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%TCIDATA{Created=Wed Oct 08 12:09:49 1997}
%TCIDATA{LastRevised=Thu Oct 30 15:27:37 1997}

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\title{Market Crashes Without External Shocks\thanks{%
This is a minor revision of the previous version of December 1996. We thank
Robert J. Aumann, Avraham Beja, Yaakov Bergman, Marcel Brachfeld, Yoram
Halevy, Andreu Mas-Colell, Abraham Neyman and Dov Samet for interesting
discussions on these topics.}}
\author{Sergiu Hart\thanks{%
Center for Rationality and Interactive Decision Theory; Department of
Economics; and Department of Mathematics; The Hebrew University of
Jerusalem, 91904 Jerusalem, Israel. e-mail: hart@math.huji.ac.il.
http://www.ma.huji.ac.il/\symbol{126}hart. Research partially supported by
grants of the U.S.-Israel Binational Science Foundation and of the Israel
Academy of Sciences and Humanities.} \and Yair Tauman\thanks{%
Recanati School of Business, Tel Aviv University, 69978 Tel Aviv, Israel and
Department of Economics, SUNY at Stony Brook, Stony Brook, NY 11794, U.S.A.}}
\maketitle

\begin{abstract}
It is shown here that market crashes and bubbles can arise without external
shocks. Sudden changes in behavior coming after a long period of
stationarity may be the result of \emph{endogenous }information processing.
Except for the daily observation of the market, there is no new information,
no communication and no coordination among the participants.
\end{abstract}

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\section{Introduction}

Market crash is usually considered an indication that the fundamentals of
the economy have changed and recession is around the corner. This however
need not be so. For instance, in October 1987 Wall Street lost over 20\% of
its value in one day, but this was not followed by a recession. Moreover, in
the days preceding the crash, there were no significant external events or
``bad news'' that could justify the dramatic price fall.

We argue here that market crashes (and, similarly, market bubbles) may well
be the result of information processing by the participants --- and \emph{%
nothing else}. Moreover, in terms of market observables, it looks as if
nothing is really changing. Still, underneath the surface, there is a
gradual updating of information by the participants. Then, at a certain
point in time, this causes a sudden change of behavior.

Specifically, the phenomenon we describe here has to do with the
step-by-step advance in levels of ``mutual knowledge'' (what one knows about
what the other knows ...). Each trading day increases this level through the
daily market observables --- prices, quantities traded, etc. --- which are
common knowledge. However, the behavior does not necessarily change with the
level of mutual knowledge. The behavior may be constant for all levels up to
a certain critical level --- where a jump occurs. Such a phenomenon has been
exhibited in Geanakoplos and Polemarchakis [1982].\footnote{%
For other situations with similar behavior, like the well-known puzzle of
``the 40 cheating wifes and the missionary,'' see the survey of Geanakoplos
[1994].} In their paper two agents communicate information to each other by
repeatedly announcing and revising their posteriors. It is shown there that
the agents can repeat exactly the same opinions, yet still manage to
communicate relevant information. Here, we fit their example to our model
and show that there is no need for any direct communication between the
traders; observing the market suffices.

We emphasize, first, that the phenomenon exhibited here is completely
endogenous. Except for the daily observation of the market, there is no new
information --- whether pertinent or irrelevant (like ``sunspots''); also,
there is no communication nor any coordination among the participants ---
whether expressed or tacit. And second, the stationary unchanging behavior
of the market for arbitrarily long periods of time is no sign that nothing
is happening. Underneath the surface, completely unobservable, information
is being processed by the participants --- which ultimately leads to a
sudden change of behavior.

The behavior of the agents in the example will turn out to resemble rules
actually used by traders in the market. For instance: ``I am buying every
day; but, if others keep selling every day, then at some point I will start
selling too.'' Some of the so-called ``technical analysis'' is indeed of
this kind. Intuitively, if others are willing to sell all the time, then the
buyer will at some time have to take this fact into account: perhaps his
assessment is incorrect after all. The framework of this paper allows to
make such arguments precise.

We present here a simple example. However, the phenomenon we highlight is
robust and general. One may change almost every feature of the example (like
the state space, the probabilities, the number of traders, the daily rules
of behavior, the prices, and so on), without qualitatively affecting the
result.

To summarize: No exogenous ``shocks'' are needed to explain sudden
departures from stationary behavior; these may well be due to the
participants updating their information, based on the market observables.
This information processing is however not observable --- there is no change
in behavior --- up to the point where it generates an abrupt switch of
behavior: a ``crash.''

\section{The Example}

There are two traders. Day after day they keep trading --- one is selling
and the other is buying. Then, at some time, the updating of information
leads one of them to reverse his behavior --- from buying to selling. This
happens in the absence of any exogenous influence.

Let $I$ be the set of all states of the world. Assume $I$ contains nine
states, say, $I=\{1,2,\ldots ,9\}$. We assume a common prior probability
assessment,\footnote{%
This ``common priors'' assumption plays no role here; it is used for
simplicity only. See Remark (5) in Section 3 below.} namely the uniform
distribution $P(w)=1/9$ for all $w\in I$.

The private information of each trader is described as usual\footnote{%
For a formal treatment, see Aumann [1995].} by a partition of the state
space: Two states belong to the same block if and only if the trader cannot
distinguish between them. Call the two traders Alice and Bob. Alice's
partition is 
\[
A_{1}=\{1,2,3\},\quad A_{2}=\{4,5,6\},\quad A_{3}=\{7,8,9\},
\]

\noindent and Bob's partition is 
\[
B_{1}=\{1,2,3,4\},\quad B_{2}=\{5,6,7,8\},\quad B_{3}=\{9\}. 
\]

\noindent The interpretation is as follows: Trader Bob, for example, cannot
distinguish between states $1,2,3$ and $4$, nor between $5,6,7$ and $8$. If
state $9$ is the true state, then Bob will know that for sure. If, however,
state $1$ is the true state, then Bob will only know that it is either $1$
or $2$ or $3$ or $4$ (but not $5,6,7,8,9$).

Consider the event $E = \{1, 5, 9\}$. For instance, assume that $E$ is the
event of a ``bad'' outcome (e.g., the company earnings will go down).
Suppose that each one of the two traders behaves each day according to the
following rule:

\[
\left\{ 
\begin{array}{ll}
\mathrm{SELL,} & \mathrm{{if~the~probability~of}~}E\mathrm{~{is~}}0.3\mathrm{%
~or~more;} \\ 
\mathrm{BUY,} & \mathrm{{if~the~probability~of}~}E\mathrm{~{is~less~than~}}%
0.3\mathrm{.}
\end{array}
\right. 
\]

\noindent Of course, the relevant probability is always computed given the
current information.

The manner in which the cutoff point of $0.3$ (or, for that matter, the
whole policy) is determined, is irrelevant to the analysis here; in
particular, we abstract away from the stock price and the quantities.%
\footnote{%
For concreteness, one may assume that ``sell'' actually means ``sell the
quantity $q$ for the price $p$,'' and ``buy'' means ``buy the quantity $q$
for the price $p$'' (where $q$ and $p$ are the same for both decisions).
Since, as we will see below, every day (up to the ``crash'') Alice will sell
and Bob will buy, the price need not change. If so desired, varying prices
and quantities can be added to the model. This will make the analysis more
complex, but---once the traders' policies are defined appropriately---will
not affect the phenomenon we exhibit.} Also, we note that it does not matter
whether both traders use the same behavior strategy.

Assume that the true state of the world is $w_{0}=1$. Initially, Alice
assesses the probability of $E$ to be $1/3$ (since at $w_{0}=1$ she knows
that the true state is either $1$ [which belongs to $E$], or $2$ or $3$
[which do not belong to $E$]) --- therefore she gives an order to sell; Bob
assesses this probability to be $1/4$ (he knows the state is $1,2,3$ or $4$)
--- therefore he gives an order to buy. So a transaction takes place.

We will show that this will happen not only on the first day, but also on
each one of the first four days: the assessments of the two traders for the
probability of $E$ remain $1/3$ and $1/4$, respectively. On the fifth day,
however, there is a sudden and major change: Both assessments become $1/3$
and both traders give orders to sell. So, a ``crash'' occurs after four
seemingly ``quiet and normal'' days.

Let us see this in detail (see Table 1). Let $t=1,2,\ldots $ denote the
``days,'' and $I^{t}$ the common knowledge information at time $t$, before
the traders choose their actions.

On day $t=1$, we have $I^{1}=I$. The assessments of Alice are 
\[
P(E\mid A_{i}\cap I^{1})=P(E\mid A_{i})=1/3,\quad i=1,2,3,
\]

\noindent and those of Bob are

\[
P(E \mid B_j \cap I^1) = P (E \mid B_j) = \left\{ 
\begin{array}{ll}
1/4, & \mathrm{if} ~j = 1, 2, \\ 
1, & \mathrm{if} ~j = 3.
\end{array}
\right. 
\]

\noindent Since $w_{0}=1$, the current information is $A_{1}$ for Alice and $%
B_{1}$ for Bob, so Alice sells and Bob buys. Note that Bob only computes $%
P(E\mid B_{1})$ (he knows that the true state is in $B_{1}$). However, for
the sequel, one also needs to know what Bob would have computed --- and done
--- in the other states as well.

On day $t=2$, it is common knowledge that Bob bought on the previous day,
therefore it is common knowledge that $B_{3}=\{9\}$ did not occur.\footnote{%
Note that, at $w_{0}=1$, both traders initially knew that the state is not 9
(so this was \emph{mutually known}); however, it was \textbf{not} \emph{%
commonly known}. To see it, let $F=\{1,2,\ldots 8\}$ and let $K_{A}F$ be the
event ``Alice knows $F$.'' Similarly, $K_{B}K_{A}F$ is the event ``Bob knows
that Alice knows $F$,'' etc. Then $K_{A}F=\{1,2,\ldots ,6\}$, $%
K_{B}K_{A}F=\{1,2,3,4\},K_{A}K_{B}K_{A}F=\{1,2,3\}$ and $%
K_{B}K_{A}K_{B}K_{A}F=\phi $. Thus, it is never the case that Bob knows that
Alice knows that Bob knows that Alice knows that the state is not $9$.
Hence, in no state of the world --- in particular, at $w_{0}=1$ --- is $F$
common knowledge.} Therefore, the new common knowledge information is $%
I^{2}=\{1,2,\ldots ,8\}$. The new assessments are:

\[
P(E \mid A_i \cap I^2) = \left\{ 
\begin{array}{ll}
1/3, & \mathrm{if} ~i = 1, 2, \\ 
0, & \mathrm{if} ~i = 3,
\end{array}
\right. 
\]

\noindent for Alice, and 
\[
P(E\mid B_{j}\cap I^{2})=1/4,\quad j=1,2, 
\]

\noindent for Bob. Again, at $w_0 = 1$ a transaction takes place: Alice
sells and Bob buys.

The new common knowledge information on day $t=3$ is $I^{3}=\{1,2,\ldots ,6\}
$ (since Alice would not have sold at $t=2$ if the state was in $A_{3}$),
and we have on day 3: 
\[
P(E\mid A_{i}\cap I^{3})=1/3,\quad i=1,2,
\]

\noindent and 
\[
P(E \mid B_j \cap I^3) = \left\{ 
\begin{array}{ll}
1/4, & \mathrm{if} ~j = 1, \\ 
1/2, & \mathrm{if} ~j = 2.
\end{array}
\right. 
\]

\noindent Thus there is a transaction on day 3, and on day 4 it is common
knowledge that the state is in $I^{4}=\{1,2,3,4\}$ (since $B_{2}$ is
commonly ruled out), and thus 
\[
P(E\mid A_{i}\cap I^{4})=\left\{ 
\begin{array}{ll}
1/3, & \mathrm{if}~i=1, \\ 
0, & \mathrm{if}~i=2,
\end{array}
\right. 
\]

\noindent and

\[
P(E\mid B_{j}\cap I^{4})=1/4,\quad j=1. 
\]

\noindent Finally, on day 5 we get $I^{5}=\{1,2,3\}$ (since $A_{2}$ is
commonly ruled out),

\[
P(E\mid A_{i}\cap I^{5})=1/3,\quad i=1, 
\]

\noindent and

\[
P(E\mid B_{j}\cap I^{5})=1/3,\quad j=1. 
\]

\noindent But now both Alice and Bob send orders to sell and a ``crash''
occurs.

What is happening in this example is the following. Initially, both Alice
and Bob know that the state is\footnote{%
Alice even knows more: She knows that the state is not $4$.} $1,2,3$ or $4$.
However, this fact is $\emph{not}$ common knowledge among them. For example,
from Bob's point of view, the state could well be $4$, in which case Alice
would have known that it is either $4,5$ or $6$. So Bob does \emph{not} know
that Alice knows that it is $1,2,3$ or $4$. As time goes by, the trading
increases the common knowledge (see the last column of Table 1), until, on
day 5, it reaches its conclusion: it is common knowledge that $w=1,\;2$ or $3
$.

\section{Remarks}

We conclude with a number of remarks.

(1) We have assumed throughout that $w_{0}=1$; i.e., a ``bad'' state (in $E$%
) is the true state. In the end (day 5), both traders indeed want to sell
(the two traders, even by pooling their information, cannot distinguish
between states $1,2$ and $3$; they both sell since the probability
assessment of $E$ ends up being $1/3$). However, exactly the same behavior
would have resulted if the true state were $w_{0}=2$ or $w_{0}=3$ --- which
are ``good'' states (not in $E$). Also, note that if $w_{0}=4$, then a
``bubble'' occurs (i.e., both buy) at $t=4$.

(2) A similar example consisting of $(n+1)^{2}$ (instead of $9$) states will
yield $2n$ days where transactions occur, and a ``crash'' on day $2n+1$ (see
Aumann's example, at the bottom of page 97 in Geanakoplos and Polemarchakis
[1982]). So a ``crash'' can be preceded by an arbitrarily long period of
time where trading occurs normally and nothing seems to change.

(3) If the state space is finite (or, more precisely, if the two information
partitions are finite), then there can be only finitely many instances of
information updating, after which the two assessments necessarily agree (the
latter, under the assumption of common priors; this is the ``Agreement
Theorem'' of Aumann [1976]).

(4) We have made the example as simple as possible; in particular, there are
only two traders. One may of course deal with more traders; one easy way is
to have two types of traders, A (like Alice) and B (like Bob).

(5) It is important to emphasize that, even though the example seems to be
very ``special,'' it is not. For instance, any small changes in the prior
probabilities --- including making these priors different for the two
traders --- will not affect it. What matters is only that the posterior
probabilities computed by the two traders remain above and below $0.3$,
respectively.\footnote{%
Actually, even the cutoff point of $0.3$ may change with time (by adjusting
the trading policies).}

(6) One may add to the model the stock price (changing according to market
demand and supply), allow for ``limit'' orders, varying quantities, and so
on. Defining trading policies accordingly --- in such a manner that the
information deduced remains the same --- will however preserve the
phenomenon we exhibit (recall also Footnote 4 above).

(7) In our example, Alice is initially better informed at $w_{0}=1$ than
Bob: She knows that Bob's information is $B_{1}$. Therefore she can
construct ahead of time the whole process as in Table 1. This can easily be
remedied by taking $w_{0}=5$, or, if one wants the same number of periods
before the crash, a bigger example as in (2) above.

(8) The only way for an order to be executed in our example is for the two
traders to give opposite orders. When this is taken into account by the
traders, their decisions may be affected. For example, at $t=4$, Bob reasons
as follows: The only case where my ``buy'' order will be executed is when
Alice sells; but that happens only when the state is $1,2$ or $3$ --- and
then I should sell, not buy! This is correct when Alice and Bob are the only
two traders in the market. Assume therefore that, besides the two ``major''
informed traders Alice and Bob, there are additional traders, and
transactions do not necessarily take place between the two. Still, each one
observes the moves of the other (they are, after all, the major players in
this market). Then the phenomenon in the example works as presented.

\begin{thebibliography}{9}
\bibitem{}  Aumann, R.J. [1976], ``Agreeing to Disagree,'' \emph{The Annals
of Statistics} 4, 1236--1239.

\bibitem{}  Aumann, R.J. [1995], ``Notes on Interactive Epistemology,''
Center for Rationality DP-67, The Hebrew University of Jerusalem (mimeo).

\bibitem{}  Geanakoplos, J. [1994], ``Common Knowledge,'' \emph{Handbook of
Game Theory}\textit{, }Volume 2, R.J. Aumann and S. Hart (eds.), Elsevier /
North-Holland, 1437--1496.

\bibitem{}  Geanakoplos, J. and H.M. Polemarchakis [1982], ``We Can't
Disagree Forever,'' \emph{Journal of Economic Theory} 28, 192--200.
\end{thebibliography}

\end{document}