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.
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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.
*