Asset Pricing and Earnings Fluctuations in a Dynamic Corporate Economy
William Gordon Ritter
We give a new predictive mathematical model for macroeconomics, which deals specifically with asset prices and earnings fluctuations, in the presence of a dynamic economy involving mergers, acquisitions, and hostile takeovers. Consider a model economy with a large number of corporations $C_1, C_2, \ldots, C_n$ of different sizes. We ascribe a degree of randomness to the event that any particular pair of corporations $C_i, C_j$ might undergo a merger, with probability matrix $p_{ij}$. Previous random-graph models set $p_{ij}$ equal to a constant, while in a real-world economy, $p_{ij}$ is a complicated function of a large number of variables. We combine techniques of artificial intelligence and statistical physics to define a general class of mathematical models which, after being trained with past market data, give numerical predictions for certain quantities of interest including asset prices, earnings fluctuations, and merger/acquisition likelihood. These new models might reasonably be called ``cluster-size models.'' They partially capture the complicated dependence of $p_{ij}$ on economic factors, and generate usable predictions.
mergers.tex
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\begin{document}
\begin{frontmatter}
\title{Asset Pricing and Earnings Fluctuations in a Dynamic Corporate Economy}


\author{William Gordon Ritter}

\address{{\small Harvard University Department of Physics} \\
{\small 17 Oxford St., Cambridge, MA 02138} \\
{\it Email: ritter@fas.harvard.edu} \\
{\it Phone: 857-919-2572, Fax: 617-495-0416}}

\begin{abstract}
We give a new predictive mathematical model for macroeconomics,
which deals specifically with asset prices and earnings
fluctuations, in the presence of a dynamic economy involving
mergers, acquisitions, and hostile takeovers. Consider a model
economy with a large number of corporations $C_1, C_2, \ldots,
C_n$ of different sizes. We ascribe a degree of randomness to the
event that any particular pair of corporations $C_i, C_j$ might
undergo a merger, with probability matrix $p_{ij}$. Previous
random-graph models set $p_{ij}$ equal to a constant, while in a
real-world economy, $p_{ij}$ is a complicated function of a large
number of variables. We combine techniques of artificial
intelligence and statistical physics to define a general class of
mathematical models which, after being trained with past market
data, give numerical predictions for certain quantities of
interest including asset prices, earnings fluctuations, and
merger/acquisition likelihood. These new models might reasonably
be called ``cluster-size models.'' They partially capture the
complicated dependence of $p_{ij}$ on economic factors, and
generate usable predictions.
\end{abstract}

\begin{keyword}
Cont-Bouchaud model\sep asset pricing\sep macroeconomic
dynamics\sep econophysics

% PACS codes here, in the form: \PACS code \sep code
\PACS 89.65.Gh
\end{keyword}
\end{frontmatter}



\section{Introduction} \label{sec:intro}

A number of recent papers in theoretical macroeconomics have
explored the utility of mathematical models coming from
statistical physics. These theories typically model \emph{agents}
as vertices in a graph. Links between agents typically represent
some form of coordinated behavior, in which clusters of agents
have similar action on the market. The simplest and most
successful model along these lines is the percolation model of
Cont and Bouchaud \cite{cb}, which entails a random communication
structure between agents. Given two agents $i$ and $j$, denote by
$p_{ij}$ the probability that they are linked (\emph{linked} means
these two agents have the same belief of the future movements of
the asset price, and are similarly bullish or bearish). In order
to obtain a mathematically tractable model, Cont and Bouchaud
assume that $p_{ij} = p$ is a constant, independent of $i$ and
$j$. Thus the communication structure is a random graph, with
connected components assumed to have the same action on the
market. Such clusters are known to form in real financial markets,
and they contribute to \emph{herd behavior}, a phenomenon which is
well-documented but not completely understood by theorists.
However in real financial markets, clusters certainly do not form
randomly; we therefore define in the present paper a model which
attempts to capture this non-randomness.


In the Cont-Bouchaud model, at every time interval, each cluster
randomly selects to buy (probability $a$), sell (probability $a$)
or sleep (probability $1-2a$). The parameter $a$ lies in the range
$0 \leq a \leq 0.5$ and is called the \emph{activity} of the
agents. The asset pricing structure of the Cont-Bouchaud model is
very simple: aggregate excess demand, i.e. the sum of all orders,
is the sole driving force of the asset price. Excess buying drives
the price up, while excess selling drives it down. The price
adjustment is therefore given by a price impact function
\be \lab{price-impact}
    \ln P(t+1) - \ln P(t) = \f{1}{b} \sum_{\te{agents } i} D_i(t),
\ee
where $P(t)$ denotes the price at time $t$, $D_i(t)$ is the demand
of agent $i$ in period $t$ for the asset, and $b > 0$ is a
liquidity parameter describing how much excess demand is necessary
to move the asset price by one unit. The sign of $D_i(t)$
indicates buying or selling. In the Cont-Bouchaud model, $D_i(t)$
does not depend upon $i$. For each discrete time $t$, and for all
$i$,
\[
    D_i(t) = \begin{cases} +1 & \te{ with probability } a, \\
            -1 & \te{ with probability } a, \te{ and} \\
            0 & \te{ otherwise}
            \end{cases}
\]
This structure has been shown to mimic actual asset price dynamics
quite closely \cite{Stauffer}.

Since a link is allowed to form between any pair of investors, the
CB model is equivalently described as  mean field theory of
infinite-range bond percolation\footnote{See Section 8.4 of
\cite{Cardy} for background.}. There is a critical value of $p$,
denoted $p_c$ and known as the \emph{percolation threshold}, at
which an infinite-range cluster can be formed. The change in
log-price is proportional to $[\sum_{s \geq 1} s (n_s^+ - n_s^-)]$
where $n_s^+$ is the number of buying clusters of size $s$, and
$n_s^-$ is defined analogously for selling clusters. As discussed
on p.~157 of \cite{Cardy}, this number should be related to $\d /
\d Q |_{Q=1}$ of the one-point function of the $Q$-state Potts
model, although this connection has not been explored in the
literature.



In real financial markets, $p_{ij}$ is very far from being
constant; in fact $p_{ij}$ is a complicated function of a large
number of variables\footnote{We defer a discussion of which
variables might be involved to Section \ref{sec:ai}.}. The
principal goal of the present paper is to give a new, predictive
model for macroeconomic dynamics involving agents for which
$p_{ij}$ is not constant. A cornerstone of this new model,
discussed in Sec.~\ref{sec:ai}, is a procedure which effectively
determines $p_{ij}$ from past market data.

Although there does exist an analog of $p_{ij}$ in the new model,
and the new model does apply to asset pricing, it is not a direct
generalization of the Cont-Bouchaud model, and is substantially
more detailed. In our model, an individual site now represents a
single unit of wealth. Agents in our model are firms (such as
corporations or mutual funds) represented by clusters of sites.
The size of the cluster is a crucially important variable
representing the future earnings of the firm. The expectation of
future earnings (EFE) may of course fluctuate as a function of
time, and the amount of these fluctuations and which firms
experience them are, like $p_{ij}$, complicated functions of a
large number of variables.

In a real-world economy, fluctuations in asset prices are
partially the result of rational investors following sound
economic principles and seeking to optimize monetary gain, and
partially the result of other factors, psychological or
environmental, which by assumption do not lend themselves to
scientific analysis and, for modeling purposes, may be viewed as
random noise. The beauty of the new model introduced here lies in
the fact that it works the same way. Fluctuations in the EFE of a
firm, as well as the probabilities of mergers and acquisitions,
are determined by a process which simultaneously incorporates both
predictable economic trends, and the effect of randomness.


Monte Carlo time is a measure of efficiency of the simulation of a
complex system; however in this paper, we take the approach that a
Monte Carlo simulation of a lattice statistical system is a
particular type of cellular automaton, and in simulations of
dynamical systems using cellular automata, the ``time''
parameterizing the sequence of steps which update the cellular
automaton is often identified with physical time. This
identification of Monte Carlo time and physical time is key in
what follows.

By way of introduction, we illustrate the simplest possible
example of an interaction between two clusters of the same size.
In later sections, we of course wish to generalize, allowing a
large number of clusters of arbitrary sizes, and a much more
detailed analysis of their interactions.

Imagine a hypothetical fledgling economy, in which there are a
large number of consumers who demand a particular asset, but no
producers of this asset. This is represented by a large
two-dimensional lattice, with a minus sign on each lattice site to
denote an absence of producers\footnote{In physics, this is called
the ground state of the Ising model}. It will turn out that the
dimension of the lattice must always be at least two in order to
be useful to us; we defer a discussion of this point.


Since there is a demand without a corresponding supply, the system
is not in equilibrium, and so two firms are created with the
intent of producing this asset. Initially, the firms are
comparable in size and difficult to distinguish in other respects,
so analysts estimate the future earnings of each firm in the next
quarter to be 4, in some appropriate units.

One lattice site represents one unit of expected future earnings,
so each of these firms is denoted by the presence of a $2 \times
2$ matrix of plus signs occupying four lattice sites.
\[
    \epsfig{file=twofirms,width=1in}
\]
In our model, distance between connected regions in the lattice
plays the role of likelihood for a merger. If a horizontal merger
between the two firms in this hypothetical example is considered
likely, we place them at a distance of only one lattice site from
each other as in the figure. These simple binary variables which
take their values in the set $\{+,-\}$ are called \emph{spins} due
to an analogy with physics.


We define a \emph{link} to be an unordered pair of adjacent
lattice sites. For example, if the lattice is the 2d integral
lattice $\Z \times \Z$, then $b = \{(i,j), (i+1,j)\}$ is a link
for all $i, j \in \Z$. We define the sign of the link to be the
product of the signs at its two endpoints; so $(i,j)=+1,
(i+1,j)=-1$ represents a negatively signed link.


We define the ``energy'' of the configuration to be
\be \label{energy}
    E = N_{+-}
\ee
where $N_{+-}$ denotes the total number of negatively signed
links. It follows that the system with two firms depicted above
has energy $E = 16$.


Now, suppose that the firms decide for various reasons to
undertake a horizontal merger. In statistical mechanics, a system
will tend to relax to a state of lower energy. There are at least
two lower energy configurations which achieve the merger,
\[
    \epsfig{file=merger,width=2.5in}
\]
one of which involves region A moving to become adjacent to region
B ($E=12$), and the other involving the creation of two additional
+1 lattice sites between regions A and B, in order to join the
regions ($E = 14$). Since after the merger, the conglomerate will
have access to all of the original clients of \emph{both} firms,
the expected future earnings of this new firm will be 8 or 10
units, depending on the details of the merger as in the previous
figure.


A process is called \emph{energetically favorable} if it leads to
a state of lower energy. It is now clear that any positive
multiple of \er{energy} defines an energy operator in such a way
that the merger of two nearby firms is an energetically favorable
process. The toy model involving two firms is too simple to
describe an actual economy, but the lattice described here forms
the basis for a classic model of statistical physics (the Ising
model), and the energy \er{energy} is one term in the full energy
operator of that theory; see equation \er{nrg-plusminus}. In
subsequent sections, we will develop the full model which entails
this as a special case. We will construct a cellular automaton
which converges to equilibrium, and describe one method of making
this into a predictive simulation of a large-scale economy.

A key point is the following. In the $E = 14$ example above, two
`spins' are flipped from $-$ to $+$ in the region between the two
firms, as was necessary for the regions to merge. But a predictive
model must tell us where to flip the next spin. In a large lattice
with many firms, how would the site of the next flip be chosen? In
Sec.~\ref{sec:ai}, we outline a procedure for making this choice,
involving a Markov model which learns from past behavior of the
economy.

We give a simple example to illustrate the behavior of this Markov
model. Suppose now that there are three firms $A,B$, and $C$, and
as in the previous example, we give them all EFE $= 4$. However,
these firms differ in other characteristics, and in particular,
the book values are given by $v_A = 1, v_B = 2,$ and $v_C = 10$.
Also suppose that this hypothetical economy has a long history, in
which economists observe that, in each past case involving one
very wealthy firm (high book value) and a number of other firms
with relatively small book value, the wealthy firm always acquired
the least expensive of its competitors. It is then reasonable to
expect, given this trend, that $C$ will acquire $A$. Of course,
this trend behavior is not completely certain (trends are
sometimes broken), so the model should be probabilistic, while
still retaining sensitivity to past economic trends.

All of the above goals are simultaneously accomplished by the
following strategy: choose the location of the next spin flip to
be between $A$ and $C$ (because $A+C$ is the most likely merger
based on past trends), but flip the spin conditionally with a
probability distribution that depends on the energy.




\section{The Ising Model and Monte Carlo Iteration}

In this section we introduce important terminology and define the
basic model with which the rest of the paper is concerned. The
model will be generalized and applied to economics in Section
\ref{sec:economics}.



\subsection{The Ising Model}

The Ising model was originally introduced as a model for
ferromagnetism, and has since been generalized and applied to many
different fields, ranging from finance to biology. See
\cite{Cardy} for a review. The system is based on a lattice with
${\cal N}$ sites in $d$ dimensions, with a binary ``spin''
variable $\sigma_i \in \{\pm 1\}$ at each site $i$. While our
examples are given in $d=2$ to simplify the diagrams, for a
realistic simulation it is likely that $d > 2$ is necessary. This
technical issue is discussed in an appendix.

The model is defined by its energy function, which in the absence
of an external magnetic field, is taken to be
\be \label{eq:energy}
    E = -J \sum_{\<ij\>} \s_i \s_j
\ee
where $J>0$ is a coupling that determines how strongly nearby
spins tend to align.

As in the physics literature, $\sum_{\<ij\>}$ denotes a sum over
all pairs $(i,j)$ of sites which are nearest-neighbors in the
lattice. We let $N_{+-}$ denote the number of unlike
nearest-neighbor pairs in the lattice, and let $N_{++}$ or
$N_{--}$ denote the number of like nearest-neighbor pairs which
are $+$ or $-$, respectively. It is not hard to see that
\er{eq:energy} can be rewritten as
\be \label{nrg-plusminus}
    E = -J(N_{++} + N_{--} - N_{+-}) = -J ({\cal N} - 2N_{+-})
\ee
Therefore, the energy consists of a ground state contribution $-J
{\cal N}$ plus the positive constant $2J$ times \er{energy}.

A lattice has the defining property that each site has the same
number of nearest-neighbors. This number is denoted by $q$ and is
called the \emph{coordination number} of the lattice. If we wish
to embed the lattice in a plane using regular polygons, there are
exactly three such tessellations of the plane:
\[
    \epsfig{file=tessel,width=2.5in}
\]
These generate regular lattices with $q = 3,4,$ and 6
respectively.

The thermodynamic behavior of a canonical ensemble is governed by
the Boltzmann weight,
\[
    p(\s) \propto e^{-E(\s) / kT}
\]
where $k$ is Boltzmann's constant, and $T$ is the temperature. The
combination $1/kT$ occurs frequently, and so it is given the name
$\beta$. This $\beta$ has no relation to the coefficient in the
linear Capital Asset Pricing Model, which measures the volatility
of a security relative to its asset class. We will work in units
for which Boltzmann's constant $k = 1$, in which case $\beta$ is
inverse temperature.

The ensemble average $\<M\>$ of an observable $M$ at temperature
$T$ is given by
\[
    \<M\> = Z^{-1} \sum_\s M(\s) e^{- \beta E(\s)}
\]
where $Z = \sum_\s e^{-\beta E(\s)}$ is the \emph{partition
function}. The notation $\sum_\sigma$ denotes a sum over the
$2^{\cal N}$ possible states. It is not possible to calculate such
an average by enumeration of all possible states, since an $L
\times L$ square lattice entails $2^{L^2}$ states. However,
ensemble averages can be accurately estimated with Monte Carlo
methods.

A naive Monte Carlo estimate would draw configurations $\s^{(1)},
\ldots, \s^{(n)}$ from the uniform distribution and calculate
\[
    \<M\>
    \approx
    \left(
        \f{1}{n}
        \sum_{j=1}^n M(\s^{(j)})
        e^{-\beta E(\s^{(j)})}
    \right)
    \times
    \left(
        \f{1}{n} \sum_{j=1}^n e^{-\beta E(\s^{(j)})}
    \right)^{-1}
\]
The energy $E$ is an extensive quantity, which means there will be
huge fluctuations in $e^{-\beta E}$, and naive Monte Carlo is not
a useful method for this model.

However, if $\s^{(1)}, \ldots, \s^{(n)}$ are drawn from the
Boltzmann distribution rather than the uniform distribution, the
naive Monte Carlo estimate is replaced with
\be \lab{mcb}
    \<M\> \approx \f{1}{n} \sum_{j=1}^n M(\s^{(j)}) \, .
\ee
Assuming that we have an efficient way to generate a sequence of
Monte Carlo states that satisfy the Boltzmann distribution, then
\er{mcb} is a remarkable improvement. We will now describe how to
generate a sequence of Monte Carlo states according to the
Boltzmann distribution; this is known as the \emph{Metropolis
algorithm}. It was first published in \cite{met}, and subsequently
generalized and applied to many fields. For example, a similar
algorithm \cite{Mac} gives a Monte Carlo calculation of the ground
state wave function in quantum mechanics. See also \cite{ceperley}
and references therein.


\subsection{The Metropolis Algorithm}
\label{sec:metropolis}


The Metropolis algorithm was proposed as an algorithm to simulate
the evolution of a system in a heat bath towards thermal
equilibrium. From a given state $i$ of energy $E_{i}$, generate a
new state $j$ of energy $E_{j}$ by a small perturbation. If the
proposed new state $j$ has smaller energy than the initial state,
then make $j$ the new current state; otherwise accept state $j$
conditionally, with probability
\[
    A_{ij} = \exp\big(\!-\beta (E_{j}-E_{i})\big)
\]
After a number of iterations, we would expect to visit states of
different energies according to the canonical distribution. This
can be rigorously justified by showing that the sequence of states
visited by the algorithm forms an ergodic Markov chain with the
canonical distribution as a stationary distribution.

In more detail, the Metropolis algorithm is a stationary Markov
chain on the state space, which generates a new state $x'$ from
the current state $x$ as follows:
\begin{enumerate}
\item   Select a candidate state $x^*$, in which all components
other than the $k$th are the same as in $x$, while $x^*_k$ is
chosen at random from a proposal distribution $S_k(x,x_k^*)$.

\item Accept the candidate state with probability $A(x,x^*)$;
otherwise, reject it and retain the current state.
\end{enumerate}

We will always assume the proposal distribution is
\emph{symmetric}, in the sense that $S_k(x,x_k^*) = S_k(x^*,x_k)$
whenever $x_i^* = x_i$ for all $i \ne k$. Clearly, a sufficient
condition for the Markov chain to be ergodic is that $S_k(x,x_k')$
is nonzero for all $x_k' \ne x$ and $P(x)$ is nonzero for all $x$.
\emph{Detailed balance} is a sufficient (but not necessary)
condition for existence of equilibria, which says
\[
    A(x,x') P(x) = A(x', x) P(x')\,.
\]
Whether or not the limit of the Metropolis algorithm satisfies
detailed balance depends on the specific acceptance function.

For the pure Ising model, $S_i(x_i, x_i')$ is given by the delta
function $\delta(x_i' - x_i)$, which means the candidate state is
always obtained by flipping spin $i$ to its opposite value. The
acceptance function originally chosen by Metropolis {\it  et al.}
is
\be \lab{A1}
    A(x,x') = \min(1, P(x')/P(x))
\ee
where $P(x)$ is the canonical Boltzmann distribution with respect
to some energy function. For this choice of $P(x)$,
\be
    A(x,x') = \min(1, e^{-\beta \Delta E}), \quad
      \Delta E = E' - E
\ee
Suppose that state $x'$ is obtained from state $x$ by flipping the
spin at location $i$. Then
\begin{eqnarray*}
    \f{1}{J} E &=& -\sum_{j \in nn(i)} \s_i \s_j
    \ - \
    \sum_{\te{links $(k,l)$ not} \atop \te{involving i}} \s_k \s_l
    \\
    \f{1}{J} E' &=& -\sum_{j \in nn(i)} (-\s_i) \s_j
    \ - \
    \sum_{\te{links $(k,l)$ not} \atop \te{involving i}} \s_k \s_l
\end{eqnarray*}
where $nn(i)$ denotes the set of nearest-neighbors of site $i$.
Therefore,
\be \lab{deltaE}
    \Delta E = E' - E = 2 J \sum_{j \in nn(i)} \s_i \s_j
\ee
In particular, evaluating the acceptance probability for any one
spin flip only requires calculation of $q$ links.


The probability of accepting state $x'$ is therefore found to be
\be \lab{acc}
    A(x, x')
    =
    \min\Big(
    1,
    \exp\big(-2K \s_i  \sum_{j \in nn(i)} \s_j\big)
    \Big)
\ee
with $K = \beta J$. This is perhaps the simplest acceptance
function for which detailed balance holds. However, a large class
of acceptance functions are known for which the Metropolis
algorithm satisfies detailed balance. Aside from \er{acc}, the
most commonly used one is the Boltzmann acceptance function
\be \lab{A2}
    A(x,x') = P(x') / (P(x) + P(x'))\, ,
\ee
which, for distributions defined by an energy function, yields
\[
    A(x,x') = \f{e^{-\beta E'}}{e^{-\beta E} + e^{-\beta E'}}
    = \f{1}{1 + e^{\beta \Delta E}}
    \, .
\]
where $\Delta E$ was calculated in \er{deltaE}. It is easy to show
that both \er{A1} and \er{A2} lead to systems satisfying detailed
balance.


\subsection{Cellular Automata} \label{sec:CA}

Generally speaking, a \emph{cellular automaton (CA)} consists of a
$d$-dimensional lattice of ``cells,'' each in one of a finite
number of states. The grid can be in any finite number of
dimensions. Time is also discrete, and the state of a cell at time
$t$ is a function of the state of a finite number of cells called
the ``neighborhood'' at time $t-1$. In the traditional
interpretation, this function is not considered to contain any
random variables, and thus the time evolution of the system is
deterministic. However, an important generalization (and the one
which we will use) is the case in which the state-update function
is stochastic; such a system is termed a \emph{probabilistic
cellular automaton (PCA)}.

Every cell is assumed to have the same rule for updating, based on
the values of neighboring cells. Each successive application of
the rules to the whole grid is called a ``generation.'' The number
of updates in a Monte Carlo simulation is sometimes referred to as
\emph{Monte Carlo time}. However, units of Monte Carlo time simply
label generations in the CA, and in general, cellular automata
provide a natural language in which to describe simulations of
lattice systems.

The Ising model with energy \er{eq:energy}, evolved forward in
time by the Metropolis algorithm (defined with the acceptance
function \er{acc}), clearly forms a probabilistic cellular
automaton. Henceforth, we refer to this PCA as the
\emph{Metropolis-Ising system}.




\section{Application to Economics}
\label{sec:economics}

\subsection{Assumptions} \label{sec:assumptions}

To apply the Metropolis-Ising cellular automaton described in the
previous section to economics, we make the following assumptions.
\begin{enumerate}
\item \label{axiom1}\emph{(Economic interpretation)}
Lattice sites represent units of wealth. A connected
region of ${\cal E}$ spin-up $(+)$ sites represents a company with
expected future earnings equal to ${\cal E}$. Spin-down $(-)$
sites represent vacancies.

\item \emph{(Initial state)} The lattice is initialized with a state in
which each currently existing company in the economy under
consideration\footnote{The global economy is certainly too large
to effectively apply this model. By the ``economy under
consideration'' we mean a particular sector of the economy which,
from the point of view of mergers, and acquisitions, may be
considered as approximately independent of other sectors. This
assumption does \emph{not} entail zero trade with the other
sectors.} is denoted by a connected region of appropriate size,
proportional to that firm's expected future earnings.
Additionally, the distance $d(C_i, C_j)$ between companies $C_i$
and $C_j$ is proportional to a current estimate of merger
likelihood.

\item \label{axiom3}\emph{(Time evolution)}
The predicted time evolution after the initial state is given in
discrete steps, with state $x_{i+1}$ determined from state $x_i$
by conditionally flipping the spin at lattice site $\tau_i$, with
probability determined by the Metropolis function
\[
    \min(1, e^{-\beta \Delta E}),
\]
or a different acceptance function leading to detailed balance,
such as \[ (1+e^{\beta\Delta E})^{-1}. \] Here, $\beta$ denotes
the inverse temperature; an appropriate value of the temperature
for use in economics is determined in Section \ref{sec:temp}. The
location of the site $\tau_i$ is also crucial, and is determined
by an artificial intelligence algorithm (Section \ref{sec:ai})
which learns from past market behavior.

\item \label{axiom4}\emph{(Nearest-neighbor interaction)}
The energy function $E$ includes a nearest-neighbor interaction
$\ds -J \sum_{\<ij\>} \s_i \s_j$.
\end{enumerate}


Axiom \ref{axiom1} as it stands is appropriate for the intended
application to earnings fluctuations. It will be modified in
Section \ref{sec:assetprice} when we consider the application of
our methods to asset pricing.

Axiom \ref{axiom4}, the nearest-neighbor interaction, needs some
explanation. Based on Axiom \ref{axiom1}, connected regions of
spin-up $(+)$ sites represent firms, which are the agents in this
model; it must be statistically possible for mergers to occur and
for small clusters to be randomly created, and this is true of the
energy function defined in Axiom \ref{axiom4}. A more
sophisticated justification for Axiom \ref{axiom4} is that the
Ising Hamiltonian is the simplest of a class of lattice
interactions which have a phase transition and associated critical
point. This is important because it means that there is an
adjustable parameter ($K = \beta J$) in the theory in terms of
which the correlation length is an unbounded
function\footnote{More precisely, it is unbounded for an infinite
lattice. For a finite lattice, it can grow to at most the size of
the lattice. See Ref \cite{Cardy} for a further discussion.}.
Since the correlation length is the typical size of clusters, it
follows that without an Ising-like phase transition, there might
be no temperature for which we have the typical cluster size
observed in the economy. That is to say, equation \er{eq:Tc} might
have no solution. The energy function $-J \sum_{\<ij\>} \s_i \s_j$
is merely the simplest in a large class of known energy functions
which guarantee existence of a phase transition, and hence
existence of solutions to \er{eq:Tc}.

In real-world financial markets, the expected future earnings
referred to in Axiom \ref{axiom1} tend to fluctuate partially
according to predictable economic trends, and partially due to
unpredictable events, which can be viewed as randomly occurring.
The model proposed in this paper is appropriate in this sense, as
it is a stochastic model in which the sizes of connected regions
can experience random fluctuations. However, like the merger
probabilities, these fluctuations in future earnings are also
partially determined by intelligently modeling past economic
trends; see Section \ref{sec:ai} for further discussion of this
point.

Just as the original Ising model is only an approximate
description of a ferromagnet, there are a number of reasons why
the Metropolis-Ising model described here can only be a rough
approximation to a real-world economy. When two connected regions
in the lattice merge, the size of the new region formed is (at
least for a few units of Monte Carlo time) roughly the algebraic
sum of the sizes of the two constituent regions. This corresponds
in microeconomic theory to the assumption that under a merger or
acquisition, the expected future earnings of the conglomerate is
given approximately by the algebraic sum of the EFE's of the two
merging firms, which may or may not be the case depending on the
details of the merger.

However, in spite of these approximations, the model, with its
given macroeconomic interpretation, has a number of advantages.
One advantage is that the Metropolis algorithm, which defines the
Ising cellular automaton, was originally invented in order to
facilitate numerical simulation of the Ising model in various
dimensions. In our interpretation, Monte Carlo time corresponds to
real time in the economy, but the natural amenability to computer
simulation remains.\footnote{There is an extensive literature on
the ``critical slowing-down'' effect for Metropolis-type
simulations. Stated precisely, the problem is that the
autocorrelation time diverges as a power-law with increasing
system size, near the critical point. In other words, the
computing time required to generate an independent configuration
increases superlinearly with the system volume. Fortunately, in
the economic application described here, we are not concerned with
generating independent configurations or approaching equilibrium;
our sole concern is that the system continue to correctly model
the economy, and this is guaranteed (within the approximations we
make) by the intelligent choice of spin update sites through the
use of our Markov model. Thus critical slowing down is
\underline{not} a problem!}





\subsection{The Critical Temperature} \label{sec:temp}

We now discuss the notion of \emph{temperature}, one of the free
parameters of the Ising model, and determine the correct
``temperature range'' for the current economy.

At temperatures well above the critical temperatures, the spin
arrangement converges in Monte Carlo time to a nearly random
arrangement, independent of the starting state, and fluctuates
quickly. In physics terminology, above the critical temperature
there is a single thermodynamic state with zero magnetization.
This corresponds to a nightmarish economy in which firms of any
size form, merge, split, and terminate frequently and randomly;
the number of companies in existence tomorrow has no correlation
with the number today. These values of the temperature do not
describe any real-world economy.

Below the critical temperature, there are two thermodynamic states
(the ``up spin'' state with positive magnetization and the
negative magnetization ``down spin'' state) and the system stays
in one or the other depending on how the spins are initialized.
This corresponds to a completely socialist economy in which either
there are no firms, with all industry controlled by the
government, or all corporations have merged into a single firm.

The actual economies of most countries fall somewhere between
these two limiting cases. Near the critical temperature in the
Metropolis-Ising model we expect large clusters of spins with the
same orientation, which fluctuate, but very slowly. Near-critical
temperature corresponds, under our interpretation of the clusters
as companies, to the behavior of a large free-market capitalist
system such as the U.S. economy or global economy.\footnote{This
shows in particular that the one-dimensional Ising model is
useless for the particular macroeconomic interpretation advocated
here, as it has no phase transition.}


The typical size of clusters is called the \emph{correlation
length} $\xi$, which is maximal at the critical temperature for a
finite system, and which diverges at $T_c$ for an infinite system.
Section \ref{sec:assumptions} detailed a procedure for predicting
the time evolution of an economy, within the bounds of a simple
lattice model. This predictive procedure is defined in terms of a
parameter $\beta =$ inverse temperature, so to use the model, we
need a numerical value for the temperature. In the predictive time
evolution, we know the state of the economy now, which (when drawn
on a lattice) contains a number of clusters. Assume that we have
calculated the average cluster size of this initial state, and
denoted this number by $\xi_0$. This is also called the
correlation length of that state. Near criticality in the $d=2$
Ising model, the correlation length displays the power-law
behavior $\xi \approx (4|K_c - K|)^{-1}$, where $K = \beta J = J /
T$ (see \cite{Pathria}, p.387).


These observations indicate that a good choice for the temperature
is $T_0 = J / K_0$, with $K_0$ defined as either of the two
solutions to the equation
\be \lab{eq:Tc}
    \f{1}{4|K_0 - K_c|}
    \ = \ \xi_0 \, .
\ee
In words, eqn.~\er{eq:Tc} instructs us to determine the
temperature needed for the Metropolis update procedure based on
the apparent temperature of the initial state. The latter is
calculated from (an approximation to) its correlation length.

The value of $K$ at criticality is determined by a simple
self-duality argument (\cite{Pathria}, p.~383). Its numerical
value is
\[
    K_c = \f12 \sinh^{-1} 1 = \f12 \ln(\sqrt{2} + 1) \simeq 0.4407
\]
If we work at subcritical temperature, $T_0 < T_c$, we then have
explicitly
\[
    K_0 = \f{1}{4\xi_0} + 0.4407
\]


 Both the Metropolis acceptance function \er{acc}
and the Boltzmann acceptance function \er{A2} depend on $\beta$
and $J$ only through the combination $K = \beta J$.


\subsection{The Markov Model} \label{sec:ai}


This is the most important section in the paper, as here we
discuss how to use the Metropolis-Ising model to obtain
predictions for real-world financial markets. As we shall see, a
Markov-model determination of the update location, $\tau_i$, is
the key.


The Metropolis-Ising model, as presented above, can certainly be
considered as a qualitative model for macroeconomics. It allows
for an initial configuration to evolve forward in time, while
tending to a state of thermodynamic equilibrium. The evolution is
governed by an energy function $E$, together with one adjustable
parameter $K = \beta J$ which depends on the coupling and the
temperature, and this energy function has the property that
mergers of economically aligned firms tend to give a state of
lower energy, and for that reason are statistically likely. Also
likely (both in the economy and in our model) is the random
formation of new companies with small market share. The model does
not support spontaneous creation of very large firms, and neither
does the economy. In summary, the properties of time evolution in
this model all agree qualitatively with the properties of an
arbitrary, generic large-scale corporate economy, but the model as
it stands is too general to give specific properties of one
particular economy.



The problem of over-generality of a model is common in physics.
Einstein's equations for gravity admit thousands of known
solutions, and only a fraction of these correspond to physically
possible configurations.

For similar reasons, the Metropolis-Ising model will be of very
limited use in modeling real-world economies unless it can be
trained to take advantage of the fact that history repeats itself.
The world's financial markets are known to exhibit various cycles,
or patterns (such as bubble economies) which are self-similar, in
the sense that the $(n+1)$st bubble can be compared with the first
$n$ bubbles and certain similarities emerge.


We will show in this section that it is possible to program the
Metropolis-Ising model with historical data in such a way that a
particular pattern of corporate mergers or splitting which
occurred in the past is more likely to occur in a similar pattern
again. The technique is borrowed from artificial intelligence.

A \emph{Markov model of order $n$} is a semi-random walk through a
state space $\Sigma$. Given $i-1$ states, the $i^{\te{th}}$ state
is chosen based on a probability distribution over $\Sigma$ which
may depend on the previous $n$ states, but not on states further
back. A common choice for the probability distribution is
described as follows. For simplicity, consider $n=2$, so a
second-order Markov model. Suppose that we have a sequence $(x_i)$
of length $k \gg 2$ which represents behavior we wish to mimic.
For each ordered pair $(a, b) \in \Sigma^2$, we define a histogram
$p_{a, b}$ over $\Sigma$ by $p_{a, b}(c) = $ conditional
probability, given that $a,b$ were the $i$th and $(i+1)$st
elements in the sequence, that $c$ is the $(i+2)$nd element. The
process of determining $p_{a, b}$ for all $(a, b) \in \Sigma^2$ by
iterating over the elements of $(x_i)$ is called \emph{training}.
This clearly extends to general $n$. A typical application of this
technique is to fill in a missing note in a Mozart piano concerto,
by first training the Markov model on all of Mozart's other works.

More generally, let $S$, $T$, and $U$ be any sets. One might have
two sequences $(s_i) \subset S$ and $(t_i) \subset T$, and a third
sequence $f_i$ taking values in $U$, which depends on the first
two in some complicated way that we wish to mimic. (Equivalently,
there is a complicated function $f(i,s,t)$ on $\N \times S \times
T$ which is unknown; all that is known are its values $f(1, s_1,
t_1), f(2,s_2, t_2), \ldots$) For each ordered pair $(a,b) \in S
\times T$, define a histogram $p_{a,b}$ on $U$ such that
$p_{a,b}(u)$ equals the number of times $f_i = u$ given that $s_i
= a$ and $t_i = b$. It is actually this more general kind of
second-order Markov model which we will use. In our application,
$S = T = $ the space of possible \emph{key statistics} of a
company (defined below), and $U = $ the space of lattice
triangles. The histograms could each be normalized so that
$\sum_{u \in U} p_{a,b}(u) = 1$, in which case the $p_{a,b}(u)$
may be viewed as probabilities.

The first step in applying this to the Metropolis-Ising model is
that a computer must generate, from past financial data about the
number of companies as a function of time (measured in discrete
time-slices separated by $\delta t$) and their relative amounts of
wealth, a sequence of Ising model configurations $x_1, x_2,
\ldots, x_N$ in such a way that

\begin{enumerate}
\item \emph{(Single-site updates)}
The state $x_{i+1}$ differs from $x_i$ by the flip of a single
spin. Denote the location of this spin by $\tau_i$.

\item \label{subseq}\emph{(Historical Accuracy)}
There exists a subsequence $x_{i_1}, x_{i_2}, \ldots$ such that
$x_{i_j}$ correctly represents a snapshot of the market at time
$t_0 + j \cdot \delta t$.

\item \emph{(Present Accuracy)}
The last configuration, $x_N$, represents the current state of the
economy, so $x_{N+1}, x_{N+2}, \ldots$ are unknown, as are
$\tau_N, \tau_{N+1},$ etc.
\end{enumerate}

Condition \ref{subseq} needs some explanation. We divide time into
intervals of length $\delta t$; since earnings are often reported
at the end of each quarter, it probably makes most sense for
$\delta t$ to correspond to one fiscal quarter. The amount of
merger activity and earnings fluctuations for the various firms in
an economy, per unit time, clearly changes over time. This
activity is directly related to the number of spin flips needed to
propagate the lattice from one time interval to the next. From
state $x_{i_0}$, corresponding to time $t_0$ (the earliest date
for which data is available), suppose that $n$ spin-flips are
needed to reach the state of the economy at time $t_0 + \delta t$.
In this situation, our notation is such that $i_1 = i_0 + n$.

It is now possible to train a Markov model or neural network using
the sequence of histories $\{ x_i \}$; we will describe the Markov
model approach. What exactly should be the data sets (elements of
$\Sigma$ in our previous notation) on which to train the AI? The
naive choice, to train on the Ising configurations themselves,
will certainly be useless since the exact positions of the
companies on the grid is arbitrary; the model is invariant under
translations and rotations. A more sophisticated training approach
is necessary.

To this end, we define the notion of \emph{key statistics} for a
company. By this we mean any collection of purely numerical data
about the company which an analyst might consider when studying
mergers involving that company, or when studying its earnings.
These variables include, but are not limited to: stock valuation
measures, profitability, return on equity, revenue, earnings
growth, total cash, total debt, and many other indicators. Using
more indicators might make the model's predictions more accurate,
but too many variables will lead to an undesirable Markov model,
having a large number of distinct histograms $p_{a,b}(u)$ which
are all trivial. For a company $C$, we denote its key statistics
by $\S(C)$; this is a vector in $\R^{{\cal K}}$ where ${\cal K}$
is the number of key variables.

If it occurs in a periodic or semi-periodic fashion that a company
with key statistics $\S$ tends to merge with a second company (key
statistics $\S'$) whenever all other market conditions are equal,
then that is an economic trend. In what follows, we determine a
Markov model which will certainly capture such a trend if it
exists. If there is no relation between key statistics and the
prediction of a merger, then the problem is completely intractable
and it would be impossible even in principle for a model to be
developed; we do not take this view.



The Markov model we propose is an artificial intelligence
technique which chooses the position of the next Metropolis update
point intelligently, based on learning of past market behavior. By
assumption, for all $i \geq 1$, configuration $x_{i+1}$ differs
from $x_i$ by a single spin-flip; denote the site of this flip as
before by $\tau_i$. The fundamental problem is then to determine
the locations of $\tau_N, \tau_{N+1}, $ and so forth. Knowledge of
the $\tau_j$ for $j \geq N$ tells us how to calculate the $x_j$ as
well, since $x_j$ is determined from $x_{j-1}$ by calculating the
value of the acceptance function $A$ and (probabilistically,
according to the Boltzmann weight) flipping site $\tau_{j-1}$.


To train the Markov model, for each $i$ we identify the unique
pair of companies $(A_i, B_i)$ that minimizes the sum of the
squares of the distances from $\tau_i$, i.e. choose $A_i, B_i$ to
minimize
\be \label{min}
    d(A_i, \tau_i)^2 + d(\tau_i, B_i)^2.
\ee
and order the pair $(A_i, B_i)$ so that
\be \label{a<b}
    d(A_i, \tau_i) < d(\tau_i, B_i)
\ee
We note that if $\tau_i$ is located at the boundary of a company
$C$, this is directly related to fluctuations of the earnings
potential of that company. In that situation, conditions \er{min}
and \er{a<b} conspire to ensure that $C$ will be the first element
of the ordered pair that is chosen.

\[
    \epsfig{file=ABtau,width=3in}
\]

For each such pair of companies, we will train the Markov model on
the ordered pair $(\S(A_i), \S(B_i))$. Since this is not as simple
a Markov model as the Mozart missing-note example discussed above,
it is nontrivial to define what we mean by \emph{training}. Let
$p, q$ denote central points in regions $A_i, B_i$ respectively.
With $\tau_i$, these form a triangle $\tri pq\tau_i$. Since
scaling the entire model is equivalent to a change of monetary
units, we are only interested in the similarity class of this
triangle, which may be encoded, for example, with two angles.

In the histogram notation from our generalized second-order Markov
model, this entails setting
\be \label{histf}
    f(i,A_i,B_i) = \tri p\,q\,\tau_i \, .
\ee
Two companies will never have exactly the same key statistics;
when the Markov model is matching pairs $(\S(A_i), \S(B_i))$, it
must use a slightly sophisticated matching function which compares
the key statistics of two companies and returns ``true'' if they
are similar, or lie in the same range\footnote{If certain key
statistics are more indicative than others, the comparison
function may also take the form of a linear combination of the
various key statistics with some weights inserted.}.

Now suppose that we are in the situation outlined above: we have a
sequence of Ising lattice configurations $x_1, x_2, \ldots, x_N$
which correctly represent the past history of the economy (within
the accuracy of the lattice approximation). Our task is to compute
$x_{N+1}$, for which we need $\tau_N$, the site of the next flip.

The new update site $\tau_N$ is chosen so that its position
relative to the two closest companies mimics the relative
positions of update sites in the past history of the economy. A
simple geometric argument involving similar triangles supplies the
correct notion of ``relative position.''

For each pair $(A,B)$ of companies on the lattice, compute the key
statistics $\S(A), \S(B)$, and look up the resulting ordered pair
$(\S(A), \S(B))$ in the histogram $f$ formed by the Markov model.
Select the pair with the best match. By definition of the
histogram \er{histf}, this returns a triangle which we will denote
$\tri_{AB}$. Denote by $p'$ and $q'$ the central points of $A$ and
$B$ respectively. There is now a unique point $\tau_N$ such that
$\tri p'q'\tau_N$ is similar to $\tri_{AB}$.

\[
    \epsfig{file=ABtau2,width=2in}
\]


\subsection{Asset Pricing} \label{sec:assetprice}

Our model is immediately applicable to asset pricing. In fact, the
same asset price structure as in the Cont-Bouchaud model makes
sense here as well, under the assumption that the amount of
trading in this asset type by each cluster is proportional to the
size of the cluster.

Previously, we wanted to estimate fluctuations in expected future
earnings of a firm\footnote{This is very important, since any
reasonable stock valuation method would have to depend on $EFE$.},
and so we worked under the approximation that, immediately
following a merger, the expected future earnings of the
conglomerate is the algebraic sum of the expected earnings of the
two merging firms. It is always the case that when two clusters in
a lattice cellular automaton merge, the size of the combined
cluster is approximately the sum of the sizes of the constituents,
for the first few generations after the clusters combine. Thus,
the cluster-size models advocated in this paper can only be used
to estimate \emph{locally additive functions}\footnote{``Locally''
here means local in time, and refers to the fact that the
functions fluctuate in time, but are additive immediately
following the merger.}. The $EFE$ was a locally additive function,
and thus it made sense to identify its value with the number of
sites forming the cluster, in some appropriate units of wealth. In
asset pricing models (such as the Cont-Bouchaud model) which are
driven by aggregate excess demand, the locally additive function
we wish to estimate is now the demand of each agent for the asset,
since in such models the price change is a linear function of the
demand for the various agents, as in eqn.~\er{p-i}. It is worth
noting that \er{eq:Tc} gives a good estimate of the temperature to
use, regardless of which locally additive function we are
modeling.



Although not necessarily always true, it is a reasonable
approximation that a conglomerate formed by a \emph{recent} merger
of firms $A$ and $B$ will carry out an amount of trade which is
the algebraic sum of the two firms' trading before the merger. As
before, we also assume that at every time interval, each cluster
randomly selects to buy (probability $a$), sell (probability $a$)
or sleep (probability $1-2a$), and aggregate excess demand is the
driving force of the asset price.

The price adjustment is, as before, given by a price impact
function
\be \lab{p-i}
    P(t+1) - P(t) = \f{1}{b} \sum_{\te{clusters } i} D_i(t)
\ee
where $P(t)$ denotes the log price at time $t$, $D_i(t)$ is the
demand of cluster $i$ in period $t$, and $b > 0$ is a liquidity
parameter describing how much excess demand is necessary to move
the asset price by one unit. The demand $D_i(t)$ is defined as
plus or minus the size of cluster $i$, with the sign determined as
before: $+1$ with probability $a$, $-1$ with probability $a$, and
zero otherwise. The departure from the Cont-Bouchaud model lies in
the fact the structure of the clusters is no longer random. The
number and sizes of the various clusters will fluctuate in an
economically meaningful way, determined by artificial intelligence
from past market behavior based on analysis of key statistics.
These fluctuations generate important corresponding implications
for the asset price.

Since $D_i(t)$ for an individual site can take values $-1, 0, +1$,
this asset pricing system suggests that the appropriate
statistical physics model for this application is, in fact, the
\emph{Blume-Capel model} \cite{BC,Cardy}. The Blume-Capel model
(sometimes called the \emph{tricritical Ising model} due to the
point in its phase diagram at which three lines of second-order
phase transitions meet) is a generalization of the Ising model in
which the spin variable on each lattice site can take three
possible values $s = -1, 0, +1$. In physics, this is the classical
version of a quantum spin-one magnet.

A typical configuration of the Blume-Capel model is a lattice with
various connected regions of plus or minus sites, separated by
sites $r$ with $s(r) = 0$, which may be thought of as vacancies.
The Blume-Capel model, and its associated Metropolis cellular
automaton (see Sec.~\ref{sec:CA}), can be thought of as a model
for mergers and acquisitions if we introduce assumptions similar
to those of Sec.~\ref{sec:assumptions}. It is now fruitful to let
connected regions of plus or minus denote firms (or agents,
abstractly), leaving the zeros as vacancies. The Blume-Capel model
becomes an asset pricing model reminiscent of Cont and Bouchaud
when we declare that, at each time interval, the plus sites buy
the asset, the minus sites sell the asset, and the vacancies
sleep. It becomes a \emph{predictive} asset pricing model (which
takes into account mergers/acquisitions and fluctuations in
demand) when we train it with past market data using a Markov
model of the type introduced in Sec.~\ref{sec:ai}.

The Metropolis algorithm becomes slightly more complicated for the
Blume-Capel model, because even after the update site is chosen,
the spin can `flip' to one of several possibilities. We assert
that a reasonable Metropolis procedure, which satisfies detailed
balance, is the following. Let $x_1^*, \ldots, x_k^*$ denote the
different possible choices for the new state ($k=2$ for the
Blume-Capel model). If one of the $x_i^*$ has lower energy than
the current state, then switch to a new state which minimizes
energy among the candidate states. Otherwise, choose the candidate
state with the minimum $|\Delta E|$, and conditionally switch to
that state with probability $\exp(-\beta \Delta E)$.





\section{Conclusions}

We have defined a class of new predictive mathematical models for
\emph{locally additive} economic functions (such as demand for an
asset, or expected future earnings), which predicts the future
behavior of that function based on past economic trends involving
mergers, acquisitions, and fluctuations in the function itself.
This model builds upon previous models of statistical physics,
Monte Carlo simulation, and artificial intelligence. Since demand
for an asset is among the class of allowable functions, this
immediately gives a new asset pricing model.

The Metropolis-Ising cellular automaton was seen to be a
qualitative model for macroeconomic dynamics. It allows for an
initial configuration to evolve in time, approaching a state of
equilibrium, with evolution governed by an energy function $E$.
The energy function we use is familiar from statistical physics,
and has the property that mergers of economically
aligned\footnote{This means that the corresponding clusters are
close together in units of lattice spacing.} firms lead to states
of lower energy, and hence are statistically likely. Also likely
(both in the economy and in our model) is the random formation of
new companies with small market share and small short-term
expected earnings. The model has one adjustable parameter $K =
\beta J = J/kT$. The model does not support instantaneous creation
of firms with very large market share, and neither does the
economy. In summary, the properties of time evolution in this
model agree qualitatively with the properties of an arbitrary,
generic large-scale corporate economy.

A more important property of the model is the following. Two firms
$C_i$ and $C_j$ are more likely to merge if the next Metropolis
update point is chosen between the respective clusters. The
position of this update point, which mostly determines the
probability of a merger $p_{ij}$ (see also Sec.~\ref{sec:intro}),
is chosen by a Markov model which effectively reproduces past
economic trends. In real financial markets, and in this Markov
model, $p_{ij}$ is partially a complicated function of a large
number of variables which are called \emph{key statistics}, and
partially a function of purely random events. The same reasoning
applies to predict the fluctuation in expected future earnings of
a firm. The EFE is, like $p_{ij}$, a time-dependent pseudorandom
variable which is partially determined by a complicated function
of key statistics.

Our model is immediately applicable to asset pricing. In fact, the
same asset price structure as in the Cont-Bouchaud model makes
sense here as well, under the assumption that the amount of demand
for this asset type by each cluster is proportional to the size of
the cluster. In this application, it makes more sense to consider
a slight generalization of the Ising lattice, which is known in
physics as the Blume-Capel model \cite{BC}. The asset pricing
model which this entails represents a dramatic generalization of
the work of Cont and Bouchaud: the structure of the clusters is no
longer random. The number and sizes of the various clusters in
this new model will fluctuate in an economically meaningful way,
with corresponding implications for the asset price.

The model is explicitly computable. At no point does it involve
iteration over the $2^{{\cal N}}$ possible states of the Ising
system; the acceptance function $A(x,x')$ is computed only from
data involving the neighbors of a single lattice site.


\section*{Acknowledgements} The author is grateful to Daniel
Jafferis for helpful discussions.

\appendix

\section{Appendix: Geometric Limits} \label{sec:geom}


In our model, firms likely to merge must be in close proximity on
the lattice. The following scenario may certainly arise: a certain
sector of the economy is dominated by a single large firm, with a
number $N$ of smaller firms competing for the remaining market
share.
\[
    \epsfig{file=circles,width=2in}
\]

The large firm will acquire one or more of the small firms by the
mechanisms described elsewhere in this paper. In what follows, we
will assume that the regions are roughly spherical. This means
that $N$ small regions (radius $r'$) must be in close proximity to
one larger region (radius $r$) in the graph. In any dimension,
there is a geometric limit on $N$, which is determined by a very
simple sphere-packing bound, derived below. Of course, any such
limit is an artificial restriction placed by the model, not a real
restriction on the economy; fortunately, our estimates show that
this problem is resolved by increasing the dimension $d$ of the
lattice.

For this estimate, we may approximate an integral lattice of rank
$d$ by the continuum limit $\R^d$. In $d=2$, the situation
depicted in the figure, the number of circles of radius $r$ we can
pack around a fixed circle of radius $R > r$ is bounded as
\[
    N_2 \leq \pi \f{R+r}{r},
\]
since the sum of the diameters $2 N r$ should not exceed the
circumference of a circle with radius $R+r$. A similar argument in
$d=3$ yields
\[
    N_3 \leq
    4\left(\f{R+r}{r}\right)^2.
\]
For concreteness, suppose the small companies are 1/100 the size
of the large one. Then $(R+r)/r = 11$, and we have approximately
$N_2 \leq 35, \ N_3 \leq 484$. Thus, in three dimensions, if a
given firm is considering making an acquisition of a smaller firm
with one-hundredth its wealth, there can be at most 484 such
smaller firms. This may not be enough, but these bounds continue
to grow in higher dimensions. We therefore assert that for
sufficiently large dimension, the geometry contains sufficiently
many degrees of freedom for this kind of economic modeling.


%==================================================
\begin{thebibliography}{widest-label}

\bibitem{cb} R. Cont and J.P. Bouchaud, Macroeconomic Dynamics {\bf 4}, 170 (2000)
\href{http://www.arXiv.org/abs/cond-mat/9712318}{{\tt
cond-mat/9712318}}.

\bibitem{Stauffer} Stauffer, D. ``Percolation Models of Financial
Market Dynamics,'' Adv. Complex Systems {\bf 4} no.~1, 19--27.

\bibitem{Cardy}  Cardy, John L.~
\emph{Scaling and renormalization in statistical physics},
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\bibitem{met} N.~Metropolis {\it et al.}, J. Chem. Phys. {\bf 21} 1087 (1953)

\bibitem{Mac} McMillan, W.~L., {Phys. Rev. A}, {\bf 138}, 442
(1965)

\bibitem{ceperley} D.~M.~Ceperley, ``Metropolis Methods for
Quantum Monte Carlo Simulations,''
\href{http://www.arXiv.org/abs/physics/0306182}{{\tt
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\bibitem{BC} M. Blume, Phys. Rev. {\bf 141}, 517 (1966). \newline
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\bibitem{Pathria}  Pathria, R. K.
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\end{thebibliography}




\end{document}

cond-mat0404189.extracted_bib
\begin{thebibliography}{widest-label}

\bibitem{cb} R. Cont and J.P. Bouchaud, Macroeconomic Dynamics {\bf 4}, 170 (2000)
\href{http://www.arXiv.org/abs/cond-mat/9712318}{{\tt
cond-mat/9712318}}.

\bibitem{Stauffer} Stauffer, D. ``Percolation Models of Financial
Market Dynamics,'' Adv. Complex Systems {\bf 4} no.~1, 19--27.

\bibitem{Cardy}  Cardy, John L.~
\emph{Scaling and renormalization in statistical physics},
Cambridge University Press, 1996.


\bibitem{met} N.~Metropolis {\it et al.}, J. Chem. Phys. {\bf 21} 1087 (1953)

\bibitem{Mac} McMillan, W.~L., {Phys. Rev. A}, {\bf 138}, 442
(1965)

\bibitem{ceperley} D.~M.~Ceperley, ``Metropolis Methods for
Quantum Monte Carlo Simulations,''
\href{http://www.arXiv.org/abs/physics/0306182}{{\tt
physics/0306182}}.

\bibitem{BC} M. Blume, Phys. Rev. {\bf 141}, 517 (1966). \newline
                  H. W. Capel, Physica {\bf 32}, 966 (1966);
                                    {\bf 33}, 295 (1967);
                                    {\bf 37}, 423 (1967).

\bibitem{Pathria}  Pathria, R. K.
\emph{Statistical Mechanics}, 2nd ed. Oxford; Boston:
Butterworth-Heinemann, 1996.

\end{thebibliography}