Large Deviations Theory of Increasing Returns

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Simone Franchini and Riccardo Balzan

Sapienza Università di Roma, Piazza A. Moro 1, 00185 Roma, Italy

An inﬂuential theory of increasing returns has been proposed by the economist W. B. Arthur in the

’80s to explain the lock-in phenomenon between two competing commercial products. In the most

simpliﬁed situation there are two competing products that gain customers according to a majority

mechanism: each new customer arrives and asks which product they bought to a certain odd number

of previous customers, and then buy the most shared product within this sample. It is known that one

of these two companies reaches monopoly almost surely in the limit of inﬁnite customers. Here we

consider a generalization [G. Dosi, Y. Ermoliev, Y. Kaniovsky, J. Math. Econom. 23, 1–19 (1994)]

where the new customer follows the indication of the sample with some probability, and buy the

other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane and

Sudderth, and includes several models of physical interest as special cases, like the Elephant Random

Walk, the Friedman’s urn and other generalized urn models. We provide a large deviation analysis of

this model at the sample-path level, and give a formula that allows to ﬁnd the most likely trajectories

followed by the market share variable. Interestingly, in the parameter range where the lock-in phase

is expected, we observe a whole region of convergence where the entropy cost is sub-linear. We

also ﬁnd a non-linear differential equation for the cumulant generating function of the market share

variable, that can be studied with a suitable perturbations theory.

arXiv:2210.12585v4 [math.PR] 19 Jun 2023

CONTENTS

I Main results 2

I. Introduction 3

II. Relation with HLS urns 4

III. Relation with other models 8

IV. Zero-cost trajectories 9

V. Trajectories of the DEK model 14

II Methods 15

VI. Large deviations 16

VII. Scaling of the entropy inside the sub-linear region 20

VIII. Cumulant generating function 23

IX. Scaling of the master equation 25

X. Perturbations theory for k=1 26

XI. Perturbations theory for k=3 29

Acknowledgments 31

References 31

Part I

Main results

I. INTRODUCTION

It is known that certain economic markets - especially the technological ones - show increasing returns

[1–3], a positive feedback phenomenon where if a company gains some initial advantage (even small) is

more likely to get even more in the future, eventually dominating the market share in the long run - this

phase is also called lock-in into a monopolistic state. To understand the origin of this effect, a simpliﬁed

market model has been introduced in the ’80s by the economist W. B. Arthur [4] in the framework of its

Increasing Returns theory (IRT) [2, 3]. Let consider two competing companies that launch a new kind

of product roughly at the same time (as practical example we could think about two smart phones in the

early 2000s). Suppose that these products are roughly equivalent, such that there is no practical reason for

choosing one over the other, we can imagine that a buyer will base his decision in part on personal opinions

(personal tastes, ideologies, advertising, etc.) and in part on those of other people that already purchased

one of the products. Then, let us consider a simpliﬁed situation in which the new customers are imperfectly

informed about the products, so that they will make their choices by looking at the number of adopters who

are already using it [3–5, 7]. An alternative hypothesis that gives the same effect is to consider positive (or

negative) externalities in adoption [5, 7]. In both cases, we consider the additional rule that any new adopter

will choose the technology used by the majority of the sample only with a certain probability, and the other

technology otherwise [5, 7].

This scenario has been considered by G. Dosi, Y. Ermoliev and Y. Kaniovsky (DEK, 1994) [5], the pro-

posed model is as follows: consider a binary vector that represents the individual choices of the customers,

XN:={XN,1,XN,2, ...,XN,N},(1)

with XN,n∈ {0,1}and Npotential size of the market. This vector represents the full history of the market

evolution, from the ﬁrst sell to full saturation, when the maximal number of customers is reached. The

variable XN,nrepresents the choice of the n−th customer, we arbitrarily associate the value one to the ﬁrst

product and zero to the second. The total number of customers of the ﬁrst product will therefore be

ΓN,n:=∑

m≤n

XN,m(2)

the market share of the ﬁrst product up to the n−th customer is represented by the variable

xN,n:=ΓN,n

n=1

n∑

m≤n

XN,m.(3)

Then, the choice of the next customer XN,n+1is determined by the following rule: ﬁrst, sample kprevious

customers, where kis an odd integer (this to avoid inconclusive outputs from the poll). Then, if the sample

is found to have more customers that bought the ﬁrst product, the variable XN,n+1will be equal to one

with a probability p, and will be zero otherwise. On the other hand, if more customers owning the second

product are found in the sample, XN,n+1will be zero with probability p, and one otherwise. Notice that the

new customers follow the majority of the polled sample with probability p, that in some sense quantiﬁes

the trust of the newcomers in the behavior of their predecessors: hereafter we will call p trust parameter,

although it may also reﬂect more practical constraints, such as a requirement for compatibility with the

technology adopted by the polled customers. For p=1 the DEK model describes a market where the

customers always buy the product owned by the majority of the sample, i.e., the original version introduced

by Arthur et al. (1983) and Arthur (1989) [3, 4]: some sample trajectories of this process for p=1 and k=3

are in Figure 2a of G. Dosi et al. (2017) [7]. Concerning the initial conditions, we will distinguish of two

kinds: we introduce τ∈[0,1]the fraction of customers that made their choices already (market saturation

parameter): in this paper we will consider an early start in the market at some ﬁxed number of customers

M<∞, also called virgin market condition, that in the limit of inﬁnite customers is equivalent to a debut in

the market approximately at τ=0 (and does not affect the LDT theory for N→∞) and a late start M =τN

(a product that enters in the market when the saturation is already macroscopic), that strongly inﬂuences the

distribution of the ﬁnal share also at the LDT level.

II. RELATION WITH HLS URNS

In this paper we develop a Large Deviations theory (LDT) for the DEK model for any pand kby

adapting results from the Hill Lane and Sudderth (HLS) urn model [15, 16], a very general model for which

a mathematically rigorous LDT has been recently developed [18], and that includes the DEK model as

special case. An HLS urn process [10, 11, 15–18] is a two color urn process controlled by a functional

parameter π(x)that we call urn function (actually adoption function in Ref. [3]), where the new step XN,n+1

is one with probability π(xN,n)and zero otherwise. The relation between IRT and HLS urns is well known

since the very beginning, in fact, this model has been introduced independently by HLS (1980) and then

also by Arthur et al. (1983) within just three years. The urn function that describes the DEK model can be

determined as follows: start with k=1, the probability of extracting an owner of the ﬁrst product is x, then

their total number will increase with probability

π1(x):=px + (1−p) (1−x) = (1−p)+(2p−1)x,(4)

that is a linear urn function. In case k=3: the probability of increasing the owners of the ﬁrst product is

that of extracting two positive and one negative, plus that of extracting three positive, that is [5]

3(x):=x3+3x2(1−x) = 3x2−2x3,(5)

then, the corresponding urn function is [5]

π3(x):=p3x2−2x3+ (1−p)1−3x2−2x3= (1−p)+(2p−1)3x2−2x3(6)

and cannot be reduced to the linear case k=1. In general, the probability of ﬁnding a positive majority

when extracting an odd number kof steps is [5]

k(x):=∑

h>k/2

h!(k−h)!xh(1−x)k−h(7)

where the hsum runs from (k+1)/2 to k. Follows that the urn function that describes a DEK model with

k>2 extractions per step is [5]

πk(x):=pP

k(x)+(1−p)(1−P

k(x)) = (1−p)+(2p−1)P

k(x),(8)

this is a k−th degree polynomial, and is therefore non-linear for all non-trivial values of the trust parameter

In case of a virgin market start, the convergence properties of the HLS urns with any continuous urn

functions have been studied in [3–5, 15–18], ﬁnding that the points of convergence of xN,Nalways belong

to the set of solutions of

π(x) = x,(9)

and that these solutions are stable only if the derivative of the urn function in those points is smaller than

one, i.e., if the π(x)crosses xfrom top to bottom (down-crossing). For the DEK model with k=1, the

urn function π1(x)crosses xat 1/2 for any value of p<1, and therefore 1/2 is the only possible point of

convergence for the associated share xN,N, see Figure 1. This imply that xN,Nconverges to 1/2 almost surely

lim

N→∞xN,N=1/2,a.s.(10)

for all values of p<1 and of the initial condition xN,Ma phase diagram for the DEK k=1 is shown in

Figure 3. This model does not show the lock-in phenomenon, although there is still a value of pwhere the

dynamics is expected to slow down (see Section VIII).

In the DEK with k>2 one can see the appearance of the lock-in phase above some critical pc. For k=3

the Eq. (9) is a third degree equation, and can be solved with the well known formula. In general, we ﬁnd

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LargeDeviationsTheoryofIncreasingReturnsSimoneFranchiniandRiccardoBalzanSapienzaUniversitàdiRoma,PiazzaA.Moro1,00185Roma,ItalyAninfluentialtheoryofincreasingreturnshasbeenproposedbytheeconomistW.B.Arthurinthe’80stoexplainthelock-inphenomenonbetweentwocompetingcommercialproducts.Inthemostsimplifiedsi...

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