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Highlights

Contents lists available atScienceDirect

Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage:www.elsevier.com/locate/chaos

Heterogeneity and the (de)stabilizing role of rationality

Fausto

Cavalli

a,1

_,

_Ahmad

_Naimzada

a,2

_,

_Marina

_Pireddu

b,∗ Q1

a_{Departmentof Economics, Management and Statistics, University of Milano-Bicocca, U6 Building, Piazza dell’Ateneo Nuovo 1,}

20126 Milano, Italy

b_{Departmentof Mathematics and Applications, University of Milano-Bicocca, U5 Building, Via Cozzi 55, 20125 Milano, Italy}

a r t i c l e

i n f o

Article history:

Available online xxx

a b s t r a c t

In this paper we study oligopolies of generic size consisting of heterogeneous firms, which adopt best response adjustment mechanisms with either perfect foresight (rational firms) or static expectations (naive firms). Assuming an isoelastic demand function and larger total costs for the rational firms, we focus on the local stability of the Nash equilibrium. We show that, with respect to the oligopoly composition, described in terms of the fraction of rational firms, different scenarios are possible. We find that a high rationality degree may not always guarantee stability, in particular when rational firms have sufficiently larger marginal costs. In fact, in this situation, increasing the fraction of rational firms can even introduce instability. Besides the usual scenarios in which replacing some naive firms with rational ones leads to a stabilization of (or at least keeps unchanged) the dynamics, we provide a family of situations, characterized by costs ratio favorable to naive firms, in which equilibrium loses its stability when naive firms are replaced by rational ones. The results we present are both analytical and simulative.

© 2015 Published by Elsevier Ltd.

1. Introduction

1

In modern game theory, rationality concerns the capabil-2

ity of a player to collect information and to use it to compute 3

his optimal strategy with respect to the other players’ strate-4

gies. In an oligopolistic Cournotian competition, in which few 5

firms controlling the market compete in the amount of out-6

put they produce, a firm is fully rational if it is able to fore-7

cast its competitors’ strategies and to compute the produc-8

tion level that allows maximizing its profit. However, such a 9

high degree of rationality is not always easy or realistic to 10

achieve, since it implies a perfect knowledge of both mar-11

ket and competitors’ strategies (perfect foresight). Even in the 12

∗ _{Corresponding author. Tel.: +39 0264485767; fax: +39 0264485705.}

E-mail addresses:fausto.cavalli@unimib.it(F. Cavalli),ahmad.naimzada@ unimib.it (A. Naimzada), marina.pireddu@unimib.it, marina.pireddu@ gmail.com(M. Pireddu).

1_{Tel.: +39 0264485879; fax: +39 0264483085.} 2_{Tel.: +39 0264485813; fax: +39 0264483085.}

first formal theory of oligopoly developed by Cournot in 1838 13 (see [1]), the firms are supposed not to know their com- 14 petitors’ next period production levels, and so they use the 15 so-called “static” expectations, namely they assume that the 16 other firms will produce in the next period the same quan- 17 tity of the previous one. If all the firms were supposed to 18 be fully rational, they would be able to choose their optimal 19 choice in one shot, simultaneously achieving the Nash equi- 20 librium. Conversely, if some firms are not fully rational, they 21 have to gradually adapt their production level, giving rise to 22 a dynamical adjustment mechanism. Under suitable condi- 23 tions, such dynamics can be convergent to the equilibrium, 24 but in general they can also give rise to unstable, both pe- 25 riodic and chaotic, complex output level trajectories. Since in 26 the framework in which all the firms are fully rational (which 27 is actually static) the equilibrium is always stable, it is natural 28 to assume a stabilizing role for the rational firms. 29 Considering oligopolies of variable size, several authors 30 (see for instance the contributions by Palander in [2], 31 Theocharis in [3], Bischi et al. in [4], Lampart in [5], 32

http://dx.doi.org/10.1016/j.chaos.2015.05.017

0960-0779/© 2015 Published by Elsevier Ltd.

Matsumoto and Szidarovski in[6], Naimzada and Tramon-33

tana in[7]) showed in various economic contexts that the 34

equilibrium stability can severely change when the oligopoly 35

size increases, having the size a general destabilizing role. 36

However, the previous contributions concerned homoge-37

neous oligopolies, i.e. oligopolies in which all firms adopt the 38

same decision rule to choose their production levels. Con-39

versely, the works concerning heterogeneous oligopolies, in 40

which at least two firms adopt different decision mecha-41

nisms, mainly focused on studying different coupling of dis-42

tinct decision mechanisms in oligopolies of fixed reduced 43

sizes, usually duopolies or triopolies. We mention the papers 44

by Agiza et al.[8–12], Angelini et al.[13], Bischi et al.[14–16], 45

Matsumoto[17], Cavalli et al.[18–20]and Tramontana[21]. 46

Our contribution aims to investigate the role of oligopoly 47

size and composition for heterogeneous competitions. The 48

heterogeneity we investigate concerns the degree of ratio-49

nality of the firms, as we consider two different informa-50

tional capabilities. Both the kind of firms we consider use 51

best reply mechanisms to choose their strategy, but rational 52

firms have full informational and computational capabilities 53

to solve the resulting optimization problem and possess per-54

fect foresight, while naive firms are not able to predict such 55

strategies, and assume instead static expectations. We focus 56

on an economy characterized by an isoelastic demand func-57

tion and we consider linear total costs for all the firms. More-58

over, we assume that naive firms, due to a limited rationality 59

degree, may adopt a more cautious behavior, and so they do 60

not immediately choose the production level they computed 61

using the static expectations best response, but they more 62

prudently adapt their strategy toward the expected profit 63

maximizing production level. 64

The present work belongs to a research strand which in-65

cludes[22], in which the same behavioral rules were stud-66

ied in an economy characterized by a linear demand func-67

tion. Moreover, in[22] also the possibility for the firms to 68

switch the adjustment mechanisms was considered. We re-69

mark that, in the existing literature, the study of heteroge-70

neous oligopolies of generic size can be found in the works 71

by Anufriev et al.[23](where the role of heterogeneity in 72

learning is investigated in a Bertrand oligopoly), Banerjee and 73

Weibull[24](where an evolutive game with agents hetero-74

geneous in the rationality degree is considered), Droste et al. 75

[25](where an infinite population of firms is studied with re-76

spect to the possibility to switch among different decisional 77

mechanisms), Gale and Rosenthal[26](in which experimen-78

tation and imitation behaviors are analyzed). 79

Conversely, in our contribution we study equilibrium sta-80

bility in an oligopoly consisting of firms which can be differ-81

ent with respect to the degree of rationality, the technology 82

and the adopted adjustment mechanism. The main question 83

we address is the following: does an increase in the number 84

of rational firms always improve the equilibrium stability? 85

To try to give an answer, for any oligopoly size, we consider 86

all the possible compositions of firms heterogeneous with 87

respect to the rationality degree, parameterizing such com-88

positions through the fraction

ω

of rational firms. This gives 89

rise to a multidimensional discrete dynamical system, whose 90

equilibrium stability we investigate on varying

ω

. 91

Our setting is similar to those studied by Hommes et al. 92

in[27]and Bischi et al. in[28], which focus on oligopolies 93

of generic sizes in which the firms can adopt different 94 decisional mechanisms. Indeed, in [27]the authors study 95 oligopolies of rational and naive firms which are heteroge- 96 neous just with respect to the rationality degree. Moreover, 97 naive firms, at each time step, play the best response to the 98 average global strategy, and not to the actual strategy of each 99 other firm. Such assumptions only allow for stabilizing or un- 100 conditionally stable scenarios. Conversely, in[28]the authors 101 study oligopolies in which all the firms are boundedly ratio- 102 nal, as they can be either naive or use a local monopolistic ap- 103 proximation rule, extending to heterogeneous oligopolies the 104 investigations about the effect on the equilibrium stability 105 of such bounded rationality mechanisms, previously known 106 for homogeneous settings only[4,5]. Moreover, both in[27] 107 and[28]the evolutionary fraction setting, in which firms can 108

switch among heuristics, is studied too. 109

Our main result concerns the existence of different possi- 110 ble behaviors with respect to the variation of the fraction of 111 rational agents, as increasing their fraction

ω

in an oligopoly 112 of given size may have, besides neutral or stabilizing, also a 113 destabilizing role. To the best of our knowledge, the latter be- 114 havior has not been observed in the models studied in the ex- 115 isting literature, in which increasing the fraction of rational 116 players can only have a stabilizing (or at least neutral) effect, 117 as for example in[22,27]. This is mainly due to the presence 118 in our model of two stability thresholds, with respect to the 119 oligopoly composition, which occur when marginal costs of 120 rational firms are different from those of naive firms. Con- 121 versely, in the examples investigated in the existing litera- 122 ture, stability is always regulated by a single threshold. In 123 particular, also in[22], where a linear demand function is 124 considered, stability is regulated by a unique threshold, even 125 if the considered decision mechanisms are the same as in the 126 present work. This suggests that the ambiguous effect on sta- 127 bility of the oligopoly composition is due to the presence of a 128 nonlinear demand function. Conversely, both in the present 129 work and in[22], when total costs are identical for all the 130 firms, we recover the usual unambiguous stabilizing role for 131

rational firms. 132

We show that the existence of two stability thresholds al- 133 lows for the following possible scenarios, depending on the 134 parameters configuration, and in particular on the value of 135

the costs ratio. 136

• Unconditionally stable/unstable scenario: in this case, the 137 equilibrium stability is unaffected by the fraction of ratio- 138 nal firms, and the trajectories converge/do not converge 139 to the equilibrium for any composition. If the dynamics 140 do not converge, both periodic and chaotic dynamics are 141

possible. 142

• Stabilizing scenario: in this case, when the fraction of ra- 143 tional firms is too small, the equilibrium is unstable, but 144 its stability can be recovered replacing some naive firms 145 with rational ones, namely increasing

ω

. It is a classical 146 situation in which increasing the overall degree of ratio- 147 nality leads the dynamics to stabilization. 148

can occur for any oligopoly size, but only if naive firms are 154

technologically more efficient than rational firms. 155

A theoretically possible fourth framework is the mixed 156

scenario, in which the fraction of rational firms has a further 157

ambiguous role, since when it is too small, the equilibrium is 158

stable, in an intermediate region of values for

ω

stability is 159

recovered and then, if the number of rational firms is further 160

increased, stability is lost again. In this work, we mainly focus 161

our attention on dynamics involving strictly positive produc-162

tion levels. Under such hypothesis, we prove that the mixed 163

scenario is actually impossible. If however we relax the posi-164

tivity hypothesis and we consider also null production levels, 165

we are able to show through simulations that the mixed sce-166

nario is possible and that the destabilizing scenario occurs 167

for a wider parameters setting. 168

We remark that the previous scenarios are found in a 169

context of general heterogeneity among the firms, which in-170

volves the informational endowment (different rationality 171

degrees for the firms), the technology (different marginal 172

costs) and the adjustment mechanisms (naive firms are as-173

sumed to cautiously adjust their production level). In partic-174

ular, the destabilizing nature of the third scenario cannot be 175

completely ascribed to the influence of rationality, as other 176

heterogeneity aspects are involved. With this respect, we find 177

that the destabilizing scenario is possible only if marginal 178

costs are suitably unfavorable to the rational firms, while it 179

cannot occur if a similar, but unfavorable to naive firms, level 180

of heterogeneity in technology is considered. Moreover, in or-181

der to disentangle the effect of heterogeneity in the adjust-182

ment mechanisms, we investigate the consequences of as-183

suming the same cautious adjustment rule for all the firms 184

and we find that the destabilizing scenario still occurs. 185

The remainder of the paper is organized as follows. In 186

Section 2we describe the oligopolistic competition we want 187

to study and we present the model. InSection 3we perform 188

the local stability analysis and we discuss the possible aris-189

ing scenarios. InSection 4we report some simulative results. 190

InSection 5we investigate the effects of assuming the same 191

cautious adjustment mechanism for all the firms. InSection 6

192

we draw some conclusions and we outline future research 193

developments. Finally, in Appendix Awe collect some tech-194

nical results used for the proofs inSection 3. 195

2. Oligopolistic Cournot game

196

In this section we present the oligopolistic market we 197

want to study and the model through which it can be de-198

scribed. Since the firms we consider are not all endowed 199

with full rationality, the resulting model is represented by 200

a discrete dynamical system in which, at each time pe-201

riod, the firms repeatedly interact on the same market to 202

maximize their profits. In this first part of the section, af-203

ter describing the market, we present the oligopolies we 204

focus on and the Nash equilibrium of the corresponding 205

oligopolistic Cournotian competition. Moreover, we show in 206

Proposition 2.1how the components of the Nash equilibrium 207

vary when modifying the total number of firms and the frac-208

tion of rational firms. Then we enter into detail about the de-209

cisional mechanisms and the dynamical model inSection 2.1. 210

We consider an economy characterized by a homoge- 211 neous market controlled by N≥ 2 firms producing quantities 212 qi, i = 1, . . . , N, of the same good, for which the price function 213

is given by 214

p

(

Q

)

= 1

Q, (2.1)

where Q=N

i=1qiis aggregate demand. Function(2.1)is the 215 same isoelastic one used for example by Puu [29], micro- 216 founded on Cobb–Douglas preferences. We suppose that firm 217 i= 1, . . . , N faces the linear total cost function 218

C

(

qi

)

= ciqi+ Ci,

where ci> 0 ∈

{

cR, cN

}

, i = 1, . . . , N, are two possibly dif- 219 ferent (constant) marginal costs and Ci> 0 ∈

{

CR,CN

}

, i = 220 1, . . . , N, are two possibly different fixed costs. The choice to 221 use subscriptsR, N , which will refer to rational and naive 222 agents, will become more clear in the next section. In par- 223 ticular, in agreement with the existing literature, we assume 224 that C_R> CN, as the higher informational costs incurred by 225 rational firms are reflected by larger fixed costs. The possible 226 difference in the marginal costs allows instead considering 227 technological heterogeneity among the firms. 228 Without loss of generality, we order the firms so that the 229 first

ω

N firms, indexed by i_{= 1, 2, . . . ,}

ω

N_{, have marginal} 230 costs c_R, while the remaining N

(

1−

ω

)

firms, indexed by 231 i=

ω

N+ 1, . . . , N have marginal costs cN. In particular,

ω

232 represents the fraction of firms belonging to the first group. 233 We want that each group consists of at least one firm, so we 234

impose that 235

1

N ≤

ω

≤ 1 −

1

N. (2.2)

In the previously described setting we have a game, where 236 the players are represented by the N oligopolists, the feasible 237 strategies are the nonnegative production levels qi≥ 0 and 238 the payoff functions are given by the profit functions 239

π

i= pqi− ciqi− Ci, i = 1, . . . , N.

We only consider the situation in which all firms have non 240 null production levels, so that the resulting oligopoly will al- 241 ways be composed by N firms, and we focus on the inter- 242 nal Nash equilibrium, namely, the equilibrium consisting of 243 strictly positive strategies. A simple but tedious computation 244 shows that such internal equilibrium is characterized by 245

q∗_i =

(

cNN

(

1−

ω

)

− cR

(

N

(

1−

ω

)

− 1

))(

N− 1

)

N2

(

_c N

(

1−

ω

)

+ cR

ω

)

2 = q ∗ R, i= 1, . . . ,

ω

N (2.3) and 246 q∗_i =

(

cRN

ω

− cN

(

N

ω

− 1

))(

N− 1

)

N2

(

_c N

(

1−

ω

)

+ cR

ω

)

2 = q ∗ N, i=

ω

N+ 1, . . . , N. (2.4)

Introducing the cost ratio 247

r= cR

cN , (2.5)

it is easy to see that quantities(2.3)and(2.4)are both strictly 248

positive provided that 249

N

ω

− 1

N

ω

< r <

N

(

1−

ω

)

N

(

1−

ω

)

− 1, (2.6)

which means that, for a fixed oligopoly size N and composi-250

tion

ω

, only for sufficiently similar values of marginal costs 251

for all firms both the production levels are nonnegative. We 252

shall maintain(2.6)for the remainder of the paper. Notice 253

that the interval given by(2.6)is always nonempty. In partic-254

ular, we stress that the upper bound on r imposed by(2.6) in-255

creases with

ω

. For instance, when

ω

_{= 1 −}1

Nthe above

con-256

dition imposes no upper bound on r, while for

ω

≤ 1 −2

Nthe

257

right inequality in(2.6)implies that r< 2. Similarly, the lower 258

bound on r increases with

ω

. For instance, when

ω

_{= 1 −}1

N,

259

we need r> 1 − 1

N−1, while

ω

= 1/N does not impose any 260

lower bound conditions on the cost ratio. We notice that the 261

more unfavorable the cost ratio is to a particular group of 262

firms, the more numerous that group shall be to preserve the 263

positivity of the equilibrium. In what follows, we will focus 264

on sets of oligopolies which, having the same size N, can dif-265

fer for their composition

ω

. In particular, for a given oligopoly 266

size N and cost ratio r, we will consider 267

FN,r=

{ω

: q∗i > 0, i = 1, . . . , N

}

,

which represents the family of all the oligopoly compositions 268

for which the equilibrium is strictly positive. The number of 269

oligopoly compositions with positive equilibrium is then rep-270

resented by the cardinality of the setFN,r, namely by #

(

FN,r

)

.

271

From(2.6)and the previous considerations, for r= 1 we have 272

thatFN,r=

{

1/N, . . . , 1 − 1/N

}

, while for r < 1 (respectively

273

r> 1) we have that FN,r=

{

1/N, . . . , n

(

r

)

/N

}

(respectively

274

FN,r=

{

1− n

(

r

)

/N, . . . , 1 − 1/N

}

), where n(r) is a suitable

in-275

teger in

{

1, . . . , N − 1

}

which depends on r and represents the 276

size of the family. For example, if we wanted to consider the 277

complete family of heterogeneous oligopolies of a given size 278

N, namely the family consisting of all the possible compo-279

sitions

ω

_{= 1/N, . . . , 1 − 1/N, we would need to restrict our} 280

attention to the most restrictive upper and lower constraints 281

on the cost ratio. A straightforward computation shows that 282

this implies 1_{− 1/}

(

N_{− 1}

)

_{< r < 1 + 1/}

(

N_{− 2}

)

. 283

In the next result we investigate how q∗_N and q∗_R vary 284

when modifying

ω

and N. 285

Proposition 2.1. It holds that:

286 • if r_{< 1, then}∂q∗N ∂ω < 0 and∂ q∗_N ∂N < 0 ; 287 • if 1_{< r <} Nω+N−2 Nω , then∂ q∗_N ∂ω > 0 and∂q ∗ N ∂N < 0 ; 288 • if r> Nω+N−2 Nω , then∂ q∗_N ∂ω < 0 and∂q ∗ N ∂N > 0 ; 289 • ∂q∗N ∂ω = 0 for r = 1 or r = NωN+N−2ω , and ∂ q∗_N ∂N = 0 for r = 290 Nω+N−2 Nω ; 291 • if r< N(1−ω) 2N−Nω−2, then∂ q∗_R ∂ω > 0 and ∂ q∗_R ∂N > 0 ; 292 • if_2NN_−N(1−_ωω)_{−2< r < 1, then}∂q∗R ∂ω < 0 and∂ q∗_R ∂N < 0 ; 293 • if r> 1, then∂q∗R ∂ω > 0 and∂q ∗ R ∂N < 0 ; 294 • ∂q∗R ∂ω = 0 for r = 1 or r =2NN(−N1−ωω)−2, and ∂ q∗_R ∂N = 0 for r = 295 N(1−ω) 2N−Nω−2. 296

Proof. Direct computations show that

297

∂

q∗ N

∂ω

=

(

N− 1

)(

cR− cN2N

(

_c

)(

cNN− 2cN+ cNN

ω

− cRN

ω

)

R

ω

+ cN− cN

ω

)

3 and 298

∂

q∗_N

∂

N = −

(

c_NN− 2cN+ cNN

ω

− cRN

ω

)

N3

(

_c R

ω

+ cN − cN

ω

)

2 .

Recalling the definition of r in(2.5), the desired conclusions 299 on q∗_Neasily follow. Similar computations allow to derive the 300 desired conclusions on the behavior of q∗_R, too.  301

Hence, q∗_Nis increasing with

ω

when 1< r < Nω+N−2

Nω and 302 decreasing with

ω

for the remaining values of r, while q∗_N is 303 increasing with N when r_>Nω+N−2

Nω and decreasing with N 304 otherwise; q∗_Ris increasing with

ω

for r< N(1−ω)

2N−Nω−2 or r> 1, 305 and decreasing with

ω

for the remaining values of r, while 306 q∗_Ris increasing with N for r< N(1−ω)

2N−Nω−2and decreasing with 307 N otherwise. When the marginal costs for all firms coincide, 308 i.e., when r= 1, then q∗

Nand q∗Rare not influenced by

ω

, but 309 both of them are decreasing with the number of firms. In fact, 310

in this particular case it holds that 311

q∗_N= q∗

R=

N− 1

cN2 .

2.1. The decision mechanisms 312

If all the firms were endowed with full rationality, namely 313 they had complete informational and computational capabil- 314 ities, the firms would simply choose at once the Nash equi- 315 librium(2.3)and(2.4). Conversely, in this work we assume 316 that only a subset of the firms are fully rational, so that they 317 have complete knowledge of the market and they are able to 318 optimally respond to the other firms strategies. In addition 319 to this, they are endowed with perfect foresight, so that they 320 exactly forecast the next time production levels of their op- 321 ponents. We will refer to such firms as to rational firms. To 322 encompass their high informational capability, we associate 323 rational firms to the first group of firms, which have larger to- 324 tal costs. Conversely, the firms belonging to the second group 325 lack in perfect foresight and we assume for them static ex- 326 pectations (naive firms). This setting gives rise to a dynamic 327 adjustment of the production levels, in order to maximize the 328 one-period profits, and the resulting oligopoly turns out to 329 be heterogeneous in terms of the decision mechanisms. This 330 means that, if we denote by qe_{j, t+1},i the production for period 331 t+ 1 of firm j expected by firm i, we have 332

qe_{j, t+1},i = qj,t+1, 1 ≤ i ≤

ω

N, 1 ≤ j ≤ N, (2.7)

for the rational firms and 333

qe_{j, t+1},i = qj,t,

ω

N+ 1 ≤ i ≤ N, 1 ≤ j ≤ N, (2.8) for the naive ones. We remark that

ω

now also represents 334 the fraction of rational firms. Condition(2.2) assures that 335 the considered oligopoly is always heterogeneous. Since for 336 N= 2 we have only one possible heterogeneous composition 337 consisting of a rational and a naive firm, in what follows we 338 will focus on the case N> 2, in which it is meaningful to 339 investigate the effect of varying the oligopoly composition. 340 Moreover, the case with N_{= 2 firms has already been stud-} 341

ied in[20]. 342

Let us now specify the strategies for the two groups of 343 firms. At each time, all the players try to choose their pro- 344 duction level in order to maximize their one-period profits. If 345 Q_te,i₊₁is the total output expected for the next period by firm i 346

Q_te₊₁,i =

N



j=1, j=i

we have that its expected profit at t_{+ 1 is} 347

π

e i, t+1= qi,t+1 N j=1, j=iqej, t+1,i + qi,t+1 − csqi,t+1− Cs,

where s= R for i = 1, . . . ,

ω

N and s= N for i =

ω

N+ 348

1, . . . , N, we can write that the expected marginal profit is 349 350

∂π

e i, t+1

∂

q_i,t+1 =

(

Q_te₊₁,i − qi,t+1

)

(

Q_te,i₊₁

)

2 − cs, s = R, N . (2.9) First of all we notice that, thanks to perfect foresight as-351

sumption(2.7), at each time step, all the rational firms have 352

the same best response to the strategies of the other agents, 353

namely, if i is a rational firm, we have that q_{i,t+1= qR,t+1} 354

for 1 _{≤ i ≤}

ω

N, from which we then find Q_te₊₁,i _{= Q}t+1= 355

Q_{R,t+1+ QN ,t+1= N}

ω

q_R,t+1+N_i=ωN+1q_{i,t+1, where we} in-356

troduced the aggregate production quantities Q_Rand Q_N of 357

the rational and naive firms, respectively. Inserting the ex-358

pression for Q_te₊₁,i into the right hand side of(2.9)and impos-359

ing the first order condition, we find 360

(

N

ω

q_R,t+1+ QN ,t+1− qR,t+1

)

= cR

(

N

ω

qR,t+1+ QN ,t+1

)

2. Solving with respect to q_R,t+1, we get two solutions, one of 361

which is negative. Hence the only admissible solution is given 362 by 363 q_R,t+1= f

(

Q_{N ,t+1}

)

=Nw−1−2cRN

ω

QN ,t+1+



(

Nw−1

)

2_{+ 4c} RN

ω

QN ,t+1 2cRN2

ω

2 , (2.10) where f :

(

0, +∞

)

→ R is a positive function provided that 364

Q_{N ,t+1}< 1/cR. In what follows, we will mainly focus on

dy-365

namics in which the production levels stay strictly positive. 366

If not, the best response to QN,t+1would be qR,t+1= 0.

367

Conversely, if

ω

N+ 1 ≤ i ≤ N is a naive firm, the assump-368

tion of static expectations(2.8)gives Q_te₊₁,i ₌N_{j=1, j=i}q_{j, t+} 369

qi,t+1, which, inserted in(2.9), provides

370



N  j=1, j=i qj, t



= cN



N  j=1, j=i qj, t+ qi,t+1



2_.

Solving with respect to q_i,t+1, we find two solutions, one of 371

which is negative. 372

Calling Q_−i,t the aggregate quantity produced by all the 373

firms but the ith naive one, the other solution is given by 374

qi,t+1= h

(

Q−i,t

)

=



Q_−i,t

cN − Q−i,t, (2.11)

where h :

(

0_{, +∞}

)

_{→ R, Q−i,t → h}

(

Q_−i,t

)

_{, is the best} re-375

sponse of the ith naive player provided that Q_−i,t< 1/cN,

376

otherwise qi,t+1= 0. We remark that Q−i,tis actually a func-377

tion of q_{j, t}with j_{= i, that is,} 378 Q_−i,t = Qt− qi,t= ωN  j=1 qj,t+ N  j=ωN+1, j=i qj,t =

ω

Nq_R,t+ QN ,t− qi,t. (2.12)

Inserting the expression for q_R,tfrom(2.10)into(2.12)we 379

obtain 380

Q_−i,t =

ω

N f

(

Q_{N ,t}

)

+ QN ,t− qi,t, (2.13)

and consequently(2.11)becomes 381

q_i,t+1=



ω

N f

(

Q_{N ,t}

)

+ QN ,t− qi,t

c_N

−

(

ω

N f

(

Q_{N ,t}

)

+ QN ,t− qi,t

)

. (2.14) Naive firms, due to their bounded rationality, act in an un- 382 certainty context, as they are aware that adopting static ex- 383 pectations is not able to guarantee that the production level 384 they choose for the next period actually provides their next 385 period optimal profits. Then, a more cautious behavior can 386 be expected for naive firms, due to the reduced degree of 387 confidence in their own expectations. For a discussion about 388 firms behavior we refer to the book of Sterman[30]. Exam- 389 ples of cautious adjustment mechanisms for naive firms can 390 be found in[6,29]. Hence, we assume that naive firms do 391 not immediately choose the production level they computed 392 using the static expectations best response, but they more 393 prudently adapt their strategy toward the expected profit 394 maximizing production level. To model such effects, we sup- 395 pose that the naive firms adopt the following adjustment 396

mechanism 397

q_i,t+1= qi,t+

σ

(

h

(

Q−i,t

)

− qi,t

)

, i =

ω

N+ 1, . . . , N, (2.15) where

σ

:_{R → R is a function describing the possible pro-} 398 duction variation. This function is differentiable, strictly in- 399 creasing and bounded, with

σ

(

0

)

= 0, so that it preserves 400 the steady states of the best response functions for the naive 401

players in(2.11). 402

The resulting model is then obtained considering the iter- 403 ates of the N

(

1−

ω

)

-dimensional system generated by(2.15). 404

We will call g the associated map. 405

We underline that, since

ω

can assume the values 406 1/N, 2/N, . . . ,

(

N− 1

)

/N, the dimension of(2.15)can vary be- 407

tween N_{− 1 and 1.} 408

We notice that the oligopolistic competition we study is 409 described by the evolution of the strategies of both ratio- 410 nal and naive players, namely by the

ω

N identical equations 411 in(2.10)and the

(

1−

ω

)

N equations in(2.15). On the other 412 hand, the dynamical system that has to be analytically in- 413 vestigated is made up by the

(

1₋

ω

)

N-dimensional system 414

(2.15)only. We remark that it is easy to prove that when the 415 strategies of the naive players converge, they necessarily con- 416 verge to q∗_N (and, consequently, the strategies of the rational 417

players converge to q∗_R). 418

3. Stability analysis 419

In order to perform the local stability analysis for model 420

(2.15), at first we need to distinguish between the case

ω

< 421 1− 1/N, in which the model consists of a system of N

(

1− 422

ω

)

equations, and the case

ω

_{= 1 − 1/N, in which the model} 423

consists of a single equation. 424

In the framework with

ω

< 1 − 1/N, we need to evaluate 425 the N

(

1₋

ω

)

_{× N}

(

1₋

ω

)

Jacobian matrix J₌

(

J_{i j}

)

₌

∂

qjgiat 426 the Nash equilibrium, where girepresents the r.h.s.of each 427 equation in(2.15). Recalling(2.13)–(2.15), we have that 428

∂

qigi= 1 +

σ

₍

_h

₍

_Q

−i

)

− qi

)(

h

(

Q−i

)

∂

qiQ−i− 1

)

= 1 +

σ

₍

_h

₍

_Q

and, for i_{= j,} 429

∂

qjgi=

σ

(

h

(

Q−i

)

− qi

)

h

(

Q−i

)

∂

qjQ−i

=

σ

₍

_h

₍

_Q −i

)

− qi

)

h

(

Q−i

)(

ω

N f

(

QN

)

+ 1

)

, where 430 f

(

QN

)

= 1 N

ω



(

1− N

ω

)

2_{+ 4NQ} NcR

ω

− 1 N

ω

.

We denote by J∗=

(

J_{i j}∗

)

the Jacobian matrix J evaluated at the 431

Nash equilibrium, and, noticing that h

(

Q_−i∗

)

= q∗

N, we have 432 J∗_ii= 1 +

σ

₍

₀

₎₍

_ω

_Nh

₍

_Q∗ −i

)

f

(

QN∗

)

− 1

)

= a (3.1a) and, for i= j, 433 J∗_{i j}=

σ

₍

₀

₎

_h

₍

_Q∗ −i

)(

ω

N f

(

QN∗

)

+ 1

)

= b. (3.1b) This means that J∗matrix evaluated at the equilibrium has 434 the structure 435 J

(

q∗

)

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

a b b . . . b b a b . . . b .. . . .. ... .. . . .. ... b . . . b b a

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(3.2)

Since it is a particular case of circulant matrix (see for in-436

stance[31]), it allows us to explicitly compute the eigenval-437

ues of(3.1). 438

Proposition 3.1. The Jacobian matrix of System (2.15) evalu-439

ated at the Nash equilibrium has eigenvalues 440

λ

1= 1 −

σ

(

0

)(

h

(

Q_−i∗

)

+ 1

)

, with multiplicity N

(

1−

ω

)

− 1, and 441

λ

2 = 1 −

σ

(

0

)(

h

(

Q_−i∗

)

− Nh

(

Q_−i∗

)(

1−

ω

)

×

(

N

ω

f

(

Q_N∗

)

+ 1

)

+ 1

)

,

which is simple. 442

Proof. If the elements of the first row of a circulant matrix

443

are c1, . . . , cN(1−ω), its eigenvalues are given by 444 ˆ

λ

m= N(_1−ω) j=1 cj

ρ

mj−1, m = 1, . . . , N

(

1−

ω

)

where

ρ

m= exp

(

2

π

i

(

m− 1

)

/

(

N

(

1−

ω

)))

are the N

(

1−

ω

)

445

distinct complex roots of unity[31], with i representing the 446

imaginary unit. In(3.2)we have that c1= a and cj= b for

447

j= 2, . . . , N

(

1−

ω

)

, so that we can rewrite 448 ˆ

λ

m= a + b N(_1−ω) j=2

ρ

j−1 m , m = 1, . . . , N

(

1−

ω

)

.

For m= 1,

ρ

m= 1 and hence ˆ

λ

1= a +

(

N

(

1−

ω

)

− 1

)

b. Con-449

versely, if we consider m> 1, we have that

ρ

m= 1 and so we

450 can write 451 ˆ

λ

m= a + b

ρ

m N(_1−ω) j=2

ρ

j−2 m . For m_{= 1 we have} 452

ρ

m N(_1−ω) j=2

ρ

j−2 m =

ρ

m

ρ

N(1−ω)−1 m − 1

ρ

m− 1 =

ρ

N(1−ω) m −

ρ

m

ρ

m− 1

where, since

ρ

mN(1−ω)= 1, we can conclude that 453

ρ

mN_j=2(1−ω)

ρ

mj−2= −1 and hence ˆ

λ

m= a − b for m ≥ 1. 454 Using(3.1a)and(3.1b), we can conclude. _ 455 Let us then consider the case with

ω

= 1 − 1/N. Since 456 now Q_{N ,t= q}N,t, by(2.13)system(2.15)reduces to the one- 457

dimensional equation 458

qN,t+1= g

(

qN,t

)

= qN,t+

σ

(

h

(

ω

N f

(

qN,t

))

− qN,t

)

, (3.3)

from which we obtain 459

g

(

q∗_N

)

= 1 +

σ

₍

₀

₎₍

_h

₍

_ω

_{N f}

₍

_q∗

N

))

ω

N f

(

q∗N

)

− 1

)

.

We have the following result: 460

Proposition 3.2. Setting k=

σ

₍

₀

₎

_{, the local stability of the 461}

Nash equilibrium requires that 462

z

(

ω

)

= N

(

r− 1

)(

kr− k + 4

)

ω

2₊

₍

_{4N+ 4r − 2Nk} −8Nr + 2Nkr + 4

)

ω

+ Nk − 4 < 0. (3.4)

Proof. Firstly, we show that the equilibrium stability 463 is guaranteed in both the one-dimensional and in the 464 multi-dimensional case by imposing

λ

2> −1. In the one- 465 dimensional framework, the local stability of(3.3)requires 466

|

g

(

q∗_N

)

|

< 1. We notice that g

₍

_q∗

N

)

is equal to (3.1a) and 467 it is easy to see that, for

ω

= 1 − 1/N, we also have

λ

2= 468 a in (3.1a), so that the equilibrium stability in the one- 469 dimensional case is guaranteed by |

λ

2|< 1. For the multi- 470 dimensional case we need_{−1 <}

λ

s< 1, s = 1, 2. However, 471 we shall show that it holds

λ

2 <

λ

1 < 1 for each 1/N ≤ 472

ω

< 1 − 1/N and that

λ

2 < 1 for

ω

= 1 − 1/N, so that the 473 only stability condition to be imposed is

λ

2> −1 for each 474 1/N ≤

ω

≤ 1 − 1/N, which leads to(3.4). 475

To this end, we notice that, since 476

Q_−i∗ =

ω

Nq∗_R+

(

N

(

1−

ω

)

− 1

)

q∗_N = cN

(

N− 1

)

2

N2

(

_c

N+ cR

ω

− cN

ω

)

2

and recalling(2.5), we have 477

h

(

Q_−i∗

)

=N

ω

(

r− 1

)

+ 2 − N 2

(

N− 1

)

.

Since for

ω

≤ 1 − 2/N by(2.6)it follows that r< 2, we have 478 h

(

Q_−i∗

)

_{< 0. Moreover, since N ≥ 2, we have} 479

h

(

Q_−i∗

)

+ 1 =N

ω

(

r− 1

)

+ N 2

(

N− 1

)

> 0,

which allows concluding that−1 < h

₍

_Q∗

−i

)

< 0. This means 480

We notice that 484 cR

ω

(

N

(

2−

ω

)

− 1

)

− cN

(

1−

ω

)(

N

ω

− 1

)

> 0, because 485 r>

(

1−

ω

)(

N

ω

− 1

)

ω

(

N

(

2−

ω

)

− 1

)

is satisfied thanks to the lower bound on r given by(2.6), as 486 N

ω

− 1 N

ω

−

(

1−

ω

)(

N

ω

− 1

)

ω

(

N

(

2−

ω

)

− 1

)

=

(

N

ω

− 1

)(

N− 1

)

N

ω

(

2N− N

ω

− 1

)

≥ 0,

so that we can rewrite(3.5)as 487

f

(

Q_N∗

)

= − N+ 2r − 2Nr − N

ω

+ Nr

ω

N

(

ω

+ N

ω

+ r

ω

− N

ω

2_{− 2Nr}

ω

_{+ Nr}

ω

2_{− 1}

)

. Moreover, from(3.5)it follows that

488

1+

ω

N f

(

Q_N∗

)

=√1

_η

> 0.

The last relation, together with h

(

Q_−i∗

)

< 0, allows con-489

cluding that

λ

2<

λ

1< 1, for each

ω

= 1/N, . . . , 1 − 2/N. 490

Inserting the expressions of f

(

Q_N∗

)

and h

(

Q_−i∗

)

in

λ

2, af-491

ter some algebraic manipulations, we find 492

λ

2= −

Nk

(

r

ω

−

ω

+ 1

)

2

2

(

ω

+ N

ω

+ r

ω

− N

ω

2_{− 2Nr}

ω

_{+ Nr}

ω

2_{− 1}

)

+ 1, which for

ω

= 1 − 1/N gives

493

λ

2− 1 = −k

(

Nr− r + 1

)

2 2

(

N− 1

)(

rN− 1

)

+ 1< 0.

Hence,

λ

2< 1 for all

ω

= 1/N, . . . , 1 − 1/N and thus the Nash 494

equilibrium is locally asymptotically stable if

λ

2> −1, which 495 can be rewritten as 496 N

(

1− r2

₎

ω

2_{− 2}

₍

_r

₍

_{1− N}

₎

_{+ 1}

₎

ω

₋

₍

_{N− 2}

₎

2N

(

1− r

)

ω

2_{− 2}

(

_{N+ r − 2Nr + 1}

)

ω

_{+ 2} > 1 − 2 k.

Rearranging the last inequality leads to(3.4).  497

Solving(3.4)with respect to

ω

, we can obtain the stabil-498

ity of the Nash equilibrium on varying the oligopoly compo-499

sition. 500

When r= 1 or k = 4/

(

1− r

)

, since stability is governed 501

by a second degree inequality, we have, at least in principle, 502

two stability thresholds. Such framework is investigated in 503

Proposition 3.3, whose proof simply requires to solve(3.4)

504

with respect to

ω

. We stress that when r= 1(3.4)reduces to 505

a first degree inequality and thus we find a unique threshold 506

¯

ω

=

(

Nk− 4

)

/

(

4N− 8

)

for

ω

, above which the system is lo-507

cally asymptotically stable and below which it is not stable. 508

Similarly, when k= 4/

(

1− r

)

(which is possible only if r< 509

1), we again have a first degree inequality, which gives the 510

unique threshold ¯¯

ω

=

(

N+ r − 1

)

/

(

N+ Nr − 2Nr2_{+ r}2_{− 1}

₎

_, 511

above which we again have that the system is locally asymp-512

totically stable. 513

We shall return on the framework with r= 1 after hav-514

ing analyzed the more interesting and richer scenario with 515

different marginal costs. Due to the similarity with the case 516

r= 1, we will not consider the framework with k = 4/

(

1− r

)

517 anymore. 518 Let us introduce 519

ω

1= −2N − 2r + Nk + 4Nr − Nkr − 2 − √

N

(

r− 1

)(

kr− k + 4

)

ω

2 = −2N − 2r + Nk + 4Nr − Nkr − 2 + √

N

(

r− 1

)(

kr− k + 4

)

(3.6)

and notice that

ω

1 ≤

ω

2 provided that

> 0 and N

(

r− 520 1

)(

kr− k + 4

)

> 0, namely if either r > 1 or r < 1 and k > 521 4/

(

1− r

)

. Conversely, when r< 1 and k < 4/

(

1− r

)

, we have 522

ω

2≤

ω

1. 523

Proposition 3.3. When the marginal costs of the rational firms 524 are larger than those of the naive ones (r_{> 1) and}

_{> 0, the} 525 local stability at the Nash equilibrium requires that

ω

1<

ω

< 526

ω

2. Otherwise, if

≤ 0 the Nash equilibrium is always unstable. 527 Conversely, when the marginal costs of the rational firms are 528 smaller than those of the naive ones (r< 1) and

> 0, the local 529 stability at the Nash equilibrium requires that 530

ω

1<

ω

<

ω

2 (3.7)

if k> 4/

(

1− r

)

, or 531

ω

<

ω

2 ∨

ω

>

ω

1 (3.8)

if k< 4/

(

1− r

)

. Otherwise, for

≤ 0, the Nash equilibrium is 532 always unstable if k> 4/

(

1− r

)

and always locally asymptoti- 533 cally stable if k< 4/

(

1− r

)

. 534 We notice that, for r> 1, condition

> 0 is equivalent to 535

k< k

=

(

4N2−4N + 1

)

r2+ 2

(

−2N2+ N + 1

)

r+ N2− 2N + 1 2Nr

(

r− 1

)(

N− 1

)

.

It is not difficult to see that, for any r> 1 and N ≥ 3, it holds 536 that k > 0 and thus the previous condition is always fulfilled 537 by some positive k. Conversely, when r_{< 1 the sign of}

is 538 less clear and it is studied inLemma 6.7. 539 In the remaining part of this section, we identify the oc- 540 currence (or the impossibility to occur) of some significant 541 frameworks, depending on particular parameter settings. For 542 the sake of clarity, we split the results concerning r> 1 and 543

r< 1. 544

•Case r> 1 545

In the next propositions, we prove the existence of un- 546 conditionally stable/unstable and stabilizing scenarios for 547 oligopolies of any size. In particular, we show that these 548 scenarios can occur for complete families of oligopolies, in 549 which each heterogeneous composition has a strictly posi- 550 tive equilibrium, provided that the marginal cost ratio satis- 551

fies r< 1 + 1/

(

N− 2

)

. 552

We stress that results similar toPropositions 3.4–3.6hold 553 also for N_{= 3 and N = 4, but under different conditions on} 554 the parameters. For sake of brevity, we will not report such 555 statements. Moreover, we remark that, for the sake of sim- 556 plicity, we provide just sufficient conditions for the occur- 557 rence of the various scenarios, which are possible for other 558

parameters conditions as well. 559

Proof. To have the unconditionally stable scenario we need

566

that the whole interval of possible compositions be a subset 567

of the stability interval, namely

ω

1< 1/N < 1 − 1/N <

ω

2. 568

From Lemmas 6.1and6.4, it is sufficient that k_{< min}

{

k2, 569

k

}

= k2by(6.7), provided that r< 1 + 1/

(

N− 2

)

.  570

Proposition 3.5 (Unconditionally unstable scenario). For any

571

N≥ 5 and for sufficiently large values of k, there exist fami-572

liesFN,rof oligopolies such that #

(

FN,r

)

= N and for which the

573

equilibrium is unstable for all the compositions

ω

∈ FN,r. For

574

such families the fraction of the firms with perfect foresight does 575

not influence the stability of the steady state. 576

Proof. To be in the unconditionally unstable scenario in

577

principle we have three possibilities. We can have

< 0, 578

or equivalently k _{> k} . We could have

ω

2 ≤ 1/N, which, 579

however, fromLemma 6.3, is not possible if r< 1 + 1/

(

N− 580

2

)

. Finally, we can have

ω

1≥ 1 − 1/N which, from (6.9)of 581

Lemma 6.2, happens for k4< k < min {k3, k } and r< 1 + 582

1/

(

N− 2

)

.  583

Proposition 3.6 (Stabilizing scenario). For any N≥ 5 and if 584

k is neither too small nor too large, there exist familiesFN,rof

585

oligopolies such that #

(

_FN,r

)

= N and for which the equilibrium

586

is stable only if the number of rational firms is sufficiently large 587

(namely

ω

> ¯

ω

, for suitable ¯

ω

). For such families an increase in 588

the fraction of rational firms leads to a local stabilization of the 589

steady state. 590

Proof. To have the stabilizing scenario we need 1/N <

ω

1< 591

1_{− 1/N <}

ω

2, inequalities that are considered inLemma 6.1 592

in condition (6.2), in Lemma 6.2 in condition (6.8), in 593

Lemma 6.4in condition(6.15). Recalling(6.11), we have that 594

the desired chain of inequalities holds true for k2< k < k4, 595

provided that r< 1 + 1/

(

N− 2

)

.  596

Theoretically, we may have two other possible frame-597

works: a destabilizing scenario, in which

ω

1< 1/N <

ω

2< 598

1_{− 1/N, and a mixed one, in which 1/N <}

ω

1<

ω

2< 1 − 599

1/N. In both cases, suitably increasing the number of rational 600

firms would lead to instability. However, through arguments 601

similar to those used in the previous propositions, it can be 602

proved that, for any N≥ 5, if we impose

ω

2< 1 − 1/N, it is not 603

possible to have #

(

F_N,r

)

> 3, as the equilibrium strategies q∗

R

604

of oligopolies with

ω

= 1/N, . . . , 1 − 4/N would be actually 605

null. This means that if we want to focus on dynamics involv-606

ing strictly positive strategies, we should limit our attention 607

to families consisting of subsets of FN,r=

{

1− 3/N, 1 − 2/

608

N_{, 1 − 1/N}

}

. We have the following result, whose proof is a 609

straightforward consequence ofLemmas 6.5and6.6. 610

Proposition 3.7 (Destabilizing scenario). For each N≥ 5, it 611

is possible to find suitable values of r_{> 1 and k > 0 so that} 612

FN,r=

{

1− 2/N, 1 − 1/N

}

and in which the Nash equilibrium is

613

stable for

ω

= 1 − 2/N and unstable for

ω

= 1 − 1/N. 614

•Case r_{< 1}

615

When the marginal cost of the rational firms is smaller 616

than that of the naive ones, the classical stabilizing and 617

unconditionally stable/unstable scenarios arise again. Since 618

those situations have already been analytically investigated 619

for r > 1 and are quite predictable, we avoid providing 620

detailed evidence of their occurrence, which can be ob- 621 tained by arguments similar to those used in the previous 622 framework. We will only report some simulative results in 623

Section 4. More interesting is to investigate the possibil- 624 ity to have a destabilizing or mixed scenario. The following 625 proposition excludes such occurrence. First, we notice that to 626 have a significant destabilizing scenario we necessarily need 627 a family of at least two oligopolies (FN,r=

{

1/N, 2/N

}

) and, 628 therefore, recalling the considerations about the equilibrium 629 positivity beforeSubsection 2.1and(2.6), we need r> 1/2. 630 For the mixed scenario, we would actually need a family of 631 at least three oligopolies (FN,r=

{

1/N, 2/N, 3/N

}

), so that r> 632 1/2 is indeed necessary (even if not sufficient). 633

Proposition 3.8. Increasing

ω

has neither a destabilizing nor a 634

mixed effect. 635

Proof. To have that increasing

ω

leads to instability, we 636 would need either z(

ω

) to be convex and

ω

2< 1 − 1/N or 637 z(

ω

) to be concave and

ω

2> 1/N. Both cases are not possi- 638

ble due toLemmas 6.8and6.9. _ 639

•Discussion of the previous results 640

We make some considerations about the previous propo- 641 sitions. First, we notice that the scenarios described are valid 642 for any oligopoly with N≥ 5. We stress that in the uncon- 643 ditionally stable/unstable and stabilizing scenarios, both for 644 r_{< 1 and r > 1, we can always consider complete families} 645 of oligopolies, as the equilibrium strategy remains positive 646 when we vary

ω

between 1/N and 1− 1/N. Such scenarios 647 are the same that occur when r_{= 1, i.e., when the marginal} 648 costs of the rational and naive firms coincide, and that we 649 recovered in[22]using a linear demand function, even for 650

different marginal costs. 651

Indeed, when r_{= 1 we recall that we found a unique sta-} 652 bility threshold ¯

ω

=

(

Nk− 4

)

/

(

4N− 8

)

for

ω

. Hence, if ¯

ω

< 653 1/N, the Nash equilibrium is locally asymptotically stable for 654 all

ω

_{= 1/N, . . . , 1 − 1/N and thus we are in the uncondition-} 655 ally stable scenario; if 1/N < ¯

ω

< 1 − 1/N, the Nash equilib- 656 rium is unstable for

ω

below ¯

ω

and locally stable above it, so 657 that we are in the stabilizing scenario; finally, if ¯

ω

_{> 1 − 1/N,} 658 the Nash equilibrium is unstable for all

ω

and thus we are in 659 the unconditionally unstable scenario. No other frameworks 660

may occur. 661

Conversely, when r> 1 inProposition 3.7we prove the 662 existence of a scenario which is not possible with identi- 663 cal marginal costs or with a linear demand function. The 664 situation described by this proposition is rather interesting 665 and we want to stress further it through an example. Let us 666 choose for instance N= 10 and let us consider suitable cost 667 ratio r and k for which the conclusions ofProposition 3.7are 668 valid. Then we have that if we consider from 1 to 7 ratio- 669 nal firms, all the rational firms production levels would con- 670 verge to zero, so they would actually leave the market and 671 the resulting oligopoly would simplify into homogeneous 672 oligopolies of naive firms. Considering the oligopoly with 673