DUOPOLY AND CHAOS THEORY: NEW DIRECTIONS FOR RESEARCH AND PUBLIC POLICY? Julian L. Simon INTRODUCTION In his popular survey Chaos (1987; see also Hofstadter, 1985, chapter 16, and Peterson, 1988, chapter 6; for a survey focusing on economics, see Baumol and Benhabib, 1989), Gleick notes that chaos theory "offered a fresh way to proceed with old data, forgotten in desk drawers because they had proved too erratic" (p. 304). That aptly characterizes the results of a rich non-linear simulation of duopolistic competition published in 1973 by Simon, Puig and Aschoff. A second look at that model and its output from the standpoint of chaos theory opens a new door to exploration of realistic duopoly behavior in particular, and of economic phenomena in general. Combined with the point of view of Hayek and Schumpeter with respect to the natures of various kinds of competition, as well as the point of view of chaos theory, the results of that model might imply very different governmental policies toward the structure of competition in industrial and service markets (as distinguished from perfect competition among farmers and on stock markets) than does neo-classical theory. The implications - especially that fewer competitors may be better than more - are startling, and some are at first thought quite uncongenial to this writer. (Perhaps they will become more congenial to me and to others after living for a while with the idea that atomized competition does not promise greater progress.) The paper also discusses some lines of research in economics that are likely to be made fruitful by recourse to chaos theory, A NOTE ABOUT CHAOS THEORY Perhaps a few lines of general explanation would be useful for those who (like the author) have presumed that chaos theory describes interesting behavior of pure numbers, but is unlikely to have valuable application to economics (and indeed was likely to be simply one more fad for economics): The term "chaos" is unfortunate. The behavior comprehended by chaos theory is entirely determined (though stochastic disturbances can be brought in as desired) in the sense that the theorist produces the observed results by systematic variation of a set of simple (non-linear) equations. Each time a run is made with a given set of parameters, exactly the same results are produced. (This is an operational rather than metaphysical view of the notion of determination). Such systems are the very opposite of systems which are described by the Law of Large Numbers1. They contain a very few variables, each of which is important to the outcomes, in the sense that a change in a variable strongly influences the results. That is, the results are wholly predictable even though counter-intuitive. In ordinary everyday use the term "chaos" suggests exactly the opposite. A more appropriate term is "complex". Indeed, the astonishing aspect of chaos theory is that very simple systems produce, with some settings of the parameters, very complex patterns of results, rather than moving toward the stable equilibria which are the staple of conventional and classical thought in both physics and economics. The difference between random and chaotic behavior comes out sharply in phase-space diagrams. "Truly random data remains spread out in an undefined mess. But chaos - determined and patterned - pulls the data into visible shapes" (Gleick, pp. 266- 7). But in some cases the subjective impressions of the kinds of data produced by the two sorts of processes, and to some extent the processes themselves, are not so easily distinguished. (See footnote 2 below.) Chaos theory at first seemed appealing to some economists as a way of illuminating movements in stock markets. "Economists looked for recognizable strange attractors in stock market trends but so far [as of the time Gleick was writing] had not found them" (p. 307). Later there follow some reasons why one should not expect chaos theory to fit stock market data. DUOPOLY AND CHAOS THEORY Duopolistic competition would seem a priori likely to exhibit the behavior characteristic of chaotic systems. This has been suggested earlier in context of duopoly games by Rand (1978) and by Dana and Montrucchio (1986). Key elements of duopoly, which are also those of chaotic systems, are as follows: 1) Economic theory suggests that duopoly competition is powered by two major forces, acting in opposed directions, rather than by a multitude of small forces. 2) Casual observation and industry studies suggest an absence of smooth equilibrium process, and instead the presence of sharp variation at best loosely related to outside forces. 3) The simulation work described below displays sensitivity to initial conditions. 4) Economic theory and observation both suggest that duopoly price behavior is influenced by its history (unlike price behavior in securities markets). Each of these four elements will now be discussed briefly. 1. The choice of a price by a given reference firm at a given moment is influenced by two major opposing forces. In the one direction pushes the short-run opportunity for immediate gain by undercutting (secretly, if possible) the price of the competitor firm (Stigler, 1964). In the other direction pushes the awareness of the longer-run benefit of refraining from starting a price war. That is, the system is both driven and damped, in Gleick's words (p. 43). Several variables influence the balance of the two opposing forces and therefore should influence the character of the non- equilibrium behavior. These include the rate of learning of how the competitor firm behaves, the vigor of response of the firms to each other's price-cutting, the discount factor for the time value of money, and the importance of fixed costs. These variables are absent from standard analytic models of duopoly, which explains why and how the models produce equilibrium results. But casual observation makes it obvious that duopoly prices are not stable, and studies show that duopoly prices are more variable than monopoly prices (Stigler, 1964; Simon, 1969; Primeaux and Bomball, 1974; Primeaux and Smith, 1976). This suggests that absence of these variables leads to a false impression of reality, the impression that the outcome of duopolistic competition is a single stable equilibrium level. And it also suggests that a richer model of duopoly should be the appropriate vehicle for studying the behavior of duopoly prices.2 The model of price competition in duopoly published in 1973 by Simon, Puig, and Aschoff contains the "richer" variables mentioned above (as does the model of advertising competition in duopoly of Simon and Ben-Ur, 1982). The structure of the model is summarized in Table 1, and the structure is given in Appendix 1. The results (repeated in Table 2 here) show that all of these variables do indeed influence the final price level. This multi-outcome functional dependence upon a variety of variables, rather than a single price-level outcome a la Cournot and practically all other duopoly theory, was the central conclusion of that paper. Tables 1 and 2 The market-share outcomes also varied from run to run. Those data were never analyzed systematically, but many of the findings about prices were also true of market shares. And in the related study of advertising competition by Simon and Ben-Ur, advertising budgets also showed dependence upon initial conditions, great variation in outcomes, and in general, the behavior to be expected of a non-linear chaos-producing system. In contrast, price behavior in stock markets surely is influenced by a great many variables, most of them (by definition) relatively unimportant in influence. This is the recipe for "true" random behavior, that is, behavior which cannot be shown to arise as a deterministic result of a few simple equations (or even many complex equations). Hence chaos theory should not be expected to illuminate stock-market pricing, in my view. 2. The secret cheating and the price see-saws of commodity cartels, and the price wars among gas stations and airlines, are perhaps more dramatic than the happenings in other markets. But they are closer to the nature of most ordinary markets than is the behavior of securities markets. Beyond any doubt, the impression left by all analytic duopoly theory - that a market moves toward a stable equilibrium and remains at that equilibrium unless disturbed by external changes - is a misleadingly false picture of reality. 3. The price-level results are sensitive to initial conditions -- the price level at which the trial starts, the distribution of market shares, and the initial expectations about how the competitor will respond to various pricing decisions, as seen in Table 2. 4. Not only the price level, but whether the price level is stable or cyclical or chaotic is influenced by the parameters. This is the result which ties the work to chaos theory. Some settings of the parameters lead to a high-level cooperative price equal to the monopoly price, others to a low-level price corresponding to the perfect-competition level, still others to a variety of in-between prices. Some of the in-between price patterns are stable, others cyclical, and still others never reach any identifiable pattern and hence may be considered chaotic, even a very large number of trials. These data on the number of trials required to reach equilibrium, found in the middle column for each of the six panels of output results fit the classic pattern of chaotic behavior. Here are some specifics: a) Runs 1-3 differ only in the market share held by firm A, and the runs generally go more quickly to stability with a 50-50 initial distribution than with an unbalanced distribution. The same is true in most other otherwise-similar pairs or triplets of runs in the table. Indeed, in run 8 three of the five initial expectation patterns produce runs which never reach an equilibrium with 30-70 initial market shares but do reach an equilibrium with initial equal market shares. b) The initial expectations of a firm about how its competitor will respond to price changes affect the number of periods required to reach stability, even though these expectations themselves are updated in a learning process. This may be seen in almost any run. c) Even the initial price affects the process. This is because the cost function is the same in all runs, and hence a change in the initial price is not just an overall change in scale. The same is true with changes in the cost function with the initial price kept the same. But this is more than an "initial condition" change. The same is true of changes in the discount factor, response speed, speed of probability update, price increment, and market share revision factor.) It was contrary to our expectations at the time that these in-between price patterns often did not move toward some stable equilibrium, even an observable cyclic equilibrium, as cybernetic theory had led us to believe would occur. This confounding of expectations apparently is common among those who have worked in a variety of fields with systems which they later discovered jibed with chaos theory. And like many of them, we did not know what to make of our results, or how to analyze them in a systematic manner. 4. Efficient-market price behavior is unaffected by events of the past, in the sense that all earlier information is assumed to have been utilized fully. This is consistent with a zero correlation of movements over time. But duopoly price behavior is certainly affected by the participants' memories of how the other competition behaved in the past, and hence one would expect continuity in the sense of positive correlation of price movements from t to t+1. IMPLICATIONS OF THE RESULTS In physics and chemistry and engineering, if I understand correctly, the main focus of interest in chaos-oriented studies of non-linear processes is the course of the procession from stability to the ultimate point of chaos. But in economics the focus of interest should be different. As in meteorology, the important question is which kinds of initial conditions alter the final state. And for policy purposes in economics, one wants to know which conditions one might alter in order to influence the results in a desired fashion. Before considering particular conditions, however, it is worth discussing which results may be of interest. Again, the central phenomenon in chaos theory is the pattern of variability rather than the final level reached by prices (or advertising budgets, or whatever). And variability is at the heart of Austrian economics. Whereas neo-classical economics has no interest in market variability for its own sake, Austrian economics points to benefits of non-stability, especially the incentives to innovate and to advertise (there is private incentive for neither in perfect markets) and the incentive for new firms to enter the market. We could draw from this the conclusion that in an Austrian context, policy-induced alterations which induce more instability - even turbulence - offer benefits (which might need to be weighed against costs, of course), and might be considered desirable changes. Judging instability to be substantive and valuable, rather than an excrescence in theory or empirical work and an outcome to be avoided in actual economic life, is a radical shift in thinking. Whereas "perfect competition" is the sign of a well- working market in neo-classical thinking, Austrian economics could interpret such a stable market as an invitation to industrial stagnation. Austrian economics - both the Schumpeterian and the von-Mises-Hayek varieties - might prefer competition among a few firms if it leads to more innovation and more new entry, and hence greater long-run growth, even if there is a short-run reduction in allocative efficiency and temporarily there are above-average returns to sellers and too little production, as compared to perfect competition with many sellers. And whereas the non-economist might lament the lack of "orderly" competition, the Austrian might cheer it. It is also relevant for Austrians that because individual competitors in a "perfect" market are not likely to innovate, innovation is often done by government agencies in such industries as agriculture. A policy which led to fewer competitors for the sake of greater instability and opportunity might therefore have the side effect of less government, then. All this implies examining the results of the model for the conditions of intervention that might produce more market variability. The results in Table 2 immediately make possible such an inquiry without recourse to the techniques used for exploration of how the process proceeds toward the endpoint of turbulence. These conclusions may be drawn from Table 2: 1. Duopoly produces more variability than does monopoly, in contradiction to the theory of the kinky demand function (which was the original point of departure for this work). This is seen in the fact that, unlike the stable monopoly solution, stability is not reached quickly in many or most conditions of realistic oligopoly, and even an ultimate stability is reached only with special conditions in duopoly. The same sort of comparison can immediately be made against perfect competition. Some theorists in the past - especially Schumpeter - have for this very reason advocated a system which promotes competition among a few rather than among many competitors. But the work at hand is more specific in suggesting duopoly as the optimum number of competitors. Even adding just a third firm produces much greater stability than does duopoly, bringing the results much closer to perfect competition than to duopoly. This is shown in an analysis of analogous models with three rather than two competitors (Simon and Puig, 1990). The implication of this conclusion is rather shocking from the points of view of both regulators and neo-classical economists: increasing the number of competitors beyond two is not necessarily to be desired, assuming that the two can be kept from colluding so as to effectively constitute a monopoly. 2. In one of his most famous passages, Adam Smith asserted that communication among sellers - no matter how innocent the occasion - leads to price fixing. And the results in Table 2 support Smith; firms that begin with cooperative expectations arrive at higher and more stable prices than do firms that began with more cutthroat expectations. This provides some justification for a law against price fixing. Indeed, to the surprise of many of those who usually are sympathetic to his thinking, Hayek favors such law, though neo-classicals tend not to (though Hayek does not give the promotion of instability and subsequent discovery as the reason for his position. 3. Initial equality of size and market power among the competitors leads to less volatility than does inequality. (Compare otherwise-similar runs that begin with 50-50 market shares against those that begin with 30-70 splits.) Again, if one seeks more instability in the market, this finding points to a state of affairs quite the opposite of what regulators ordinarily strive for - equality of market power. 4. Tax policies such as those affecting the allowable rate of depreciation of plant and equipment affect the rate of investment. And the rate of investment affects the cost function, and especially the relative role of fixed cost. If analysis shows that greater fixed cost leads to more variability in the market, faster tax write-off might be adopted. This issue cannot be examined with the existing results because data are lacking, but new work can easily examine this question. 5. The speed of market response, and the extent of the "exploitable advantage" about which the 1973 paper theorized, is affected by the amount of consumer information. These factors in turn affect the amount of variability as well as the outcome price level. Analyses of the effects of various parametric conditions could then throw light on policies about the provision of consumer information with an eye to their effect upon the extent of variability. EMPIRICAL TESTING OF DUOPOLY AS A CHAOS SYSTEM Brock (1990) properly emphasizes the importance of empirical connections and tests if chaos theory is to be accepted as useful in economics. And he is not hopeful of valid tests for the applications that have been most discussed until now, securities markets and business cycles. "While at an an anecdotal level it seems obvious that chaos and instability are common in interactive social behavior systems such as economics, it is much harder to document the presence of chaos and instability in economic data. To my knowledge, although claims have been made, this has not been done in a scientifically convincing way" (p. 447). The outlook is more hopeful in duopoly theory, however. It is possible to find many occurrences of retail duopoly (or other oligopoly) behavior within particular industries; the price behavior of gasoline stations on the same corner or stretch of road is an example. One could chart the outbreak of price wars between pairs of stations, and test (by, say, a runs test, or other tests developed more recently specifically for such purposes; see Brock and Dechert, 1990) whether or not the time intervals between wars are random. Such an inquiry suggests a related inquiry into another general concept of non-linear dynamical systems, fractal-like self-repetitive nestedness of similar behavior at different hierarchical levels. The sort of test described in the paragraph above could first be applied to gasoline refiners, then to wholesalers, and then to retailers, to see whether small-fleas- atop-bigger-fleas patterns appear. It is reasonable that random shocks as well as endogenous deterministic variation at, say, the level of international oil supplies would be transmitted downward from level to level. DISCUSSION 1. One might expect non-random distribution of episodes of conflict and cooperation in all struggle and play systems, not only in economic systems. The wrestling of two puppy dogs may occur with non-random periodicity in a cycle during which they expend energy, accumulate energy, exhaust the energy in play or fight, then rest, then fight again, and so on. Perhaps the same is true of countries and border wars. The pattern of accumulating and expending makes intuitive sense, and hence there exist in the system the key elements of a driving force and a damping force, connected by dynamic changes. Seen from a different angle, one might expect chaotic behavior in all biological systems that accumulate and then discharge - for example, the frequencies of eating and defecating, and of sex behavior. 2. The basic duopoly model referred to here has connections to the tit-for-tat cooperation theory of Axelrod (1984). Indeed, the pattern of results seen in Table 2 for the different regimes of original expectations constitutes evidence that a tit-for-tat outlook - as seen in the "cartel" (better called "tit for tat") probabilities - leads more quickly and more frequently to a cooperative outcome than does any other expectation regime. (This matter will be explored in more detail in a separate paper.) 3. If chaos theory does indeed illuminate realistic competition in most markets in a modern society, then it not only can take advantage of the theoretical insights of Austrian economics, but it also offers a powerful theoretical tool for the theoretical development of Austrian economics, which has heretofore suffered relative to neoclassical economics in that regard. Austrian economics accepts as a crucial datum that almost all markets are out of equilibrium almost all the time, rather than considering the departures from equilibrium as temporary, unimportant, and annoying deviations from the model. Those deviations are the driving forces of development in this view, because they represent opportunity for new entrants and for existing firms to make changes in order to achieve competitive advantage (see Schultz, 1975). There has already been some related formal work. Simulation work by Eliasson and others (Eliasson, 1988) finds that an absence of variability at the level of the firm renders the economy unstable. In his words, "Simulation analysis with the Swedish M-M [micro to macro] model suggests that a significant variation over time, and diversity over time in regard to performance and price structures of individual agents, must be present for the model to generate stable macroeconomic performance. If not, small disturbances can be very disruptive" (p. 172). The linkup between chaos theory and Austrian economics is made stronger by the interpretation of both in terms of information. The informational role of price movements, of price differentials among locations, and (perhaps most important) among technologies, which then give rise to economic actions, is one of the main threads in the economic thinking of Hume, Smith, Menger, Hayek, and Friedman. On the chaos theory side, the notion that chaotic behavior contains much information (in contrast to behavior which conforms to the common-sense view of order, which contains little information) is entirely consistent with information theory. Indeed, the phase-space diagrams which are a crucial analytic tool of chaos theory are derived from information theory. An information-theory view also explains why chaos theory does not illuminate stock-market data. By definition, an efficient market is one in which the available information is rapidly used in totality and its results are therefore quickly exhausted; this explains the observed lack of correlation in price changes from one short period to the next. In contrast, in markets with few competitors, the relevant information is largely not available publicly, is learned only gradually and exceedingly imperfectly, and is acted upon with mechanical rules rather than more pure profit-maximization. Hence there is a historical element of persistence in imperfect markets which is consistent with chaos theory; the opposite is true with efficient markets. 4. Though chaos theory has had much of its development at the hands of physicists, it is more in the spirit of biological than of classical physical thinking. "Evolution is chaos with feedback", says chaotician Ford (in Gleick, p. 314). This also is more consistent with the spirit of Austrian economics, at least with respect to the role of variation and market "discovery" of successful innovations in economic development (see Hayek, 1960, early chapters), than with the spirit of neoclassical economics. 5. Gleick observed that "those studying chaotic dynamics discovered that the disorderly behavior of simple systems acted as a creative process" (p. 43, italics in original). Indeed, there is a impressive analogy between this mathematical description and the psychology of creativity; the artist or scientist or entrepreneur must both have the expansive freeness to produce new forms, but also the discipline to control and harness the results of the productive process. But there is more than analogy in economics and business: a departure from equilibrium represents actual opportunity, and the self-adjusting market mechanism represents actual constraint upon the possibility of exploiting the opportunity. So this view of market process does jibe with the Austrian view that markets are fecund and creative, and hence discover improvements in ways of doing things. 6. Though we were able to publish the 1973 paper in a prestigious journal (after several rejections from other prestigious journals, it must be said), the profession paid little attention to the results. We assumed that this was because the results were unsatisfying theoretically, leading to many outcomes rather than a single outcome. It was also the case that this result was a challenge to the validity of the standard analytic lines of inquiry into duopoly, all of which almost necessarily omitted the variables that bring out chaotic behavior. Indeed, as Dudley Dillard presciently noted at the time in his 1974 summary of the important developments in economics in The Encyclopedia Brittanica Yearbook: Simulation was another computer-based technique that came into wider use during 1973. An article in the July 1973 issue of The Review of Economic Studies, "A Duopoly Simulation and Richer Theory: An End of Cournot," employed a computer-simulated model in an attempt to give a definitive solution to one of the oldest problems of economic theory. In 1838 a French economist, Antoine Augustin Cournot, published the classical "solution" to the duopoly problem; that is, the determination of price and output in a market with only two sellers. The authors of the 1973 article contended that no single, definitive solution can result from duopoly; the specific solution depends on the conditions and numerical specification in any particular duopoly case. They concluded: "Finally, the Cournot question should be considered dead, and analytic attempts to answer it or to expand it should be abandoned as a waste of time" (p. 365). Old issues in economic theory never die easily, however, and one could safely predict that the Cournot question would be back in the journals in the form of further "contributions to knowledge." Perhaps even more important, our work did not seem to open up and point toward new analytic lines which aspiring researchers could fruitfully develop, because of the analytic intractability of the system. This inevitably discourages interest in a topic. But wedding this approach to duopoly with chaos theory may well point to exciting lines of work (though it may also point to "sophisticated" but arid lines of work). AN OFFER The purpose of this article is to exhibit some of the patterns observed in duopoly simulation in order to demonstrate that the system does indeed fit into the context of chaotic behavior. We also hope that this display will whet the interest of some readers who might then analyze the results using the techniques applied to chaotic behavior in other fields, such as phase-space analysis, analysis of periodicity using Feigenbaum techniques, and the extraordinary sequential plotting devices that have been developed. The model has recently been re- programmed by Carlos Puig for the personal computer, and we would be pleased to share the program with those who might wish to cooperate in advancing this line of work. SUMMARY AND CONCLUSIONS This paper discusses the application of chaos theory to ordinary markets, which - unlike agricultural and securities markets - characteristically have only a few sellers and non- identical goods. Duopoly is the specific example considered. A richly specified simulation model produces results which are consistent with the patterns of chaos - sensitivity to initial conditions, wide variety of results ranging from immediate stability to continuing instability with no immediately identifiable patterns. The structural features of the system also are consistent with chaos theory - non-linearity in variables, with both damping and forcing influences. Implications for public policy, for theory, and for future research are suggested. duopchao 9-186 article0 9-10-90 REFERENCES Axelrod, Robert, The Evolution of Cooperation (New York: Basic Books, 1984). Baumol, William, and Jess Benhabib, "Chaos: Significance, Mechanism, and Economic Applications", The Journal of Economic Perspectives, vol 3, Winter, 1989, pp. 77-106. Brock, William A., "Chaos and Complexity in Economic and Financial Science", in George M. von Furstenberg (ed.), Acting Under Uncertainty: Multidisciplinary Conceptions (New York: Kluver, 1990), pp. 423-450. --- and W. D. Dechert, "Nonlinear Dynamical Systems: Instability and Chaos in Economics", in Werner Hildenbrand and Hugo Sonnenschein (eds.), Handbook of Mathematical Economics, Vol 4, forthcoming, 1990 Dana, Rose-Anne, and Luigi Montrucchio, "Dynamic Complexity in Duopoly Games", Journal of Economic Theory, vol 40, October, 1986, pp. 40-56. Eliasson, Gunnar, "Schumpeterian Innovation, Market Structure, and the Stability of Industrial Development", in Horst Hanusch (ed.), Evolutionary Economics: Applications of Schumpeter's Ideas (New York: Cambridge U. P., 1988), pp. 151- 198. Friedman, James, Oligopoly Theory (New York: Cambridge, 1987) Hofstadter, Douglas R., Metamagical Themas (New York: Basic, 1985) Peterson, Ivars, The Mathematical Tourist (San Francisco: Freeman, 1988) Primeaux, Walter, and Mark Bomball, "A Reexamination of the Kinky Oligopoly Demand Curve", Journal of Political Economy, July/August, 1974. --- and M. Smith, "Pricing Patterns and the Kinky Demand Curve", The Journal of Law and Economics, Vol. XIX, April, 1976. Rand, David, "Exotic Phenomena in Games and Duopoly Models", Journal of Mathematical Economics, vol 5, 1978, pp. 173-184. Schultz, Theodore W., "The Value of the Ability to Deal with Disequilibrium," Journal of Economic Literature, vol 13, September 1975, p. 827ff. Simon, Julian L., Carlos M. Puig, and John Aschoff, "A Duopoly Simulation and Richer Theory: An End to Cournot", The Review of Economic Studies, XL(3), July, 1973, pp. 353-366. ---, and Joseph Ben-Ur, "The Advertising Budget's Determinants in a Market with Two Competing Firms", Management Science, vol 28, May, 1982, pp. 500-519. ---, "A Further Test of the Kinky Oligopoly Demand Curve", American Economic Review, vol 49, December, 1969, pp. 971-975. Stigler, George J., "A Theory of Oligopoly", Journal of Political Economy, vol 72, February, 1964, pp. 44-61. FOOTNOTES 1 The relationship between the two concepts comes out very nicely in the pseudo-random number generator used in computers. This is a perfectly deterministic chaos-type function that produces a sequence of numbers indistinguishable from numbers produced by a true physical random number generator. But the sequence is the same each time the function starts with the same "seed". By changing the seed "randomly", each run is also made different, in addition to being indistinguishable from a series produced with a physical process in accord with the Law of Large Numbers. That is, the principles are different but the outcomes are similar. Similarly, in the model described in the paper, each run made with the model produces the same result whenever it is repeated, as is the case with the data of chaos theory. There is no stochastic variation even though the results of many runs are not distinguishable with the naked eye from what consider ordinary or "true" random behavior. 2 Academic specialties are not quick to take in new ideas, of course. This is the message of Thomas Kuhn's well-known work. With respect to duopoly theory, see the quote from Dillard in the text. Gleick details how the specialties resisted chaos theory (p. 38). Eventually, however, it happened that (as Gleick quotes Kuhn) "The profession can no longer evade anomalies" (p. 315), and chaos theory began to make its way into the mainstream. duopchao 9-186 article0 September 10, 1990 APPENDIX: DESCRIPTION OF THE PRICE MODELS This Appendix describes the original price models, taken directly from the 1973 article by Simon, Puig, and Aschoff. The advertising models in Simon and Ben-Ur (1982) are quite analogous. The choice of a price for a given period by a single duopolistic competitor is conceptually simple. The firm faces a decision tree which consists of (a) the future possible price acts by the firm and the competitor; (b) the firm's assessments of the probabilities of the competitor's possible acts conditional on its own acts, probabilities which may be estimated from general experience without knowledge of the competitor's cost functions; (c) the expected market response to the two competitors' prices taken together; and (d) the firm's expected net revenue, discounted, in each period for each possible outcome. These items summarize all that is relevant to the price decision and, together with the firm's choices of appropriate allowances for risk and time preference for money, provide the basis for rational profit- maximizing decisions in the context of dynamic programming. If a decision tree tells which move firm i (say A, our reference firm), will choose at time t, then a sequence of endogenously determined decision trees (past events influencing its probability structure) represents the pattern of behavior over time in a duopolistic market. This is the behavior we have modeled here. We (a) construct a decision tree for A, choose the best price for it, and then move it to that best price for period t = 0; (b) construct a decision tree for B conditional upon A's price choice in t = 0 as well as upon B's knowledge of A's prior price-making behavior, choose a best price for B, and move it to that price; (c) construct a new tree for A based on B's just-previous price choices and A's knowledge of B's past behaviour; (e) and so on for as many moves as needed. The decision trees facing the competitors are probabilistic in the sense that a competitor does not know for certain the responses of the other competitor; rather, she estimates a probability distribution. We simulate the competitor saying to herself, "If I lower my price this period, there is a 70 per cent chance that my competitor will also lower his. If I then response to that by moving back to the old price, there is only a 50 per cent chance that he will too. If...If...If..." Nevertheless, our models are deterministic in that we never cast dice to determine what happens. There is no element of chance in the path traced out; two runs with the same parameters produce the same results. The events portrayed by a firm's forward-looking decision tree will be referred to as "conjectured moves, "conjectured choices", and so on. These are to be distinguished from the "real moves", "real prices", and so on, which the firms actually make in the simulation. Notation: g = index of any unique choice or state; i = a firm, A or B; i' = the other firm in the industry, the one not being referred to as i; Cit = cost of producing goods sold by firm i in period t; Dit = sales of firm i in t at a given price, i.e. firm demand when viewed ex ante; >< = Summation Sign. Interactive Structure Within the Models A basic characteristic of a duopoly model is the relationship between the competitor's decisions. In models SEQMATCH and SEQCUT the competitors change price sequentially, only one competitor in each period, rather than both changing price in the same period. The sequential aspect is an operational representation of secret price-cutting, the essence of which is that some time passes before a price cut is detected. In models CONKNOW and CONIG the firms choose their prices concurrently (simultaneously) in each period. In CONKNOW each firm assumes that the competitor will know the firm's price choice before the competitor choose its price--which in fact the competitor will not. In model CONIG, firm i expect its competitor to be ignorant of i's price in period t before it sets its prices in period t. Instead, when i sets its price in t, it expects i' in its next move to react to i's previous price change in t - 1, and vice versa for i' when it sets its price. In CONIG the imagined sequence of events is the same as the actual sequence. The Conditions Influencing the Choice of Price Except for the competitor-response function, all elements of the firm's situation are assumed known with certainty; that is, the ex-post functions are the same as the ex-ante functions. The price alternatives. In models CONKNOW and CONIG, a firm may choose to raise its price 6 per cent (2 per cent in a few runs), or keep price the same, or lower it 6 per cent (2 per cent). (The number of alternatives was so severely limited because of the limits of computing power.) In model SEQMATCH a firm has the option of matching its competitor's prices at any time, rather than being limited to a 6 per cent change. This price-matching option seems the more realistic. In SEQCUT the firm can also undercut the competitor's price cut, the other options being to match or to remain at the same price. In SEQMATCH and SEQCUT the stimulus is the level rather than the price behaviour of the competitor as in the other models. In models CONKNOW and CONIG the notion is, e.g. "If I raise my price, the probability is L that he will keep his price the same". In SEQMATCH and SEQCUT the notion is "If my price is above his, the probability is L that he will raise his to match mine". The cost function. For each firm, the conventional realized and conjectured total cost function is some fraction or multiple of Cit = 55 (Dit - 0.8)3 + 60 (Dit - 0.8) + 96. (1) Sales by each firm. The conjectured demand and the realized sales of a firm are influenced both by the industry demand function, as described below, and by the share of the total market the firm captures. The firm's market share is in turn determined by two factors: (a) its price and the competitor's price in the period in question; and (b) its market share in the previous period. That is, we assume that the market is imperfect and there is a considerable measure of lagged price effect. If A's price is lower (higher) than B's, A takes (loses) share from (to) B. Specifically, whichever firm's price in t is lower, its market share in t equals its market share in t - 1 plus an increment in share which is a function of the ratio between the prices PA,t and PB,t. Consider, for example, the case in which PA,t is lower than PB,t. The increment Y is calculated as follows: Y = H [log(PB,t/PA,t)]; l > Y > 0; H = 0.66 (0.22 in some trials) (2) Firm A's total market share is as follows, if it is lower than its competitor's: MA,t = MA,t-1 + YA,B (MB,t-1) (3) The other firm's share is calculated in a similar manner (MB,t = l - MA,t). This gives the firm with the lower price an increment in market share equal to a proportion of the other's previous market share. Some runs use an algorithm in which the increment in market share is a proportion of whichever market share was lower in the previous period, as in equation (3a). For the case in which PA < PB MA,t = MA,t-1 + Y . min (MA,t-1, MB,t-1). (3a) The industry demand function. The sum of the firms' sales is a function of the weighted average of their prices. This describes a market that is not perfect; some customers buy from the higher-price firm, as is observed in the world, and total demand is affected by both firm's prices. The firms' prices are weighted with their market shares in the prior period. >< Di,t = 39 [PA,t MA,t-1 + PB,t MB,t-1]-0.7 (4) i=A,B The conjectured competitor-response functions. Each possible alternative that a firm may contemplate choosing may be followed by one or another price response by the competitor -- either a raise, a drop, or no change. The firm's decision is crucially affected by the estimated likelihoods of those responses. The competitor-response functions (the subjective probability distributions of various possible competitor reactions) conjectured by the firms for any future period are changed in each period on the basis of three factors: (a) the initial conjectured competitor- response functions assigned to the firm as input data; (b) the past responses of the competitor to the firm's behaviour; (c) an adjustment factor W that determines how much influence each actual competitor response has upon the conjectured response functions. There are five competitor-response-function set-ups with which trials4 in each run are started. In the case of the "Sweezy" set up, the conjectured probabilities of a price raise, stay-same, or drop by competitor firm i' contingent on a price raise by firm i are 0, l.0, and 0 respectively; the "Sweezy" probabilities of a raise, stay-same, or drop by firm i' contingent on firm i staying-same or dropping, are respectively 0, l.0, 0; and 0, 0, 1.0. For the other four initial set-ups, the sets of probabilities, in the same order as above, are: "Uncertainty": 0.33, 0.33, 0.33; 0.33, 0.33, 0.33; 0.33, 0.33, 0.33; "Cartel": l.0, 0, 0; 0, l.0, 0; 0, 0, 1.0; "No-move opponent" or "Cournot": 0, 1.0. 0; 0, 1.0, 0; 0, 1.0, 0; "Partly-Sweezy": 0.5, 0.5, 0; 0.1, 0.8, 0.1; 0, 0.5, 0.5. (The probabilities for SEQCUT differ slightly from the aforegoing, and for SEQCUT and SEQMATCH the probabilities refer to price levels rather than to price actions.) An actual firm's conjectured competitor-response function does not remain the same over time. The firm may "learn". The model therefore revises the conjectured probabilities in light of the firm's experience with competitive reactions in the following fashion. Consider any action-and- reaction pair, e.g. a price rise by i followed by no change on the part of i'. Assume i's prior probability estimate was 0.40 that i' would keep his price the same, conditioned on i's price rise. Because i' actually did keep the same price, i's probability estimate that i' will keep price the same the next time i thinks about raising his price will be higher than 0.40. The ex-ante probability of the event that did occur, call it Lg,ex-ante is revised in accordance with the following rule: Lg,ex-post = Lg,ex-ante + W (1 - Lg,ex-ante) (4) where W is a fixed constant between 0 and 1, usually 0.25. For example, if A's ex-ante period-t probability of the response that occurred was 0.40, A's estimate of the probability of that event ex-post period t will be Lg,ex-post = 0.40 + 0.25 (l - 0.40) = 0.55 The ex-ante probabilities of events that did not occur are lowered in total by the amount that Lg,ex-ante is raised, and the total drop is distributed in proportion to their ex-ante sizes. Only the three probabilities of the competitor's responses contingent on the price choice that the firm actually did make are changed in each period. This scheme may be viewed as having some rough analogy to the Bayesian procedure. The choice rule. The firm chooses that price alternative in each period which maximizes its discounted expected net revenue, using standard backward-induction dynamic programming. Conjectured gross revenue5 in each period is the expected market share multiplied by industry demand multiplied by price. The discount factor used in most of the runs is 0.60. The decision tree is calculated for three separate conjectured periods, and then the expected net revenue for each third- period branch is assigned to twelve periods following, to take account of a long horizon (l5 periods) without increasing the number of branches, hence keeping within reasonable computation limits; this is the operational equivalent of assuming no price change from the third to the fifteenth period. The cost of capital is set so high in order that the last 12 periods' weight would not be excessive. A trial is stopped after 100 periods unless "stability" is not reached by then. A trial is judged stable when all variables cease to move, or when prices cease to move and one market share continues for many periods toward zero as a limit, or where a cycle clearly emerges. Cycles with long periods and wide amplitudes are hardest to judge for stability, of course. But in most cases, the results reached stability well short of 300 periods. "Stability" may be indicated by three successive identical peak or trough prices, with identical numbers of periods between the two pairs of peaks. Figure 1 FOOTNOTES FOR APPENDIX 4"Trial" refers to a single simulation experiment with a given response-function set-up and a given set of parameters. "Run" refers to a set of five trials with various response-function set-ups and a common set of parameters. 5An allowance for the greater riskiness of one alternative compared with another has been built into the SEQMATCH and SEQCUT models, substituting a utility function and certainty equivalents for dollar revenues. We made only a few runs with risk aversion, however. 3. The in-between price patterns are often characterized by the absence of any clear pattern within 3 Tennis racket, subjective randomness, pseudo-random numbers 99 In many a certain amount of judgment was involved as to when the run reached stability, and how to characterize that stability in a single number. The results of the original rus filled half of a large storeroom, and were disposed of when the space was needed for other purposes. But runs can be repeated now as needed.