Journal of Economic Dynamics and Control 17 (1993) 327-353 . North-Holland

Competition, collusion, and chaos

Donald C. KeenanUnirerritt' vi Geo gia, Athens, GA 30602, USA

Mike J. O'BrienSprint . Burlingame, CA 94010, USA

Received October 1990 . final version received March 1992

In this paper, we present results which show that cartel formation can still be observed ina completely deterministic model where myopic firms compete through price in finite time . The firmsare spatially differentiated and take turns with their neighboring firms in setting prices . Among thedifferent possible dynamic scenarios . we focus attention on those where cooperative and competitivebehavior continue to coexist throughout time . The continual interaction of cooperative andcompetitive behavior results in complex dynamic regimes . Because the set of prices which firmschoose will be discrete, a formal equivalence is established between the model's dynamics and thetheory of cellular automata .

1 . Introduction

It is widely recognized that collusive behavior can arise in a repeated non-cooperative game from the strategic interactions of patient firms with long,usually infinite planning horizons . While similar results can be obtained withina finite horizon through considerations of incomplete information [Kreps et al .(1982)], such `tacit collusion' remains in general highly sensitive to the length ofthe planning horizon. Indeed, in the basic Bertrand model of price competition,the set of equilibria collapses to the competitive equilibrium as soon as thehorizon becomes finite .

In this paper, we present results which show that, despite the maintainedassumption of myopic firms, cartel formation can still be observed in a com-pletely deterministic model in which firms compete through price in finite time .The firms are spatially differentiated and take turns with their neighboring firmsin setting prices . As a result of this formulation and the assumed myopia, all

Correspondence to : Donald C . Keenan, Department of Economics, University of Georgia, Athens .GA 30602, USA .

0165-1859/93/$05 .00 c 1993-Elsevier Science Publishers B .V . All rights reserved

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D.C. Keenan and M . .1 . O'Brien, Competition . collusion, and chaos

strategic considerations are eliminated . Nonetheless, the resulting dynamics areshown to be capable of sustaining considerable collusive activity .

Among the different possible dynamic scenarios, we focus attention on thosewhere cooperative and competitive behavior continue to coexist throughouttime. This is made possible by the spatial extent of our model, which allows localregions to follow different standards of behavior . The continual interaction ofcooperative and competitive behavior results in complex dynamic regimes .'Collusion takes the form of unstable local cartels that are limited in bothduration and extent of their market, but arise continually through time . Inparticular, one prominent set of reactions by firms is interpreted as the unstabledynamics proposed by Edgeworth as a solution to the Bertrand paradox .A connection is made to this and the notion of Stackelberg warfare .

Because the set of prices which firms choose will he discrete, a formalequivalence is established between the model's dynamics and the theory ofcellular automata . By this means, standard analytical tools are brought to bearon the problem of dynamic oligopoly . The propensity for environments tosupport collusive outcomes is interpreted in terms of exact measures of 'self-organization' .

2. Spatial price competition

2.1 . Background

In reviewing Cournot's (1838) seminal analysis of a quantity-setting oligopoly,Bertrand (1883) proposed instead that firms compete through price, LikeCournot, Bertrand maintained the assumption of myopic firms . In this formula-tion, Bertrand found that the only possible equilibrium has firms charging theirlowest sustainable price, thus yielding the competitive outcome . This is oftenconsidered paradoxical, since it might be expected that, with few enough firms,collusive behavior would arise instead .

Edgeworth (1897) sought to escape this paradox by evoking the possibility ofcapacity constraints, or of diseconomies of scale in general . The competitiveoutcome would then no longer he an equilibrium, since when other marginal-cost pricing firms do not cover the market, a firm has an incentive to raise itsprice against the residual demand . With no price equilibrium possible, Edgeworthargued that prices would instead cycle about,

It is difficult to evaluate the possibility of Edgeworth cycles in the standardBertrand formulation . Reaction functions are not well-defined there, given thatfirms want to lower price below rivals by as little as possible, but that no single

I Most other economic models known to exhibit dynamic complexity also rely on a high rate ofdiscount . However, see the paper by Woodford (1989) .

D.C. Keenan and M.J. O'Brfen, Competition, collusion, and chaos

329

choice ever accomplishes this. As in the work of Maskin and Tirole (1988), thisdifficulty will be prevented here by restricting attention to a discrete number ofpossible prices .'

2 .2 . The economic model

The model employed is that of a fixed number of firms equally spaced alonga ring, conveniently thought of as a circular city. Consumers are locatedsymmetrically about the city and purchase goods from their neighboring firmsin response to the prices posted by these two firms . Consumers are assumed tohave symmetric preferences between the goods of their two neighboring firms, sowith the symmetric distribution of consumers, each firm in the city faces theidentical environment.'

Each firm maximizes its current profits in a completely myopic fashion, sothat its pricing decision only depends on the prices posted by its immediateneighbors, with whom it shares a market on either side . In order that a reactionto neighboring prices be well-defined, it is supposed that adjacent firms alternateevery other period in setting prices. Thus, one may think of the odd-numberedfirms as moving in odd periods and the even ones in even periods . This device,introduced by Cyert and DeGroot (1970), is interpreted by Maskin and Tirole(1988) in terms of price rigidities resulting from short-run commitments . For thepresent, this asynchronicity is to be the only difference among the otherwiseidentical firms .

Note that the myopia we assume assures that any observed cartel activity isnot being supported by the usual folk-theorem argument for repeated games,wherein short-run deviations from collusion arc punished by long-run retali-ation. Instead, cartel activity will be attributable to spontaneous self-organiza-tion, in the manner to be described .

2.3 . Dynamic oligopoh, as a cellular automaton

Since time as well as the firms' common set of prices will be discrete, theresulting dynamics are those of a finite cellular automaton [Wolfram (1986)] .

2 1n an earlier version of this paper an alternative formulation of our model is developed, whereany price is allowed, but firms choose to charge only p and p . That formulation, unlike the standardBertrand model, has consumers who do not regard the goods of differing firms as perfect substitutesand so their demand does not change discontinuously .

'We have taken an intermediate position between the Hotelline_ spatial approach and thequality characteristic approach to product differentiation . It would he possible to take a purelyHotelling approach by regarding the goods of differently located firms as perfect substitutes, exceptfor transportation costs, and generating demand behavior from the spatial distribution of con-sumers . It would also be possible to remove all actual spatial considerations by regarding firms asbeing located at different quality levels of their goods . One might then want to arrange the firmsalong an interval, rather than a circle, but this does not greatly affect the analysis .

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This follows from the fact that firms on every site have identical reactions ofthe form

Pr , I =1 (Pi -t , Pi * t )

(1)

where p; is the price at time t of the firm at the ith site . Such locally determineddynamics arc exactly what distinguish a cellular automaton (CA) . The only CAcondition formally being violated is that sites should move synchronously .However, it is sufficient to restrict attention to, say, the odd-numbered firms,and write their reduced reaction functions synchronously as

P .,=I(f(P-,P),f(P,P'`))_O(P-,P,P'2),

(2)

i odd, t odd .

The indirect effect of a firm's past prices on those chosen by its immediateneighbors in the previous time period allows the firm's reduced reaction func-tion to include the firm's own past actions .

Whether in the reduced or unreduced form, the reaction functions of ourmodel will be 'reflection-symmetric' . That is, due to the symmetric distributionof consumers . a firm's choice of price depends only on the two neighboringprices and not on which of the two neighboring firms is charging which price .

Notice that because strategic considerations are absent, mixed play neveroccurs, and so the dynamic rules arc entirely deterministic . The only factor notendogenous to the model is the historically given initial price configuration .'Despite this deterministic setup, seemingly stochastic behavior will be seen toemerge .

3. Cellular automata

3.1, Types

Since our economic model is of the form of a cellular automaton, somediscussion of this class of dynamic processes will prove useful .

4 Because of the lagged role of commitments, historically given forms of behavior may persistthrough lime within a particular region . [See the discussion of the role of history in Kreps andSpence (1984) or Fudenberg and Tirole (1986) .] As time passes, however, a firm is indirectlyinfluenced by the past actions of an increasing domain of firms, and thus the behavior of a particularregion may he transformed through interaction with other regions. Despite the consequent sensitiv-ity to the initial price configuration, it is a common feature of cellular automata processes thata particular rule will exhibit the same qualitative behavior for all but an exceptional set of initialconfigurations [as discussed by Wolfram (1983) and shown by Hurley (1990)] .

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Cellular automata were introduced by Von Neumann (1966) in his pioneeringwork on self-organizing behavior. CA rules are broadly classified into fourcategories, according to the dynamic behavior of their reaction functions [Wolf-ram (1986) and Hurley (1990)] . 5 Type I rules arc those that rapidly lead froma typical initial price configuration to a steady state configuration where eachfirm's price is constant over time . Thus, the effect of a local perturbation in theinitial price configuration completely disappears over time for a type I rule .

Type 11 phenomena arise when the rule sends initial configurations into pricecycles. The global cycle consists of a number of neighborhoods in differentregimes, each on its own subcycle . Thus, the effect of a local perturbation in theinitial configuration remains localized over time in a limited neighborhood ofthe firms whose prices were initially changed .

Type III rules are the most common category of CA rules and those to whichwe devote the most attention. Within an infinite cellular automaton, that is onewith an infinite line of firm sites, type III limiting behavior is aperiodic . Unliketype I and 11 rules, the dynamics exhibit sensitive dependence on initial condi-tions . That is, nearly identical configurations of prices will evolve over time intoincreasing different configurations . This occurs despite reaction functions beingin general globally noninvertible mappings that over time progressively reducethe number of possible price configurations . It is this contractive property ofnoninvertible dynamics that allows for self-organizing behavior. As a conse-quence of this continual contraction, an attractor emerges which represents thelimiting behavior implicit in any initially disordered price configuration belong-ing to the basin of that attractor. On the other hand, it is the divergence ofconfigurations within these `strange' attractors for type III dynamics that allowsself-organizing structures to be complex (neither steady states nor cycles) .

Rules of the remaining type I V category are somewhat rarer and do not occurfor elementary cellular automata .' We do not explicitly consider this type rulefurther .

3 .2 . Statistical measures of complex behaviorComplex dynamics are most easily dealt with by statistical analysis, despite

the previously noted fact that our oligopoly model is entirely deterministic . The

'While Wolfram's classification is entirely empirical, the recent analysis of Hurley providestheoretical support for the validity of the distinction between the various types of cellular automata .

'For type IV rules, the effect of a local perturbation in an initial configuration may growthroughout time, but whether it does so and the extent to which it does so seem unpredictable . It isthought that most type IV rules are computationally irreducible, so that no more efficient meansexists for determining the effect of an initial perturbation than to simulate the rule . Whethera perturbations effect dies off is then undecidable in the sense that there is no assured means ofanswering such a question in a finite number of steps . It is furthermore thought that some if not alltype IV rules are capable of acting as universal computing d evices. so that each would be capable ofexhibiting any specific dynamic behavior .

3 32

D. C. Keenan and M . J . O'Brien, Competition, collusion, and chaos

local expansiveness of the rule is described by the Lyapunov exponent [Packard(1985)]. This is a measure of the time-averaged rate of divergence of initiallynearby price configurations, and describes the rate of local information propa-gation . It is found by measuring the asymptotic rate at which a change in theprice of a single firm propagates to affect other firms over time . There is botha lefthand and a righthand Lyapunov exponent, A L and A s , in general [Wolfram(1986)], but they coincide as i . for rules, such as ours, with reflection symmetry .The i. will be nonzero only for type III or type IV systems .

A more global means of describing the rule's complexity is in terms of entropy[Grassbergcr (1986), Milnor (1986), or Wolfram (1984)]. One defines themeasure-theoretic information content :

HM (X, T) _ - Y p(ax .T) log, R(ax ' T ) ,

(3)

where p signifies the rule's asymptotic, invariant probability measure thatassigns the eventual likelihood to any particular length T time sequence a' Tof price configurations for X consecutive firms . One may similarly definea topological information content H(X, T), where possible configurations arenot weighted by their probability, and so, unlike the measure content, thetopological content may be influenced by price configurations of arbitrarily lowprobability .

Given the information content, one obtains the asymptotic entropy measures

h", = lim (H„ (X + 1, T) - H„ (X, T)),

h;, = lim (H„(x . T+ 1)-H„(X, T)),

h„=lint lim(H„(X+1,T+1)-H„(X,T)) .X-, I-,

(4)

These are referred to as the spatial measure entropy, the temporal measureentropy, and just the measure entropy, respectively . The correspondingtopological entropies-h`, h'. and h-are defined in the same fashion . Spatialentropies measure the complexity of the asymptotic configurations of a rule, sothat the topological spatial entropy is the dimension of the rule's attracting set[Eckmann and Ruelle (1985), Wolfram (1984)] . Both kinds of spatial entropieswill be zero for a type I rule, since the limiting price configuration is a singlesteady state. Temporal entropies describe the asymptotic time series of a singlefirm, and so such entropies are zero-valued for type II and hence type Idynamics, where behavior repeats itself. Finally, the overall entropy entails

D.C. Keenan and M .d . O'Brien, Competition, collusion, and chaos

333

both spatial and temporal considerations, so that the topological entropy givesthe global mean rate of information creation in the sense that initially similarprice configurations increasingly separate and so become distinguishable fromone another .

No measure-theoretic entropy is ever larger than its correspondingtopological entropy, and assumes the same value only when configurationsare equi-probable . The maximum entropy occurs with a random mapping, inwhich case both the spatial and temporal entropies are unity .' Thus, onehas

OghuSh`51, 0<_hi<_h`<_I, 0<_h„<h .

(5)

In view of our discussion, the four possible types of rules may then besuccinctly described in terms of these statistical measures as follows :

Note that this definition of types is only strictly applicable to infinite cellularautomata, since for a finite CA the dynamics must eventually enter a cycle, giventhat the number of possible price configurations is finite . These entropies arenonetheless of considerable importance for finite cellular automata, since theirpurpose is to characterize the dynamic differences among rules, and thesedistinctions survive in a finite context . The intent in using entropies defined foran infinite cellular automaton when doing our economic analysis is merely toseparate the effect of the reaction function's dynamics from the constraintsimposed by the global structure of the circular city .

In the remainder of this paper, the statistical measures defined in this sectionwill be determined for those elementary reaction functions we develop andshown to be meaningful measures of behavior .

In addition, the various statistical measures are seen to he related by

It,, !_2h, .

hi.<h„<(d,.+i.,,)h ;, .

See Wolfram (1986) .

Type l

Typell

hN=hN=i.=O,

i.=0,h,',>0, 111,=0,(6)

Type III hN>0, h;,>0, i>0,

Type IV h's undefined , A z 0 .

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D.C. Keenan and M. .l. O'Brien . Competuion, collusion, and clmos

4 . Edgeworth dynamics

4.1. The role of increasing costs

In this section, we consider our framework in the simplest case of just twoprices : a higher 'cartel' price P and a lower 'competitive' price P . Such a model,where each firm takes on one of but two (k = 2) states, is referred to as anelementary cellular automaton .

In addition to Bertrand-style consumers who always seek the lower priceof the two firms in their neighborhood, there may also be consumers who donot regard the two firms' goods as perfect substitutes . For definiteness,we suppose that such consumers behave exactly oppositely to the Bertrandconsumers and so always purchase a fixed amount at the firm nearest tothem. In contrast to this captured market, a firm gets the Bertrand consumersof one side only if its price is below that of the other firm to that side, and inthe case of a tying price, the firms split the market . These two types ofconsumers, together with a specification of the firm's cost function, providesufficient flexibility to obtain the elementary CA reaction functions weconsider .

An Edgeworth cycle in our context would have a firm reacting to neigh-boring low prices P by raising its price up to P . Note, however, that this maynow occur entirely for demand reasons, without regard to the diseconomiesof scale considered by Edgeworth . It may simply be that the revenuefrom charging one's captured consumers the high price P exceeds getting thelower price p from them together with half the Bertrand consumers on eitherside.

The other part of the Edgeworth process has the higher price P being loweredto the competitive price P, so that a price cycle is set up . Even were there onlyBertrand consumers present, however, it would not follow with discrete pricesthat a firm need react to both its neighbors charging the cartel price P by itselfcharging the competitive price p. For the case of discrete prices, the firm mayprefer the higher price P from half the Bertrand consumers together withwhatever captured market it has to getting a lower price p from all the Bertrandconsumers and the captured market .

Edgeworth proposed that the reaction of charging high price P when facedby low prices p be explained by capacity constraints on the part of the rivalfirms . If such constraints could be present for rival firms, then they wouldcertainly also be present for the given firm when it considers charging the lowprice P in reaction to the rival firms charging P . This is, after all, the situationwhere this firm faces the greatest demand any firm will ever encounter . The samecapacity constraints Edgeworth suggested to support the reaction P whenneighboring firms both charge p also support reaction P when both neighborscharge P.

D.C. Keenan and M . J . O'Brien, Competition, (ollusion, and chaos

335

We examine the case where p = f(PPp) is indeed part of the firm's reactionfunction." This condition is obviously necessary if local high-price cartels are toarise. Further, as will now be seen, it is only with this reaction that Edgewortheconomy of scale considerations need again play a role .

Given that neighboring firms both charging p lead a firm to also charge p,then if an Edgeworth cycle is to be set up, it is necessary that a high price p onone side and a low price P on the other lead the firm to adopt the low price p .This, however, can only be the case with diseconomies of scale. Indeed . ifmarginal cost is constant, then the assumption that neighbors both chargingp or both charging p lead the firm to charge p logically requires that the firm alsoselect p when neighbors charge opposite prices . This follows since, in terms ofprofit, the latter situation is but a convex combination of the first two situations,given that marginal costs are constant and the revenue function is additive . Forthe reaction function being considered to occur, the cost function must not alsobe additive- It must instead satisfy the condition

iC(x(plp)) + 4C(x(jIP))-CLx($I P') + ix(Pl p))

< ~'(x(PIPU+zC(x(Plp))-C(zx(PIP)+ x(plp)L

(7)

where, say, x(PIP) is the total demand from both sides for a firm chargingP when it faces firms both charging price p . This condition requires some degreeof convexity of cost, given that x is decreasing in its first argument .q

It is possible in our model for the reaction p =f(P, P) to be explained entirelyby demand elasticity considerations and for the reaction P = f(p, p) to also be soexplained. However, as we have seen, the only way to have both restrictions holdand yet have p =,f(P, p), so that recurrent dynamics occur, is for increasing costconsiderations to come into play . Because the reaction being discussed requiresdiseconomies of scale and because the rule yields unstable motion as envisionedby Edgeworth, we will refer to the reaction function

p=.f(p,f), P=f(p,p), P=f(P.p)

(8)

as the Edgeworth process .

'Taking the simple quadratic cost function, one sees that the condition holds ifx0, P) - x(p,P)>x(p .A)-x(p,P), which one would expect to he the case . It will certainly hold forthe situation with Bertrand-style as well as captured consumers, so long as their demand schedule isdownward-sloping .

'Notationally, by, say, p= f(P . p), we mean that when, say. p~ - ' = p and p';" = F', thenn;,,=p

33 6

D C. Keenan and MN.J. 0 Brien, ('onipe noon . collusion and chaos

4.2. Obtaining Edgeu'orth dynamics in the discrete price model

Rather than simply provide a parametrization for which our circular citymodel yields the Edgeworth process, we present a short description of theproperties used to assure this outcome . Since it is sufficient for our purposes, welet the total demand x of a firm's captured consumers be fixed, as is the totaldemand x of the Bertrand consumers to one side of a firm . We let marginal costhave the constant value c < up to some quantity to be discussed .L°

When other firms charge p' there needs to be sufficient captured demandrelative to the portion of Bertrand consumers that total demand is not veryelastic going from price p to p . Since one obtains none of the Bertrand con-sumers when charging p, the condition in terms of profit Ti in order that thehigher price p be chosen is that [ '

17 (xpI3)) _ (p - c)x

>

(9)

11(X(/31/3))=(p- c)( +z) .

On the other hand, when one of the rival firms charges p and the other chargesp, then one obtains half of the Bertrand consumers to one side when chargingp and an additional half from each side when charging price p. We require thatthis larger portion of Bertrand consumers relative to captured demand causestotal demand between p and p to become sufficiently elastic so that

n(1x(P Ip) + + x(p,lp)) _ (p - c)( ; + 12 x)

>

(10)

n(zx(JIV) + ix(plp)) _ (p - C) (X + ?x)

This then assures that the lower price p is chosen when the rival firms arccharging opposite prices.

Let the marginal cost rise to c = p near x = x + 2'x ; for definiteness, let therise occur exactly at this quantity . Going from the case where rival firms charge

"In the actual construction of reaction functions. we always have increasing costs occur as thejump at some quantity of marginal cost from a constant value c < p to c = p. This has the advantagethat a firm is always willing to absorb the demand it encounters, so that one does not need toaccount for spillover demand . It also helps to explain where the particular price p comes from .

"The reader may simply assume c = 0, so that profit comparisons become revenue comparisons .We maintain nonzero marginal cost c since in section 6 we will want to lessen costs in the loweroutput range in order to rationalize a second kind of reaction function .

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337

opposing prices to where they both charge P, there will then be no change inprofit for the given firm if it has chosen low price P, since the additional demandbelongs in the high-cost range where the marginal cost coincides with price P .As a result, when the given firm instead charges high price P, the additionaldemand 2x that arises going from the case where neighboring firms chargeopposing prices to where both charge P can easily prove large enough sothat

17(x(PIP)) _ (P - c) (x + x)

>

(11)

II(x(fl 1 )) = n(2xwP) + 2x(P[P)),

despite II(2x(fIp) + zx(P I P)) < II(x(PI P)) . In such a case, the firm will switchto also choosing high price P whenever rival firms raise their prices to P . It is herethat increasing costs serve their role in limiting the profitability of dropping tothe low price p when the other firms are keeping price high .

The conditions we have imposed can be satisfied for a range of parametervalues of nontrivial size . Thus the Edgeworth process can be generated by ourmodel in a robust way .

5 . Analysis of Edgeworth dynamics

5 .1 . Infinite cellular automata5 .1 .1. Formal dynamics

We consider the Edgeworth process, employing the techniques introduced todescribe complex automaton rules . Using the reduction to alternate sites only,described in eq. (2), the Edgeworth process is found by a standard method ofenumeration [Wolfram (1983)] to be `rule 90' among elementary synchronousrules. This rule is the simplest example of a linear or additive rule, where theprice reactions are specified to be the sum of the earlier neighboring states,modulus k [Martinet al. (1984)] . That is,

4)(p'-2, p , , p ,+2 ) = p,-2 + p,+2

(mod 2),

where in an entirely formal manner we reverse the natural labels of states . s o thatthe cartel price is P 0 and the competitive price is P

I .It can be shown that, for an infinite cellular automaton, additive rules act as

surjective maps of configurations . This implies that all possible configurationscontinue to occur over time . Furthermore, configurations continue to occur

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D.C. Keenan and M .J. O'Brien, Competition . collusion, and chaos

with equal probability starting from a random initial configuration [Milnor(1986)] . It thus follows that

h g' =h„=1 .

(13)

with the corresponding topological entropies also being unity .' 2 The Edgeworthrule is thus maximally chaotic both spatially and temporally, and in terms of justthese statistics cannot be distinguished from a truly random mapping . Nonethe-less, inspection of a typical dynamic development (fig. 1) convinces one that thebehavior is not truly random ."

5 .1 .2. Cartels

The most prominent regularity of the Edgeworth process consists of thesequences of adjacent high-price firms that organize and persist in the fashionof a local cartel . These cartels, however, are unstable and unravel over timeas competition at the fringes of the cartel cause the marginal firms toabandon the cartel . Thus, such cartels appear as triangles, as they spring upand then die off .

As we have just seen, such temporal-spatial behavior is not reflected in eitherthe spatial or temporal entropies separately, though this lack of randomness isrevealed by the Edgeworth process entropy

h„=2,

(14)

which combines consideration of time and space . The topological entropy hasthis same value and indicates that, quite unlike a random mapping, it takes thehistory of but two adjacent sites to reconstruct the entire development of aninfinite cellular automaton following the Edgeworth rule.

Given that it is the high-price sequences that are of interest in our analysis ofcartels, we introduce some other measures that distinguish Edgeworth's rulefrom random behavior . We define Q(i) as the density of high-price sequences ofexact length i. so that the two firms bordering the sequence are of low price . Lind(1984) shows this distribution to he of the form

QU) - 4 (2) - '

"It is easily seen that the l .yapunov exponent ;. for the Edgeworth process is unity, since by (12)a price change by one firm cannot help but change the prices of neighboring firms in the next period .

i 3 Given an arbitrary initial price configuration, t hen . as described in the caption of fig . 1, thedynamic evolution of prices is obtained by successive application of rule 90 [eq . (12)] to each firmsite .

D.C. Keenan and M .J . O'Brien, Competition . collusion . and chaos

339

Fig_ t . Edgeworth dynamics (leader-leader dynamics) .

Evolution of the one-dimensional cellular automaton where alternate firms are allowed to changeprices at alternate times according to Edgeworth dynamics [eq . (8)] . Firms charging the'competitive' low prices p are represented as white rectangles ; firms charging the 'cartel' high price1) are represented as black rectangles . The configuration of the cellular automaton at successive limesteps is shown on successive lines, going down the page . The prices of firms are initially uncorrelatedwith one another and are taken to he fi or P with probability 1 ;2 . The evolution is shown for 120firm sites for 300 periods . Site 1 and 120 arc treated as neighbors, producing a circular city ofsites . While Edgeworth dynamics are maximally chaotic, it is nevertheless apparent that cartelspersist over time (black triangles) more commonly than random chance would dictate . Whilean infinite chaotic cellular automaton does not cycle, the fact that a fixed number of sites(120) are used produces a cycle, as indicated by the fact that, for example, lines A and B

are identical .

As a spatial measure, this density is again identical to that of a random initialconfiguration, Q(i) _ (#)''' 2 . However, turning once more to both spatial andtemporal considerations, it is clear for the Edgeworth process that the densityT(i) of cartel triangles of length i is of the same basic form as Q(i), since

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1).C' . Keenan and At . J . O'Brien, Competition, collusion, and chaos

a high-price sequence always persists over time as a triangle . More precisely, weobtain

T(i)=is(2) .

(16)

On the other hand, triangular configurations for random mappings must be ofa far lower density, on the order of T(i) = (D"', where /3(i) is proportional to i 2 .(This follows since a high-price sequence at one moment of time would thenindicate nothing about the next time step, and so a 'cartel' of i original firms isbut the coincidental occurrence of high prices throughout a triangular region .) Itis thus formally established that cartels persist tinder the Edgeworth process ina manner not accountable for by random occurrence and that this nonrandombehavior can be measured using the statistical measures of complexity we haveintroduced .

5.2 . Finite cellular automata

The preceding discussion precisely concerns infinite cellular automata, wherethe reaction function's effect is isolated from that of the environment. Theprimary difference in the finite case is that, rather than remaining aperiodic, typeIII dynamics must eventually enter into cycles . Because the additivity propertyallows an algebraic analysis, precise results are available concerning these cyclesand their transients in the case of an Edgeworth process [Martin et al . (1984)] .The upshot is that the maximum period of cycles as well as the number ofthese cycles grows on average exponentially with n, the number of pairs offirms . Most cycles have periods near their maximum . Transients, on the otherhand, grow at most linearly with n . For all these reasons, most configurationsappear on a cycle of near the maximum period . Thus, as the number of firmsgrows large, it becomes increasingly difficult in a set amount of time to distin-guish the finite case from the infinite aperiodic case, and the entropy measures ofthe infinite cellular forms become increasingly accurate descriptors of the finitecase .

6. Leader-follower dynamics

6.1 . The rationale

While rather complete results are available for additive processes such asEdgeworth's dynamics, additive CA are quite exceptional within the class oftype III processes in that they exhibit maximum spatial entropy . Nonadditivecellular automata are almost always contractive maps that force greater self-organization than do the surjective maps of additive cellular automata . Thus,

D.C. Keenan and M.J. O'Brien, Competition, collusion, and chaos

341

with nonadditive reaction functions, cartels will arise more consistently thanwith the Edgeworth process .

Our framework can rationalize any number of nonadditive reactions whenmore than two prices arc permitted, but to obtain reactions of this kind ofbehavior within the two-price formulation we need to relax the assumption ofidentical firms . The simplest way to do this is to let alternate firms differ fromone another. We continue to assume that, say the odd firms, follow Edgeworth'sdynamics, but now the even ones have the reaction function

g(P,P)=P, g(P,P)=P, op, P)=P-This can occur, for instance, if the even firms still face the same demand as theodd ones but their costs are less than those of Edgeworth firms at lower levels ofoutput [less than x(p I P)] . Such a low-cost firm prefers to keep prices down, andso attract customers, in those situations where it has not reached its low-costcapacity .

6.2 . The dynamics

When we look at the reduced dynamics for the odd firms,

P4+2 =f(g(p,,-2,p,),

g(P`t, P`t' 2 )),

i odd, t odd ,

(18)

we find it is 'rule 18' among elementary cellular automata (fig . 2b). On the otherhand, looking at the reduced dynamics of the even firms,

Pr+2 = g(f(P', -2 , P;),f(P ;, P4 +2)),

i even, t even ,

(19)

it is found to be 'rule 126' (fig . 2c).Whether one examines the reduced dynamics in isolation or their conjunction

in the unreduced dynamics (fig . 2a), it is apparent that significant organizationby firms into cartels is occurring . At the same time, however, forces of competi-tion are undermining the cartels, and the resulting dynamics are chaotic, ratherthan degenerating into either perfect competition or a global cartel (typeI phenomena). This behavior is reflected in the statistical measures of rule 18 orrule 126 [Wolfram (1986)], which, as one would expect, yield identical results :

Rule 18 or 126

h;,=z, )t;,=z, h„=l, i=1 . (20)

Entropies smaller than those of rule 90 indicate that possible configurationsare being eliminated over time as the dynamics induce considerable cartel

342

D.C. Keenan and .4l. .f. O'Brien. Competition, collusion, and chaos

formation . Nonetheless . these entropies do not vanish, indicating that thetension between competition and collusion persists over time .

6.3 . Leader-%b/finv r dvnwnic~s vs . leader-leader Slackelherg warfcn'e

Both the Edgeworth firms with reaction function (18) and their neighboringfirms with reaction function (19) share the feature that they will support a cartelfrom within but will undercut it at the fringe . The Edgeworth firms differ fromtheir neighbors in that the former will abandon a competitive regime and raise

Fig. 2a . Leader follower dynamics .

Evolution of the cellular automaton according to leader-follower dynamics : odd sites evolveaccording to eq . (H), even sites according to eq . (17), at alternate tines . The initial configuration isidentical to that used in fig . I . However, the degree of self-organization is much greater than withEdgeworth dynamics, as evidenced by the more frequent occurrence and large size of the cartelswhich persist over time (triangles) . As in fig . I, due to its finite nature the cellular automaton has

begun cycling, as indicated by the identity of lines A and n .

D.C. Keenan and M.J . O'Brien, Competition, collusion, and chaos

343

Fig. 2b . Price leaders in leader-follower dynamics .The evolution process of only the odd sites of fig . 2a . The dynamics are those of rule 18 in the

standard method of enumeration [Wolfram (1983)] .

~-

_

.~ J

Fig. 2c. Price followers in leader-follower dynamics .The evolution process of only the even sites of fig . 2a . The dynamics are those of rule 126 in the

standard method of enumeration [Wolfram (1983)] .

344

D.C. Keenan and M . .l . O'Brien, Competition, collusion, and chaos

price, whereas the latter will continue to support the competitive regime . In thissense, we refer to the Edgeworth firms as price leaders and the other firms asprice followers .

With rule 90, all firms are price leaders and so it is not surprising that theoutcome is maximally chaotic, in the form of Stackclberg (1934) warfare . On theother hand, when price leaders alternate with price followers (fig . 2a), then incontrast to the Stackelberg outcome, the result is one of more regular behavior,supporting extensive collusive activity . 14

6.4. Historically differing regimes ofbehaviorThough not obvious from casual inspection of fig . 2a, there is a sense in which

leader-follower dynamics is divided into differing regimes of behavior[Grassberger (1983)] . Take the point of view of the price leaders, that is,examine the reduced dynamics of rule 18 in fig . 2h, reproduced in fig . 3 . The onlytime the leader-follower dynamics differ from the case of Edgeworth leader-leader dynamics is when a price-following firm faces a competitive neighbor-hood where the rival firms charge p . However, such competition becomes scarce,occurring only along the barriers between the two regions of behavior . Withinthe regions, price-leading firms alternate in setting the cartel price . This situationperpetuates itself, so that price-following firms within the region need never facea competitive environment . The difference between the two regions is merelyone of timing in the setting of the high price by alternate price leaders .

Since competitive price-following never arises within the regions, the behaviorthere is indistinguishable from Edgeworth dynamics in these same circumstan-ces. While it is evident that more organization is occurring within these regionsthan with typical Edgeworth dynamics, this is because the prevailing conditionsrestrict the types of situations that either kind of firm will face . These restrictionsare a result of the contractive self-organizing nature of the rule.

Indeed, the difference between Edgeworth and leader follower dynamicsoccurs, of necessity, at the barrier between regions, where price followers mayface the competitive situation where neighbors charge p. The reaction p bya price follower does not disturb the alternating cartel-price arrangement insidethe nearby regions, whereas the reaction fl that would occur with Edgeworthdynamics sets in motion an expanding sequence of low-price disturbances thatwrecks this alternating high-price arrangement by the price leaders and mixesthe regions out of existence .

The only way the barrier between regions may disappear with leader- followerdynamics is if one region completely collapses and the barriers on either side

1° As might he expected, followerfollower dynamics reduces all firms 1o the competitive low priceand so all entropies to zero .

D.C. Keenat find M .J. O'Brien . Competition . collusion, and chaos

345

Fig . 3 . Regions for price followers in leaderfollower dynamics .This figure is identical to fig . 2h (i .e ., it represents evolution based on rule I8), except that in one of itstwo type regions rectangles corresponding to p are colored grey . This is done to show that . for rulet8 evolution, it is only along the border of these two type regions that two price-following firms,both charging p, will he neighbors . It is only when two such firms charging f are neighbors thatleader- follower dynamics differ front leader leader dynamics . The obviously more organized natureof the inside of the regions as compared to fig . I is solely a result of the reduction there in the number

of possible conditions faced by a firm .

annihilate one another, as occurs in the top portion of fig . 3. Such a collapseoccurs only when an entire region cartelizes and then this cartel unravels underthe pressure of the outside mode of behavior on either side . Only when there isa single region will its complete cartelization persist over time, in the form ofa global cartel . Thus, large-scale cartelization of an entire region of behavior hasan all-or-nothing aspect to it, in that the presence of any competing mode ofbehavior will instead result in the complete elimination of the cartelized regime .Nonetheless, the competitive outcome is even more unstable and so cartelscontinue to arise over time .

346

D.C. Keenan and M . J . 0 Brien, Conpetliot . collusion, and chaos

Since barriers at each moment of time are being newly affected by the pastactions of one additional firm to each side, their movement is a random walkwith diffusion coefficient z [Grassberger (1983)]." Given only a finite number offirms, the wandering barriers would then tend to annihilate one another overtime, so that a single pattern of behavior would emerge . This is limited only bythe fact that the oligopoly may enter a price cycle before this need happen .

6 .5 . Dependence on nest, information and scale invariance

It has been seen that while a rational firm's reaction function must depend onneighboring firms' previous, now fixed prices and not on its own previous, nowvariable price, the reduced reaction function can indeed involve the firm's ownpast behavior. Nonetheless, in the case of Edgeworth dynamics, that is rule 90[eq . (8)], it turns out that even the reduced dynamics do not depend on a firm'sown past behavior . By continuing the process of reduction, one sees that amongall the firms whose past actions might be influencing a particular firm's choice,only the outermost ones actually do. These firms' prices constitute newlyarriving relevant information .

The total dependence of the choice of price on new information and not onown past history represents an absence of inertia, which serves to explain theextreme variability and apparent randomness of Edgeworth dynamics . Thiscomplete dependence on newly arrived information shows in extreme form thedistinction of type III chaotic dynamics from type I or II dynamics, wherea firm's state throughout time depends only on a limited neighborhood of firms .The newly arriving influence of increasingly distant firms allows type III dy-namics to have regions continually mix cooperative and competitive behavior,without ever completely settling into one or the other mode .

As just seen, Edgeworth dynamics appear the same for any uniformly spacedsampling of representative firms, no matter the scale . This indicates the self-similar, fractal nature [Mandelbrot (1983)] of these chaotic systems ." Forleader-follower dynamics there is still much self-similarity . though it is more

`The competitive low-price barrier persists until it is absorbed into a cartel containing an evennumber of price leaders (not to be absorbed is to encounter a cartel of length zero) . This carteldwindles until the competitive low-price barrier reappears at its bottom The original length of thecartel is dictated by those price leaders of the two neighboring regimes who are not of necessitycharging a high price . Since these alternate price-leading firms satisfy Edgeworth dynamics, so thatspatially their price configuration is random, it follows that the position of the barrier going into thecartel is random . Therefore the position at which the cartel dies and the harrier reappears is alsorandom with respect to the barrier's initial position. The distribution is clearly a binomial one, andso, for an infinite cellular automaton, the barrier follows a random walk .

"See the discussion of Mandelbrot (1983) in the context of cotton prices .

D.C. Keenan and M.J. O'Brien, Competition, collusion, and chaos

347

limited in the small due to the greater degree of local self-organization . That is, ittakes an additional reduction of, say, rule 126, the price follower's dynamics,before it appears self-similar, becoming, in fact, rule 90 . Note that this invarianceof scale only applies within a relevant range, limited in the small by firm size andin the large by the finiteness of the economic environment ."

7. Multiple price environments

Collusion is but one example of a standard of behavior, and interest naturallyfocuses on how different standards, which in our model have a regional charac-ter, can persist and interact over time . Besides cartels, we have considered thetwo out-of-phase regions of behavior that become apparent in the leader-follower model after a certain amount of transformation . When a firm is allowedmore than two price levels, such coexistent regions of differing behavior canarise in striking fashion, which are quite apparent without having to firsttransform the patterns (see figs . 4 and 5), 18 As the number of prices increases,however, the set of possible dynamic rules rapidly explodes in size. Therefore,the particular examples presented in this section are meant not so much to beimportant in and of themselves, as to display significant features characteristic ofbroad classes of chaotic rules when in multiple price environments ."

7.1 . Stable sets

In their interpretation as standards of behavior, our dynamic regimes beara certain resemblance to Von Neumann-Morgenstern stable sets [Greenberg

"If the reductions are done on rule I8, the price leader's dynamics, then, as indicated in earlierdiscussion, there will be two regions, one that appears as the Edgeworth dynamics and the other asa high-price cartel .

"The reaction function used to generate fig . 4 is identical to that of fig . I when p' is relabeled P,with the addition of

if p,T=P .

1(P,YI=V .

1(P.P ) -P .The reaction function used to generate fig . 5 is identical to that of fig. 3 when p is relaheled P and p isrelabeled p, with the addition of

PP . P)=P

1lP•P ) - P

J(P.P)=P •

(odd firmsl,

00, P)=p,

sup. PI=P,

91P,P)=P .

even firms)

"Except for the Edgeworth cycle in reaction to the three possible combinations of two prices, allother combinations of reactions for any number of prices can be obtained from the demand sidealone, when confining attention to discrete prices . Revenue values supporting the desired reactionscan always be found, and upon selecting prices this gives demand values . Moreover, it is known thataggregate demand values can he arbitrary at a finite number of prices [Shafer and sonnenschein(1982)] . Therefore, with the exception of the two-price Edgeworth cycles, whose rationalizationrequires nonlinearitics in the cost function, any reaction function can he rationalized using thedemand side only.

348

D.C. Keenan and ,L1.,1. 0 Br(en . (o npetirion, collusion, and thaw

(1986)] . Like stable sets, each regime consists of a set of possible configurationsand there are in general several such regimes possible . Each regime displaysinternal stability in the sense that, beginning with a configuration withinthe regime, one obtains only further configurations within that regime.Furthermore, if the regime is not to be transient, they must also be externallystable in the sense that competing regimes must not collapse the given regimeat its edges. While the cartels of chaotic dynamics display internal stability,they lack external stability . It is this failure that causes them to be onlycomponents of large regimes, rather than being themselves enduring standardsof behavior .

If, on the other hand, there are to be several coexisting behavioral regimes,a particular pattern of conduct must not display too strong external domina-tion: that is, it must not consistently erode the neighboring regimes . This balanceof power between regimes is rather delicate . Even if there is average balance, sothat regions only expand and contract in an apparently random fashion, wehave already seen that with sufficient time one arbitrary regime will come todominate. Once again, however, such long-run behavior may be cut short by thecycle that must occur with a finite number of firms .

7 .2 . Coexisting chaotic regimes

In fig. 4 we have allowed three price levels, with firms being identical_ Withthree prices, it becomes possible to have two chaotic regions, each consisting oftwo prices, where the high price is shared in common by the regions . There isa large amount of symmetry in the chosen reactions, so that the two regions arcof the same basic form, the only difference being that the low price unique to oneregion has its role replaced in the other region by that region's unique low price .The symmetry assures balance along the barrier between the two regions, whichthen expand and contract in an apparently random fashion. As with theleader follower dynamics, a region is only eliminated when it completely car-telizes and then deteriorates ."'

As seen in fig. 4, the continued existence of a region of behavior which doesnot include a particular price does not imply that a firm that is at a given time

20The existence of the two regions is supported by a barrier of cartel prices . To see this, considerthe following . Suppose a high-price firm arises with one type region on one side and the other typeon the other side. Either the firm continues to adopt the high price next period, continuing thisbarrier, or it does not. Also, either the combination on both sides that leads to a noncartel price itselfinvolves a high price, in which case the barrier is continued to that side, or it does not. Examinationof the reaction function shows, however, that those combinations not containing a high price andnot leading to a high price cannot have been created, given that one of the neighbors who mustcontribute to their existence is the original high-priced firm . Thus, the barrier between regions, oncestarted, must continue .

D.C. Keenan and M . .1 . O'Brien, Compela on, collusion, and chaos 349

Fig. 4. Differing chaotically collusive regimes .

This figure is the evolution fur identical firms of a rule which is a combination of two differentEdgeworth dynamics. The two types of Edgeworth regions have the same cartel price, but a differentlower price, so that there are three prices in all. This figure illustrates the fact that the behavior ofregions can be quite complex . For example, as can be seen in the upper right portion of the figure .a region can be eradicated by an opposing region which is itself eradicated at a later time by a regionidentical to the first one. This implies that a given site can be a member of several regions at differenttimes in the evolution. Since time periods A and B are identical, the figure exhibits a cycle havingboth regions of behavior. "Pile value of each site is initially uncorrelated and is taken asp, p, or p with

probability 1 ;3 .

a member of the regime will never charge that price . This follows since the firmmay at some time pass out of this regime . The persistence of a region does notimply the persistence of this standard of behavior by any particular firms, butrather that the behavior continues to be passed on to some collection of firms .

3 50

D. C. Keenun rail 4/ . .l . 0 ffiiei, Competition, coNusion, rold ehnos

Fig . 5 . Competitive and chaotically collusive regimes .The dynamics in this figure involve three prices and firm of alternating type. Now, white representsthe intermediate price p. whereas the low price p is represented in grey . Firms act as either leaders orfollowers with respect to the intermediate and high prices p and p . This shows tip as chaotic regionsof the same type as fig . 2a . In addition, however, the low price p is stable, so that there are alsoregions of competitive behavior . While firms remain in one region or another for all time, theirinclusion in any particular region is the result of the rules' interaction with the initial conditions andnot of differences in the firms themselves . The value of each site is initially uncorrelated and is taken

as ji, pk or p with probability U3 .

7 .3 . Competitive regimes coexisting n ith chaotic regimes

In fig. 5 we also allow three prices, but now firms again differ from theirneighbors . Thereby we can generate two regions of behavior, one of type I andone of type 111 .

The dynamics arc those of the leader-follower type with regard to the two

higher prices p and P . In the presence of the low price p, though, a leader willalways raise the price to the greater value of his two neighbors, whereas

D.C. Keenan and M. J . O'Brien, Competition, collusion, and chaos

35 1

a follower will always adopt the low price. As a result, the low price is stable andso purely competitive regimes coexist with the leader-follower regimes ofrecurring collusive activity . Observe that the particular firms that constitute thecompetitive regime are fixed over time, though this results from inheritedcircumstances and not from any inherent behavior special to just these firms .

This figure is particularly useful for considering the effect of different sizecities, since each chaotic region may be interpreted in this fashion. The onlydifference from a city proper is that there is a boundary of always competitivefirms, rather than there being a circular city closing in on itself. As might heexpected, the simulations show that narrower regimes lead to more rapidself-organization and cycling ."

The external balance between the competitive and collusive regimes of fig . 4 isof a more subtle form than our general discussion of external stability mighthave indicated. The competitive price is not robust against both other prices, butit does not need to be since matters arrange themselves, due to the self-organizing nature of the rule, so that cartelized firms charging p and thecompetitive firms charging p never come into contact . They are, instead, separ-ated by a pair of firms charging the intermediate price p . Potentially price-following firms react to one neighbor charging p and the other p by themselvescharging p, which provides the outer edge of the competitive regime . Theadjacent, potentially price-leading firms react instead to the situation of oneneighbor charging p" and the other q by themselves charging p, which serves toset up the given reaction of their aforementioned competitive neighbors . Finally,these firms charging p are in turn substantiated by their price-following neigh-bors to the other side who react to one firm charging p and the other notcharging p by charging p as well . Since the price p does not arise from theleader-follower environment on the outside, the annunciated scheme of reac-tions is self-supporting and secures a regime of competitive prices within . Thus,external stability does not require that a standard of behavior dominate allpossible alternatives, but rather that on average it prevails over the alternativesthat actually arise . Which alternatives actually arise can depend delicately onthe overall character of the dynamics .

While all the transient cartels in our examples have been triangular, suchprecision is by no means necessary. With sufficient prices, cartels may grow andcontract in complicated fashions . In particular, the steady decline of our triangu-lar cartels is due to their contact with a noncartel price that continuallyundercuts the cartel by causing fringe firms to abandon the cartel and adopt thatnoncartel price . Instead, a noncartel price in contact with a cartel might lead to

21 Compared to a true circular city, though . this effect is attenuated by the boundary condition,which acts to continually inject newly competitive forces, rather than reflect the already processedinformation coming out of the other side of a self-enclosed city .

352

D.C ;. Keenan and M . J . O'Brien, Competition, collusion, and chaos

an advancing sequence of different noncartel prices at the edge of the unravelingcartel that eventually ends with the cartel price . At this point, the cartel will haveceased unraveling and will have expanded by a firm site . Depending on the priceof the newly neighboring firm it then encounters, the cartel may expand furtheror begin another sequence of contractions . While this scenario would appear toallow for a wide variety of shapes of cartels, depending on the sequence ofimpinging prices, simulations show that highly organizing rules will arrange thatcartels actually encounter a very small number of different sequences of prices .Typically then, after a short transient period due to the initial conditions, onlya very small number of different shaped cartels will occur .

8. Conclusion

The classical Bertrand model of price competition by myopic firms does notpermit collusive activity to arise. We have altered that model in a number ofregards in order to weaken competitive forces . In particular, we have introducedspatial product differentiation, explicit dynamics, and increasing-cost considera-tions. On the other hand, we have maintained the assumption that firms aremyopic. It is relaxation of this assumption which is typically the critical elementin permitting game-theoretic explanations of collusion : so, in maintaining thisassumption, it is established that the collusive activity observed in this paper isof a different nature, arising from spontaneous self-organization .

Given our changes, we have demonstrated that cartel formation can besupported by a variety of dynamic reactions obtainable from our model . Wehave also shown that a natural extension of Edgeworth dynamics to our modelresults in collusion and competition chaotically interacting throughout time .The formal analysis of the economic model has been shown to be identical to thatof a cellular automaton . Measures of complex behavior developed for cellularautomata have been shown to be meaningful measures of the self-organizingtendencies of our economic model . The reduction of conflict via the introductionof follower firms was reflected in the reduction of these entropy measures .The leader-leader and leader follower dynamics we have examined were

obtained by introducing significant nonlinearity in tastes and technology . Thesenonlinearities are at the root of the chaotic dynamics that have been the focus ofthis analysis .

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