')} = {\mu_i(\omega)\over
 \sum_{\omega''\in \Pi_i(\omega')} \mu_i(\omega'')}
\end{equation}
where $R_i(\omega',k')$ is the set of $k$ such that $(\omega, k)\in \bar
\Pi_i^N(\omega', k')$.  (Note that this is independent of $\omega$ as
long as $\omega\in \Pi_i(\omega')$.)
To show this, first consider $k' \le j(i,\omega')$.  For $k'$ in this
range, the left--hand side of (\ref{eq}) is
 \begin{eqnarray*}
  \sum_{k=1}^{2N} \bar\mu^N[(\omega,k)\mid \bar \Pi^N_1(\omega',1)] 
    &=& 
  {\sum_{k=1}^{j(i,\omega)} \alpha z_k(\omega) \over
  \sum_{\omega''\in \Pi_i(\omega')}\sum_{k=1}^{j(i,\omega'')} \alpha
    z_k(\omega'')}\\
    &=& 
  {\mu_i(\omega)\over \sum_{\omega''\in \Pi_i(\omega')}
   \mu_i(\omega'')}, 
 \end{eqnarray*}
as was to be shown.  
 
For $k'$ strictly between $j(i,\omega')$ and $(N-1)I + j(i,\omega') +1$, 
the event $\bar \Pi_i^N(\omega', k')$ 
takes the form $\bar \Pi^N_i(\omega', (n-1)I + j(i,\omega')+1)$ for some $n$ in
$\{1,\ldots, N-1\}$.  Analogously to the above, for $\omega\in \Pi_i(\omega')$, 
 \[ \sum_{k=1}^{2N} \bar\mu^N[(\omega,k)\mid \bar \Pi^N_1(\omega',(n-1)I + j(i,
  \omega') + 1)]
 = {\sum_{k=(n-1)I+j(i,\omega)+1}^{nI + j(i,\omega)} \alpha
   z_k(\omega) \over 
   \sum_{\omega''\in \Pi_i(\omega')}\sum_{k=(n-1)I + j(i,\omega'') + 
   1}^{nI + j(i,\omega'')} \alpha z_k(\omega'')} \]
After cancelling $\alpha$ from numerator and denominator, the
numerator is
 \[ (\beta+1)^{n-1} \sum_{k=j(i,\omega) + 1}^{j(i,\omega) + I}
   z_k(\omega)
 = (\beta+1)^{n-1} \left\{ \sum_{k=j(i,\omega)+1}^I \ell_k(\omega) +
   z_{I+1}(\omega) + (\beta + 1)\sum_{k=2}^{j(i,\omega)}
   \ell_k(\omega) \right\} \]
where, to cover the case of $j(i, \omega) = 1$, we treat
$\sum_{k=2}^1 x_k = 0$ for any $x_k$ sequence.  (The denominator is
analogous, where $\omega''$ replaces $\omega$ and we sum over
$\omega''$ in $\Pi_i(\omega')$.)  Substituting for
$z_{I+1}(\omega)$ on the right--hand side and rearranging gives
\begin{eqnarray*} 
 \lefteqn{(\beta+1)^{n-1}\left\{\sum_{k=j(i,\omega)+1}^I \ell_k(\omega) 
    + (\beta +1)\sum_{k=1}^{j(i,\omega)} \ell_k(\omega) 
    - \max(\omega)\right\}}\hspace{2in}\\
  &=& (\beta+1)^{n-1}\left\{\sum_{k=1}^I \ell_k(\omega) + \beta
   \sum_{k=1}^{j(i, \omega)} \ell_k(\omega) - \max(\omega)\right\}\\
  &=& (\beta+1)^{n-1} \beta \sum_{k=1}^{j(i, \omega)} 
      \ell_k(\omega)\hspace{1.2in} \\
  &=& (\beta+1)^{n-1} \beta  \mu_i(\omega).
\end{eqnarray*}
Hence
 \[ \sum_{k=1}^{2N} \bar \mu^N[(\omega,k)\mid \bar \Pi^N_1(\omega',(n-1)I + j(i,
  \omega') + 1)]
 = {\mu_i(\omega)\over \sum_{\omega'' \in \Pi(\omega')}
  \mu_i(\omega'')}, \]
as was to be shown.~\qed
 
I now use this lemma to show
that for any $\omega'\in \Omega$, $n^{\rm th}$ order 
beliefs at state $(\omega', 1)$ in ${\cal
M}^N$ are the same as $n^{\rm th}$ order beliefs at $\omega'$ in ${\cal M}$ 
for all $n\le N$.  I show this by demonstrating the stronger statement
that for all $n\le N$, all $i$,
all $\omega'\in \Omega$, and all $k'\le (N-n)I + j(i,
\omega')$, $\imath$'s $n^{\rm th}$ order beliefs at $\omega'$ in ${\cal M}$ 
are the
same as his $n^{\rm th}$ order beliefs at $(\omega',k')$ in ${\cal M}^N$.
This is shown by simply establishing that all relevant events in $\bar
\Omega^N$ are conditionally $\Omega$--measurable 
and then appealing to Lemma~\ref{meas-match}.
 
The proof is by induction on $n$.  So first consider $n=1$.  
Note that the value of $s$ at a state
$(\omega, k)$ is independent of $k$.  That is, for any value of $s$,
the event that the parameter takes on this value is 
$\Omega$--measurable.  Hence by Lemma~\ref{meas-match}, for all $i$,
all $\omega'$, and all $k'\le (N-1)I+j(i,\omega') + 1$, $\imath$'s
first--order beliefs at $(\omega',k')$ must be the same as his 
first--order beliefs at $\omega'$.
 
So suppose we have established that for all $i$, all
$\omega'\in \Omega$, and all $k'\le (N-n)I + j(i,
\omega')$, $\imath$'s $n^{\rm th}$ order beliefs at $(\omega',k')$ in 
${\cal M}^N$ are the same as his $n^{\rm th}$ order beliefs at $\omega'$
in ${\cal M}$ for $n=1,\ldots, \bar n -1$ where $\bar n -1 < N$.
I now show that this implies that the same is true for $n=\bar n$.  
 
So fix any $i$, any $\omega'$, and any $k'\le (N-\bar n)I + j(i,\omega')$.
Recall that $\imath$'s $\bar n^{\rm th}$ order beliefs are a probability
distribution on tuples consisting of $s$ and the $n^{\rm th}$ order
beliefs of the other players for all $n < \bar n$.  Fix any such tuple
and let $E$ denote the event where the parameter and lower order beliefs
take on this value.  Suppose 
$(\omega, k)\in E~\cap~\bar \Pi^N_i(\omega', k')$.  Clearly, then, 
at every $(\omega, k'')$, we have the same value of the parameter since
its value is independent of $k$.  Note from the construction of
partitions, that we must have $k\le (N-\bar n)I + j(i,\omega')$.
Since $j(i,\omega') \le I$ for all $i$, we have
 \[ k\le (N-\bar n)I + j(i,\omega') \le (N - \bar n)I + I < (N - (\bar n
        -1))I + j(j,\omega').\]
By the induction hypothesis, then, for any
$j\ne i$, $\jmath$'s lower order beliefs at $(\omega,k)$ 
must be the same as his lower order beliefs at $(\omega, k'')$ for all
$k'' \le (N- \bar n + 1)I + j(j,\omega)$ since both must match his
lower order beliefs at $\omega$ in the original model.  Since, as shown
above,
 \[ (N- \bar n)I + j(i,\omega) < (N - \bar n +1) I + j(j,\omega), \]
this implies that $E$ is conditionally $\Omega$--measurable for $i$ at 
$(\omega', k')$.  Hence by Lemma~\ref{meas-match}, $\imath$'s $\bar
n^{\rm th}$ order beliefs at $(\omega', k')$ in ${\cal M}^N$ must equal
his $n^{\rm th}$ order beliefs at $\omega'$ in ${\cal M}$, completing
the proof.~\qed
 
\section{Proof of Lemma~\ref{wc=wc}}
 
Suppose $u^*$ is weakly consistent and generates a countable 
belief--closed subspace $U$.  I define a partitions model which is equivalent to
it and is consistent.  So let $\Omega=U$.  Define $f$ by setting $f(u)$
equal to the projection of $u$ onto $S$ for each $u\in U$.  Define
player $\imath$'s partition by setting $\Pi_i(u)$ equal to the set of
$u'\in U$ such that the projection of $u'$ onto the type space for $i$
is the same as the projection of $u$.  In other words, $u'\in \Pi_i(u)$
if and only if $\imath$'s type at $u'$ is the same as his type at $u$.
Obviously, this gives a partition for each player.
 
To define priors, first fix a probability distribution $\phi_i$ on
$\Pi_i$ for each $i$ where we require $\phi_i(\pi) > 0$ for all $\pi\in
\Pi_i$.  Then define $\imath$'s prior probability on $u$ to be
$\phi_i(\Pi_i(u))$ times the probability his type at $u$ puts on $u$.
It is easy to see that this definition gives a partitions model which is
equivalent to $U$.  Also, by weak consistency, we know that every player
$i$ at every world $u$ puts strictly positive probability on $u$.  Hence
$\mu_i(u) > 0$ for all $i$ and all $u\in \Omega$.  Hence the partitions
model is weakly consistent.~\qed
 
\section{Proof of Lemma~\ref{near-consis}}
 
In light of Mertens--Zamir's Theorem 3.1, 
we only need to show that
given any world 
generating a finite belief--closed subspace, there is another
arbitrarily nearby which is weakly consistent.  Let $u^*$ denote 
a world which generates a finite belief--closed subspace and let $U$
denote this subspace.  Note that the finiteness of $U$
for each player $i$, every $u\in U$,
and every $k$, $\imath$'s $k^{\rm th}$ order beliefs at $u$ 
have a finite
support.
 
So fix a sequence $\{\varepsilon_n\}$ converging to zero from above.
For each $u\in U$, I construct a sequence of worlds $\{u^n(u)\}$ as
follows.  Let $s(u)$ be the
parameter value at $u$ (the projection of $u$ onto $S$).  Let $s(u)$ also
be the parameter value at $u^n(u)$ for all $n$ and all $u$.  
For any player $i$, let
$\imath$'s first--order beliefs at $u^n(u)$ be a convex combination of 
his first--order beliefs at $u$ and a
degenerate distribution with probability 1 on $s(u)$ with weight $1 -
\varepsilon_n$ on the former and $\varepsilon_n$ on the latter.
   
Second--order beliefs are slightly more complex.  Player
$\imath$'s second--order beliefs at $u^n(u)$ give the pair consisting of 
$s(u)$ and the true first--order beliefs at $u^n(u)$ probability equal
to $\varepsilon_n$ plus $1-\varepsilon_n$ times the probability $\imath$'s
second--order beliefs at $u$ gave to the pair $s(u)$ and the true 
first--order beliefs at $u$.  For the pair consisting of
$s(u')$ and the first--order
beliefs at $u^n(u')$, player $\imath$ gives probability $1 -
\varepsilon_n$ times the probability his second--order beliefs at $u$ 
gave to
$s(u')$ and the true first--order beliefs at $u'$.  Probability 0 is
given to any other pair.
 
We proceed analogously to this for higher levels.  That is, player
$\imath$'s $k^{\rm th}$ order beliefs at $u^n(u)$ give the tuple
consisting of $s(u)$ and the true lower order beliefs at
$u^n(u)$ probability equal to $\varepsilon_n$ plus $1 - \varepsilon_n$
times the probability his beliefs at $u$ gave to the pair consisting of
$s(u)$ and the true lower order beliefs at $u$.  His beliefs
give the tuple consisting of $s(u')$ and the true lower order
beliefs at $u^n(u')$ probability equal to $1 - \varepsilon$ times the
probability his beliefs at $u$ gave $s(u')$ and the true lower 
order beliefs at $u'$.  Probability 0 is given to any other tuple.
 
It is easy to show that the following construction on the types space is
equivalent.  We simply define $u^n(u)$ to be the world with parameter
$s(u)$ and where the type of player $i$ is the one with beliefs equal to
a convex combination of probability 1 on $u^n(u)$ and a distribution,
say $\delta^n_i$, described shortly, with weight $\varepsilon_n$ on the
former and $1 - \varepsilon_n$ on the latter.  The distribution
$\delta^n_i$ simply gives $u^n(u')$ the same probability that $\imath$'s
type gave $u'$ at world $u$.
 
By this construction, we see that these beliefs do indeed satisfy common
knowledge of coherence in the sense of
Brandenburger and Dekel~\cite{BrandenburgerDekel1993}, so $u^n(u)
\in {\cal U}$ for all $n$ and all $u\in U$.  The construction shows that
each $u^n(u)$ has the property that every
player believes $u^n(u)$ possible at $u^n(u)$ and that 
for each $n$, $U^n = \{u^n(u)\mid u\in U\}$ is 
belief--closed.  Hence for every $n$ and $u\in U$, $u^n(u)$ is weakly
consistent.
 
It is not hard to show by induction that $u^n(u) \rightarrow u$ as 
$n\rightarrow \infty$.  To see this, note that for any player $i$ and
any $u$, $\imath$'s first--order beliefs along the sequence $\{u^n(u)\}$
always have a finite support.  For any point, say $s'$, in the support of
$\imath$'s first--order beliefs at $u$, there is a sequence of points, say $\{s^n\}$
in
the supports of his beliefs at $\{u^n(u)\}$ converging to $s'$ and the
sequence of probabilities given to $s^n$ converges to the
probability given $s'$ at $u$.  This is, of course, sufficient for
convergence in the weak$^*$ topology.  Hence first--order beliefs for
every player converge.  But then this establishes precisely the
analogous property for second--order beliefs, so that for every player
$i$, $\imath$'s second--order beliefs along the sequence $\{u^n(u)\}$
converge to his second--order beliefs at $u$, etc.  Hence $u^n(u)$
converges pointwise --- and therefore in the product topology --- to
$u$.~\qed
 
\vfill\eject
 
\begin{thebibliography}{99}
 
\bibitem{Aumann1976} Aumann, R., ``Agreeing to Disagree,'' {\it Annals
of Statistics}, {\bf 4}, 1976, pp.~1236--1239.
 
\bibitem{Aumann1987} Aumann, R., ``Correlated 
Equilibrium as an Expression of
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\end{document} 
 
 
 
 


  Define
 $$\min(\omega) = \min\{\mu_1(\omega),\mu_2(\omega)\},$$
 $$\resid(\omega) = \max\{\mu_1(\omega), \mu_2(\omega)\} -
  \min(\omega),$$
and
 $$\beta = 1 + \max_{\omega\in \Omega} {\resid(\omega)\over
  \min(\omega)}.$$ 
Let
 $$\bar \Omega ^k = \Omega\times \{1,\ldots,2^k+1\}.$$
Naturally, let $\bar f^k(\omega,j) = f(\omega)$.  The common prior
$\bar \mu^k$ is given by 
 $$\bar\mu^k(\omega,j) = \alpha z_j(\omega)$$
where $\alpha > 0$ is a constant to be determined and $z_j(\omega)$
is defined by
 $$z_1(\omega) = \min(\omega)$$
 $$z_2(\omega) = \resid(\omega)$$
 $$z_3(\omega) = \beta\min(\omega) - \resid(\omega)$$
and for $j\ge 4$,
 $$z_j(\omega) = \beta z_{j-2}(\omega).$$
If all the $z_j$'s are positive, then we can choose $\alpha$ so that
the $\bar \mu^k$'s sum to 1 and ensure that this is a probability
distribution.  By construction, $\min(\omega) > 0$ and
$\resid(\omega) \ge 0$ for all $\omega$.  Hence this is a legitimate
probability distribution as long as
 $$\beta\min(\omega) \ge \resid (\omega)$$
which is guaranteed for all $\omega$ from the definition of $\beta$.
 
The partitions are as follows.  First, fix any $\omega'$ with
$\mu_i(\omega') = \min(\omega')$.  Let
\[ \bar \Pi^k_i(\omega', 1) =
  \left\{(\omega,j)\left| \begin{array}{l} \omega\in  \Pi_i(\omega')
     ~~\mbox{and either}~~\\   
    ~~~~(1)~\mu_i(\omega) > \min(\omega)~\mbox{and}~j=1~{\rm
      or}~j=2\\
    ~\mbox{or} ~(2)~\mu_i(\omega) = \min(\omega)~\mbox{and}~j=1
   \end{array} \right.\right\} \]
For $\jmath' \le 2^{k-1}$, let 
\[ \bar \Pi^k_i(\omega', 2\jmath') 
    =\bar \Pi^k_i(\omega', 2\jmath'+ 1) = 
  \left\{(\omega,j)\left| \begin{array}{l} \omega\in  \Pi_i(\omega')
     ~~\mbox{and either}~~\\   
    ~~~~(1)~\mu_i(\omega) > \min(\omega)~ \mbox{and}~ j=2\jmath' +
        1~\mbox{or}~j=2\jmath'+2\\  
    ~~~\mbox{or} ~(2)~\mu_i(\omega) = \min(\omega)~\mbox{and}~j=2\jmath'
        ~\mbox{or}~j=2\jmath'+1
     \end{array} \right.\right\} \]
This completely defines $\bar\Pi^k_i(\omega',j)$ for all values of
$j$ whenever $\mu_i(\omega') = \min(\omega')$.
 
Now suppose $\omega'$ with $\mu_i(\omega') > \min(\omega')$.  In this
case,
\[ \bar \Pi^k_i(\omega', 1) 
   = \bar \Pi^k_i(\omega',2) =
  \left\{(\omega,j)\left| \begin{array}{l} \omega\in  \Pi_i(\omega')
     ~~\mbox{and either}~~\\   
    ~~~~(1)~\mu_i(\omega) > \min(\omega)~\mbox{and}~j=1~{\rm
      or}~j=2\\
    ~\mbox{or} ~(2)~\mu_i(\omega) = \min(\omega)~\mbox{and}~j=1
   \end{array} \right.\right\} \]
For $1 < \jmath' \le 2^{k-1}$, let
\[ \bar \Pi^k_i(\omega', 2\jmath'-1) 
    =\bar \Pi^k_i(\omega', 2\jmath') = 
  \left\{(\omega,j)\left| \begin{array}{l} \omega\in  \Pi_i(\omega')
     ~~\mbox{and either}~~\\   
    ~~~~(1)~\mu_i(\omega) > \min(\omega)~ \mbox{and}~ j=2\jmath' -
        1~\mbox{or}~j=2\jmath'\\  
    ~~~\mbox{or} ~(2)~\mu_i(\omega) = \min(\omega)~\mbox{and}~j=2\jmath'-2
        ~\mbox{or}~j=2\jmath'-1
     \end{array} \right.\right\} \]
This completely defines $\bar\Pi^k_i(\omega',j)$ for all values of
$j\le 2^k$ whenever $\mu_i(\omega') > \min(\omega')$.  To conclude,
then, let
 $$\bar \Pi^k_i(\omega', 2^k+1) = \left\{(\omega,j)\mid
  \mu_i(\omega) > \min(\omega)~\mbox{and}~j = 2^k+1 \right\}.$$  
 
\begin{claim}
For any $\omega\in \Omega$, beliefs at state $(\omega, 1)$ in $(S,
\bar \Omega^k, \bar f^k, \{\bar \Pi^k_i, \bar\mu^k\}_{i\in {\cal I}})$ are
the same as the beliefs at $\omega$ in $(S, \Omega, f, \{\Pi_i,
\mu_i\}_{i\in {\cal I}})$ at least up to order $2^k -1$.
\end{claim}
 One way to show this is to construct a ``tree'' for each
player giving the events of $\bar \Omega^k$ which are relevant to
determining his beliefs.  Intuitively, the tree for player 1 
begins with an initial node labelled
$\{(\omega_1,1), (\omega_1, 2), (\omega_2,1)\}$ since it is 1's 
beliefs at this event
that we are interested in.  The immediate successors of this node are labelled
by the events in 2's partition that 1 thinks possible given his event.  That is,
there are two immediate successors, one labelled 
$\{(\omega_1, 1), (\omega_2,1), (\omega_2,2)\}$ and one labelled
$\{(\omega_1,2), (\omega_2,3), (\omega_2,4)\}$.  Each of these nodes is labelled
by the collection of events from 1's partition that 2 thinks possible 
at the given event.  For example, $\{(\omega_1,1), (\omega_2,1), (\omega_2,2)\}$
has two immediate successors, one labelled
$\{(\omega_1,1), (\omega_1,2), (\omega_2,1)\}$ and one labelled
$\{(\omega_1,3), (\omega_1,4), (\omega_2,2)\}$.  More generally, given any node
labelled $E\in \bar \Pi^k_i$, $i=1,2$, the set of immediate
successors of $E$ is labelled by the set of events 
 \[ \{E'\in \bar \Pi^k_j\mid E'=\bar \Pi^k_2(\omega),~{\rm for~some}~
  \omega \in E\} \]
where $j\ne i$.  That is, there is exactly one immediate
successor for each element of this set and each label is used exactly
once.  The tree for player 2 has $\{(\omega_1,1), (\omega_2,1),
(\omega_2,2)\}$ as the label for its initial node with successors
defined and labelled analogously.  (See Figure 1 for the first few
levels of the tree for player 1.)
 
We can use this tree to calculate a player's hierarchy of beliefs.
To determine player $\imath$'s first--order beliefs, we
take the prior and condition on the unique event which labels his initial node.
More specifically, player 1's initial node is labelled
$\{(\omega_1,1),(\omega_1,2),(\omega_2,1)\}$.
