International Journal of Industrial Organization
16 (1998) 415–444
Adoption of network technologies in oligopolies
Paul Belleflamme*
CITA, University of Namur, rue Grandgagnage 21, B-5000 Namur, Belgium
Received 30 October 1997; accepted 4 December 1997
Abstract
We analyze a two-stage non-cooperative game where the firms choose first to adopt
(either simultaneously or sequentially) one of two network technologies, and then compete
on the market. The two-stage procedure and the assumption that firms have heterogeneous
tastes with respect to the technologies lead to a novel treatment of network externalities. In
particular, as the network of some firm enlarges, the change in this firm’s payoff is shown
to depend both on the newcomer’s identity and on the composition of the networks and, as a
result, is not necessarily positive.  1998 Elsevier Science B.V.
Keywords: Network externalities; Standardization; Oligopoly
JEL classification: C72; D43; D62; L13
1. Introduction
Firms contemplating technology adoption decisions can face two generic
situations in which coordination might be valuable. One involves the decision to
join some communication network; because the value to join such a network
increases with the number of firms doing the same choice, this situation is said to
exhibit positive-feedback effects. This communication paradigm applies for all
communication carriers, including both abstract (e.g., computer languages, Electronic Data Interchange message standards,...) and concrete (e.g., television
* Tel.: 132 81 724961; fax: 132 81 724967; e-mail: [email protected]
0167-7187 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved.
PII S0167-7187( 98 )00003-4
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
transmission signals, computer operating systems,...) communication standards (see
¨ 1994).
Auriol and Benaım,
Another situation involves the choice of some durable good (hardware) that
requires the consumption of complementary goods (software); in such a case, the
utility that a firm derives from the acquisition of the durable good increases with
the availability of complementary goods, which may itself depend (in case of
economies of scale in production) upon the size of the installed base of users of the
durable good. Similar positive-feedback effects are thus observed again. This
hardware /software paradigm applies to many markets: computer hardware and
software, credit-card networks, durable equipment and repair services, video
games,... (see Katz and Shapiro, 1985).
Over the last decade, a growing literature has investigated the coordination
problems raised by such situations, referring to the positive-feedback effects they
exhibit as (respectively, direct and indirect) network externalities. This literature
has stressed how network externalities raise technical problems (equilibrium may
not exist, or multiple equilibria may exist) and affect market performance (the
fundamental theorems of welfare economics may not apply); recent surveys of this
literature are found in Katz and Shapiro (1994); Besen and Farrell (1994);
´
Economides (1996); Matutes and Regibeau
(1996).
In this literature, network externalities are usually assumed to be monotonically
positive (i.e., the payoff of some agent i (weakly) increases when a different agent
joins the group of agents choosing the same strategy as i, whatever the size of this
group) and anonymous (i.e., only the number of agents choosing the same strategy
matters, not their identity).
Some recent contributions however depart from this common approach. In
contrast with the latter assumption, some papers consider local (rather than
anonymous) interaction models where individuals have a higher incentive to
coordinate their choice of strategy with people in their ‘neighborhood’ than with
people ‘far away’ from them (see, for example, David and Foray, 1992; Ellison,
1993; Goyal and Janssen, 1993; Oechssler, 1994; Dalle, 1995; Goyal, 1995).
Accordingly, network externalities are still assumed to be monotonically positive
but only within communities of individuals; the degree of interaction between
communities therefore becomes a critical factor for assessing the overall diffusion
of some particular standard (or technology, or social norm).
Regarding the former assumption, three recent papers have questioned it by
introducing strategic concerns which possibly undermine the value attached by
individuals to coordination. First, Matutes and Padilla (1994) study banking
competition in the presence of Automatic Teller Machine (ATM) networks. In this
environment, banks face the following dilemma: on the one hand, they have an
incentive to share their ATM networks because depositors are willing to accept
lower interest rates on their deposits in order to have access to a larger network;
but, on the other hand, compatible banks become better substitutes for each other
and suffer from the resulting price rivalry in the market for deposits.
Secondly, Economides and Flyer (1995) examine the incentives of firms, which
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
417
produce goods exhibiting network externalities, to form coalitions based on
adherence to common technical standards. Because the value of the goods
increases with the size of sales, firms have incentives to be in large coalitions of
compatible goods that share the same technical standards. However, this incentive
contrasts with the traditional incentive to differentiate each product and be a
dominant player in a particular market ‘niche’. The authors analyze the interaction
of these opposite incentives in the creation of technical standards coalitions.
Despite no inherent differences in the features of the products and no cost
differences, they find that often at equilibrium the market is highly concentrated
with coalitions of widely varying sizes charging very different prices.
Finally, Bloch (1995) analyses an oligopoly with linear demand in which, in a
first stage, firms form associations in order to decrease their costs, and in a second
stage, compete on the market. In such a setting, the enlargement of an association
induces two simultaneous contrasting effects on its members’ profits: on the one
hand, the members benefit from the reduction in their own marginal cost, but on
the other hand, they suffer from increased competition due to the reduction in their
competitors’ marginal costs. As a result, there is a critical size after which the
admission of new members in an association has an adverse effect on the initial
members’ profit (implying that network externalities become negative once a
critical network size is reached).
The present analysis shares this intuition. I consider the following model: in
order to produce differentiated final goods, firms choose, in a first stage, between
two available technologies whose marginal cost linearly decreases with the number
of firms adopting them; then (as in Bloch’s analysis), payoffs are determined by
computing the Nash equilibrium of a Cournot game in which the firms’ cost
structures result from previous technology choices. My model however departs
from Bloch’s work in a couple of important ways. First of all, the first stages of
the two games differ. Bloch models the formation of associations as a noncooperative sequential game in the spirit of Rubinstein (1982)’s alternating-offers
bargaining game. ‘‘One of the firms, the initiator, proposes a cooperative
agreement. All the prospective members of the association respond in turn to the
offer. If all firms accept the offer, the cooperative agreement takes effect (...). If
one of the firms rejects the offer, the proposed association is not formed and the
firm that rejected the offer becomes the initiator in the next round.’’ It must be
stressed that in this procedure, firms can exclude other firms from the association
they form. This explains why in equilibrium, firms form two asymmetric
associations, with the dominant association comprising roughly three-quarters of
industry members. In contrast, in my setting, networks result from simultaneous
adoption by all firms of one of the available technologies; the stability notion is
that of Nash equilibrium, which can be interpreted as free mobility: each firm is
free to choose the network it prefers. As a result, when (as assumed by Bloch) the
firms have the same preferences, the second-stage competition does not prevent
the firms from adopting all the same technology at the equilibrium of the game.
Secondly, while Bloch focuses on the symmetric case, I introduce diverging
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
tastes for compatibility among firms. This heterogeneity entails a larger set of
possible equilibria: depending on the firms’ preferences for the two technologies,
the equilibrium size of the networks can range from zero to all firms, with the
restriction (see Proposition 2) that all firms joining a particular network at the
equilibrium have a higher preference for this network than all firms joining the
other network. With heterogeneous firms, the two networks might thus coexist at
the equilibrium (and this is all the more likely that the second-stage competition is
fierce). It is important to note that (as in Bloch’s analysis) such situations lead to a
loss in social welfare: I show indeed that the optimal partition of the firms involves
overall adoption of the same technology (one or the other according to the industry
mean preference; see Proposition 4).
To sum up, the present analysis has two special features which lead to a novel
treatment of network externalities: one the one hand, there are conflicting effects
on a firm when joining a network and, on the other hand, firms are heterogeneous
in the way they benefit from the network effects. As a consequence, one observes
that as an additional firm joins the network to which firm i belongs, the change in
firm i’s payoff (i) depends both on the newcomer’ s identity and the composition of
the networks, and (ii) is not necessarily positive. In other words, the usual
properties of positive externalities and of anonymity do not apply to the payoffs
accruing from the use of the network technologies. Yet, interestingly, these two
properties apply to the firms’ incentive to join a network: this incentive increases
with the number of firms present in the network to be joined (see Lemma 2).
In this respect, it is instructive to relate my results to the seminal contributions
on the adoption of technologies exhibiting network externalities. The model in
Farrell and Saloner (1985) is closest to mine but differs in two respects: besides
the fact that the payoffs are assumed to satisfy the properties of positive
externalities and of anonymity, it is further assumed that the firms decide
sequentially which technology to adopt. In Section 5, I also assume sequential
choices and I examine thereby whether Farrell and Saloner’s results (regarding the
issue of excess inertia) are robust to the introduction of a competition effect
between the firms. Interestingly, it turns out that they are.1
1
Three other important papers considering the adoption of network technologies are Katz and
Shapiro (1986); Farrell and Saloner (1986); Katz and Shapiro (1992). As in Farrell and Saloner (1985),
the payoffs are assumed in these three papers to satisfy the properties of positive externalities and of
anonymity; moreover, consumers are assumed to be homogeneous. These assumptions make the basic
framework more tractable and allow to address issues that cannot be easily tackled here. In particular,
these papers explicitly consider the timing aspect: because at a given time not all technologies are
available and / or not all users have the same choice opportunities, there are delays in building networks
and inefficiencies might result. The two papers of Katz and Shapiro also address two issues that are
abstracted away in the present setting: the pricing and compatibility choices by sponsoring firms. These
issues are further analyzed in Farrell and Saloner (1992) where, in contrast with the previous papers
(but in accordance with what is assumed here), consumers have heterogeneous preferences for the
technologies. The introduction of a competition effect between consumers is likely to affect the results
obtained in these four papers; fully analysing these issues is however beyond the scope of the paper.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
419
The remainder of the article is organized as follows. In Section 2 I give a
description of the network technologies. Section 3 solves the second stage of the
game. Section 4 is devoted to the solution of the first stage of the game,
considering that technology choices are made simultaneously; this section also
compares the private and social points of view. Section 5 considers instead that
technology choices are made sequentially; this allows to address the issue of
excess inertia (with either complete or incomplete information), in the spirit of
Farrell and Saloner (1985). Section 6 concludes.
2. The model
The adoption of network technologies is modelled as a two-stage game. In the
first stage, a set of n firms choose simultaneously between two available network
technologies. This generates a partition of the set of firms into two networks, the
size of which contributing to determine the firms’ marginal costs. Then, in the
second stage, the firms produce differentiated goods and compete on the market in
a Cournot fashion.
2.1. Description of the network technologies
Suppose that two technologies are available (call them X and Y). These
technologies exhibit network externalities: the more they are adopted the more
benefits they offer to their users. The presence of network externalities is modelled
in the following way. Let N 5 h1,2,...,nj denote the set of firms in the industry.
First-stage decisions generate a partition of the set of firms into two ‘networks’:
N 5 X < Y (with X > Y 5 [) where X [respectively Y] is the network of
technology X- [respectively Y-] users; the cardinality of these two subsets is given
respectively by x and y 5 n 2 x. Each technology is characterized by a marginal
cost that linearly decreases with the number of firms adopting this technology.
That is, when N 5 X < Y, marginal costs are given by:
c xi 5 c 2 x(1 2 ui ), ;i [ X
y
c j 5 c 2 yuj 5 c 2 (n 2 x)uj , ; j [ Y;
(1)
(2)
(where uk [[0,1] ;k[N, and c.n in order to guarantee positive marginal costs).
It is assumed that the firms have different preferences for the two technologies.
These preferences are represented by a value of u between 0 and 1 and are
common knowledge; when the networks have the same size, a firm k with uk ,
[respectively .] 0.5 prefers technology X [respectively Y]. Without any loss of
generality, let us rank the firms by increasing values of u and assume that
0<u1 <u2 < ? ? ? <un <1. Firm 1 is thus the strongest ‘X-lover’ and firm n the
strongest ‘Y-lover’.
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This formulation assumes that the firms are heterogeneous regarding the benefits
they derive from the network effects generated by the technologies.2 This
assumption could be justified as follows. Think of X and Y as, for example, two
computing platforms.3 These platforms exhibit positive-feedback effects through
learning-by-using mechanisms: the more a platform is adopted, the more important
is the flow of information from the customers to the producer and, as a result, the
better is the design of the platform.4 Yet, as pointed out by Bresnahan and
Greenstein (1996), ‘‘[p]urely technical progress is rarely sufficient to make an
invention economically important. Users, through their own experimentation and
discovery, make technology more valuable.’’ The authors call this activity coinvention. Specifically, in the present example, the co-invention activity is the
whole process of integrating computers into the existing business systems; this
process calls for ‘‘developing new organizational forms, new job definitions for
computer-using workers, even new relationships between the [firm] and its
suppliers and customers.’’ Because firms have different abilities to achieve this
process, they value differently the technical spillovers induced by increased
adoption of a particular platform.
The two-stage game will now be solved for its subgame-perfect Nash equilibria
(SPNE) in pure-strategies by the method of backward induction. A SPNE
specifies, for each firm, a choice of technology followed by a choice of quantity. I
start by solving the last stage of the game: market competition.
3. Second-stage equilibrium
Once firms have adopted one or the other technology, they compete on the
market with a marginal cost that depends on the size of the network to which they
belong. This section derives the firms’ payoffs, for an arbitrary partition of the set
of firms, assuming that the firms compete by setting quantities.5 In order to
facilitate the exposition, the results are first derived for constant marginal costs;
they are then extended to take account of the cost structures described in the
preceding section.
2
An alternative formulation would be to introduce heterogeneity not as a multiplicating factor of the
network effect, but as an additive factor of the constant term. An appendix available from the author
shows that this assumption lead to qualitatively similar results than the ones derived here.
3
As defined by Bresnahan and Greenstein (1996), ‘‘[a] computing platform is a reconfigurable base
of compatible components on which users build applications. Platforms are most readily identified with
their technical standards, that is, engineering specifications for compatible hardware and software’’.
4
This is all the more true when the product is in its early stage of development; see Foray (1995).
5
A similar analysis can be carried out by assuming instead that the firms compete by setting prices;
qualitatively identical results are obtained.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
421
3.1. Traditional Cournot oligopoly with n asymmetric firms
Suppose there are n Cournot competitors (indexed by i51...n) with individual
constant marginal cost equal to c i . Each firm i produces a differentiated product,
qi , sold at price pi . The demand functions for the firms’ products are assumed to be
generated by a representative consumer with a quadratic surplus function:
O q 2 ]12 SO q 1 d O Oq q D 2O p q
n
U(q1 ,q2 ,...,qn ) 5 a
n
n
2
i
i
i 51
i 51
n
i
i 51 j ±i
j
i
i
i 51
with a.0, and 0<d <1.
The parameter d indicates the strength of product differentiation: as d approaches 1, the products become closer substitutes; as d approaches zero, the
products become more differentiated, and in the limit (d50), the demands are
independent. Maximizing consumer’s surplus yields the linear inverse demand
schedule pi 5a2qi 2do j ±i q j in the region of prices where quantities are positive.
The demand schedule, for d ±1, is then given by qi 5 a 2 b pi 1d o j ±i pj , with
a
1 1 d(n 2 2)
d
a 5 ]]]]; b 5 ]]]]]]; d 5 ]]]]]].
1 1 d(n 2 1)
(1 2 d)[1 1 d(n 2 1)]
(1 2 d)[1 1 d(n 2 1)]
Firm i seeks to maximize its profit by choosing quantity qi , taking as given the
quantity produced by its competitors (denoted by Q 2i ). Solving for the first-order
condition, one obtains firm i’s reaction curve:
1
qi (Q 2i ) 5 ] a 2 c i 2 d
2
S
O
j ±i
D
qj .
(3)
Summing up the n reaction curves, and using the fact that o ni 51 Q 2i 5(n21)
o
qi ;(n21)Q, gives the total quantity produced:
n
i 51
O
n
na 2 i 51 c i
Q 5 ]]]].
2 1 d(n 2 1)
(4)
Substituting Eq. (4) into Eq. (3) gives the equilibrium quantity for firm i:
O
(2 2 d)a 2 [2 1 d(n 2 2)]c i 1 d
j ±i c j
q *i 5 ]]]]]]]]]]].
(2 2 d)[2 1 d(n 2 1)]
(5)
Simple substitutions show that the equilibrium profit for firm i is equal to the
square of the quantity produced. One observes that firm i’s equilibrium quantity
(and thus profit) will increase if its own marginal cost decreases and / or if the
marginal cost of some of its competitors increases. The marginal rate of
substitution between these two effects can be expressed by the following ratio:
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
422
S
2 ≠q i*
r(d) 5 ]]
≠c j
DYS D
≠q *i
d
]]
5 ]]]].
≠c i
2 1 d(n 2 2)
(6)
One observes that r(d) is an increasing function of d (it goes from zero for d50 to
n 21 for d51); this means that as the products become less differentiated (i.e., as d
approaches 1), the negative effect on a firm’s quantity (or profit) stemming from
an improvement of the competitors’ situation becomes relatively more important
than the positive effect stemming from an improvement of the firm’s own
situation. This ratio will turn up frequently in the analysis to follow.6
3.2. Extension to network technologies
In what follows, I will define several variables that will depend on the partition
of the set of firms resulting from the firms’ first-stage decisions. To shorten the
notation, these variables will be parameterized solely by the subset of firms
adopting technology X (the so-called ‘X-network’). One such variable (that proves
very useful for expository purposes) is E(X) which denotes the total network
effects enjoyed by the firms (i.e., the sum of the reductions in marginal costs due
to network externalities) for a given partition of the set of firms; that is,
E(X) ; x
O (1 2 u ) 1 (n 2 x) O
i
i [X
uj
(7)
j [N •X
The following lemma establishes that the total network effects reach a maximum
only when all firms adopt the same technology (i.e., in cases of standardization).
Lemma 1. The total network effects are maximized when all firms adopt
technology X (respectively Y) if and only if u¯ ,(respectively .)1 / 2 (where u¯ is
the mean of u ’s over N).
Proof. See Appendix A.
6
Under Bertrand competition, the ratio r(d) must be replaced by a similarly interpreted ratio, r9(d),
whose value (for d ±1) is given by:
d[1 1 d(n 2 2)]
r9(d) 5 ]]]]]]]
.
2 1 3d(n 2 2) 1 d 2 (n 2 2 5n 1 5)
Straightforward computations show that r9(d).r(d), ;d [[0,1[. A firm’s profit is thus relatively more
affected by a variation of the competitors’ position in a Bertrand setting than in a Cournot setting (this
is a standard result in the analysis of differentiated products industries; see, for example, Vives, 1985).
As a consequence, it is harder for Bertrand than for Cournot competitors to coordinate their technology
decisions and to reach standardization in the first stage of the game.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
423
Solving for the last stage of the game amounts to finding the equilibrium
Cournot profits for any partition of the set of firms that results from the first-stage
adoption decisions. This is easily done by substituting Eqs. (1), (2), (7) into Eq.
(5) derived in the general case. Simple computations lead to the following
equilibrium quantities (where b denotes the difference a2c, between the intercepts
of the inverse demand and average cost curves; it is a measure of the size of the
market):
(2 2 d)b 1 [2 1 d(n 2 1)]x(1 2 ui ) 2 dE(X)
x
q i (X) 5 ]]]]]]]]]]]], ;i [ X;
(2 2 d)[2 1 d(n 2 1)]
(8)
(2 2 d)b 1 [2 1 d(n 2 1)](n 2 x)uj 2 dE(X)
q yj (X) 5 ]]]]]]]]]]]], ; j [ N•X.
(2 2 d)[2 1 d(n 2 1)]
(9)
As before, equilibrium profits are simply equal to the square of the quantities
produced at equilibrium; these quantities therefore deserve a closer look. Considering Eqs. (8) and (9) and the definition of the total network effects E(X), one
observes that each firm’s quantity (and thus profit) is an increasing function of the
network effect the firm itself enjoys and a decreasing function of the network
effects enjoyed by its competitors. The balance between these two opposite effects
is given by the ratio r(d) defined above.
n12
Let v 5(u1 , u2 ,...,un , b, d)[R 1
denote a vector of parameters for an neconomy. In what follows, I will only consider vectors of parameters such that the
quantities produced by all firms at Cournot equilibrium are non negative, whatever
the partition of the set of firms. A sufficient condition is b>(d / 22d) maxhE(N),
E([)j (since network effects reach a maximum in either case of standardization). I
will therefore make the following assumption (denoted 326 as a mnemonics for
‘positive quantities’): b>(d / 22d)n 2 maxhu¯ , 12 u¯ j (where u¯ is the mean of u ’s
over N).The propositions that follow are thus established for a restricted domain of
parameters: V 5hv uv satisfies 326 j.7
This domain restriction allows to compare equilibrium quantities instead of
equilibrium profits in the first stage of the game. Going even one step further, I
normalize these quantities by multiplying them by (22d)[21d(n21)], by
subtracting the common parameter (22d)b, and by dividing again by 21d(n22).
The following expressions give the payoffs that will be used for solving stage 1; in
these expressions, the first term is the network effect enjoyed by the firm itself
while the second corresponds to the sum of the network effects enjoyed by the
firm’s competitors, multiplied by the ratio r(d).
7
Note that 326 becomes less stringent as d decreases. In Appendix A, I briefly discuss how the
analysis is affected when this assumption is relaxed.
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
FO
FO
;i [ X, B xi (X) 5 x(1 2 ui ) 2 r(d) x
(1 2 uk ) 1 (n 2 x)
k [X 2i
; j [ Y, B yi (X) 5 (n 2 x)uj 2 r(d) x
(1 2 uk ) 1 (n 2 x)
k [X
O u G;
l
(10)
l [N •X
O
G
ul ;
l [(N •X ) 2j
(11)
As outlined in Section 1, these payoffs do not necessarily exhibit positive
network externalities; it is indeed easy to find values of the parameters such that a
firm’s payoff strictly decreases as an additional firm joins the network to which it
belongs. The reason is the following. Take, for example, some firm i [X, some
firm j [N\X and examine how firm i’s payoff changes if firm j switches from
technology Y to X. There are four effects to consider. First, there is a positive
direct effect: firm i’s own marginal cost decreases because the X-network enlarges
[Dc i 5 2(12ui )]. Moreover, there is a positive indirect effect: by leaving the
Y-network, firm j makes the network effect enjoyed by its former ‘network mates’
go down [Dc k 5uk , ;k±j [N\X], which improves firm i’s payoff. But on the other
hand, this move also creates a negative indirect effect: by joining the X-network,
firm j makes the network effect enjoyed by firm i’s network mates go up
[Dcl 5 2(12ul ), ;l ±i [X], which reduces firm i’s payoff. Finally, the change in
firm j’s marginal cost [Dc j 5(n2x)uj 2(x11)(12uj )] entails a last indirect effect:
according to the size of the networks and to firm j’s preference for the
technologies, its cost might decrease, which will harm further firm i’s position.
Although the negative effects have a relatively lower weight [given by r(d)], they
might offset the two positive effects, yielding an overall reduction of firm i’s
payoff.
We can now turn to the resolution of the first stage of the game.
4. First-stage equilibrium
The preceding section has shown how to derive the firms’ payoffs for all
possible combinations of technology choices. The individual payoffs have now to
be compared in order to derive the equilibria in terms of technology adoption,
which will be referred to as the equilibrium partitions of the set of firms between
the two networks. Private and social points of view will also be confronted and a
simple example will illustrate the analysis.
4.1. Equilibrium partitions of the set of firms
A partition of the set of firms is part of a SPNE of the game if and only if no
firm can strictly improve on its payoff by unilaterally changing its technology
choice. The whole analysis can thus be phrased in terms of incentives for firms to
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
425
switch from one network to the other. Remarkably, as will now be shown, these
incentives contrast with the payoffs just derived in that they are independent of the
identity of the firms belonging to either network: only the size of the network
matters when a firm considers switching (the concern for other firms’ preferences
is summed up by the influence of the industry mean preference for the
technologies).
Consider a partition where the two networks coexist, that is, N5X <Y, with X,
Y ±[. Let us compute the incentives for firms in X to leave the X-network and for
firms in Y to join it. Because only the size of the X-network matters, let me adopt
the following notation; let Li (x) and Jj (x) denote, respectively, the incentive for
some X-adopter i to leave and for some Y-adopter j to join an X-network of size x.
Building from Eqs. (10) and (11), one obtains the following expressions:
Li (x) 5 B yi (X•hij) 2 B xi (X)
5 [n 1 1 1 r(d)]ui 2 x[1 2 r(d)] 2 r(d)[1 1 nu¯ ];
(12)
Jj (x) 5 B jx (X < h jj) 2 B jy (X)
5 [n 1 1 1 r(d)](1 2 uj ) 2 (n 2 x)[1 2 r(d)] 2 r(d)[1 1 n(1 2 u¯ )].
(13)
Simple comparative statics on Eqs. (12) and (13) establish the following lemma.
Lemma 2. A firm’ s incentive to leave an X-network depends (i) positively on the
firm’ s own preference for technology Y (ui ), (ii) negatively on the size of the
X-network, and (iii) negatively on the industry’ s mean preference for technology Y
( u¯ ). Conversely, a firm’ s incentive to join an X-network depends (i) positively on
the firm’ s own preference for technology X (12ui ), (ii) negatively on the size of
the Y-network, and (iii) negatively on the industry’ s mean preference for
technology X (12 u¯ ).
Before proceeding further, it is interesting to note (through easy transformations
of the expressions above) that Li (x)5 2 Ji (x21): a firm’s incentive to leave an
X-network of size x has the opposite value of its incentive to join an X-network of
size (x21).8
It is now easy to see that the partition (X, Y) of the set of firms is part of a SPNE
of the game if and only if the following two sets of conditions are met:
;i [ X, Li (x) < 0⇔[n 1 1 1 r(d)]ui < x[1 2 r(d)] 1 r(d)[1 1 nu¯ ];
(14)
8
This result is not surprising: saying that firm i is eager to leave an X-network of size x (Li (x).0), is
equivalent to saying that firm i will not make the switch back (i.e., that it has no incentive to join an
X-network of size (x21), or Ji (x21),0).
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
; j [ Y, Jj (x) < 0⇔Lj (x 1 1) > 0⇔[n 1 1 1 r(d)]uj
> (x 1 1)[1 2 r(d)] 1 r(d)[1 1 nu¯ ].
(15)
Building on Eqs. (14) and (15), we can establish the existence of an equilibrium
partition (see Proposition 1), and even characterize the set of possible equilibrium
partitions (see Proposition 2) in the family of games obtained by letting v roam
over V.
Proposition 1. For each v [ V, there exists at least one equilibrium partition of
the set of firms.
Proof. Consider first the partition Xn 5N where each firm adopts technology X. If
it is a Nash equilibrium, we are done. Otherwise, there must be some firm(s) with a
positive incentive to leave Xn . According to Lemma 2, firm n is the most likely to
meet this condition; that is, we necessarily have that Ln (n).0. Consider then the
partition Xn 21 5N\hnj. If it is a Nash equilibrium, we are done. Otherwise, there
must be at least one firm wishing to switch from one technology to the other. Since
it cannot be firm n (Ln (n).0⇒ Jn (n21),0), there is at least one firm in the
X-network which is eager to leave it; applying Lemma 2 again, we know that firm
n21 certainly does: Ln 21 (n21).0. We can repeat this procedure as long as no
equilibrium partition is found. In the worst-case scenario, we will have that
X1 5h1j is not a Nash equilibrium, meaning that L1 (1).0. But then, since
L1 (1).0⇒ J1 (0),0⇒ Jk (0),0, ;k.1 (from Lemma 2), the situation where each
firm adopts technology Y is an equilibrium partition. Q.E.D.
Note that multiple equilibrium partitions are possible (see Section 4.3 for an
illustration in the two-firm case).9 The proof of Proposition 1 indicates that in an
equilibrium partition, the firms adopting technology X are the ones with the lowest
values of u. The next proposition confirms this fact, which allows to characterize
the set of partitions of N which can qualify as equilibria for the first stage of the
game.
Proposition 2. (i) If the equilibrium partition (X, Y) yields the coexistence of the
two networks, then all firms in X have a strictly lower u than all firms in Y5N\X.
(ii) For each v in V, the equilibrium partition has the following structure: the
X-network is either empty, or it consists of the firm(s) with the lowest value of u, or
it consists of the firms with the lowest and second lowest values of u, or..., or it
consists of all firms.
9
This multiplicity results from the simultaneity of the technology choices in the first stage; it
disappears when it is assumed instead that choices are made sequentially (see Section 5).
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
427
Proof. (i) Suppose that some firm i [X and some firm j [N\X satisfy Eqs. (14)
and (15); suppose further that ui >uj . From Lemma 2, the latter inequality and Eq.
(15) imply that Ji (x)<0⇔Li (x11)>0; but from Lemma 2 again, this implies in
turn that Li (x).0, which contradicts Eq. (14) and completes the proof. (ii) The
second statement is a direct consequence of the first one: since, in any equilibrium,
all firms adopting technology X must have a lower value of u than all firms
adopting technology Y, possible equilibrium partitions are the two cases of
standardization and as many situations of network coexistence than there are
different values of u minus one (that is, for example, if ui 5u ;i [N, then only the
two cases of standardization can emerge as a SPNE of the game). Q.E.D.
Let me now consider more closely the two polar cases of standardization (i.e.,
overall industry adoption of the same technology). Using the assumption that the
firms are indexed by increasing values of u together with Lemma 2, one derives
the following conditions from Eqs. (14) and (15).
– Overall adoption of technology X is an equilibrium partition iff
O (1 2 u ).
n21
Ln (n) < 0⇔(n 1 1)un < n 2 r(d)
k
(16)
k 51
– Overall adoption of technology Y is a an equilibrium partition iff
Ou .
n
J1 (0) < 0⇔(n 1 1)u1 > 1 1 r(d)
k
(17)
k52
Some simple comparative statics on these last two equations allows to state the
following proposition.
Proposition 3. Other things being equal, the more products are differentiated, the
more complete standardization is likely to be an equilibrium partition.
Proof. Recall that r(d) is an increasing function of d. Thus, as product
differentiation increases (i.e., as d decreases), r(d) decreases, meaning that the
RHS of Eq. (16) goes up and that the RHS of Eq. (17) goes down. Hence, in both
cases, the conditions become more likely to be satisfied. Q.E.D.
This result confirms the intuition: as product differentiation increases, competition becomes less fierce and firms become relatively less sensitive to the effect on
their own payoff of their competitors’ cost structures. In other words, the firms
become relatively more concerned with network sizes than with network compositions, which makes standardization more likely to emerge. Note that the
reverse proposition also holds: it is indeed possible to show that, for a given
combination of the parameters, a SPNE involving the coexistence of the two
networks is more likely to emerge when the products become closer substitutes.
The next subsection discusses the welfare properties of equilibrium partitions.
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
4.2. Welfare results
There are different ways to gauge the performance of the subgame-perfect
equilibria of the game. We can first take the viewpoint of a first-best regulator who
imposes particular technology choices and marginal cost pricing on the product
market. Alternatively (if we believe that any regulator would be unable in practice
to enforce the first-best marginal cost principle), we can adopt a second-best point
of view: the regulator assigns a technology to each firm and let the firms compete
on the product market.
Remarkably, both welfare criteria point in the same direction: as demonstrated
in Proposition 4, social welfare (whatever the criterion used to define it) cannot
reach a maximum outside standardization. Therefore, any partitioning of the set of
firms at the SPNE of the game can be described as socially inappropriate.
Proposition 4. Whether the regulator controls both the level of output and the
technology choice of each firm or only the latter, she should always prescribe the
unanimous adoption of the same technology by all firms; this technology is X
(respectively Y) if and only if u¯ ,(respectively .) 1 / 2 (there is indifference if
ū 51 / 2.)
Proof. See Appendix A.
The intuition behind this result is the following. It has been shown above that
standardization leads to the highest network effects, that is, to the lowest marginal
costs on aggregate; because the demand system is symmetric, all prices are thus
lower, which makes consumers better off, and quantities produced are larger,
which (in spite of the decrease in prices) increases total profits. Yet, as already
emphasized, firms do not care only about their own marginal cost; they are also
sensitive to the level of their competitors’ marginal costs. Loosely speaking, each
firm would wholeheartedly favor standardization if it could be the only one to take
advantage of it. Because this condition is not met, the networks can coexist at the
equilibrium, making private and social incentives diverge. Note that private and
social incentives may also diverge when the market standardizes on the wrong
technology.
The preceding results are now illustrated for the simple case where the industry
is composed of only two firms.
4.3. Illustration: The case with 2 firms
Resolution of the second-stage Cournot competition leads to the normalized
payoffs that are summarized in the matrix of Table 1 (where the first entry in each
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
429
Table 1
Payoffs for n52 and Cournot competition
X
Y
X
Y
422d24u1 12du2 , 422d 12du1 24u2
2u1 1du2 2d, 22du1 22u2 ,
222u1 2du2 , du1 12u2 2d
4u1 22du2 , 4u2 22du1
box is firm 1’s payoff, and the second is firm 2’s and where, for expositional
convenience, all payoffs are multiplied by 2). Solving for stage 1 amounts to
finding the Nash equilibria in matrix 1. This is done graphically in Fig. 1,
Fig. 1. Possible equilibrium partitions in the two-firm cases. Equilibrium partitions are: Area I: (X, X);
Area II: (X, X) and (X, Y); Area III: (Y, Y); and, Area IV: (X, Y). The solid [dotted] lines refer to the
case where d51 [d50].
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
representing the u2 2u1 plane (with u2 >u1 ).10 Two straight lines allow to
characterize all possible equilibrium partitions in this plane.
– Schedule A(d) is the indifference condition for firm 2 when firm 1 adopts
technology X: L2 (2)5 J2 (1)50⇔u2 2du1 542d; as d decreases, A(d) rotates to
the right around point (2 / 3,1); for pairs of parameters on the right of A(d), firm 2
prefers adopting technology X.
– Schedule B(d) is the indifference condition for firm 1 when firm 2 adopts
technology Y: J1 (0)5L1 (1)50⇔6u1 2du2 52; as d decreases, B(d) rotates
downward around point (0,1 / 3); for pairs of parameters above B(d), firm 1 prefers
adopting technology X.
These two lines delimit four areas: (i) in area I [left of A(d), below B(d)],
standardization on technology X is the unique equilibrium; (ii) in area II [left of
A(d), above B(d)], the two standardization results are equilibria; (iii) in area III
[right of A(d), above B(d)], standardization on technology Y is the unique
equilibrium; and (iv) in area IV [right of A(d), below B(d)], coexistence of the two
networks (with firm 1 adopting X) is the unique equilibrium.
The results of the first three propositions are easily identified on Fig. 1. First, it
is observed that an equilibrium partition exists for each pair of parameters, and that
multiple equilibria can arise, as stated in Proposition 1. Second, it is noticed that
the situation where firm 1 is the sole adopter of technology Y cannot be an
equilibrium when u2 >u1 , confirming part (i) of Proposition 2; furthermore, part
(ii) of this proposition is confirmed in that, on the 458-line (where u1 5u2 ), only
two equilibria are possible, whereas below this line (where u2 .u1 ), three equilibria
are possible. Finally, one observes that as d increases, the areas where standardization is an equilibrium shrink (since A(d) rotates to the left and B(d) rotates
upward), as stated in Proposition 3.11
Fig. 2 compares the private and social points of view for the case with two
firms and a homogeneous product (d51). From Proposition 4, we know that
standardization on technology X [respectively Y] maximizes total surplus on the
left [respectively on the right] of the line where u1 1u2 51. Hence, there are two
areas where private and social incentives diverge: in area I [respectively II], the
market leads to incompatibility whereas standardization on technology X [respectively Y] should take place. Noteworthy is the fact that everywhere else, private
choices of technology are efficient (supposing that in the area where two equilibria
coexist, the socially optimal one is selected).
10
For simplicity, I assume that b.4, which makes sure that all equilibrium quantities are positive
whatever the values of d, u1 and u2 (in accordance with assumption 326 ).
11
Conversely, as d increases, area IV (where incompatibility obtains) expands: for d50, area IV
amounts to 22% of the total area, and for d51, to 43%.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
431
Fig. 2. Private vs. social incentives. Standardization on X [Y] maximizes social welfare below [above]
the dotted line. Private and social incentives diverge in areas I and II.
5. Sequential technology choices
In the literature on network externalities, several contributions suggest that the
coordination problems entailed by the externalities might retard innovation:
situations of lock-ins, of persistence of inefficient norms, or of conformism are
envisioned.12 A seminal article considering this issue is Farrell and Saloner (1985).
The authors consider markets where network externalities make standardization
beneficial for firms; they examine whether these standardization benefits can ‘trap’
an industry in an obsolete or inferior standard when there is a better alternative
available (what they call symmetric excess inertia). Their answer is no for cases
with complete information but yes when information is incomplete.
The present analysis can easily be extended in order to resemble Farrell and
Saloner’s models. Consider that technology Y is an old technology that all firms
have adopted before the game starts and that they envision to replace by a new
12
See, for example, Arthur (1989); David (1985); Farrell (1990); Katz and Shapiro (1992).
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
one, technology X. The main difference with the previous analysis is that it is now
assumed that the firms act sequentially in deciding whether or not to switch to the
new technology. Farrell and Saloner consider in a first model that there is complete
information about the ‘eagerness’ of each firm to make the switch; in a second
model, they consider instead that each firm only imperfectly knows the other
firms’ preferences. I shall now apply Farrell and Saloner’s methodology to my
setting; the interesting point is to check whether their results still hold in situations
where positive network externalities may not always be observed. The general
answer is that they actually do, meaning that the assumption of positive network
externalities is a too strong requirement in this particular framework.
5.1. A model with complete information
It is now assumed that before the Cournot game takes place, there are n periods
to the game: in period j, firm j decides whether to switch to the new technology
(all firms are initially with the old technology); no switch back is possible. While
this prespecified order of moves may seem artificial, it can be shown that it exactly
leads to the same equilibria than if timing was endogenous, in the sense that the
firms the most attracted by the new technology go first.13 Note that it can also be
shown that every equilibrium partition in this sequential game is also an
equilibrium partition in the simultaneous-move game (the only difference is the
following: in regions of parameters where multiple equilibria obtain in the
simultaneous-move game, the sequential-move game selects a unique equilibrium).
Farrell and Saloner express conditions for standardization on the new technology
to be the unique SPNE of the sequential-move game. Because the result does not
rely on the assumption of positive network externalities, it carries over to the
present setting. Rephrasing it with my notation gives the following proposition.
Proposition 5 (Farrell and Saloner). Suppose that, for each j,
B xj (N) . B yj (h1,2,..., j 2 1j).
(18)
Then the unique SPNE ( in the sequential game) involves all firms’ switching.
Proof. See Farrell and Saloner (1985) p. 73.
The intuition behind this result is the following: Eq. (18) ensures that each firm
j prefers to adopt the new technology X when all its predecessors have done so, if
13
I have indeed assumed that firms are ranked by increasing values of u, which represents the
intrinsic preference for technology Y; therefore, the lower u (or the index of the firm), the higher the
eagerness to switch to technology X.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
433
it believes all the rest would follow; but this is exactly what j believes since it
knows that Eq. (18) is true for all its successors and thus that they will switch if it
does; as a consequence, each firm changes.
Consider now the issue of symmetric excess inertia, which is defined by Farrell
and Saloner as a situation where each firm prefers an overall industry switch over
the status quo but it fails to happen. In my notation, the conditions for symmetric
excess inertia are the following:(C1) B xi (N).B yi ([), ;i [N;(C2) standardization
on the new technology is not a SPNE of the game.
In Farrell and Saloner’s setting, excess inertia is clearly impossible: because
they assume that the payoffs exhibit monotonically positive network externalities,
the satisfaction of conditions (C1) entails the satisfaction of Eq. (18) and
standardization on the new technology is thus an equilibrium partition according to
Proposition 5.
In contrast, we know that the payoffs derived here do not always exhibit
positive network externalities. Hence, it is possible to find constellations of
parameters such that, for some firm j, B yj (h1, 2,..., j21j).Bj (N).B yj ([), meaning
that there can be firms for which condition (C1) is met while Eq. (18) is not. Yet,
even though Farrell and Saloner’s argument breaks down, their conclusion does
not necessarily so (Eq. (18) is indeed sufficient but not necessary for overall
switch to be the unique SPNE). One has therefore to find out an alternative line of
argument to establish whether excess inertia might occur or not in the present
setting. Proposition 6 provides the answer by asserting that Farrell and Saloner’s
conclusion remains valid: the possibility of negative network externalities does not
induce excess inertia when information is complete.
Proposition 6. When all firms prefers an overall switch over the status quo
(conditions ( C1) are satisfied), the unique SPNE involves all firms’ switching,
meaning that there can be no symmetric excess inertia.
Proof. See Appendix A.
5.2. A model with incomplete information
Suppose now—more realistically—that firms are uncertain whether they would
be followed if they switched to the new technology. This uncertainty is modelled
by the following game with incomplete information which, for the sake of
simplicity, is restricted to two firms.
Each firm is identified by its type, ui , which indicates its preference for the
change to standard X (higher values meaning lower preferences, as assumed
above). It is assumed that types are a priori (independently) drawn from the
uniform distribution on the unit interval. Information is incomplete: each firm
knows its own type but ignores the other firm’s type. The timing of the game has
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
two periods, 1 and 2. Each firm has the opportunity to switch at time 1 or time 2 or
not at all; reswitching is ruled out. Payoffs accrue at the end of date 2.
In Farrell and Saloner’s analysis, payoffs from the adoption of technologies
(what they call ‘benefit functions’) are given exogenously and supposed to obey
the following four requirements: (R1) networks are beneficial; (R2) firms with a
higher intrinsic preference for the new technology X are more eager to switch,
both unilaterally and if the other firm also switches; (R3) unilateral switching is
worthwhile for at least one possible type of firm, and (at the other end of the
spectrum) there are some types who would rather remain alone with the old
technology than join the other firm with the new technology; (R4) if a firm with a
given intrinsic preference for the new technology prefers a combined switch to X
than remaining alone with technology Y, then so do all firms with higher intrinsic
preferences for the new technology.
In my setting, the formation of the payoffs has been endogenized by introducing
a Cournot game after the firms’ technology decisions. It is easily checked,
however, that these payoffs still obey requirements (R2)–(R4). The main
difference stems thus from the possible violation of requirement (R1) for
combinations of types where negative network externalities are observed. An
additional difference comes from the fact that each firm’s payoff now depends on
both firms’ types and not only on its own type (as assumed by Farrell and Saloner).
Remarkably, despite these differences, Farrell and Saloner’s main proposition
carries over to the present setting: the authors show that for benefit functions
satisfying requirements (R1)–(R4), a unique ‘symmetric bandwagon equilibrium’
(see definition below) exists; Proposition 7 states that this conclusion survives even
when requirement (R1) is relaxed in the way described above.
To establish this result, some definitions are needed. First, a bandwagon
strategy for a firm is defined by a pair (a, b ) with 0, a , b ,1 such that: (i) if
ui < a, the firm switches at time 1; (ii) if a ,ui < b, the firm does not switch at
time 1, and then switches at time 2 if (and only if) the other firm switched at time
1; and (iii) if ui . b, the firm never switches. Then, a symmetric bandwagon
equilibrium is defined to be a perfect Bayesian Nash equilibrium in which each
firm plays a bandwagon strategy with the same pair (a, b ).
To determine the values of the cutoffs a and b, there are three ‘actions’ to
consider: 14
– a 1 : switch at time 1;
– a 2 : do not switch at time 1, switch at time 2 if opponent switched at time 1;
– a 3 : do not switch at time 1 nor at time 2.
14
An action indicates a move for period 1, and a move for period 2 according to what the other firm
did in period 1.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
435
Table 2
Outcomes of the game when the opponent plays a bandwagon strategy
Firm i’s actions
0<uj < a
a ,uj < b
b ,uj <1
a1
a2
a3
X,X
X,X
Y,X
X,X
Y,Y
Y,Y
X,Y
Y,Y
Y,Y
Let u i (a k , ui ) be the expected benefit to a firm i of type ui when it uses action a k
and when its opponent is using a bandwagon strategy (a, b ). Table 2 is helpful for
deriving the expected benefits for the three actions: each cell indicates the outcome
of the game (the first entry is firm i’s choice of technology, and the second firm
j’s) according to the action played by i (in row) and to the type of j (in column).
Consider, for instance, action a 1 : firm i switches at time 1, which induces its
opponent to switch too unless its type is close from unity. Therefore, firm i will
earn B xi (hi, jj) if its opponent’s type is lower than b and B xi (hij) otherwise. Its
expected benefit for this action is thus equal to:
E B (hi, jj)du 1 E B (hij)du .
b
u i (a 1 ,ui ) 5
0
1
x
i
j
x
i
b
j
By a similar reasoning,
u i (a 2 ,ui ) 5
E B (hi, jj)du 1 E B ([)du ;
u i (a 3 ,ui ) 5
E B (h jj)du 1 E B ([)du .
a
0
1
x
i
j
a
0
y
i
a
j
1
y
i
j
a
y
i
j
We can now express the indifference conditions between adjacent actions:
u i (a 1 ,a )5u i (a 2 ,a ), and u i (a 2 , b )5u i (a 3 , b ). Solving this system of equations (call
it system Sab ) for its two unknowns gives the values of the cutoffs a and b as
functions of the degree of product differentiation (d). Table 3 displays these
Table 3
Values of the cutoffs according to the degree of product differentiation
d
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a
b
0.419
0.667
0.422
0.654
0.426
0.640
0.430
0.627
0.435
0.615
0.439
0.602
0.444
0.589
0.450
0.577
0.455
0.564
0.462
0.552
0.469
0.540
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
values.15 Plugging the values of the cutoffs into the expected benefit functions for
actions a 1 to a 3 (and for the other possible actions),16 one checks that firm i’s best
reply is the bandwagon strategy (a, b ) (that is, action a 1 yields the highest
expected benefit for values of ui lower than a, a 2 does so for values of ui between
a and b and a 3 for values of ui larger than b ).
I am now in a position to state the result outlined above.
Proposition 7. With the cutoffs a and b defined as the solutions of system Sab , a
unique symmetric bandwagon equilibrium exists in the game with sequential
technology adoption under incomplete information.
Noteworthy is the fact that, as d increases (i.e., as competition becomes fiercer
between the two firms), a increases while b decreases: the region of types where
action a 2 (the ‘wait and see’ action) is played therefore narrows down; in other
words, fewer types postpone their technology choice to the second period. The
intuition for this result is the following: when competition is fiercer, a firm is more
harmed by the compatibility benefits it provides to its rival by joining its network;
firms are thus less eager to coordinate their technology choices and they are more
likely to commit themselves without waiting to see what the rival does.
The symmetric bandwagon equilibrium entails symmetric excess inertia. There
indeed exist pairs of types (u1 ,u2 ) such that ;i51,2, (i) ui , b (meaning that the
firms do not switch), and (ii) B ix (h1,2j).B iy ([) (meaning that each firm would be
better off were the switch made). Farrell and Saloner clearly explain the intuition:
‘‘both firms are fencesitters, happy to jump on the bandwagon if it gets rolling but
insufficiently keen to set it rolling themselves’’. There is also asymmetric excess
inertia; that is, situations where the switch is not made (because ui , b, i51,2)
even though the sum of benefits is higher when standardization takes place on the
new rather than on the old technology (B x1 (h1,2j)1B x2 (h1,2j).B y1 ([)1
B 2y ([)⇔u1 1u2 ,1). Finally, cases of (asymmetric) excess ‘momentum’ (where
the switch occurs although the sum of benefits would be higher in the status quo)
are also observed.
The interesting feature here is that market failures of these three kinds become
less likely as competition between the firms gets fiercer. This result has two main
causes: first, as already explained, an increase in d leads to a decrease in b and to
15
These values are computed for b54, which makes sure that the market is large enough for an
interior solution to exist in the second stage, whatever the firms’ types and technology choices (see
above). It can be shown that for higher values of b, the values of the cutoffs are not changed by more
than 0.005.
16
There are two more actions to consider if switch back to the old technology is precluded: (a 4 ) do
not switch at time 1, switch at time 2 if opponent did not switch at time 1; and (a 5 ) do not switch at
time 1, switch at time 2 (whatever the opponent did at time 1). If switch back is allowed, three more
actions have to be considered.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
437
an increase in a ; second, it can be checked that the areas where both firms prefer
standardization on one technology rather than on the other narrow down as d
increases (see Fig. 3).
6. Concluding remarks
This article analyzes the adoption of network technologies in an asymmetric
oligopoly with linear demand. First, a two-stage game in which simultaneous
technology decisions precede a market competition game is solved for its
subgame-perfect Nash equilibria. The interesting feature of this game is the
potential conflict between size and composition of the networks: loosely speaking,
firms are eager to join large networks but not with anybody. Firms are indeed
heterogeneous regarding their taste for compatibility and are rival on the final
market; hence, because a firm’s profit depends on its relative competitive position,
a firm might be reluctant to share the benefits of compatibility with rivals that
value compatibility more than it does.
I characterize the subgame-perfect Nash equilibria for the family of games
induced by the possible constellations of parameters (satisfying an assumption
regarding the non-negativity of second-stage quantities). I show that an equilibrium always exists and that in any equilibrium partition of the set of firms between
the two technologies, the network of each technology is made of the firms with the
highest intrinsic preference for this technology. Private and social incentives
towards standardization are also compared; I establish that these incentives may
diverge: standardization maximizes the total surplus level but does not necessarily
emerge as an equilibrium partition of the set of firms. It must be stressed that
co-existence of the networks (and thus market failure) is all the more likely at the
equilibrium of the game that (i) firms are heterogeneous, and (ii) competition is
fierce.
The model is then extended by considering sequential technology decisions, and
can thereby be compared with Farrell and Saloner (1985). The comparison reveals
that Farrell and Saloner’s results (regarding the issue of excess inertia) still hold in
situations where positive network externalities may not always be observed.
Noteworthy is the fact that, under incomplete information, market failure becomes
less likely as second-stage competition gets fiercer.
The method described in the paper can certainly be generalized to other
environments where heterogeneous agents care not only about the number of
agents making the same choice as they do, but also about the identity of these
agents. In this respect, a parallel has certainly to be drawn between my analysis
and the one by Axelrod et al. (1995) which studies the formation of competing
alliances to sponsor technical standards (the basic assumptions in this paper are
indeed that the utility of a firm for joining a particular standard-setting alliance
increases with the size of the alliance but decreases with the presence of rivals in
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P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
Fig. 3. Excess inertia in the game with incomplete information. From d50 (above) to d51 (below), the
areas where excess inertia and excess momentum are observed narrow down.
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
439
the alliance, especially close rivals; the 1988 efforts of nine computer companies
to create and sponsor Unix operating systems standards provide an illustration of
this intuition). Similarly, it would be interesting to reconsider the recent literature
on economic geography 17 and investigate whether positive locational externalities
depend only on the number of firms locating close to each other or also on their
identity.
Another natural direction for further research would be to set up a dynamic
version of the model. Repeating the two-stage game over time would indeed allow
to examine whether the constitution of networks can be used as a barrier to entry
or as a means of excluding other firms from the market (assumption 326
regarding the positivity of the quantities produced at Cournot equilibrium should
therefore be lifted and some switching cost for the firms changing their technology
decision should be introduced).
Acknowledgements
This article is based on Chaps. 2 and 3 of my Ph.D. Thesis at the University of
Namur. Financial support from the Belgian F.N.R.S. is gratefully acknowledged. I
would like to thank my advisor, Louis Gevers, for his guidance. Discussions with
´
´
Francis Bloch, Jacques Cremer,
Pierre Regibeau,
Jacques-François Thisse,
François Maniquet, Philippe De Donder and Eric Toulemonde have been very
valuable. I am also grateful to the Editor and two anonymous referees for their
constructive criticisms on an earlier draft.
Appendix A
Proof of Lemma 1
The first step of the proof consists in showing that E(N),E([)⇔u¯ ,1 / 2; this
is obvious since E(N)5n 2 (12 u¯ ), and E([)5n 2 u¯ .
Our task is now to show that ;X ,N, E(X)<maxhE(N); E([)j. Suppose first
that u¯ .1 / 2 and suppose that E(X).E([). This inequality rewrites as x 2 .nxu¯ 1
no i [Xui . Letting w5o i [Xui and invoking the fact that u¯ .1 / 2, we derive from the
2
previous expression that 2x 2nx1nw.0, which is satisfied provided that
]]]
x . (1 / 4)fn 1Œn 2 1 16nwg.
(19)
We also know that n2x>o j [N \Xuj 5nu¯ 2o i [Xui .n / 22w. This implies that
17
See for examples Audretsch and Feldman (1996); Schmutzler (1995).
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
440
x , n / 2 1 w.
(20)
One easily checks that Eqs. (19) and (20) are compatible if and only if w.n / 2.
But in this case, Eq. (20) would imply that x.n, which is impossible. Therefore,
ū .1 / 2 implies that E(X)<E([). (The proof follows exactly the same lines for
ū ,1 / 2.) Q.E.D.
Proof of Proposition 4. When we compare the first-best output profile (obtained
under marginal cost pricing) with the equilibrium quantities under Cournot and
Bertrand competition, we can observe some regularities in the analytical expressions. It is possible to represent these different quantities with the following
generic expressions:
B
;i [ X,qi (X) 5 ]]]][Ab 1 (A 1 Bdn)x(1 2 ui ) 2 BdE(X)];
A(A 1 Bdn)
(21)
B
; j [ X,q j (Y) 5 ]]]][Ab 1 (A 1 Bdn)(n 2 x)uj 2 BdE(X)];
A(A 1 Bdn)
(22)
with the variables A and B defined as in Table 4.
For a given partition of the set of firms (N5X <Y), the total surplus level is
defined as the sum of the consumer’s surplus and of the total industry profits. As
far as the consumer’s surplus is concerned, we easily obtain the following
expression by plugging the inverse demand schedule into the quadratic surplus
function:
O q (X) 1 (d / 2) OO q (X)q (X).
n
U(X) 5 (1 / 2)
n
2
i
i 51
i
j
i51j ±i
Regarding total industry profits, it is easy to show that they are equal (i) to zero
under marginal cost pricing, (ii) to the sum of the square of the quantities
produced under Cournot competition, and (iii) to the sum of the square of the
quantities produced, divided by b, under Bertrand competition. The following
generic expressions accommodates these three cases:
O q (X).
n
Ptot (X) 5 (1 /B)[A 2 B(1 2 d)]
2
i
i 51
Table 4
Common representation of the quantities
A
B
Cournot
Bertrand
First-best
22d
1
(12d)[21d(2n23)]
11d(n22)
12d
1
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
441
We are now in a position to give a generic expression of the total surplus level;
rearranging terms, we have
W(X) 5 T 1 [bn 1 E(X)] 2 1 T 2 [nC (X) 2 E(X)2 ],
(23)
where C (X)5x 3 [sX 2 1(12 u¯X )2 ]1(n2x)3 [sY 2 1 u¯Y 2 ], with sK 2 and u¯K standing,
respectively, for the variance and the mean of u over subset K (K5X,Y), and
where the variables T 1 and T 2 are defined as in Table 5 (note that both T 1 and T 2
are positive ;d,n).
The first step of the proof consists in showing that W(N). (,,5) W([)⇔u¯ ,
(.,5) 1 / 2. Observing that C (N)2E(N)2 5C ([)2E([)2 5n 4 sN 2 , we note that
W(N) and W([) only differ through the first term of Eq. (23); therefore, the
difference between W(N) and W([) has the same sign as the difference between
E(N) and E([); and we know, from Lemma 1, that this difference is positive
(negative, nul) if and only if u¯ , (., 5) 1 / 2.
In a second step, we have to show that ;X ,N, (i) u¯ <1 / 2⇒W(X)<W(N), and
(ii) u¯ .1 / 2⇒W(X),W([).
Consider first statement (i) and suppose it is violated; that is, W(X).W(N).
From Lemma 1, we know that u¯ <1 / 2⇒E(X)<E(N); we also know that both T 1
and T 2 are positive. Therefore, using Eq. (23), we can rewrite the previous
inequality as
nC (X) 2 E(X)2 2 n 4 s N2
]]]]]]]
. [2bn 1 E(N) 1 E(X)](T 1 /T 2 ).
E(N) 2 E(X)
(24)
The remainder of the proof consists (i) in deriving an upper bound for the LHS
(denote it L up ) and a lower bound for the RHS (denote it R low ), and (ii) in showing
that L up ,R low , meaning that even in the most favourable configuration, Eq. (24)
cannot be met. I will not go here into the details of these last two steps. Let me
just mention the following results: (1) The upper bound for the LHS is reached for
x5n21 and ui 50, ;i [N; hence, L up 5(n21)3 /(2n21). (2) By virtue of
Table 5
Common representation of the total surplus level
T1
T2
Cournot
3 1 d(n 2 1)
]]]]2
2n[2 1 d(n 2 1)]
32d
]]]2
2n(2 2 d)
Bertrand
[1 1 d(n 2 2)][3 1 d(n 2 4)]
]]]]]]]2
2n[1 1 d(n 2 1)][2 1 d(n 2 3)]
[1 1 d(n 2 2)][3 1 d(3n 2 4)]
]]]]]]]
2n(1 2 d)[2 1 d(2n 2 3)] 2
First-best
1
]]]]
2n[1 1 d(n 2 1)]
1
]]]
2n(1 2 d)
442
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
assumption 326, 2bn.(Bdn 3 ) /A. (3) E(N)1E(X) reaches a minimum for x5n / 2
and ui 51 / 2, ;i [N, and is then equal to (3n 2 ) / 4. (4) Using the two previous
results and the expressions from Tables 4 and 5, one computes that R low is equal to
[n 2 (n12)(4n13)] / [8(n11)2 ] in the Cournot case, to [n 2 (2n21)] /(3n21) in the
Bertrand case, and to (3 / 4)n 2 in the first-best case. (5) One checks that the latter
three expressions are higher than L up 5(n21)3 /(2n21).
The proof of statement (ii) follows exactly the same lines. Q.E.D.
Proof of Proposition 6. Simple manipulations allow to establish that if conditions
(C1) are satisfied, then all firms have necessarily a value of u strictly less than
1 / 2, which implies the following two findings:
1. Ji (x).0, ;i [N and ;x>(n21) / 2;
2. if the sets V and W are such that W ,V #N and (n2v2w)[n1122(v2w)]<
0 (with v,w standing, respectively, for [V, [W ), then B xi (V ).B yi (W ), ;i [(V \W ).
To establish that overall switching is the unique SPNE of the game, we have to
show that when all predecessors have switched, each firm’s best response is to
switch as well. From the first finding, we easily conclude that this is necessarily
true for the firms with an index higher than (n21) / 2. For the firms playing earlier
in the game, things are however more complex. Take first firm k, with k5(n21) /
2 [respectively n / 2] if n is odd [respectively even]. If k switches, the previous
argument applies: all k’s successors will switch as well and k will end up with a
payoff equal to B xk (N). If k does not switch, we have to examine firm (k11)’s
decision. In this particular subgame (where firms 1 to k21 have switched and firm
k has not), a simple backward induction argument allows us to say that if one of
the remaining firms prefers to switch although its predecessors (from firm k on)
have not switched, then these predecessors necessarily prefer to switch as well. In
other words, firm k11 could prefer not to switch only if it knows that all its
successors will not switch either. In this case, firm k11 compares B xk 11 (N\hkj) to
B ky 11 (h1,..., k21j). But we can use the second finding (with v5n21 and
w5k21) to show that even in this situation, firm k11 prefers to switch. Now,
this implies that firm k will earn a payoff of B yk (N\hkj) if it does not switch, which
is clearly inferior to B kx (N) since, from the first finding, Jk (n21).0. We have thus
established that firm k decides to switch when all its predecessors have done so,
and we can replicate the argument up to firm 1. Q.E.D.
Discussion of assumption 326
When assumption 326 is not satisfied, the market size (measured by b5a2c)
might be not large enough; in other words, it can happen that, given their
competitors’ technology decisions, some firms are unable to produce any positive
P. Belleflamme / Int. J. Ind. Organ. 16 (1998) 415 – 444
443
quantity whatever the technology they choose. In such a case, the satisfaction of
Eqs. (14) and (15) does no longer guarantee the existence of an equilibrium
partition; one or several firms are indeed forced to leave the market as long as their
competitors do not change their technology choices. As the number of firms in the
industry decreases, a new second-stage equilibrium takes place amongst the
remaining firms and technology decisions have to be re-assessed with respect to
this new equilibrium.
The difficulty to analyze such a situation stems from the fact that if some
remaining firm now finds it optimal to switch to another technology, some of the
previously excluded firms might be able to re-enter the market, which modifies
again the size of the industry.
For the two-firm case, the problem can rather easily be dealt with (in particular,
it appears that the same equilibrium partitions obtain even if b50; assumption
326 is therefore unnecessary). But as soon as there are more than two firms in
the industry, the model becomes hardly tractable without the simplifying assumption 326.
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