From: John Conover <john@email.johncon.com>
Subject: Equity Markets
Date: Sun, 17 Nov 1996 21:48:46 -0800
Hi Dave. You had ask the question whether fractal analysis of the
equity markets included such things as market "moods," "nervousness,"
"sentiments," and "beliefs" of the investors. Yes it does-in point of
fact, fractal analysis assumes that these inductively rationalized
"beliefs" are the "engine" that make the markets work. The how and why
is a bit complicated. The attached is from Chapter 2, "Fractal
Analysis of Various Market Segments in the North American Electronics
Industry," John Conover, 1995, so there are some copyright issues. So
please limit distribution.
I choose to present the issues from a game-theoretic tautology,
instead of fractal analysis, since the logic is easer to follow. There
is only one equation-the equation for compound interest. Pay
particular attention to the footnotes regarding self-referential logic
systems-that is the key. The understanding as to why the prisoner's
dilemma has no solution, unlike the game of Mora, is a key
point. There are references in the bibliography to delve further into
the issues.
John
Generalization
Consider the general equation for a fractal, which is also known
as a random walk, or Brownian motion.
R = R (1 + (f * F )) .................................. (1
n + 1 n n n
Where R is the value of the capital, or cumulative returns, of an
investment in the n'th time interval, f is the fraction of the
capital placed at risk, in the n'th interval, and F is a function
of a random variable. A typical, illustrative application would be
for a simple tossed coin game. Equation 1 is the recursive
discreet time formula for the exponential function. If F has as
many wins as losses, one should, obviously not play the
game. However, if F has more wins than losses, say P = wins /
(wins + losses) , then one's wager fraction, f, should be f = 2P -
1. This will maximize the exponential growth of the capital. The
issue discussed is the extensibility of Equation 1 to other than
simple games, for example the equity markets. This will be done
with an analysis of a two person mixed strategy game, Mora. Then
the analysis of a two person game, the prisoner's dilemma, where
it can be shown that there is no complete and consistent strategy
will be presented. Finally, a full multi-agent market where the
strategies are, by necessity, inductive and incomplete will be
discussed. (Note that in Equation 1, if F = 1 for all n, then the
equation becomes the simple compound interest formula.)
To reiterate the general concepts presented so far, a fractal is a
cumulative sum of a random process. In the literature, it is
sometimes called a Brownian motion, or "random walk," process
since, at any time, the next element in the process time series is
a random increment added to the current element in the time
"We emphasize that in Brownian motion it is not the position
of the particle at one time that is independent of the
position of the particle at another; it is the displacement of
that particle in one time interval that is independent of the
displacement of the particle during another time interval."
This is a subtile concept. Note that the term "cumulative sum"
really means that in any time interval, the position of the
particle is dependent only on the position of the particle in the
previous time interval, and a random displacement. But the
position in the previous time interval was dependent only on the
position in the time interval prior to that, and another
displacement, and so on, ie., to make a fractal process, we need
only know where the particle is at the current time, and add a
displacement to it, for each interval in time. The subtilty is
that we need only know where the particle is, and not where it has
been to calculate where it will be.
This section will use this concept, and expand the concept of the
random process to include game-theoretic issues by introducing
iterated two player mixed strategy games, then a simple
self-referencing game where no formal strategy can exist, and
finally multi-player games, where the random process is generated
by the inconsistency of the self-referential, inductive reasoning
among the players. In all cases, the iterated time series of such
games will be argued to be fractal, in nature.
The Game of Mora
A simple coin tossing game was analyzed previously. In this
section, those concepts will be expanded to include games of
strategy. The game of Mora, following [pp. 434, Bronowski], is
very old, (being mentioned in Sanskrit,) and is played between
two players and, in its simplest version, goes as follows. The
two players move simultaneously. Each shows either one or two
fingers, and at the same time guesses whether the other player
is showing one or two fingers. If both players guess right, or
both guess wrong, no money changes hands. However, if only one
player guesses right, the player wins from the other as many
coins as the two players together showed fingers. The possible
outcomes of any game are as follows if your call is right, and
your opponent's wrong:
1 Guessing your opponent will show 1 finger and showing
finger you will win 2 coins.
2 Guessing your opponent will show 2 fingers and showing 1
finger you will win 3 coins.
3 Guessing your opponent will show 1 finger and showing 2
fingers you will win 3 coins.
4 Guessing your opponent will show 2 fingers and showing 2
fingers you will win 4 coins.
The game is fair, but a player who knows the right strategy
will, with average luck, win against one who does not. The
right strategy is to ignore courses 1) and 4), and to play
courses 2) and 3) in the ratio of 7 to 5, ie., the right
strategy is, in any 12 iterations of the game, to play course
2) on the average 7 times, and course 3) on the average 5
times. Obviously, your opponent must not know which course you
are going to play, so the two courses must be intermixed
randomly.
The game is zero-sum, meaning that what one player wins, the
other looses. The mathematical method by which the best
strategy was found is called game theory. However, it is not
hard to verify that the strategy is effective by calculating
what happens when your opponent counters by using course 1),
2), 3), or 4), above. Namely, if your opponent chooses course:
1 Course 1), will, on the average, win 7 times out of 12,
and will win only 2 coins for each win; whereas losses
will occur 5 times out of 12, and those losses will be 3
coins for each loss-making an average loss of 1 coin in 12
iterations of the game.
2 Course 2), will have no coins change hands, since either
both players are right, or both are wrong.
3 Course 3), will have no coins change hands, since either
both players are right, or both are wrong.
4 Course 4), will, on the average, win 5 times out of 12,
and will win 4 coins for each win; whereas losses will be
occur 7 times out of 12, and those losses will be 3 coins
for each loss-making an average loss of 1 coin in 12
iterations of the game.
As in previously analyzed in the coin tossing game, the
objective of each player is to maximize the number of coins
won over many iterations of the game, ie., to maximize the
cumulative returns of the game. Note that each player's
capital, will fluctuate, depending on the outcome of a
particular iteration-and that fluctuation will be random, and
either 0, 2, 3, or 4 coins. We would expect that the time
series representing the fluctuations in a player's capital to
be a random walk, which could be represented by a formula
similar to Equation 1.
It is often convenient to represent the game as a table, which
lists all the possibilities of the courses for both players,
and how much the each player would win or loose for each
course, ie., a "payoff matrix," where one player's
alternatives are represented by the columns in Table 1, and
the other player's alternatives are represented by the
rows. The payoff to a particular game solution is the
intersection of the row and column of the course played by the
two players.
Table 1, The Game of Mora, Payoff Matrix.
+--------------------------------------+
|Finger, Guess | 1,1 | 1,2 | 2,1 | 2,2 |
+--------------+-----+-----+-----+-----+
| 1,1 | 0 | 2 | -3 | 0 |
| 1,2 | -2 | 0 | 0 | 3 |
| 2,1 | 3 | 0 | 0 | -4 |
| 2,2 | 0 | -3 | 4 | 0 |
+--------------+-----+-----+-----+-----+
The optimal strategy for a game as simple as Mora can be
derived by game-theoretic methodology[1] [pp. 56, Luce],
[pp. 441, Hillier], [pp. 419, Dorfman], [pp. 209, Saaty],
[pp. 127, Singh], [pp. 435, Strang], [pp. 258, Nering],
[pp. 67, Karloff], [pp. 105, Kaplan], but in many games of
interest, the rules are too complicated, and may even change
over time[2]. In these scenarios, the strategy can be derived
empirically, over time, using "adaptive control" computational
methodologies. For example, if the strategy of Mora was not
known, the optimal ratio of courses could be determined by
varying the ratio, and observing the effect on the cumulative
reserves over many iterations of the game. Note that such a
methodology can be problematical since your opponent may be
doing the same thing. An example of such a scenario is
presented in the next section.
Prisoner's Dilemma
A simple mixed strategy zero-sum game was analyzed in the
previous section. In the game of Mora, the optimal strategy
does not depend on how your opponent plays the game over
time. The prisoner's dilemma game is qualitatively
different. It is also one of the most commonly studied
scenarios in game theory[3] [pp. 94, Luce], [Poundstone:PD],
[pp. 262, Waldrop], [pp. 262, Casti:C], [pp. 295, Casti:AR],
[pp. 199, Casti:PL] [pp. 297, Casti:SFC], [pp. 439, Strang],
and [pp. 155, Kaplan] [pp. 170, Davis].
The rules of the game are simple. There are two players, and
each player has only two choices for each iteration of the
"game," and those choices are to chose either "A" or "B." If
both players pick "A," then each wins 3 coins. If one picks
"A," and the other "B," then the player picking "B" wins 6
coins, and the other player gets nothing. However, if both
players pick "B," then both win 1 coin.
The payoff matrix for the prisoner's dilemma game is shown in
Table 2, where, as before, one player's alternatives are
represented by the columns, the other player's alternatives
are represented by the rows. The payoff to a particular game
solution is the intersection of the row and column of the
course played by the two players.
Table 2, The Prisoner's Dilemma Game, Payoff Matrix.
+-------+-----+-----+
|Choice | A | B |
+-------+-----+-----+
| A | 3,3 | 6,0 |
| B | 6,0 | 1,1 |
+-------+-----+-----+
The prisoner's dilemma is not a zero-sum game-neither player
can ever loose any money. So there is an incentive to always
play. The choice "A" is known as a "cooperation strategy," and
the choice "B" is known as the "defection strategy" for each
player. It is a very subtile and devious game. Here is why,
and the logic you would go through. Just before you played an
iteration of the game, you would think:
1 If you choose "A," there are two possible scenarios:
i If your opponent chooses "A," you would get 3 coins,
and your opponent would get 3 coins.
ii If your opponent chooses "B," you would get 1 coin,
and your opponent would get 6 coins.
2 If you choose "B," there are also two possible
scenarios:
i If your opponent chooses "A," you would get 6 coins,
and your opponent would get nothing.
ii If your opponent chooses "B," you would get one
coin, and your opponent would get one coin.
Note that by choosing "A," the best you could do is to win 3
coins, and the worst is to win nothing. But, by choosing "B,"
the best you could make is 6 coins, and the worst is one
coin. It would appear, at least initially, that "B," is the
best choice, irregardless of what you opponent does.
But now the logic of the game gets subtile. Your opponent will
determine the same strategy, and will never play "A." So you
both make one coin with every iteration of the the game. But
you could make 3 coins-if you cooperated, by both playing "A."
But if you do that, there is an incentive for either player to
play "B," if he knows the other player is going to play "A,"
and thus make 6 coins. And we are right back where we
started. Indeed, a very diabolical game.
It is an important concept that you will be basing your
decision whether to cooperate, ie., choose "A," or defect,
ie., choose "B," based on how you think your opponent is going
to play. But your opponent's decision will be based on
consideration of how you are going to play. Which, in turn,
will be based on how you think your opponent will play, ad
infinitum. It is circular logic, or more correctly, the game
strategy is "self-referential" [pp. 17, pp. 465, Hofstadter]
[pp. 361, pp. 379, Casti:SFC], [pp. 335, Casti:PL], [pp. 356,
Casti:AR], [pp. 84, pp. 103, pp. 215, Hodges], [pp. 101,
Penrose][4].
This presents a problem in defining an optimal strategy for
playing the game of the iterated prisoner's dilemma since no
"theory of operation" of a self-referential system can ever be
proposed that will be both consistent and complete, ie.,
whatever theory is proposed, it will not cover all
circumstances, or provide inconsistent results in other
circumstances [pp. 465, pp. 471, Hofstadter],
[Arthur:CIEAFM]. The best way to play the game is deductively
indeterminate. This indeterminacy pervades economics and game
theory [Abstract, Arthur:CIEAFM].
However, just because such problems do not have axiomatized,
provably robust solutions does not mean that good strategies
do not exist. For example, the "tit-for-tat" strategy
[pp. 239, Poundstone:PD] has been shown to be a very
effective. The objective is to avoid letting the game
degenerate into both players playing defection strategies. It
is very simple, and consists of cooperating, ie., playing "A,"
on the first iteration of the game, and then do whatever the
other player did on the previous iteration[5]. Note that it is
a "nice" strategy, (in the jargon of game theory, a "nice"
strategy is one that never defects first.) It is also a
"provocable" strategy-it defects in response to a defection by
the opponent. It is also a "forgiving" strategy-the opponent
can implicitly "learn" that there is an incentive for
cooperating after a defection[6]. An important concept of the
tit-for-tat strategy is that, unlike the game of Mora, the
strategy does not have to be kept secret. When one is faced by
an opponent that is playing tit-for-tat, one can do no better
than to cooperate. This makes tit-for-tat a stable strategy.
Unfortunately, tit-for-tat does not do so well when the
opponent occasionally defects, and then returns to a generally
cooperative strategy. Neither does it do well when the other
player is playing a random strategy. As in the case of the
game of Mora, the strategy can be derived empirically, over
time, using adaptive control computational methodologies. The
subject of "inductive reasoning" as an adaptive control
methodology is considered in the section on "Multi-player
Games."
As in the previously analyzed coin tossing game, the objective
of each player is to maximize the number of coins won over
many iterations of the game, ie., to maximize the cumulative
returns of the game. Note that each player's capital, will
fluctuate, depending on the outcome of a particular
iteration-and that fluctuation will be random, and either 0,
1, 3, or 6 coins. We would expect that the time series
representing the fluctuations in a player's capital to be a
random walk, which could be represented by a formula similar
to Equation 1[7]. Computer simulations of the co-evolving
strategies of iterated multi-player prisoner dilemma scenarios
where the individual players "learn" how to cooperate further
support the hypothesis [pp. 170, Davis].
Multi-Player Games
A simple coin tossing game was analyzed previously. In the
section describing the game of Mora, those concepts were
expanded to include zero-sum games of mixed strategy, using
the game of Mora as an example. It was shown in these types of
games, the optimal strategy does not depend on how your
opponent plays the game over time. In the section describing
the Prisoner's Dilemma, a nonzero-sum game, the prisoner's
dilemma, was analyzed and it was shown that the strategy for
the game is deductively indeterminate since the game's logic
is self-referential. The reason for this was that one player's
strategy depended on how the other player plays the game over
time. In both cases, the cumulative sum of winnings of a
player was shown to have characteristics of a random walk,
Brownian motion fractal. In this section, these concepts will
be expanded to include multi-player games, where the players
use inductive reasoning to determine a set of perceptions,
expectations, and beliefs concerning the best way to play the
game. These types of scenarios are typical of industrial
manufacturing and equity markets.
Inductive Reasoning
Paraphrasing[8] [Arthur:CIEAFM], actions taken by economic
decision makers are typically a predicated on hypotheses
or predictions about future states of the world that is
itself, in part, the consequence of these hypotheses or
predictions. Predictions or expectations can then become
self-referential and deductively indeterminate. In such
situations, agents predict not deductively, but
inductively. They form subjective expectations or
hypotheses about what determines the world they
face. These expectations are formulated, used, tested,
modified in a world that forms from others' subjective
expectations. This results in individual expectations
trying to prove themselves against others'
expectations. The result is an ecology of co-evolving
expectations that can often only be analyzed by
computational means. This co-evolution of expectations
explains phenomena seen in real equity markets that appear
as anomalies to standard finance theory [Arthur:CIEAFM],
[Arthur:IRABR].
This concept views such "games" in psychological terms: as
a collection of beliefs, anticipations, expectations,
cognitions, and interpretations; with decision-making and
strategizing and action-taking predicated upon beliefs and
expectations. Of course this view and the standard
economic views are related-activities follow from beliefs
and expectations, which are mediated by the physical
economy [Arthur:CIEAFM].
This is a very useful concept because it essentially
states that economic agents make their choices based upon
their current beliefs or hypothesis about future prices,
interest rates, or a competitors' future move in a
market. These choices, when aggregated, in turn shape the
prices, interest rates, market strategies, etc., that the
agents face. These beliefs or hypotheses of the agents are
largely individual, subjective, and private. They are
constantly tested and modified in a world that forms from
their's and others' actions [Arthur:CIEAFM].
In the aggregate, the economy will consist of a vast
collection of these beliefs or hypotheses, constantly
being formulated, acted upon, changed and discarded; all
interacting and competing and evolving and
co-evolving. Beyond the simplest problems in economics,
this ecological view of the economy becomes inevitable
[Arthur:CIEAFM].
The "standard way" to handle predictive beliefs in
economics is to assume identical agents who possess
perfect rationality and arrive at shared, logical
conclusions about the economic environment. When these
these expectations are validated as predictions, then they
are in equilibrium, and are called "rational
expectations." Rational expectations often are not robust
since many agents can arrive at different conclusions from
the same data, causing some to deviate in their
expectations, causing others to predict something
different and then deviate too [Arthur:CIEAFM].
[Arthur:CIEAFM] cites the "El Farol Bar" problem as an
example. Assume one hundred people must decide
independently each week whether go to the bar. The rule is
that if a person predicts that more than, say, 60 will
attend, it will be too crowded, and stay home; if less
than 60 the person will go to the bar. As trivial as this
seems, it destroys the possibility of long-run shared,
rational expectations. If all believe "few" will go, then
"it all" will go, thus invalidating the expectations. And,
if all believe "many" will go, then "none" will go,
invalidating those expectations. Like the iterated
prisoner's dilemma, predictions of how many will attend
depend on others' predictions, and others' predictions of
others' predictions. Once again, there is no rational
means to arrive at deduced "a-priori" predictions. The
important concept is that expectation formation is a
self-referential process. The problem of logically forming
expectations then becomes ill-defined, and rational
deduction, can not be consistent or complete. This
indeterminacy of expectation-formation is by no means an
anomaly within the real economy. On the contrary, it
pervades all of economics and game theory [Arthur:CIEAFM].
It is an important concept that this view of industrial
and financial markets address such notions as market
"psychology," "moods," and "jitters." Markets do turn out
to be reasonably efficient, as predicted by standard
financial theory, but the statistics show that trading
volume and price volatility in real markets are a great
deal higher than the standard theories
predict. Statistical tests also show that technical
trading can produce consistent, if modest, long-run
profits. And the crash of 1987 showed dramatically that
sudden price changes do not always reflect rational
adjustments to news in the market [Arthur:CIEAFM].
In this market model, inductive reasoning prevails as the
"engine" of the market since no deductive hypothesis is
possible because of the Godelian issues of
self-referential arbitrage.
It should be pointed out that inductive reasoning in such
scenarios is not an exact process, and usually relies, to
some extent, on correlation between events in the
economy. In self-referential processes, single simplex
statistical evaluations are not possible, and this can
lead to misinterpretation of the significance of the
statistics of the events [pp. 50, Casti:SFC][9].
A multi-player, self-referential model of an equities market
Suppose that throughout a trading day, agents line up to
buy or sell a stock. When a particular agents' turn comes,
the agent has the option to try to increase or decrease
the price of the stock from the transaction price of the
previous agent, (by lowering the price to sell stock the
agent owns, or raising the price to buy stock from another
agent.) The agent will have to make this decision based on
beliefs concerning the beliefs of the agents in the rest
of the market. This decision process will vary as
different agents post their transaction through the day,
based on their personal set of beliefs, cognitions, and
hypothesis concerning the market. We would expect that
the time series representing the fluctuations in a stock's
price to be a random walk, which could be represented by a
formula similar to Equation 1 [pp. 8, Arthur:CIEAFM].
Empirical analysis of many stocks tend to support the
hypothesis that stock prices can be "modeled" as a random
walk, or fractional Brownian motion fractal. Additionally,
computer models of stock market asset pricing under
inductive reasoning with many agents has been initiated
and further support the hypothesis [pp. 8, Arthur:CIEAFM].
Stability Issues
In the section describing the prisoner's dilemma the
issues of process stability were mentioned. Note that
not all processes are stable. For example, consider a
stock market scenario that historically had cyclic or
periodic increases and decreases in price. The value
at the bottom of the cycle would increase, (because
the agents in the market could exploit a "buy low,
sell high" strategy that would be predictable,) and
the price advantage would be arbitrated away, and the
cyclic phenomena would disappear. Cyclic phenomena
would then be considered as an unstable
process-similar to the El Farol Bar problem mentioned
above. However, note that if the agents in the market
believed that their financial position could be
improved by altering their investment strategy, by
buying or selling of stocks, then, as outlined in the
previous section, the stock price would fluctuate
similar to a random walk, and this would be stable
since it is a self reinforcing situation.
Extensibility Speculations
Interestingly, the arguments presented in this section
are possibly extensible into other areas. For example,
the Stanford economist Kenneth Arrow has shown that
the ranking of priorities in a group is
intransitive[pp. 1, Lenstra] [pp. 327, Luce] [pp. 213,
Hoffman]. What this means is that there exists no way
to use deductive rationality to rank priorities in a
society. If it is assumed that it is necessary to do
so, then inductive reasoning would have to be used. If
it is further assumed that such a situation is
self-referential, which seems reasonable by arguments
similar to those presented in this section, then the
same issues outlined in this section could be
applicable to social welfare issues, etc. This would
tend to imply that political issues were fractal in
nature, and the political process justified-which is
contrary to the thinking of many. The arguments
presented in [Arthur:CIEAFM], and [Arthur:IRABR] may
well be extensible into other fields of
interest. Other speculations could involve theoretical
interests in the dynamics of democratic process, legal
process[10], and organizational process[11]. There
are probably other applications[12].
As another interesting aside, the arguments presented
in this section side-stepped the issue of utility
theory.
Conclusion
In this section, it was shown that markets would be
expected to exhibit self-referential processes, which
can not be analyzed by deductive rationality. However,
when players rely on inductive reasoning to formulate
strategies to execute their market agenda, the result
is that the market will exhibit fractal
dynamics. Previously, in this chapter, it was shown
that the fractal dynamics can be exploited and
optimized. Interestingly, in some sense, there appears
to be a convergence of game-theoretic,
information-theoretic, non-linear dynamical systems
theory, and fractal/chaos-theoretic concepts.
Further, there also appears to be a convergence of
these concepts with the cognitive sciences.
Footnotes:
[1] These methodologies are often called "operations research." The
algorithm of choice used to derive the optimal game play seems to be
the "simplex algorithm"-at least for games with a small payoff
matrix. The simplex algorithm is one of a class of algorithms that are
implemented using "linear algebra."
[2] In the game of Mora, the optimal strategy does not depend on the
strategy of the opposing player. In more sophisticated games, this is
not true.
[3] The prisoner's dilemma has generated much interest since it is a
game that is simple to understand, and has all of the intrigue and
strategy of many human social dilemmas-for example, John Von Neumann,
the inventor of game theory, once said that the reason we do not find
intelligent beings in the universe is that they probably existed, but
did not solve the prisoner's dilemma problem and destroyed their
self. The prisoner's dilemma has been used to model such scenarios as
the nuclear arms race, battle of the sexes, etc.
[4] The Penrose citation, referencing Russell's paradox, is a very
good example of logical contradiction in a self-referential
system. Consider a library of books. The librarian notes that some
books in the library contain their titles, and some do not, and wants
to add two index books to the library, labeled "A" and "B,"
respectively; the "A" book will contain the list of all of the titles
of books in the library that contain their titles; and the "B" book
will contain the list of all of the titles of the books in the library
that do not contain their titles. Now, clearly, all book titles will
go into either the "A" book, or the "B" book, respectively, depending
on whether it contains its title, or not. Now, consider in which
book, the "A" book or the "B" book, the title of the "B" book is going
to be placed-no matter in which book the title is placed, it will be
contradictory with the rules. And, if you leave it out, the two books
will be incomplete.)
[5] The tit-for-tat strategy sounds like a human social strategy
between two people-as well it should. It is known to work well with
human subjects [pp. 239, Poundstone:PD]. It is also strict military
dogma, and has formed the strategy of arbitration of the complexity of
power in many marriages.
[6] Tit-for-tat is kind of a "do unto others as you would have them do
unto you-or else," strategy. The tit-for-tat strategy in human
relationships is very old. Another ancient proverb illustrating
tit-for-tat is "an eye for an eye, a tooth for a tooth."
[7] Assuming that one player, or the other, will, at least
occasionally, alter strategy in an attempt to gain an advantage-in
this case, for example, two players, each playing tit-for-tat will
"lock" in to either a defection strategy, or cooperation
strategy. This is considered a degenerate case of Equation 1.
[8] Actually, plagiarize would be a more appropriate choice of
wording. This entire section is a condensed version of the text from
[Arthur:CIEAFM] and [Arthur:IRABR].
[9] Additionally, there are issues concerning causality. Cause and
effect may not be discernable from each other.
[10] Could the legal system be optimized? Or is that an oxymoron?
[11] For example, [pp. 81, Senge] has a diagram of the sales
department process in an organization. It has the same schema as
represented in Equation 1. If it could be shown that organizational
complexity is an NP problem [pp. 313, Sommerhalder], [pp. 13, Garey],
then there there could be some reasonable formalization of the
observations presented in [Brooks] and [Ulam].
[12] Others feel a bit more epistemological about the issue-see
[pp. 178, Rucker], the chapter entitled "Life is a Fractal in Hilbert
Space."
Bibliography:
[Arthur:CIEAFM] "Complexity in Economic and Financial Markets,"
W. Brian Arthur, "Complexity, 1, pp. 20-25, 1995. Also available from
http://www.santa.fe.edu/arthur," Feb, 1995
[Arthur:IRABR] "Inductive Reasoning and Bounded Rationality," W. Brian
Arthur, "Amer Econ Rev, 84, pp. 406-411, 1994. "Session: Complexity in
Economic Theory," chaired by Paul Krugman. Available from
http://www.santa.fe.edu/arthur
[Bronowski] "The Ascent of Man," J. Bronowski, Boston, Massachusetts,
Little, Brown and Company, 1973
[Brooks] "The Mythical Man-Month," Frederick P. Brooks, Reading,
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John Conover, john@email.johncon.com, http://www.johncon.com/