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, Massachusetts, Addison-Wesley, 1982 [Casti:AR] "Alternate Realities," John L. Casti, John Wiley & Sons, New York, New York, 1989 [Casti:C] "Complexification," John L. Casti, New York, New York, HarperCollins, 1994 [Casti:PL] "Paradigms Lost," John L. Casti, Avon Books, New York, New York, 1989 [Casti:SFC] "Searching for Certainty," John L. Casti New York, New York, William Morrow, 1990 [Davis] "Handbook of Genetic Algorithms," Lawrence Davis, New York, New York, Van Nostrand Reinhold, 1991 [Dorfman] "Linear Programming and Economic Analysis," Robert Dorfman and Paul A. Samuelson and Robert M. Solow, New York, New York, Dover Publications, 1958 [Feder] "Fractals," Jens Feder, Plenum Press, New York, New York, 1988 [Garey] "Computers and Intractability," Michael R. Garey and David S. Johnson, W. H. Freeman and Company, New York, New York, 1979 [Hillier] "Introduction to Operations Research," Frederick S. Hillier, McGraw-Hill, New York, New York, 1990 [Hodges] "Alan Turing: The Enigma," Andrew Hodges, Simon & Schuster, New York, New York, 1983 [Hoffman] "Archimedes' Revenge," Paul Hoffman, Fawcett Crest, New York, New York, 1993 [Hofstadter] "Godel, Escher, Bach: An Eternal Golden Braid," Douglas R. Hofstadter, Vintage Books, New York, New York, 1989 [Kaplan] "Mathematical Programming and Games," Edward L. Kaplan, John Wiley & Sons, New York, New York, 1982 [Karloff] "Linear Programming," Howard Karloff, Birkhauser, Boston, Massachusetts, 1991 [Lenstra] "History of Mathematical Programming," J.K. Lenstra and A. H. G. Rinnooy Kan and A. Schrijver, CWI, Amsterdam, Holland, 1991 [Luce] "Games and Decisions," R. Duncan Luce and Howard Raiffa, John Wiley & Sons, New York, New York, 1957 [Nering] "Linear Programs and Related Problems," Evar D. Nering and Albert W. Tucker, Academic Press, Boston, Massachusetts, 1993 [Penrose] "The Emperor's New Mind," Roger Penrose, Oxford University Press, New York, New York, 1989 [Poundstone:PD] "Prisoner's Dilemma," William Poundstone, Doubleday, New York, New York, 1992 [Rucker] "Mind Tools," Rudy Rucker, Houghton Mifflin Company, Boston, Massachusetts, 1993 [Saaty] "Mathematical Methods of Operations Research," Thomas L. Saaty, Dover Publications, New York, New York, 1959 [Senge] "The Fifth Discipline: The Art and Practice of the Learning Organization," Peter M. Senge, Doubleday, New York, New York, 1990 [Singh] "Great Ideas of Operations Research," Jagjit Singh, Dover Publications, New York, New York, 1968 [Sommerhalder] "The Theory of Computability," R. Sommerhalder and S. C. van Westrhenen, Addison-Wesley, Reading, Massachusetts, 1988 [Strang] "Linear Algebra and it's Applications," Gilbert Strang, Third Edition, Harcourt Brace Javanovich, San Diego, California, 1988 [Ulam] "Adventures of a Mathematician," S. M. Ulam, University of California Press, Berkeley, California, 1991 [Waldrop] "Complexity," M. Mitchell Waldrop, Simon & Schuster, New York, New York, 1992 -- John Conover, john@email.johncon.com, http://www.johncon.com/

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