Non-linear dynamics and duopolistic competition: A R&D
model and simulation
Simon Whitby
a,1, David Parker
b,*, Andrew Tobias
a,1a
School of Manufacturing and Mechanical Engineering, University of Birmingham, Birmingham, B15 2TT, UK b
Aston Business School, Aston University, Birmingham, B4 7ET, UK
Received 1 December 1998; accepted 1 December 1998
Abstract
In recent years, a number of studies have considered the application of chaos theory to economics; the primary focus, however, has been the implications for stock prices, the foreign exchange market, and the macroeconomy. This paper describes a non-linear model of duopolistic competition, which focuses on a firm's expenditure and the resulting quality or technological endowment of its product. Results from computer iteration of the model are presented which indicate that chaotic outcomes are possible for a range of competing managerial policies; the associated unpredictability is due solely to the dynamics of the interaction. The study also provides the results of some initial work on how management adaptation may act to forestall chaotic outcomes.D2000 Elsevier Science Inc. All rights reserved.
Keywords:Economics; Competition; R&D; Non-linear dynamics; Chaos
1. Introduction
In recent years, a number of studies have suggested that non-linear dynamics offers a useful approach to economic modeling (e.g. Baumol and Benhabib, 1989; Day, 1987; Kelsey, 1988; Parker and Stacey, 1995; Radzicki, 1990 and the papers in Anderson et al., 1988). The approach requires a form of systems thinking which recognizes that systems evolve and adapt, persistently existing far from equilibrium. In particular, in a number of cases, systems of simple non-linear modeling equations have been shown to be capable of erratic and random-like behavior, which is in
fact deterministic; such behavior is described as
determi-nistic chaos.
This article provides a non-linear model linking R&D spending and product quality in a duopolistic industry. Results from a computer iteration of the models are pre-sented. These indicate that chaotic outcomes are possible for a range of competing managerial policies with the associated unpredictability being solely due to the dynamics of the
interaction. The study also provides some initial results on work into how management adaptation may act to forestall chaotic outcomes. The article has relevance to anyone researching or managing in the field of R&D and innovation. The results demonstrate that, based on some simple beha-vioral rules, quite complex chaotic outcomes are possible. They also show that, with an awareness of the possible outcomes from certain adaptive adjustments, management actions can prevent or remove chaotic outcomes. In particu-lar, collaboration between the duopolists may dampen erratic fluctuations and actually be welfare-enhancing in economic terms. The paper begins with a summary of previous related research into non-linear models and microeconomics.
2. Microeconomics and chaos
2.1. Chaotic systems
Chaos theory concerns systems where the relationships between variables are such that unpredictable but determi-nistic outcomes can result. Evolving chaotically, system variables are oscillatory with upper and lower bounds, the fluctuations being erratic to the extent that standard statis-tical tests cannot distinguish them from purely random data. The behavior, however, is deterministic and not a result of
* Corresponding author. Tel.: +44-121-359-3611; fax: +44-121-333-3474.
E-mail address: d.parker1@aston.ac.uk (D. Parker). 1
Tel.: +44-121-414-4263; fax: +44-121-414-7484.
outside (exogenous) factors. It is solely due to the mathe-matics of the relationship between variables endogenous to the system. In addition, such systems can have periods of regular behavior, such as equilibrium and periodicity, that arise abruptly and disappear just as suddenly.
The state of an evolving system can be represented by a number of time-dependent variables; the minimum number
of variables (n) needed to characterize a system is called its
dimension. As a system evolves, its state can be considered
to exist in a space of dimensionn called phase space, the
components of which correspond to the system's n state
variables. A geometric space with axes given by the phase space components (i.e. the system variables) provides a setting for the graphic illustration of a system's evolution. In
particular, for a dissipative system (i.e. one possessing a
mechanism to dissipate perturbations from steady motion), a steady-state behavior can take place in phase space on an
attractor. Crutchfield et al. (1986) define a system's attractors as geometric forms that characterize the system's long-term behavior in phase space. Chaotic evolutions of dissipative
systems in phase space occur onstrange attractors, which are
complicated shapes of fractal structure, first observed by Lorenz (1963). Many studies of non-linear dissipative dyna-mical systems use strange attractors to characterize chaos.
In order to predict future states of a time-dependent system, a set of modeling equations can be developed. Such
a set of equations is called adynamical system, which may
evolve continuously or discretely over time depending on the real system modeled. A continuously evolving
dynami-cal system is dynami-called aflow, while the discrete case is called a
mapping. Flows are represented by differential equations, while difference equations describe the discrete case. The
dependencies governing the evolution over time, t, of a
deterministic flow can be expressed as a set of first-order
differential equations giving the rates of change of the n
system variables,xi, in terms of the same variables, i.e.:
1
For a discrete-time-mapping, the evolutionary laws can be similarly expressed, but as difference equations giving the
values of the system variables in period t + 1(xi,t + 1) in
The most fundamental requirement for chaos is non-linearity. In the context of Eqs. (1) and (2), this essentially means that at least one state variable will appear in the
functions fi or gi (where i denotes the index of a general
variable) with a non-linear relationship to the rate or the next period value, respectively. Over and above this, precise specification of the prerequisites for chaos of a general system is difficult; there are presently no general criteria to establish the necessary and sufficient conditions for a system of differential or difference equations to be chaotic. Requirements have been established, however, regarding the minimum dimensionality of the system. Different con-ditions exist for flows and maps (Ott, 1993). For the case of first-order, autonomous ordinary differential equations, i.e. of the form of Eq. (1), the lowest dimension for which
chaos is possible (nmin) is 3. For a mapping, the minimum
required dimensionality differs depending on whether or not the mapping is invertible. Using notation as in Eq. (2), an invertible map is one whose equations, as well as giving a unique value for eachxi,t+ 1from the set ofxi,tcan also be
solved to give a unique value for eachxi,tfrom thexi,t+ 1.
For an invertible map, there cannot be chaos unlessnmin
2. For a non-invertible map, chaos is possible even for a dimensionality of one. The logistic equation introduced by May (1976) in a study of population, is a one-dimensional map often used to illustrate chaotic behavior. It is a finite difference equation with a quadratic non-linearity or hump and takes the general formxt+ 1=lxt(1ÿxt). In population
studies, xt+ 1 represents the population level and depends
on its immediate predecessor, xt. The parameter l is the
growth rate from one cycle to another andxis constrained
by a resource constraint modeled as 1ÿxt. Through
feed-back leading to period doubling (see later), the system's
evolution can become chaotic as the parameter value, l, is
increased, with chaos occurring when lis between around
3.6 and 4.
Whereas little can be said, a priori, concerning a system's capability of producing chaos, for systems shown to be chaotic some typical ``routes'' or ``scenarios'' have been identified by which the onset of chaos may be approached. In each case, the transition is effected by continually changing (usually increasing) one or more ``control''
para-meters, such aslin the logistics equation above. Normally,
only one parameter is varied, in which case the scenario is known as a codimension Ð one route to chaos. Generally, at
distinct values of the changing parameter (l) bifurcations
occur. Bifurcations involve qualitative changes in the beha-vior of the system, for example, from equilibrium to periodic or from periodic to chaotic (for a fuller explanation see, e.g. Berge et al. 1984; Eckmann, 1981; Medio, 1992;
Ott, 1993). One scenario, known as intermittency, involves
interruption of otherwise regular periodic evolution with
bursts of irregular, chaos-like behavior aslis varied above/
below a critical value. The bursts are usually seen to become
more frequent aslis increased/decreased until the original,
Another route to chaos for an initial period system, is
period doubling. As l is increased, bifurcations occur at
particular values as additional periodicities complicate the
system's cycle. The frequency of bifurcation increases asl
is progressively increased until at a finite value of l the
number accumulates to infinity; increasing l beyond this
point results in the evolution becoming chaotic. Period doubling will be described in further detail later, when considering the results of our model.
In the light of what has been described above, we can say that any possible candidate to represent a chaotic system must satisfy the non-linearity and minimum dimen-sionality requirements. In addition, the model must be oscillatory, or at least have the potential to be so, in order not to preclude the possibility of its developing from periodic to chaotic evolution.
2.2. Chaos and the firm; inventory control
If microeconomic institutions such as firms and markets are governed by linear, or simple non-linear relationships, it is possible to study them separately (adopt a reductionist methodology) without losing crucial characteristics of their behavior. Interactions with other firms or markets can be relatively easily assimilated into the models as generally well-behaved positive and negative feedback effects. Equi-librium outcomes result and uncertainty as to the model's outcome is minimized. If economic systems are significantly non-linear, however, then their behavior may be highly unpredictable, and this limits the application of both tradi-tional neoclassical market models and more recent games theoretic modeling involving Bayesian techniques.
Many of the non-linear models, which have identified the potential for chaos in microeconomics, involve discrete-time queuing or consideration of inventory. A typical example is the beer distribution model, an exercise in beer-stock management involving non-linearity and feedback (Mose-kilde and Larsen, 1988; Sterman, 1988; Van Ackere et al., 1993). Mosekilde and Larsen presented an overview of this model's behavior, producing a time series through iterations of the model and identifying chaos for certain parameter values. Sterman's approach was notably different in his use of the model to provide a simulated environment in which subjects could take part in a beer distribution game. To describe the subjects' behaviors, a general heuristic was constructed, which, on further iteration, suggested chaos for a significant minority of values.
The common feature in these inventory control models is a delay between the ordering of new stock and its delivery. Furthermore, disruption in one part of the supply chain leads to a sequence of changes in other parts; expansion of output, for example, leads to excess capacity developing when orders fall. Overall, this suggests a tendency for oscillation in supply behavior, which, for certain conditions, can become chaotic (Erramilli and Forys, 1991; Levy, 1994).
The interplay between two departments functioning se-parately and in parallel, as opposed to in sequence as in the stock management problem, was considered in a study by Rasmussen and Mosekilde (1988). This study proposed a model of a production company where resources are shared between production and marketing in accordance with a decision rule reflecting variations in inventory or the back-log of customer orders.
Other studies have suggested that the incidence of chaos in business may be much wider. For example, it has been suggested that there may be a tendency towards strongly non-linear local interactions where there are nonconvexities in production technology, due to technological indivisibil-ities for instance (Gordon, 1992). Gordon also used a variation of the logistic equation to produce a plausible model of the relationship between a company's advertising budget and its resulting sales. Further economic relation-ships subject to feedbacks and timing, which have been investigated for chaotic outcomes, are investment behavior (Mosekilde et al., 1992), the planning of teacher ± student ratios in education (Feichtinger and Novak, 1992), and advertising outlays (Baumol and Benhabib, 1989).
The effect of research and development (R&D) activity on a firm's time path has been modeled by Baumol and Wolff (1983), Feichtinger and Kopel (1993), and Kopel (1996). Baumol and Wolff focused on the relationship between R&D and productivity growth, showing chaos to be possible for certain conditions. The Feichtinger and Kopel and Kopel models, on the other hand, described a non-linear increase of sales for higher R&D allocations, employing a treatment closer to that presented here. Nota-bly, Feichtinger and Kopel placed particular emphasis on the influence of the decision-maker (i.e. the manager) whose input amounted to periodic adjustments to R&D funding in the light of current sales. This was represented by a non-linear decision rule that is well-founded in behavioral decision theory, and which also features in the model used
in this study, namely the anchoring and adjustment
heur-istic, described in Tversky and Kahneman (1974). A variety of temporal behavior was observed, including chaos.
In the above studies, the model parameters representing human behavior were kept at a constant value throughout a given evolution. This failure to allow parameter variability within evolutions effectively ignores the fact that decision-makers may adjust their responses over time; their actions may evolve through some form of iterative process, for example. There is a comment on this point in Sterman (1988, p. 174).
2.3. Chaos and oligopoly
example, by also cutting prices so as not to lose market share. Developments in oligopoly theory have shown that various other outcomes are possible depending, for exam-ple, upon the financial strength and production conditions for each of the firms. In recent years, it has been recognized that competition among a few firms can be usefully modeled in a game theoretic framework that contains feedback, time, and uncertainty. In game theoretic models of oligopoly, rivalry is studied from an equilibrium (usually a Nash equilibrium) perspective and uncertainty is dealt with by an expected profit or payoff maximization. The outcome then depends upon the assumptions made about actions and reactions at the various stages of the game (e.g. see Hargreaves Heap and Varoufakis, 1995; Phlips, 1995).
Game theory has introduced dynamism into the study of the competitive process, opening up the possibility of sufficiently non-linear reaction functions to give chaotic outcomes. Rand (1978) considered the possible conditions in which chaotic behavior might result, and more recently, Byers and Peel (1994) flagged the possible contribution the analysis of dynamic non-linear behavior might make to the study of industrial economics. To date, however, there has been limited analysis of the potential for chaos in oligopo-listic markets. Noteworthy studies include that by Dana and Montrucchio (1986), which showed how chaotic dynamical paths might arise in duopoly where there are small rates of discount, and that by Puu (1994), who considered an adjustment process for a pair of Cournot duopolists. Puu demonstrated mathematically that with an iso-elastic de-mand function and constant marginal costs, the duopoly system could exhibit periodic, and even chaotic behavior.
In Section 3, a new, non-linear model of duopolistic interaction is described in which the effect of managerial policy plays a central role. The formulation is based on the relationships specified in the R&D model of Feichtinger and Kopel (1993). However, while their study included manage-rial judgements relative to an internally determined target performance, the following extends the analysis by applying the relationships to the interaction between two separate, competing companies, the policy judgements of each being made in light of its own performance relative to the other. In addition, while Feichtinger and Kopel's treatment, along with the vast majority of model simulation studies, makes no allowance for policy adjustment over time, this study makes some attempt to include the management's capacity to adapt its strategy.
3. A basic model of duopolistic competition with non-linear dynamic interactions
The proposed model describes two manufacturers pro-ducing a similar technological product and competing through product innovation. In both cases, managerial decisions regarding the extent of an input or resource allocation are made in the light of the organization's recent
standard of achievement in terms of the resulting product, relative to that of the competition. In addition, this stan-dard is, for both firms, a direct result of the level of inputs, i.e. resources applied. In constructing the model, particular emphasis was placed on the resources that each allocates to R&D, and the resultant (measurable or subjective) product standard/quality, such as the number of features or durability. The latter are means by which discerning consumers may differentiate in a market where product standard is an important order winning criterion. Expendi-ture on R&D could also result in a change in the process of production rather than a change in the product, having implications for costs of supply; this impact on process is not modeled here. In the model, an increase in a compa-ny's resource allocation to R&D is considered to lead to an improved product standard (quality or technological en-dowment of the product), while having a product standard superior to that of the competition is deemed to constitute a competitive advantage leading ultimately to greater sales and profits. Current sales, therefore, are considered to be a direct result of the current, relative product standards of the competitors and so do not need to feature explicitly in the model. Each firm is assumed to be able to monitor readily the product standard of its competitor.
The model is structured to represent two companies, A and B, for which the resource allocations to R&D for period
t are denoted by RAtand RBt, resulting in product quality
standards denoted for A and B during period tby SAtand
SBt, respectively. Modeling R&D to impact on the product
standard in the same time period is admittedly a simplifica-tion Ð R&D might be expected to affect product standard in
t+nwherenvaries depending on the product and industry;
the simplification is introduced for ease of modeling. A notable consequence of the simplification is that for condi-tions under which the system oscillates over time, the fluctuations of the separate companies will be either in phase (peaks coincide) or in anti-phase (one's peaks coin-cide with the other's troughs).
To facilitate description, X, RXt and SXt are used to
represent either company and its associated system
vari-ables for period t, in the general case. Arising from
managerial decisions made at the end of each time period, changes to R&D allocation are modeled as dependent on the current relative quality or technological standing of the firm's products.
The interactions within the system are shown in Fig. 1, where the arrows between variables represent deterministic cause and effect Ð the ``tail'' variables affecting the ``heads,''
e.g. RAt affects SAt. The dotted arrows describe similar
effects but between tail variables and the head variables of thefollowingperiod, e.g. SAtÿSBtaffects RAt+ 1.
may have implications for costs and consequently affect prices and demand, the introduction of this feedback would complicate the analysis and is not, in any case, the relation-ship which we wished to study. Hence, in the analysis:
QAt=Ytf SAtÿSBt 3
QBt=Ytf SBtÿSAt 4
whereYtis the total market volume for periodtand QAtand
QBt are sales of A and B, respectively for that period,
making QA/Yand QB/Ythe respective market shares. As the
market is a duopoly,
QAt=YtQBt=Yt 1 5
For both companies, the R&D expenditures, RAt+1 and
RBt+1, for a given period,t+ 1, are assumed to arise from
management decisions at the end of the previous period,t,
(Eqs. (6) and (7)). The expenditure for a period t is
considered to determine directly the quality standard
achieved in production (SAt and SBt) during the same
period,t, (Eq. (8)).
curvature of the function for company X.
If the SXt in Eqs. (6) and (7) are substituted by SXt =
f(RXt), from Eq. (8), then the decision rules take the form
R Xt+1=g(RXt); expressed in this way the representation of
the decision process is seen to be non-linear. Expressions (6) and (7) are similar in form to that used by Feichtinger and Kopel (1993), and are based on the anchoring and adjust-ment heuristic described in Tversky and Kahneman (1974). The estimation of an unknown quantity by anchoring and adjustment involves a preliminary estimate given by a
known reference point (theanchor) which is thenadjusted
for the effects of other factors. Sterman (1989, p. 324) reports that ``anchoring and adjustment has been shown to apply to a wide variety of decision-making tasks'', and gives appropriate references.
For the managerial decisions in this model, the
alloca-tion of the previous period, e.g. RAt for company A, is
identified as the anchor, while an adjustment is considered to depend on the relative product standards of the two
companies. Eq. (6) for A, for example, shows RAt acting
as the anchor with an adjustment based on the advantage A
has over B in current product standard, i.e. SAtÿSBt. The
parameterscand dare used to define the two management
policies for allocation adjustment. For instance, again
considering A, a positive value of c is used to represent
a reinforcing policy of further increasing a company's
research funding when performance is good, i.e. SAt >
SBt, and reducing funds when performance is poor (SAt <
SBt). Conversely, a negative adjustment parameter reflects
a counteracting policy of reducing funds when perfor-mance is good, a poor perforperfor-mance resulting in their increase. The two policies are summarized for company A in Table 1. The modulus of the parameter is used to
model the extent to which the management alters funding
in light of the company performance, i.e. the sensitivity of
the decision-maker to performance outcomes. As the deci-sion rules stand, only the two policies are possible; the specification of a company's action when in a position of advantage necessarily implies its action when disadvan-taged and vice-versa. The formulation could be extended, however, to allow a greater range of procedures where, for example, a company could adopt a policy of reinforcement when at an advantage but take counteracting action when at a disadvantage.
For each company, the function describing the effect of various R&D fundings on product standard should show a general increase of the latter with the former, the relation-ship becoming markedly non-linear for higher R&D ex-penditures as diminishing returns set in. Furthermore, in order to investigate the possibility of complicated behavior,
it is important to consider a description of SXt=f(RXt) for
which non-linearity is apparent over the range of values of
RXt displayed by the model in practice. A function is
required, therefore, whose position and extent of curvature can be adjusted easily in the light of the range of variation
Fig. 1. Model of duopoly comprising two similar manufacturers (A and B) of technological products.
Table 1
Reinforcing and counteracting policies for company A
Allocation
Reinforcing c> 0 Attempt to reinforce advantage, i.e. increase in allocation to research
Retreat, i.e. decrease research allocation
observed on initial iterations of the model. Suitable control is obtained with the arctan form in Eq. (8) above, the shape
of which can be tuned by changing mx and nx. Together,
these parameters determine the upper and lower bound of
SXt in the model. In particular, varying nx changes that
value of product standard (SXt) corresponding to a research
allocation (RXt) of zero; SXt= 0 and RXt= 0 only coincide
for the parameter valuenx= 0. It is reasonable to expect SXt
= 0 to correspond to some positive value of RXt, reflecting
some element of fixed costs that do not impact on product standard, and this is represented in Eq. (8) by a positive
value ofnx. The general shape of Eq. (8) is shown in Fig. 2
for which the origin is (0,0) and the negative intercept on the
SXtaxis is equal to the value ofnxused.
4. Experimentation and results
In general, non-linear dynamical systems, such as the model presented here, describe temporal evolution with equations for which there is no simple solution. In such cases, investigation takes the form of computer iteration, i.e. computer simulation, which, rather than attempting to solve the equations, repeatedly implements the model and so produces actual temporal behavior for analysis. Accord-ingly, computer simulation was used to investigate the evolution of our model. Along with quantification of para-meters for Eqs. (6) ± (8), each simulation of a given system arrangement required specification of initial R&D
alloca-tions for A and B, i.e. RA0and RB0. To facilitate the study,
it was assumed that the relationship between funding for R&D and product standard, Eq. (8), was identical for the two companies. As a consequence, if one firm spends more than the other on R&D, this will imply an advantage in product standard for the same company for that period, e.g. RAt> RBt()SAt> SBt.
Often, the time paths of an evolving system are
consid-ered in phase space, a mathematical space whosen
coordi-nates correspond to the system'sn state variables (n being
the minimum number of variables needed to characterize the
system, i.e. its dimension). If n 3, or a subset of phase
space with less than three components is considered, then
this space provides a setting for the graphic illustration of a system's evolution (see Crutchfield et al., 1986 for an introduction to phase space, attractors, and the chaotic
strange attractors). However, as will be seen, for the interesting oscillatory behavior of this system, the two companies evolve either in phase or in anti-phase (i.e. with
a phase difference of p). It follows that if an appropriate
two-dimensional phase space ((RAt, RBt) or (SAt, SBt)) is
used to analyze system behavior, any oscillatory motion (including chaos) will produce an unilluminating, straight-line phase trajectory. For this model, therefore, phase space offers a less effective illustration of system behavior than that given by time series, and accordingly, the latter are used here to present the results.
The evolutionary properties of the system were found to depend on the coupling of company strategies. Particular
couplings were specified by the parameter pairings (c,d),
referred to here as the system arrangement. Arrangements
were then classified into four groups, each comprising a
different permutation of the paired signs of c and d, i.e.
(+,+), (+,ÿ), (ÿ,+), and (ÿ,ÿ), and each group containing a continuous range of system arrangements corresponding to
different moduli ofcandd.
4.1. Arrangement group(+,+)
For this class of arrangement, both companies implement reinforcing policies and incorporate a positive feedback
loop, i.e. RAt!RAt + 1 and RBt!RBt + 1, the two
inter-acting through the comparison of SAtand SBt(see Fig. 1).
When a firm is ahead in product standard, it invests more in the next period to maintain or increase its competitive advantage; when it is behind, it retreats and invests less, perhaps with a view to eventual withdrawal from the market. It was found that for this group of policy arrange-ments, the system was not prone to oscillation, but instead,
RAt and RBt repeatedly diverged, one to zero, the other
continuously increasing. As would be expected, SAtand SBt
also diverged, approaching the upper and lower limits bound by the function (8). The company gaining the advantage, therefore, spends an increasing amount on de-velopment for marginal increases in product quality/sales.
The tendency for divergence rather than oscillation can be understood in terms of the feedback structure of the system for these arrangements. Here, positive feedback in a company reflects the fact that the policy of its management
will tend to reinforce any initial difference between SAtand
SBt. It follows that for (+,+) arrangements, any initial
difference in product standard of A and B will be amplified by managerial actions in both cases and so oscillation will
not occur. Rather, SAtand SBtwill repeatedly diverge to the
upper and lower bounds of the model, as observed. Coupled with non-linearity, system oscillation is an essential condition for the onset of chaotic behavior, and the lack of oscillation for (+,+) system arrangements implies an inability to produce chaotic behavior.
Fig. 2. The relationship between R&D expenditure (RXt) and product
4.2. Arrangement group(+,ÿ)/(ÿ,+)
The groups of opposing managerial policies (+,ÿ) and (ÿ,+) are identical due to the structural symmetry of the model and while the following refers to (+,ÿ), it applies equally to (ÿ,+). For these arrangements, on occasions when the firm with the reinforcing policy is ahead in terms of product standard, this firm invests more in R&D. In such cases, the trailing competitor (with a counteracting policy) does the same in the hope of making good its present loss of competitive advantage. Equally for this arrangement group, circumstances occur for which the counteracting firm has the advantage and, true to its policy, relaxes and invests less in R&D, perhaps expecting to maintain its competitive advantage without additional R&D effort. In this case, the trailing, reinforcing competitor also invests less, representing a firm which, when trailing, sees no point in trying to compete through R&D to improve its market share.
Investigation of the model showed the potential for oscillation to depend not only on the modulus and sign of the parameters, but also on the initial relative standing of the two companies. This led to the consideration of four general scenarios.
4.2.1. Scenario 1
Arrangements were investigated for which one company
(say B) had a sensitive reinforcing policy (d = a large
positive number), while the other (A) had a less sensitive
counteracting policy (c= a smaller negative number), with
initial conditions such that the reinforcing company started
with the greater allocation (RB0 > RA0). For these
condi-tions, as for (+,+) arrangements, the model showed no
potential for oscillation or chaos. In this case, the RAtand
RBt(and so SAt and SBt) variables both increased
repeat-edly over time. As before, this tendency for monotonic variation rather than oscillation can be understood from the system's feedback structure, as determined by the policies. For these arrangements and initial conditions, the larger initial funding of B translates to a larger initial product
standard SB0. As the model is iterated and the system
evolves, B, with its sensitive reinforcing policy, will further increase funding and so amplify its advantage. At the same time, A, with its counteracting policy, attempts to remedy its disadvantageous initial position by increasing its funding. Due to the lower sensitivity of its policy, however, A repeatedly redresses B's advantage less than the extent to which B further increases it. Overall,
there-fore, SBt and SAt will diverge to the upper bound of the
model, but with SBt > SAt at all times before the upper
bound is reached.
4.2.2. Scenario 2
Further (+,ÿ) arrangements were investigated for poli-cies with relative sensitivities as above, but with initial conditions such that the reinforcing company started with
the smaller allocation (RA0 > RB0). The system again
showed monotonic behavior rather than oscillation or chaos. This behavior can be explained by the fact that here, B, with its sensitive reinforcing policy, increases its initial
disad-vantage and so directs SBt towards zero. Meanwhile, the
smaller counteracting policy of A means this company acts so as to relax on its advantage. The overall result of these
tendencies is that both SBt and SAt fall to zero, with the
former having the more sensitive policy, thus reducing the more rapidly.
4.2.3. Scenario 3
Investigations were also carried out on a third type of (+,ÿ) arrangement, somewhat different from the two considered above. In this case, the one company (B) had
a sensitive counteracting policy (d = a large negative
number), while the other (A) had a less sensitive reinfor-cing policy. Further, the initial conditions were such that the counteracting company started with a greater allocation
(RB0 > RA0), which translated to SB0 > SA0. For these
conditions, for certain parameter values, oscillatory
beha-vior was observed, shown for variables RAt and RBt in
Fig. 3(a) and (b), respectively. It is notable that while the two variables took values over different ranges, their oscillations were in phase.
The potential for chaos in non-linear systems is often investigated by varying one system parameter while the others are kept constant. The occurrence of increasingly complicated periodic behavior leading to chaos as a para-meter is varied and is well-recognized in chaotic systems. Where one parameter is varied, the scenario is known as a
Fig. 3. (a) Period 1 oscillations of RAtfor system arrangement (c,d) =
codimension-one route to chaos. Generally, at distinct
values of the changing parameter,bifurcationsoccur.
Bifur-cations involve qualitative changes in the behavior of the system, for example, from equilibrium to periodic (for a fuller explanation see e.g. Berge et al., 1984; Eckmann, 1981; Medio, 1992; Ott, 1993). Such a process was ob-served for these system conditions, as shown in Fig. 4(a), (b), and (c) for which the system's behavior was studied for
increasingly negative settings of d. These changes
corre-spond to a company (B in this case) becoming increasingly sensitive in its counteracting policy, while sensitivity at the reinforcing competitor remains unchanged.
The plots in Fig. 4 indicate classic period doubling. In
Fig. 4(a), for which policy parameterd=ÿ4, the evolution
of RBt was found to undergo two oscillations, with two
different maxima and minima, before a full cycle back to the initial value was completed. Behavior of this type is known as aperiod 2evolution, while the most basic oscillations of
Fig. 3 are termedperiod 1, the system bifurcating from the
latter to the former as the control parameter (d) is varied. In
Fig. 4(b), d has become more negative, d = ÿ4.18, and
another bifurcation has occurred so that the periodic beha-vior has become even more complicated; each complete
period now comprises four sub-oscillations: a period 4
evolution. In theory, this period doubling as a system parameter is increased/decreased will continue for even higher periods. For increasingly higher period numbers, however, the bifurcations occur for progressively smaller changes in the increasing/decreasing parameter, and there-fore, become progressively more difficult to observe.
For some non-linear systems, if the control parameter is progressed past a critical value, the motion becomes
aper-iodic, producing the phenomenon known as deterministic
chaos. Fig. 4(c) shows the model exhibiting such chaotic behavior; there is no longer a discernible repeating pattern to the series, which has become aperiodic. A comparison of
Fig. 4(c) and (d) illustrates sensitive dependence on initial
conditions (sdic), known to be a characteristic of chaotic evolutions. Both series emanate from identical system arrangements, i.e. the company policies are the same in each case, but the initial values of the evolutions, as
specified by (RA0, RB0), differ slightly: initial conditions
were (10.0, 15.0) and (10.1, 15.0) for Fig. 4(c) and (d), respectively. The marked difference in the two (chaotic) evolutions, as a result of sdic, suggests serious, fundamental limitations to forecasting, control and planning where such system arrangements exist.
A useful illustrative tool in the analysis of systems, which pursue the period doubling route to chaos, is a
bifurcation diagram. A bifurcation diagram indicates how the post-transient, limiting behavior of a system alters as one parameter is varied. One of its great strengths is that it provides a more general, global view of the system's behavior for a variety of parameter settings; in this case, company policies. More specifically, for this discretely evolving model, the bifurcation diagram shown in Fig. 5 identifies the values of a particular system variable (say
RAt) present in the post-transient evolution, plots these
values against the corresponding value of the parameter, and repeats this for a range of parameter values. The
Fig. 4. (a) Period 2 oscillations of RBtfor system arrangement (c,d) = (+0.1,ÿ4.0), initial conditions (RA0, RB0) = (10.0, 15.0). (b) Period 4 oscillations
of RBtfor system arrangement (c,d) = (+0.1,ÿ4.18), initial conditions (RA0, RB0) = (10.0, 15.0). (c) Chaotic behavior of RBtfor system arrangement
(c,d) = (+0.1,ÿ4.5), initial conditions (RA0, RB0) = (10.0, 15.0). (d) Chaotic behavior of RBtfor system arrangement as in (c), initial conditions (RA0,
evolutions have the same starting values (RA0and RB0) in
each case.
A comparison between a time series and its representa-tion in a bifurcarepresenta-tion diagram helps clarify the structure of the latter. The post-transient periodic motion of Fig. 3(a), for
example, comprises two values of RAt, which together
constitute the oscillations. Looking at the system's bifurca-tion diagram in Fig. 5, it can be seen that for policy
parameterd = ÿ3.8 (with c= 0.1) two values of RAt are
plotted, shown at f in the diagram, thereby indicating a
period 1 oscillation. It follows that the single value of RAt
plotted for policyd= ÿ3 indicates one value of RAtin the
corresponding time series, i.e. equilibrium (see g in the
diagram). In a similar way, period 2 and period 4 behaviors,
such as those shown by RBt in Fig. 4(a) and (b),
respec-tively, correspond to instances where evolutions comprise four and eight variable-values in turn, as shown on the
diagram athandi, respectively. The parameter region of the
bifurcation diagram, for which a seemingly continuous but
bounded range of RAtare present, represents chaos.
As before, analysis of the system's feedback structure sheds light on the tendency of (+,ÿ) arrangements to oscillate for these conditions. On iteration, company B, with its sensitive counteracting policy, tends to relax given its initial advantage and spends less on R&D in the next period. Simultaneously, the smaller reinforcing policy of A acts to reinforce its initial disadvantage. Due to its more sensitive counteracting policy, however, it is B, which has the stronger initial reduction, and accordingly, the system soon evolves so that RBt< RAt(and by evolution SBt< SAt). In
this state, B's counteracting policy then acts to increase RBt
while A's reinforcement has a similar but weaker self-increasing effect. Together, these effects cause evolution to
a state where SBt> SAt. This vacillation of the position of
advantage continues throughout the evolution, i.e. in these conditions the system is prone to oscillation.
4.2.4. Scenario 4
Oscillations, period doubling, and chaotic behavior simi-lar to that described above were again obtained for certain parameter values for a fourth type of (+,ÿ) arrangement. The relative sensitivities of the policies were similar to those for the conditions producing the oscillations and chaos described above, but here, the initial conditions were such that the counteracting company started with the smaller
allocation (RA0> RB0). The tendency for oscillation could
be explained by the fact that for these conditions, the sensitive counteracting B acts to remedy its initial disad-vantage, while the less sensitive reinforcing A tends to increase its advantage. As before, B's effect is the stronger,
so that the system evolves to a state where RBt> RAt, and
the policies now engender opposite effects. This change in direction of the effects due to policy occurs repeatedly, and it follows that for these conditions, there is a tendency for the system again to oscillate.
4.3. Arrangement group(ÿ,ÿ)
In this group, both companies adopt a counteracting policy. At all times, the leading company invests less in the next period while the trailing company moves to invest more. For these arrangements, the managers of both firms act so as to decrease any extant difference in their perfor-mance. Investigations were carried out on arrangements where one company, say B, had a decidedly more sensitive policy than its competitor and initial conditions such that
RA06RB0()SA06SB0). For these conditions,
oscilla-tion occurred for certain parameter values, as shown for
variables RAtand RBtin Fig. 6(a) and (b). It is interesting
that while for the (ÿ,+) group, RAt and RBt oscillated in
phase, as shown earlier, the (ÿ,ÿ) results show oscillations in anti-phase, i.e. one's maxima coincide with the other's minima. Period doubling bifurcations to chaos were ob-served as the modulus of the more strongly negative para-meter was increased (representing an increase in the sensitivity of that company's counteracting policy) as illu-strated in Fig. 6(c) ± (e). The corresponding bifurcation
diagram is shown for variable RAt in Fig. 7, while a
comparison between Fig. 6(e) and (f) shows that the
anti-phase relationship between the evolutions of RAt and RBt
(and so SAtand SBt) persists into the chaotic regime.
It was found that for (ÿ,ÿ) arrangements where the competitors adopted policies of similar sensitivity, the
tendency for oscillation was reduced and RAt and RBt
(and so with SAt and SBt) pursued a monotonic approach
to an equilibrium between RA0and RB0(and between SA0
and SB0).
Again, the system's oscillatory behavior can be under-stood from a consideration of its feedback structure. For the case when B has the notably more sensitive policy, its
corrections (i.e. reductions to RBt when in a position of
advantage, and increases when disadvantaged) will be larger than those of its competitor. Reasoning as before
suggests that for these conditions, the system is prone to oscillation where the scale of variation of the more sensitive company is substantially greater than that of the competition.
5. An adaptive model: the learning organization
The basic model outlined above describes managerial adjustments to resource allocations, but makes no allowance for any change over time in the policy concerning these adjustments. This constitutes a serious limitation of the model as, in reality, it is to be expected that the strategy adopted will change, reflecting the decision-maker's percep-tion of recent-policy efficacy. For an intelligent management (or the ``learning organization''), there will be attempts to adapt to previous shortcomings. The basic model was therefore extended to include a mechanism for this
adapta-tion process by permitting variaadapta-tion over time of the policy
parameterscandd.
5.1. Adaptation mechanisms
Two different representations of adaptation were in-cluded separately in the adaptive model, the management
in both cases contemplating a policy change every f
periods (e.g. quarterly or annually). In deciding whether to make a change, the manager (of, say, A) was considered to examine the company's performance relative to that of
the competitor over each of the f periods since the last
deliberation. The sum of these relative performances (M)
was used as an overall measure, where, at period t,
M Pt
tÿfSAiÿSBi. If M > 0, the current policy was
considered satisfactory, while M< 0 was taken to indicate
a need for change. This is admittedly a simple criterion for assessment, which makes no allowance for trends. In
Fig. 6. (a) Period 1 oscillations of RAtfor system arrangement (c,d) = (ÿ0.1,ÿ3.0), initial conditions (RA0, RB0) = (10, 5.0). (b) Period 1 oscillations of RBt
for system arrangement and initial conditions as for (a). (c) Period 2 oscillations of RBtfor system arrangement (c,d) = (ÿ0.1,ÿ3.8), initial conditions (RA0,
RB0) = (10, 5.0). (d) Period 4 oscillations of RBtfor system arrangement (c,d) = (ÿ0.1,ÿ3.99), initial conditions (RA0, RB0) = (10, 5.0). (e) Chaotic behavior
of RAtfor system arrangement (c,d) = (ÿ0.1,ÿ4.15), initial conditions (RA0, RB0) = (10, 5.0). (f) Chaotic behavior of RBt, system arrangement and initial
particular, a negative value of M would be returned for a trailing company that is steadily reducing its deficit over
the f periods. In such cases, the policy would, to some
extent, be working satisfactorily and yet the negative value
of M would precipitate its modification. Nevertheless, the
criterion does provide a basis for introducing managerial assessment of performance into the model; the effect of more complicated adaptation must await future research.
It was in the adjustment rule that the two adaptation
mechanisms differed. In one, termed proportional policy
adaptation, a change amounted to a proportional reduction in the extremity of a policy while maintaining its nature
(counteracting or reinforcing, as represented by the sign of the parameter). Specifically, this reduction was executed in the model by multiplying the appropriate company's policy parameter by a reducing factor, acor ad(0 <ac,ad< 1),
so that ck = acckÿ1 or dk = addkÿ1, where the index k
describes a particular stretch of f time periods. The same
values of ac and ad were used throughout a given
evolu-tion, i.e. for each change, policy extremity was reduced by a constant proportion. Managers adopting proportional adaptation are mindful of the extremity of the policies they change. It is notable that for this first mechanism, the nature of the individual company's policy (counteracting or reducing) does not change, and so the varying system arrangement will belong to the same group ((+,ÿ), etc.) throughout the evolution.
In the second mechanism,absolute policy adaptation, a
change was executed by adding a constant amount,bcorbd,
to the appropriate policy parameter, givingck=ckÿ1+bcor
dk=dkÿ1+bd; the sign of the constant values forbcandbd
were in each case chosen to make the initial change a reduction in policy extremity. For this mechanism, a para-meter changes by an absolute amount, independent of the parameter value. Managers adopting absolute adaptation take no account of the extremity of the policies they change. For such actions, it follows that during an evolution, the nature of a company's policy can change from counteracting to reducing or vice-versa, and the system arrangement can wander between groups. A consequence of this is that a system initially classified by one type of behavior by virtue of its arrangement/initial conditions, can, on changing arrangement group, develop to a different mode typical of that group.
Fig. 7. Bifurcation diagram for (ÿ,ÿ) system arrangements; policy parameterdvarying.
Fig. 8. (a) Evolution of RBtwithproportionaladaptation mechanism. (b)
Further evolution of RBtwithproportionaladaptation mechanism.
Fig. 9. (a) Evolution of RAt with absolute adaptation mechanism. (b)
5.2. Simulation results for the adaptive model
Examples of the system evolutions obtained from the proportional and absolute adaptation mechanisms are shown in Figs. 8 and 9. Fig. 8(a) and (b) describe typical results for
proportional adaptation;acandadwere set at values of 0.8
and 0.9, respectively with (c0,d0) at (+0.1,ÿ5.0). The early
erratic behavior (containing some initial transients) stemmed from a chaotic regime, but reductions in the size of the policy parameters effected a settling to periodic behavior for which further reductions caused decreases in amplitude, as illustrated. In a sense, the proportional adaptation mechan-ism can be thought of as leading the system through various bifurcations, from more complicated to simple behavior.
Reducing the values ofacand/oradincreases the rate with
which this simplification occurs, as does increasing the
frequency of policy adjustments (f).
Fig. 9 shows typical results for absolute adaptation with
bc andbd set atÿ0.05 and +0.2, respectively and (c0, d0)
at (+0.13,ÿ5.0). Generally, a system evolving by absolute
adaptation was not found to exhibit simply damped oscil-lations, but more complicated, less predictable behavior. Fig. 9(a) shows indications of initial damping, but
oscilla-tions become more violent in Fig. 9(b) at t = 90. This
scope for more complicated development with absolute adaptation is explained by the ability of a policy parameter to change its sign and subsequently increase in size, corresponding to a change in policy type followed by an
increase in sensitivity. The values of the evolvingctand dt
are shown in Fig. 10, with a change in sign of ct being
evident in Fig. 10(a).
6. Conclusions
In this paper, competitive interactions under conditions of duopoly have been represented by a model and support-ing computer simulations. The model related R&D expen-ditures in each of two firms working in a duopoly market to product quality or technological endowment. Expenditure by one firm, firm A, changed in the light of product quality differences in A's offering to consumers compared to the offering by B. Similarly, B's expenditure on R&D changed in the light of product quality differences between B's and A's consumer offerings. The notion that management will vary R&D spending according to competitive or relative product quality or technological endowment does not appear unreasonable. Various assumptions were made about the nature of these changes and it was shown that under certain circumstances, oscillatory, including chaotic, beha-vior could result. That is to say, behabeha-vioral outcomes could fluctuate and eventually become random-like and see-mingly unpredictable.
While the model describes innovation, linking R&D spending to product quality or technological endowment, its structure is sufficiently generic to apply when other microeconomic relationships shape competitive decision-making in the firm. In general, the model could represent a duopoly where decisions regarding the extent of an input or resource allocation are, for both organizations, made in the light of the recent standard of achievement relative to that of the competitor, and where these standards are, themselves, a direct result of previous inputs/resources applied. One obvious extension would be to levels of marketing expenditure across a duopolistic market, where marketing spending is a function of consumers' perceptions of the firm's and competitor's product offerings.
Although the basic model demonstrates the possibility of chaotic outcomes in conditions of duopoly, it is difficult to be confident that the equations provide a complete and fair representation of how managers make decisions about R&D spending in reality. Concern is shared with Allen (1994), Levy (1994), Stewart and Cohen (1994), and others regard-ing the application of simple decision rules in the modelregard-ing of human behavior. As Levy (1994) observes: ``Human agency can alter the parameters and very structures of social
systems. . .''. In addition, as they stand the equations used
are uncalibrated and any attempts at calibrations or, subse-quently, testing would present substantial difficulties. Furthermore, the attractor describing a dynamical system will change if any actors succeed in learning which para-meters are important and adapt their behavior accordingly. Towards the end of the study, some initial steps were taken to incorporate managerial attempts to adapt within the model, akin to adaptation in ``learning organizations.'' These measured changes by management could (but not necessarily would) forestall chaotic outcomes. The implica-tion, therefore, is that with a suitable awareness of the possible results from adaptive adjustments, and the
implica-Fig. 10. (a) Values ofct duringabsoluteevolution shown in Fig. 9. (b)
tions of actions when paired with those of the competition (i.e. a more global view of system behavior, such as that given by bifurcation diagrams), appropriate management actions could be taken to forestall chaotic outcomes.
Conditions of chaos will obviously be very troublesome to managers intent on forecasting markets and planning and controlling their organization: both the lack of order in behavior over time and the property of sdic reduce the possibility of effective management judgement. It follows, therefore, that for two duopolistic competitors intent on ordered trajectories, there are great benefits to be had if both have a knowledge of their combined behavior and the pairings of their individual policies over the relevant range of parameter space. If this were the case, then for both organizations, chaos could be more easily avoided and more regular time paths achieved. This implies that an aspect of co-operation (or at least honesty with a compe-titor regarding policy) would be desirable if unexpected outcomes are to be prevented. This conclusion, in turn, has implications for competition policy: collaboration between duopolists may help to dampen what otherwise would be highly erratic fluctuations in the market place (here reflected in applications of R&D resources), suggest-ing that collaboration could, in this sense, possibly be welfare-enhancing.
Turning to the possibilities for future research, an ob-vious refinement to the study would be the extension of the decision rules to allow for more flexible management policies and the inclusion of time lags between resource allocation and the resulting changes, e.g. in product stan-dard. Initial work incorporating such a time lag suggests that the system is dampened, but for reasons that are not entirely clear. In addition, the model could be amplified to include additional competition, i.e. an extension from duopoly to oligopoly. At a time when industry is moving towards ``lean supply'' manufacturing, which involves reduced time lags between consumer orders and input supplies, the possible critical effect of time lags on the behavior of economic systems has important policy implications.
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