# Group network effects in price competition

R. Soeiro  Universidade do Porto & Inesc Tec  A. A. Pinto  Universidade do
Porto & Inesc Tec

###### Abstract

The partition of society into groups, polarization, and social networks are
part of most conversations today. How do they influence price competition? We
discuss Bertrand duopoly equilibria with demand subject to network effects.
Contrary to models where network effects depend on one aggregate variable
(demand for each choice), partitioning the dependence into groups creates a
wealth of pure price equilibria with profit for both price setters, even if
positive network effects are the dominant element of the game. If there is
some asymmetry in how groups interact, two groups are sufficient. If network
effects are based on undirected and unweighted graphs, at least five groups
are required but, without other differentiation, outcomes are symmetric.

##  1 Introduction

Consider the class of two-stage games where, in the first stage, a subset of
players (e.g., firms, sellers, platforms, etc.) set a price for the service or
good they provide; in the second stage, a different set of players (e.g.,
consumers, buyers, users, etc.) decide on a (priced) option from the first
stage while subject to network effects. The equilibrium concept is that of
pure price subgame-perfect equilibrium (SPE). Examples of services range from
restaurants to online platforms; of goods, from clothing to technology.
Network effects, i.e., how second-stage players influence each other’s
choices, can be positive, negative, or both.

In general, first stage price competition is fiercer under positive network
effects, which tend to produce corner solutions (at least one firm with no
profit), while negative network effects relax competition and allow for
interior price solutions (where all firms have profit), (see Grilo, Shy and
Thisse, 2001). One of the goals of this work is to show that while this idea
holds when network effects (for each choice) depend on a single aggregate
variable (number of players on that choice), it is not necessarily true when
network effects depend on more than one aggregate variable. That is, when
there are two or more second-stage groups that influence each other
differently, the opposite may happen. The partition into groups might be as
refined as small cliques, less detailed as in peer or social groups, or even
broader as in democrats/ republicans, smokers/ non-smokers, etc.
Interestingly, two aggregate variables (groups) are enough to allow reverting
the one dimension result.111With uniform pricing, players in the first stage
pricing game always look at only one aggregate variable: demand.

A second goal of the work is a general mathematical treatment of network
effects as a determinant of equilibria, in particular, local pure price
subgame-perfect equilibria where both firms have profit (SPE+). The general
problem with finding these equilibria under positive network effects is that
the possibility of a bandwagon response in the second stage creates price
deviation incentives in the first stage. So, typically, positive network
effects are assumed sufficiently small relative to, for example, a negative
network effect (e.g., congestion), a product differentiation effect (e.g.,
Hotelling), or some lock-in variable (either in the first or second stage).
For instance, when there are no network effects, Caplin and Nalebuff (1991)
show that restrictions on consumer preferences are sufficient to ensure a pure
price equilibrium. Take that solution and perturb it with (positive) network
effects. As long as the latter are sufficiently small, one can study their
impact without disrupting the existing solution.

Our approach is, in some sense, the inverse. We consider a duopoly and start
with a general network effects function, which can depend on several aggregate
variables (e.g., groups of consumers). Sufficient conditions for a SPE+
outcome lead to (marginal) network properties on network effects. One can then
perturb this solution by introducing different variables (e.g., (intrinsic)
product differentiation) but without the need to assume a particular form (say
$\rho$-concavity and convex support). In particular, this can help assess the
appropriate function and dimension to use when modeling: how many groups are
necessary and how do they influence each other?

Let’s look at a simple stylized example with both positive and negative
network effects. Say there are two choices (e.g., restaurants) and two groups,
snobs and conformists. The former’s utility depends negatively on the number
of second-stage players on the same choice, and the latter positively.
Amaldoss and Jain (2005), for the linear case, use two commonly coupled
restrictions to guarantee a SPE+. An assumption on the distributions of
preferences of both groups and a bound on conformism. Suppose we change the
game by removing both restrictions but allowing conformists to distinguish
between snobs and other conformists (network effects, for conformists, now
depend on two aggregate variables). If the impact of snobs on conformists is
different than that of other conformists222One typical example is to take
snobs as being celebrities having a higher impact on followers than do other
followers., a SPE+ where both groups split (i.e., its members make different
choices) becomes possible. Furthermore, the resulting outcome does not depend
on how conformists distinguish others, the relative value of snobbism and
conformism, or the sizes of each group.333The last observation refers to the
linear case in question and is not necessarily completely generalizable.

Increasing the dimension can open the possibility for SPE+ outcomes, but it
can also have the opposite effect, disrupting existing equilibria. For
example, when there are only negative network effects. However, the relevant
part for constructing SPE+ is the interaction among groups that split (the set
of indifferent consumers), so the possibilities and complexity rise with the
power set of the partition into groups (which is only a problem if one wishes
to show non-existence or uniqueness). That means the type of equilibria where
some groups split, while others don’t, is always possible and works as going
back to a lower dimension (although it produces different outcomes). Thus,
increasing the detail in partitioning second-stage players into groups, in
general, creates new equilibria.

Dimension is only one part of the problem. Let’s take the example of two-sided
markets from Armstrong (2006) (the specific case with uniform pricing), with
again two groups but now only positive network effects. Two groups interact
via two platforms, like heterosexual dating agencies, nightclubs, credit cards
(consumers/ retailers), shopping malls, television channels, and, we will add,
online platforms (users/ providers, e.g., lodging, transport, food
delivery).444This serves as an example of how the relative impact of groups on
each other produces equilibria, even if some of the mentioned cases would be
more natural for the price discrimination literature. Each group’s decision
depends on the number of members of the other group that uses each platform.
Armstrong characterizes a SPE+ for the case when network effects are
sufficiently small relative to product differentiation (transport costs) using
a Hotelling specification. Without differentiation (no transport costs), such
SPE+ is not possible. Suppose now that we change the game and introduce intra-
group influences. For example, consumers may value how many other consumers a
platform attracts (e.g., as a form of quality or reliability signaling).555In
Armstrong’s example of dating agencies and nightclubs, the network effects
structure modification means to consider them bisexual. Then, a SPE+ is
possible if same-sex encounters have a higher value than different-sex ones.
We show that, in this case, if intra-group influences are sufficiently strong,
a SPE+ where both groups split among the two platforms becomes possible
(meaning it does not require product differentiation), which might seem
counterintuitive.666What might strike as counterintuitive is that introducing
intra-group influences produces an equilibrium where groups split. However,
second-stage Nash is not the issue. Intra-group influences stabilize first-
stage price competition.

As the number of considered groups increases, so does the complexity of
possible equilibria as it depends on the relative interactions among the
different groups. Some authors (e.g., Banerji and Dutta (2009)) consider a
given a priori undirected and unweighted graph, which characterizes how groups
are connected. It is an instance of a multilinear network function, with the
advantage of reducing the discussion to an adjacency matrix. Because
interactions weights are homogeneous and symmetric (either 1 or 0 for
connection), properties and the existence of SPE+ outcomes are dependent only
on the underlying structure, not the relative weights of network effects. In
particular, on the induced subgraph among groups that split. However,
homogeneity in interactions makes it harder to allow SPE+. Even if groups have
different sizes, these do not play the same role as a network effect weight
(we will see that in the more general multilinear case), forcing an increase
in dimension (more groups) to produce a solution. When a SPE+ outcome exists,
it is unique, completely symmetric, and independent of the network structure
and group sizes. The latter holds for every multilinear model where network
effects are symmetric. Thus, weights and asymmetry in connections (directed
networks) play a higher role than the underlying structure.

The technical difficulty in studying SPE under network effects is the
multiplicity of second-stage equilibria. The definition involves a second-
stage contingent plan, so this is sometimes called a second-stage coordination
problem. In a certain sense, we will argue, it is an artificial problem (in
this class of games). Our objective here is to characterize local equilibrium
outcomes, not predict play, hence multiplicity of game equilibria is not an
issue. It is a first-stage maximization difficulty: evaluating an action
(pricing) depends on whether deviations from it are profitable, but how to
evaluate price deviations when there are multiple second-stage equilibria? In
the classic definition, second-stage players (e.g., consumers) would have to
announce their choices for every possible pair of prices. However, can you
think of a practical example of such a situation? To deal with the issue, we
use a slightly different approach, resorting to the concept of the imagined
demand curve proposed by Kaldor (1934):

> The traditional "market demand curve" for a certain product is not the same
> sort of thing as the demand curve which is relevant in determining the
> actions of the individual producer. The first denotes a functional
> relationship between the price and the amounts bought from a particular
> producer. The second concerns the image of this functional relationship as
> it exists in the mind of the entrepreneur.
>
> (…)
>
> The "imagined demand curve" (…), becomes determinate as soon as it exists in
> the producer’s imagination - and since something always must exist there the
> question of indeterminateness simply does not arise in this case. If on the
> other hand, by assuming perfect knowledge we make the two coincide, not only
> do we make the analysis unnecessarily unrealistic, but we introduce
> complications (by rendering the "imagined demand curve" indeterminate)
>
> (…)
>
> The "imagined" and the "real" demand curve are thus merely required to meet
> at one point - I admit, at the critical point.

Here, the critical point is an outcome of the game: a pair of prices and a
second-stage Nash equilibrium for those prices. If at least one group splits
and no firm is at marginal cost, the implicit function theorem provides a
coordination device for the first stage. A unique continuous imagined demand
curve along which both price-setters can verify if deviations are profitable.
If the standard first and second-order conditions are satisfied, this leads to
a local SPE+.

What if, for example, we have at least one firm that always believes the
response to price is discontinuous and favorable? That is, of course, a
possibility when there are multiple second-stage equilibria, and, in that
case, it’s most likely there won’t be any equilibria where both have profit.
However, that seems to be a much stronger assumption requiring a particular
economic motivation.

We make no further assumption than the unique continuous deviation as a
coordination device. Naturally, a necessary condition for a SPE+ outcome is
negative marginal demand, but that is a characteristic of equilibria. Some
authors assume negative marginal group demand, either explicitly (e.g.,
Banerji and Dutta (2009)) or implicitly through the distribution of
preferences. However, this is an assumption, not a necessary condition, which
in particular excludes some SPE+. Removing this assumption allows SPE+, for
example, under positive network effects. For example, with two groups, the
bandwagon effect can have two different directions (one for each group) where
one compensates the other. The same dynamic may disrupt equilibria in the
negative case. SPE+ is possible as long as the sum of all groups’ marginal
demand is negative. It may be counterintuitive why some groups’ demand could
increase with price (or why first stage players imagine so). We do not aim to
discuss an economic motivation for why this may be so, but rather to point out
that it is a possibility, which produces different equilibria. Probably the
most notable discussion of a similar phenomenon is around Becker (1991)’s note
on restaurant pricing. Becker’s proposed explanation is studying particular
forms for demand curves that would produce equilibria where one restaurant has
excess demand and the other excess capacity. Karni and Levin (1994) show that
an SPE does not exist in such conditions when restaurants compete in prices.
However, Becker’s discussion is at the aggregate demand level. Our observation
is that, on a group level, positive marginal demand is sustainable at
equilibrium.

The remainder of the work is as follows. In the next section we set up the
model and present the main result. In Section 3 we discuss the results through
models from the literature, and in Section 4 we conclude. We use $\equiv$ for
definitions.

##  2 Model and main theorem.

For clarity we will use the nomenclature of firms and consumers, which form
the set of players. In the first stage two firms $a$ and $b$, simultaneous and
independently set a price for their product. In the second stage, consumers
observe the vector $\mathbf{p}\equiv(p_{a},p_{b})$ and simultaneous and
independently choose which firm to buy from. Consider a finite partition of
the set of consumers into $g\in\mathbb{N}$ groups
$\mathcal{G}\equiv\\{G_{1},\ldots,G_{g}\\}$, and let
$\mathbf{d}^{j}\equiv(d_{1}^{j},\ldots,d_{g}^{j})$ describe the number of
consumers of each group that choose firm $j$. For each player in a group $G$
that chooses firm $j$, the payoff is777We are implicitly assuming a continuous
set of consumers, and groups as a partition into measurable subsets. However,
the results also hold in the finite case. We would just need to redefine
payoffs as expected utility and work with mixed (behavior) strategies. The set
of equilibria would be more complex, but the utility function would become
multilinear. See, for example, our own work in Soeiro and Pinto (2021).

$-p_{j}+v^{j}(\mathbf{d}^{j};G),~{}~{}j=a,b,$

where $v^{j}(\cdot)$ is a $C^{2}$ function capturing the influence groups may
have on each other, i.e., network effects. Note it carries the assumption that
within each group consumers are indistinguishable. However, the function may
also encompass constant terms that represent how (all) elements of a group
value each product (may include vertical or horizontal product
differentiation), willingness to pay, or other.

Suppose each group $i$ has mass (or size) $m_{i}$. We will denote by
$\sigma_{i}\in[0,1]$ the fraction of group $i$ who choose firm $a$, hence,
$d_{i}^{a}=m_{i}\sigma_{i}$ and $d_{i}^{b}=m_{i}(1-\sigma_{i})$. We will
whenceforth use this variable to represent the strategic action of each group,
and let $\boldsymbol{\sigma}\equiv(\sigma_{1},\ldots,\sigma_{g})$ summarize
consumption.888The assumption of mandatory consumption, or total coverage, is
common in the literature and it simplifies the exposition. It is, however, not
necessary, as we will discuss in the end. Define now
$v:[0,1]^{g}\to\mathbb{R}^{g}$, a $C^{2}$ function where999It is a bit abusive
to use the same letter, but allows us to economize notation and avoid carrying
$\Delta$ around.

$v_{i}(\boldsymbol{\sigma})\equiv
v^{a}(\mathbf{m}\cdot\boldsymbol{\sigma};G_{i})-v^{b}(\mathbf{m}\cdot(\mathbf{1}-\boldsymbol{\sigma});G_{i})$

Given a pair of prices $\mathbf{p}$, whether $\boldsymbol{\sigma}$ is a
second-stage Nash Equilibrium (NE) following $\mathbf{p}$, depends on $v$. It
must hold: $v_{i}(\boldsymbol{\sigma})\leq p_{a}-p_{b}$ if $\sigma_{i}=0$;
$v_{i}(\boldsymbol{\sigma})=p_{a}-p_{b}$ if $0<\sigma_{i}<1$; and
$v_{i}(\boldsymbol{\sigma})\geq p_{a}-p_{b}$ if $\sigma_{i}=1$. As we will
assume no cost function for firms (also a simplifying assumption, not
necessary as briefly discussed in the final section), this is what determines
the game above described, thus denoted by
$\Gamma(v;g)\equiv\Gamma(v;g,\mathbf{m})$, and we will often omit
$\mathbf{m}$, as group sizes aren’t our main focus.

###  2.1 Equilibrium characterization (SPE)

In general, each game $\Gamma$ has a multiplicity of second stage Nash
Equilibria (NE). Consider the correspondence
$Q:(\mathbb{R}^{+}_{0})^{2}\to[0,1]^{g}$ where $Q(\mathbf{p})$ is the set of
second stage NE following $\mathbf{p}$. Given $\mathbf{p^{*}}$ and a
neighborhood $P(\mathbf{p^{*}})$, a _local selection_ of $Q$ is a function
$q:P(\mathbf{p^{*}})\to[0,1]^{g}$ such that $q(\mathbf{p})\in Q(\mathbf{p})$
for all $\mathbf{p}\in P(\mathbf{p^{*}})$. For each outcome, to determine if
deviations are profitable, each firm makes a local selection. Given
$\mathbf{p^{*}}$, let $P_{j}$ be a neighborhood of $p_{j}^{*}$ for each
$j=a,b$. Firm $a$ makes a local selection $q_{a}$ in neighborhood
$P_{a}\times\\{p_{b}^{*}\\}$, and similarly firm $b$ makes a selection $q_{b}$
in $\\{p_{a}^{*}\\}\times P_{b}$. Restricting each firm’s selection to a
neighborhood of its own price is not necessary but more standard to discuss
the notion of SPE and unilateral deviation. Allowing firms to make different
selections is more general than usual, but also not necessary.

Consider an outcome $(\mathbf{p^{*}},\boldsymbol{\sigma})$. In a neighborhood
$P$, (imagined) demand is

$D_{a}(p_{a};q_{a})=\mathbf{m}\cdot
q_{a}(p_{a},p_{b}^{*}),~{}~{}~{}D_{b}(p_{b};q_{b})=\mathbf{m}\cdot(\mathbf{1}-q_{b}(p_{a}^{*},p_{b})).$

Note that, for a local selection, demand is a function of only one variable:
own price. We assume firms have no costs, so profit for each firm along these
selections is $\Pi_{j}(p_{j};q_{j})=p_{j}D_{j}$. (Note that these are all
defined for a given outcome and some neighborhood of prices. At
$\mathbf{p^{*}}$, we have actual outcome profits and demand, not imagined.)

A triple $(\mathbf{p}^{*},q_{a},q_{b})$ is a local SPE in the standard way,
that is, if for all $p_{j}$ in the neighborhood of $p_{j}^{*}$ it holds $(i)$
$q_{a}$ and $q_{b}$ are local selections (from $Q$); and $(ii)$
$\Pi_{j}(p_{j};q_{j})\leq\Pi_{j}(p_{j}^{*},q_{j})$.

###  2.2 Results (SPE+)

Given a consumption profile $\boldsymbol{\sigma}$, we say that a group $i$
splits if $0<\sigma_{i}<1$, denoting by
$S(\boldsymbol{\sigma})\subseteq\mathcal{G}$ the groups that split, and
$\bar{S}(\boldsymbol{\sigma})\subseteq\mathcal{G}$ those that don’t. Thus,
$\\{S(\boldsymbol{\sigma}),\bar{S}(\boldsymbol{\sigma})\\}$ is a partition of
$\mathcal{G}$ induced by $\boldsymbol{\sigma}$, but for simplicity we will in
general omit the dependence on $\boldsymbol{\sigma}$. We say that
$\boldsymbol{\sigma}$ is a split if $S\neq\emptyset$, i.e., if at least one
group splits. The next definition is based on the observation that for
$(\mathbf{p},\boldsymbol{\sigma})$ to be a SPE+ outcome it must be that
$\boldsymbol{\sigma}\in Q(\mathbf{p})$ with at least one group that is
indifferent ($v_{i}(\boldsymbol{\sigma})=\Delta p$). As we want to look at
interior points we will focus on outcomes where
$S(\boldsymbol{\sigma})\neq\emptyset$101010This is in fact a necessary
condition for equilibria where both firms have profit. Suppose $S=\emptyset$.
If there are no indifferent consumers, then demand doesn’t change in response
to small price deviations. If there are indifferent consumers, but the groups
to which they belong don’t split (all choose the same firm), then one firm can
provoke a jump for any small price deviation. Either way it would provide an
incentive to deviations. This is general, not particular for this model. When
the option not to buy is present, it must be that, if no group is indifferent
between the two firms, there is a at least one group indifferent between
buying or not. Nevertheless, we still need Definition 1 to force all
indifferent groups out of $\bar{S}$. but also where all groups that are
indifferent between firms split.

######  Definition 1 (Stable splits).

A split $\boldsymbol{\sigma}$ is _stable_ if, given any $i\in S$: $(i)$ for
all $j\in S$, $v_{j}(\boldsymbol{\sigma})=v_{i}(\boldsymbol{\sigma})$; and
$(ii)$ for all $j\in\bar{S}$, $v_{j}(\boldsymbol{\sigma})\neq
v_{i}(\boldsymbol{\sigma})$.

As discussed above, condition $(i)$ is necessary for second stage NE.
Condition $(ii)$ is meant to focus on interior points, those where the NE
inequality is strict.

Consider now the restriction to $S$ of the Jacobian matrix of $v$, denoted
$J_{v}(\boldsymbol{\sigma};S)\equiv\left[(\partial
v_{i}/\partial\sigma_{i^{\prime}})(\boldsymbol{\sigma}),i,i^{\prime}\in
S\right]$. It can be seen as the marginal influence network among groups that
split. For each group $i\in S$ define also the Hessian matrix restriction
$H_{v_{i}}(\boldsymbol{\sigma};S)\equiv\left[\frac{\partial^{2}v_{i}}{\partial\sigma_{i^{\prime}}\partial\sigma_{j}}(\boldsymbol{\sigma}),i^{\prime},j\in
S\right]$. Let $l$ be the cardinal of $S$ (number of groups that split), and
let us define for a given split $\boldsymbol{\sigma}$, the following two
vectors, $\mathbf{k}\equiv(k_{1},\ldots,k_{l})$ and
$\mathbf{r}\equiv(r_{1},\ldots,r_{l})$ where

$k_{i}(\boldsymbol{\sigma};S)\equiv\dfrac{\sum_{i^{\prime}}C_{i^{\prime}i}(J_{v})}{\det(J_{v})},~{}~{}~{}r_{i}(\boldsymbol{\sigma};S)\equiv\dfrac{\sum_{i^{\prime}}\mathbf{k}H_{v_{i^{\prime}}}\mathbf{k}^{T}C_{i^{\prime}i}(J_{v})}{\det(J_{v})}.$

and where $C_{ii^{\prime}}(\cdot)$ denotes the $ii^{\prime}-$cofactor (for
simplicity we have omitted the dependence on $(\boldsymbol{\sigma};S)$ for all
vectors and matrices in the expression). The vectors $\mathbf{k}$ and
$\mathbf{r}$ capture the first and second order reaction to changes in
consumption among groups that split. Let finally

$K_{S}(\boldsymbol{\sigma})\equiv\sum_{i\in
S}m_{i}k_{i}(\boldsymbol{\sigma};S),~{}\textrm{and}~{}R_{S}(\boldsymbol{\sigma})\equiv\sum_{i\in
S}m_{i}r_{i}(\boldsymbol{\sigma};S).$

######  Lemma 1.

For an outcome $(\mathbf{p}^{*},\boldsymbol{\sigma})$ where
$\boldsymbol{\sigma}$ is a stable split in $Q(\mathbf{p^{*}})$, if
$J_{v}(\boldsymbol{\sigma};S)\neq 0$, there is a unique continuous local
selection $q$ such that $q(\mathbf{p^{*}})=\boldsymbol{\sigma}$. Furthermore,
$q$ is $C^{2}$, $D^{\prime}_{j}(p_{j}^{*};q)=K_{S}(\boldsymbol{\sigma})$ and
$D^{\prime\prime}_{j}(p_{j}^{*};q)=R_{S}(\boldsymbol{\sigma})$.

The response of demand to prices is given by the interdependence of groups
that split. Stable splits guarantee that $\bar{S}$ is less sensitive, so that
when the choice of a group is cohesive, it remains unchanged for small price
deviations. The proof uses an implicit function theorem, and is left for the
appendix. Let us omit the dependence on $\boldsymbol{\sigma}$ by writing
$K_{S}$ and $R_{S}$.

######  Definition 2.

A split $\boldsymbol{\sigma}$ is _realizable_ if

  1. 1.

$K_{S}<0$;

  2. 2.

$-\frac{1}{\mathbf{m}\cdot(\mathbf{1}-\boldsymbol{\sigma})}<\dfrac{R_{S}}{2K_{S}^{2}}<\frac{1}{\mathbf{m}\cdot\boldsymbol{\sigma}}.$

A split is realizable if the first and second-order conditions ensure the
split leads to a local maximum for the first stage along the unique continuous
local selection. Note that for realizable splits not to exist in a given game,
a condition much stronger than may appear at first must hold. It would be
necessary that, for every $S^{*}\subseteq\mathcal{G}$ and every
$\boldsymbol{\sigma}$ such that $S(\boldsymbol{\sigma})=S^{*}$, condition $1$
or $2$ is not satisfied. In particular, as $K_{S}$ is the sum of all $k_{i}$
for $i\in S$, consistently breaking condition $1$ imposes a restriction on all
possible combinations. For example, in the linear case, where $R_{S}=0$ and
$K_{S}$ is constant (for each $S$), it is sufficient that at least for one
group $G$ we have $K_{G}<0$. In that case, a profile where only that group
splits is a realizable split. We will discuss this further in the next
section.

Next, we state the main result. Let us first define the following price
function,

$\psi(\boldsymbol{\sigma})=\frac{1}{-K_{S}}\left(\mathbf{m}\cdot\boldsymbol{\sigma},\mathbf{m}\cdot(\mathbf{1}-\boldsymbol{\sigma})\right).$

######  Theorem 1.

Let $\boldsymbol{\sigma}$ be a realizable stable split.
If $\boldsymbol{\sigma}\in Q(\psi(\boldsymbol{\sigma}))$ then
$(\mathbf{p}^{*},q(p_{a},p_{b}^{*}),q(p_{a}^{*},p_{b}))$ is a local SPE with
profit for both firms, where $\mathbf{p^{*}}=\psi(\boldsymbol{\sigma})$ and
$q$ is the continuous local selection associated with the outcome
$(\mathbf{p}^{*},\boldsymbol{\sigma})$.

###### Proof.

Suppose $\boldsymbol{\sigma}$ is a realizable stable split in
$Q(\psi(\boldsymbol{\sigma}))$. Then, Lemma 1 is applicable. This guarantees
that (standard profit derivation along $q$) $\mathbf{p}^{*}$ is a critical
point. As it is realizable, conditions $1$ and $2$ of Definition 2 guarantee
it is a local maximum for both firms. As it is a second-stage Nash, it is a
local SPE+. ∎

##  3 Discussion and examples

Two natural questions arise from Theorem 1 (essentially amounting to whether
it is empty): $(i)$ does a given game $\Gamma$ have realizable splits? $(ii)$
is at least one of these splits stable and a solution to
$\boldsymbol{\sigma}\in Q(\psi(\boldsymbol{\sigma}))$?

The answer to $(i)$ depends on the marginal properties of $v$ and is the main
focus of this work. Question $(ii)$ amounts to whether there is a realizable
stable split solution to

$v_{i}(\boldsymbol{\sigma})=\mathbf{m}\cdot(2\boldsymbol{\sigma}-\mathbf{1})/K_{S}(\boldsymbol{\sigma}),~{}\forall
i\in S.$ (1)

The right-hand side of Equation 1 is the difference of equilibrium prices
$p_{a}^{*}-p_{b}^{*}$. Satisfying the equation means $\boldsymbol{\sigma}$ is
a second-stage NE for $\mathbf{p^{*}}$. Question $(i)$ ensures first-stage
(pricing game) solutions, while $(ii)$ second-stage (consumption game)
solutions compatible with $(i)$. This section of the work is divided into two
subsections where we discuss the above two questions through models in the
literature.

###  3.1 (i) Realizable splits.

####  3.1.1 One group, $g=1$.

Let us start by looking at games where there is only one group $\Gamma(v;1)$.
All NE splits are stable. Now, we have $K(\sigma)=m(v^{\prime}(\sigma))^{-1}$
and if $v^{\prime\prime}(\sigma)\neq 0$, then
$R(\sigma)=\frac{K^{2}v^{\prime}(\sigma)}{2v^{\prime\prime}(\sigma)}$; if
$v^{\prime\prime}(\sigma)=0$, then $R(\sigma)=0$. As such when $v$ is non-
decreasing (positive network effects) there are no realizable splits. If $v$
is decreasing for at least some point (negative network effects), realizable
splits may exist. In particular, if $v$ is linear, $K$ and $R$ are constant
and $R=0$. If $K<0$ all splits are realizable.

######  Example 1 ($\Gamma(v;1)$; Grilo et al. (2001)).

The authors consider the following case: $v^{j}(d^{j})=\alpha
d^{j}-\beta(d^{j})^{2}-t(x-x_{j})^{2}$ with mass of consumers $m$. Let us
consider the case with no product differentiation $t=0$. Then,
$v(\sigma)=(2\sigma-1)(\alpha m-\beta m^{2})$, $K=[2(\alpha-\beta m)]^{-1}$
and $R=0$. As such, whenever $\alpha<\beta m$ every split is realizable, that
is, when negative effects overcome positive ones. In that case, there is a
unique realizable split solution to Equation 1, $\sigma=1/2$. If there are
only positive network effects, $\beta=0$, there are no realizable splits.

Some variations with only one group include a dependence of $\alpha$ on the
firms. Tolotti and Yepez (2020), in the case of $t=0$, discuss
$v^{j}(d^{j})=\alpha^{j}d^{j}$. In this case
$v(\sigma)=(\alpha^{a}+\alpha^{b})m\sigma-\alpha^{b}m$. Hence,
$K=(\alpha^{a}+\alpha^{b})^{-1}$. Again, no realizable split unless negative
effects overcome positive effects. Assuming $\alpha^{a}+\alpha^{b}<0$, all
splits are realizable and stable. However, there is a unique solution to
Equation 1, $\sigma=\frac{\alpha^{a}}{\alpha^{a}+\alpha^{b}}$ leading to
equilibrium prices $p_{j}=-m\alpha^{j}$, so it is necessary that
$\alpha^{j}<0$, for $j=a,b$.

The general idea is that, with $g=1$, if positive network effects dominate,
ceteris paribus, there are no realizable splits.

####  3.1.2 Multilinear case, $g>1$.

Consider the multilinear extension for more groups,

$v_{i}^{j}(\mathbf{d}^{j})=\sum_{i^{\prime}=1}^{g}\alpha_{ii^{\prime}}^{j}d_{i^{\prime}}^{j}$

where $\alpha_{ii^{\prime}}^{j}\in\mathbb{R}$ represents how a member of group
$i$ is influenced by a member of group $i^{\prime}$ when both choose firm $j$.
Let us define
$w_{ii^{\prime}}\equiv\alpha_{ii^{\prime}}^{a}+\alpha_{ii^{\prime}}^{b}$. Note
that

$v_{i}(\boldsymbol{\sigma})=\sum_{i^{\prime}=1}^{g}m_{i^{\prime}}w_{ii^{\prime}}\sigma_{i^{\prime}}-\sum_{i^{\prime}=1}^{g}\alpha_{ii^{\prime}}^{b}m_{i^{\prime}}.$

Thus, $\frac{\partial
v_{i}}{\partial\sigma_{i^{\prime}}}=m_{i^{\prime}}w_{ii^{\prime}}$. One of the
advantages of the linear case is that $K_{S}(\boldsymbol{\sigma})$ and
$R_{S}(\boldsymbol{\sigma})$ do not depend directly on $\boldsymbol{\sigma}$
but only on $S$, the groups that split. Furthermore, $R_{S}=0$ for all $S$. We
separate $m_{i}$ from $\alpha$ because $K_{S}$ does not depend on $m_{i}$
either111111This is because the product of $m_{i}$’s cancel out in the
numerator and denominator expression for $K$. However, each $k_{i}$ depends on
$\mathbf{m}$. This will be clear in Example 2., and it is useful to work with
the matrix $W\equiv W(\mathcal{G})\equiv(w_{ii^{\prime}})$. As any $K_{S}$ is
determined by a submatrix of $W$, a multilinear game $\Gamma$ is determined by
$W$ and we can identify the specific game considered by $\Gamma(W;g)$.

######  Example 2 ($\Gamma(W;2)$).

An advantage of $g=2$ is that there are only two types of splits. _Total_ ,
where both groups split, and _singular_ , where only one group splits. Let
$\mathcal{G}=\\{G_{1},G_{2}\\}$, $\mathbf{m}=(m_{1},m_{2})$ and

$W\equiv\begin{pmatrix}w_{11}&w_{12}\\\ w_{21}&w_{22}\end{pmatrix}.$

For total splits,

$K_{\mathcal{G}}=\frac{(w_{11}+w_{22})-(w_{12}+w_{21})}{w_{11}w_{22}-w_{12}w_{21}}.$

For singular splits, $K_{G_{i}}=(w_{ii})^{-1}$. As $R_{S}=0$, realizable
splits require only that $K_{S}<0$. However, recall that $K_{S}=\sum_{i\in
S}m_{i}k_{i}(S)$. Here,

$k_{1}=\frac{(w_{22}-w_{12})m_{2}}{(w_{11}w_{22}-w_{12}w_{21})m_{1}m_{2}},~{}~{}k_{2}=\frac{(w_{11}-w_{21})m_{1}}{(w_{11}w_{22}-w_{12}w_{21})m_{1}m_{2}},$

which need not necessarily be both negative. Moreover, it shows why
$K_{\mathcal{S}}$ is independent of $\mathbf{m}$

Although realizable singular splits require negative network effects, i.e.,
negative entries in $W$, it is possible to have realizable (total) splits even
when there are only positive entries. For example, consider

$\underline{W}=\begin{pmatrix}1&2\\\ 3&5\end{pmatrix},$

Note that $k_{1}=-3/m_{1}$, $k_{2}=2/m_{2}$ and $K_{\mathcal{G}}=-1$. Hence
total splits are realizable (the unique type of realizable splits), and this
is independent of $\mathbf{m}$. If we change the sign in all entries on
$\underline{W}$, then the results reverse. Total splits would not be
realizable, but singular splits would.

######  Example 3 (Amaldoss and Jain (2005)).

The following case is considered (here we have assumed no product
differentiation): $\mathcal{G}=\\{l,c\\}$, group sizes are $m_{l}=\beta$ and
$m_{c}=1-\beta$ for $\beta\in(0,1)$, and

$v_{l}^{j}=-\lambda_{l}d^{j},~{}~{}~{}v_{c}^{j}=\lambda_{c}d^{j},$

where $\lambda_{G}>0$. Group $l$ are called snobs/ leaders and group $c$
conformists/ followers. Observe that although there are two groups, both look
at the same aggregate. Let’s look at the following modified game
$\Gamma_{\delta}(W,2)$ where

$W=\begin{pmatrix}-\lambda_{l}&-\lambda_{l}\\\
\lambda_{c}+\delta&\lambda_{c}\end{pmatrix}.$

Network effects in Amaldoss and Jain (2005) correspond to $\delta=0$. The
authors use a bound on conformism coupled with an assumption on the
distribution of preferences to ensure a SPE+ (we discuss this further in the
next subsection). The case of no product differentiation corresponds to a game
$\Gamma_{0}(W,2)$. In this case, total splits are not realizable, because the
$\det(W)=0$ so $K_{\mathcal{G}}$ is not defined. However,
$K_{l}=(-\lambda_{l})^{-1}$ and singular splits where only snobs ($l$) split
are realizable. This is similar to a game with only snobs.

In the modified game we propose, i.e., when $\delta\neq 0$, conformists
distinguish between snobs and other conformists. So there are two aggregate
variables in play as $d^{j}$ becomes $\mathbf{d}^{j}=(d_{l}^{j},d_{c}^{j})$.
If $\delta>0$ conformists prefer snobs (leaders), if $\delta<0$, conformists
prefer other conformists (to maintain the game idea we assume
$\delta>-\lambda_{c}$). Note that, for $\delta\neq 0$, we have
$K_{\mathcal{G}}=(-\lambda_{l})^{-1}$. So, total splits are now also
realizable. Interestingly, realizability will not depend on $\delta$ nor on
$\lambda_{c}$, the influence among conformists. However, although
$K_{\mathcal{G}}=K_{l}$, outcomes will differ for the two types of split. For
singular splits, if all conformists choose $a$,
$p_{a}^{*}=\lambda_{l}(1-\beta\sigma_{l})$; and if all conformists choose $b$,
$p_{a}^{*}=\lambda_{l}\beta\sigma_{l}$. For total splits,
$p_{a}^{*}=\lambda_{l}(\beta\sigma_{l}+(1-\beta)\sigma_{c})$. In both cases,
prices are proportional to the degree of snobbishness.

######  Example 4 (Armstrong (2006)).

The following structure is assumed,

$W=\begin{pmatrix}0&w_{12}\\\
w_{21}&0\end{pmatrix},~{}~{}\textrm{thus}~{}~{}K_{\mathcal{G}}=\frac{w_{12}+w_{21}}{w_{12}w_{21}},$

where $w_{12},w_{21}>0$. There are no realizable splits. The Hotelling
specification is used, with costs $t_{1}$ and $t_{2}$ for each group, which
coupled with the assumption that $16t_{1}t_{2}>(w_{12}+w_{21})^{2}$ produces
SPE+. That is when network (externality) parameters are small compared to
differentiation ones.121212Note that this is consistent with Armstrong’s
model. The author considers $m_{1}=m_{2}=1$. We get
$p_{a}^{*}=(\sigma_{1}+\sigma_{2})\frac{-w_{21}w_{12}}{w_{21}+w_{12}}<0$, thus
no equilibria, and exactly the price expression presented there if one
considers no transportation costs (observe that $w_{ij}=2\alpha_{j}$ for the
parameters $\alpha_{j}$ considered there).

Suppose we change the structure, for example, by allowing intra-group $2$
influence in the following way,

$W=\begin{pmatrix}0&w_{12}\\\
w_{21}&w_{12}+w_{21}+\delta\end{pmatrix},~{}~{}\textrm{then}~{}~{}K_{\mathcal{G}}=\frac{\delta}{-w_{12}w_{21}},$

for some $\delta>0$. Total splits are now realizable. Note that it is
necessary that the intra-group influence introduced overcomes the sum of
cross-group.

However, if one wishes a model with a zero diagonal matrix (no intra-group
influences), then at least 3 groups are necessary to guarantee the existence
of realizable splits. Suppose you have 3 groups and consider the following
structure,

$W=\begin{pmatrix}0&2r&r\\\ z&0&\frac{rz}{4r+z}-\varepsilon\\\
\frac{z}{2}&0&0\\\ \end{pmatrix},$

where $r,z>0$ and $0<\varepsilon<\frac{rz}{4r+z}$. The entries, and thus
network effects, are all positive. However, $K_{\mathcal{G}}<0$ and the total
split is realizable.

######  Example 5 (Adjacency matrix.).

In some models, a graph is given a priori representing the connections among
groups. For example Banerji and Dutta (2009) consider (for the case of
incompatible products),

$v_{i}^{j}(\mathbf{d}^{j})=\sum_{i^{\prime}\in N(i)}d_{i^{\prime}}^{j},$

where $N(i)$ are the neighbors of $i$, including itself. This is an instance
of the multilinear case, where
$\alpha_{ii^{\prime}}^{a}=\alpha_{ii^{\prime}}^{b}\equiv\alpha_{ii^{\prime}}$;
$\alpha_{ii^{\prime}}=\alpha_{i^{\prime}i}$; and
$\alpha_{ii^{\prime}}\in\\{0,1\\}$. The latter reflects whether groups $i$ and
$i^{\prime}$ are connected. Let $A$ be the adjacency matrix of the given
(unweighted and undirected) graph. As $w_{ii^{\prime}}=2\alpha_{ii^{\prime}}$,
$W=2A$, thus $\det(W)=2^{g}\det(A)$, and
$C_{ii^{\prime}}(W)=2^{(g-1)}C_{ii^{\prime}}(A)$. Hence, given $K_{S}^{*}$ in
a game $\Gamma(W,g)$, for the game $\Gamma(A,g)$ we have $2K_{S}=K_{S}^{*}$.
As such, to check if realizable splits exist, it suffices to study the
adjacency matrix.131313Note that the same holds for
$\mathbf{k}=2\mathbf{k}^{*}$

Naturally, whether realizable splits exist depends on the type of network
under consideration. In particular, for the total split we need to look at the
whole adjacency matrix, for partial splits, it is necessary to look at the
adjacency matrix of the subgraph induced by $S\subseteq\mathcal{G}$. For
example, the complete network has $\det(W_{complete})=0$, and as any induced
subgraph is also complete, $K_{S}$ is not defined for any $S$ and no
realizable split exists, whether partial or total. For the star network with
loops, $K_{\mathcal{G}-star}=\frac{1}{2}$ (for $\Gamma(W_{star};g)$), thus
total splits are not realizable.141414This can be shown by induction, but we
omit the proof here. A subgraph is either again a star network if the central
node splits (is in $S$), or a graph composed by loops only. In the latter, the
relevant matrix has only 1’s in the diagonal, leading to $K_{S}=\frac{1}{2}$.
So for the star network there are no realizable splits. In Figure 1 is an
example of a network where total splits are realizable. We have made a small
modification from a star network.

$\displaystyle A=\begin{pmatrix}0&0&1&0&0\\\ 0&1&0&0&1\\\ 1&0&1&0&1\\\
0&0&0&1&1\\\ 0&1&1&1&1\end{pmatrix}~{}~{}~{}~{}\leavevmode\hbox
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Figure 1: A network inducing realizable total splits for $\Gamma(2A;5)$.
Considering $W=2A$ we get $K_{\mathcal{G}}=-0.5$ and
$\mathbf{k}=(-1/m_{1},-0.5/m_{2},0.5/m_{3},-0.5/m_{4},1/m_{5})$.

The example in Figure 1 is not unique, nor specific to $g=5$. We did however
need to increase the number of groups to $5$ to find a solution. If $g>5$
solutions exist too and become easier to find. Moreover, $A$ could be the base
for a realizable partial split in a game with $g>5$. We have chosen a
symmetric matrix to fall within the assumption of an undirected (and
unweighted) network. However, as we will see, being unweighted forces an
increase in dimension to find an example, while being undirected produces
symmetric outcomes.

The fact that realizable splits exist in such cases might appear to contradict
the result in Banerji and Dutta (2009). Their Lemma 1 states no split can be
part of an SPE+. However, the result comes from their assumption A1 that
implies $k_{i}\leq 0$ for all $i$ and all $S$. The assumption that $K_{S}<0$
can only be obtained this way, eliminates all realizable splits in their class
of games. While it may be a reasonable economic assumption, from a
mathematical point of view it is crucial to be aware that it is in fact a
strong imposition, which eliminates a rich set of equilibria where both firms
have profit. In particular, it is an imposition on how firms imagine the
demand response. The implicit function theorem provides unique local
selections for stable splits, where it may happen $k_{i}>0$ for some $i\in S$,
but as long as $K_{S}<0$ and the split is realizable, an equilibrium is
possible. That is, equilibrium selections exist where it does not hold
$k_{i}\leq 0$ for all $i$.

###  3.2 (ii) NE solutions.

The second question arising from Theorem 1 is whether a realizable split will
produce a SPE outcome in a game $\Gamma(v;g)$. The answer is: if it is stable
and a solution to Equation 1. Whether for a given game the system induced by
Equation 1 has such a solution is a more standard existence question, but out
of scope here. Our objective is to discuss how the structure of network
effects among groups affect SPE, even if members of the same group are
indistinguishable. In this section our goal is to show that structures
producing realizable splits, also produce SPE for some classes of games.

Given $\boldsymbol{\tau}\in\mathbb{R}^{g}$ and $\varepsilon>0$, let us define

$\underline{v}_{i}(\boldsymbol{\sigma})\equiv\underline{v}_{i}(\boldsymbol{\sigma};[\boldsymbol{\tau},\varepsilon])\equiv\left\\{\begin{array}[]{lr}v_{i}(\boldsymbol{\sigma})-\tau_{i}+\varepsilon&\text{if}~{}~{}~{}\sigma_{i}=1\\\
v_{i}(\boldsymbol{\sigma})-\tau_{i}&\text{if}~{}~{}~{}i\in
S(\boldsymbol{\sigma})\\\
v_{i}(\boldsymbol{\sigma})-\tau_{i}-\varepsilon&\text{if}~{}~{}~{}\sigma_{i}=0\end{array}\right..$

A split is realizable in $\Gamma(v;g)$ if, and only if, it is realizable in
$\Gamma(\underline{v};g)$.

######  Remark 1.

Let $\boldsymbol{\sigma}$ be a realizable split of a game $\Gamma(v;g)$ and
$\mathbf{p^{*}}=\psi(\boldsymbol{\sigma})$. There is a unique
$\boldsymbol{\tau}$ such that for every $\varepsilon>0$,
$(\mathbf{p^{*}},\boldsymbol{\sigma})$ is a SPE outcome of the game
$\Gamma(\underline{v};g)$. Namely,

$\tau_{i}=v_{i}(\boldsymbol{\sigma})-\mathbf{m}\cdot(2\boldsymbol{\sigma}-\mathbf{1})/K_{S}(\boldsymbol{\sigma}).$

It is a remark because we are simply modifying the game by readjusting the
system, so that its solution is in $[0,1]^{g}$ and interior (stable split).
Network effects are the same independently of
$[\boldsymbol{\tau},\varepsilon]$, which is the focus of this work. Note that,
for example for $i\in S$, $\tau_{i}$ can be seen as coming from
$\tau_{i}^{a}-\tau_{i}^{b}$ for each group, representing product
differentiation, willingness to pay, or other constant added to the utility of
consumers (similarly for $i\in\bar{S}$ if $\varepsilon$ is included). In fact,
the remark defines a class of games $\Gamma(\underline{v};\boldsymbol{\tau})$
for every $\varepsilon>0$. If we write $\varepsilon$ dependent on $i$, the
same remark holds, and it would be natural to see it as a distribution of
preferences that induces $\underline{v}$ where some groups are the indifferent
consumers. The more standard approach, however, is to consider a distribution
for each group and $v_{i}$ as the value of consumption for the indifferent
consumer of that group. Here we are always assuming individuals of the same
group are indistinguishable a priori. Let us just emphasize what the remark
states: every realizable split is a SPE+ outcome for a class of games
$\Gamma(\underline{v};\boldsymbol{\tau})$ with network effects characterized
by $v$.

Which split is a SPE for the class $\boldsymbol{\tau}=\mathbf{0}$, if it
exists, depends on $v$. In the class of multilinear games, discussed before,
all total splits are realizable when $K_{\mathcal{G}}<0$. However, the
solution to Eq.1 is unique (it might be a real solution for the game or not),
so only one realizable split will lead to an SPE. The fact that it exists and
is unique comes from the multilinearity of the determinant operator (and
because it is not zero for realizable splits). The system of $g$ equations
induced by Eq.1 can be rewritten as the following equation for each group $i$,

$\sum_{j}\left(w_{ij}-\frac{2}{K_{\mathcal{G}}}\right)m_{j}\sigma_{j}=\sum_{j}\left(\alpha_{ij}^{b}-\frac{1}{K_{\mathcal{G}}}\right)m_{j}.$
(2)

A gross (but educated) guess (and hence, the solution if it works) is

$\sigma_{j}=\frac{1}{2}\frac{\alpha_{ij}^{b}-\frac{1}{K_{\mathcal{G}}}}{\frac{w_{ij}}{2}-\frac{1}{K_{\mathcal{G}}}}.$
(3)

It generally doesn’t work. However, recalling that
$w_{ij}\equiv\alpha_{ij}^{a}+\alpha_{ij}^{b}$, it shows the importance of
$\alpha_{ij}^{a}\neq\alpha_{ij}^{b}$ in producing price dispersion and
asymmetric demand (when no other differentiation is present). We can write the
following proposition.

######  Proposition 1.

In a multilinear game, given a group $G_{j}$, if for all $G_{i}$ it holds
$\alpha_{ij}^{a}=\alpha_{ij}^{b}$, then, in a total split SPE+ outcome,
$\sigma_{j}=0.5$.

###### Proof.

If Equation 3 is independent of $i$ then it is the solution to the system
generated by Equation 2. If $\alpha_{ij}^{a}=\alpha_{ij}^{b}$ then
$\sigma_{j}=0.5$. ∎

The proposition can be rephrased in the following way: If a member of group
$G$ has the same influence on all other players with whom it is connected
(including those of its own group), then, in equilibrium, $G$ must split in
half. The observation emphasizes the importance of (asymmetric) weights
relative to the underlying network structure and how imposing strong symmetry
properties leads to symmetric outcomes.

######  Example 6.

In the class of games based on an adjacency matrix, discussed in Example 5,
$\alpha_{ij}^{a}=\alpha_{ij}^{b}\in\\{0,1\\}$. It is also assumed the network
is undirected, i.e., we get $w_{ij}=2\alpha_{ij}=2\alpha_{ji}$. As such, the
guess leads to the solution, $\boldsymbol{\sigma}=\mathbf{0.5}$ (which is
independent of $\mathbf{m}$, $g$ and the network as long as
$K_{\mathcal{G}}<0$). Let $M=\sum_{i}m_{i}$. Equilibrium prices are
$p_{a}^{*}/M=p_{b}^{*}/M=1$.

Naturally, if one wishes an outcome where prices and demand differ while
keeping the network undirected, it is possible to choose $\boldsymbol{\tau}$
so that the modified game $\Gamma(\underline{v};\boldsymbol{\tau})$ has the
desired equilibrium.

##  4 Conclusion

How demand responds to prices determines the outcome of price competition.
However, demand is an aggregate of consumption decisions. When different
groups interact and influence each other, their response need not be
homogeneous and mimic the aggregate behavior. It is thus not necessary that
consumption jumps from corner solution to corner solution when network effects
induce bandwagon responses to price changes. Nevertheless, a plethora of
possibilities for the composition of the aggregate need not bring uncertainty.
Network effects produce an interlocking of decisions that stabilizes price
competition. These are the main points of this work.

We have considered the partition into groups given a priori, but it can be
endogeneized. One way is as a classification tool, based on the
characteristics of consumers that produce the same set of equilibria (see, for
example, Soeiro et al. (2015)). Another way is to consider the partition an
assessment by the firms, which could have an associated cost investment
decision. It would be a possible extension of the work: what happens when
firms use different partitions to decide prices?

If it is a choice of information structure, it seems natural that the
appropriate partition into groups lies somewhere between no information (just
aggregate behavior) or complete information (network effects at the individual
level). Naturally, the latter may be more accurate, but either unfeasible or
with a high cost for firms. If a firm invests in studying individual-level
network effects, it is natural it wants to price discriminate to build value
for that information. So the results obtained seem appropriate for uniform
pricing: at least two groups when the asymmetry of weights and direction is
allowed, or at least five groups for undirected and unweighted structures.

Two simplifying assumptions may concern the reader: mandatory consumption and
no costs for firms. The essential mathematical tool used is the implicit
function theorem. The result is local and, the function behind it is the
difference between buying from one or the other firm. With a third option,
this can be extended, for each group, to three sets by including the following
two splits: difference of buying or not from each firm. Removing mandatory
consumption may change some outcomes, but it will fit the implicit function
theorem without a particular problem. Introducing a cost function for firms is
even less problematic, as we have based results on second-stage interactions.
A fixed cost may eliminate some equilibria (if profits don’t cover them), and
a linear cost function provokes a translation in prices, but results follow
straightforwardly. A more general cost function will have the usual
difficulties and need their counterpart assumptions.

##  Appendix A Proof of Lemma 1

For an outcome $(\mathbf{p}^{*},\boldsymbol{\sigma})$ where
$\boldsymbol{\sigma}$ is a stable split in $Q(\mathbf{p^{*}})$, if
$J_{v}(\boldsymbol{\sigma};S)\neq 0$, there is a unique continuous local
selection $q$ such that $q(\mathbf{p^{*}})=\boldsymbol{\sigma}$. Furthermore,
$q$ is $C^{2}$, $D^{\prime}_{j}(p_{j}^{*};q)=K_{S}(\boldsymbol{\sigma})$ and
$D^{\prime\prime}_{j}(p_{j}^{*};q)=R_{S}(\boldsymbol{\sigma})$.

###### Proof.

Let $(\mathbf{p}^{*},\boldsymbol{\sigma}^{*})$ be an outcome where
$\boldsymbol{\sigma}^{*}$ is a stable split in $Q(\mathbf{p^{*}})$. Recall
that it holds

  1. (i)

if $\sigma_{i}=1$ then $v_{i}(\boldsymbol{\sigma}^{*})>p_{a}^{*}-p_{b}^{*}$;

  2. (ii)

if $\sigma_{i}=0$ then $v_{i}(\boldsymbol{\sigma}^{*})<p_{a}^{*}-p_{b}^{*}$;

  3. (iii)

if $0<\sigma_{i}<1$ then $v_{i}(\boldsymbol{\sigma}^{*})=p_{a}^{*}-p_{b}^{*}$.

Hence, for groups that satisfy $(i)$ or $(ii)$, there is some neighborhood
$N(\mathbf{p}^{*},\boldsymbol{\sigma}^{*})$ where the best response of their
members is constant. Recall that $l\equiv\\#S$ denotes the number of groups
that split, hence, the number of groups that satisfy $(iii)$. Note that $l\geq
1$ (stable split, see Definition 1). Index the groups that split by
$S(\boldsymbol{\sigma}^{*})\equiv\\{1,\ldots,l\\}$. Denote by
$\mathbf{s}^{*}\equiv(s_{1}^{*},\ldots,s_{l}^{*})$ their respective
coordinates in $\boldsymbol{\sigma}^{*}$ and let
$\boldsymbol{\sigma}^{*}_{\bar{S}}$ denote the profile of the remaining
groups. Consider now the function induced by the utility of each consumer in
any group in $S$ and given by

$\Delta u_{i}(\mathbf{p},\mathbf{s})\equiv\Delta
u_{i}(\mathbf{p},\mathbf{s};\boldsymbol{\sigma}^{*}_{\bar{S}})\equiv
v_{i}(\boldsymbol{\sigma})-\Delta p$

where $\Delta p\equiv p_{a}-p_{b}$, and define

$\displaystyle\Delta U:(\mathbb{R}_{0}^{+})^{2}\times(0,1)^{l}$
$\displaystyle\to$ $\displaystyle\mathbb{R}^{l}$
$\displaystyle(\mathbf{p},\mathbf{s})$ $\displaystyle\mapsto$
$\displaystyle\Delta u_{1}(\mathbf{p},\mathbf{s}),\ldots,\Delta
u_{l}(\mathbf{p},\mathbf{s}).$

Observe that $\Delta U(\mathbf{p}^{*},\mathbf{s}^{*})=\mathbf{0}$. By $(iii)$
and because the best response for members of groups in $\bar{S}$ is locally
constant, any outcome
$(\mathbf{p},\mathbf{s},\boldsymbol{\sigma}^{*}_{\bar{S}})$ in the
neighborhood $N(\mathbf{p}^{*},\boldsymbol{\sigma}^{*})$ such that $\Delta
U(\mathbf{p},\mathbf{s})=\mathbf{0}$, is a Nash equilibrium for consumers
given $\mathbf{p}$. Note now that $\Delta U$ is $C^{2}$ and $J_{\Delta
U,\mathbf{s}}(\mathbf{p}^{*},\mathbf{s}^{*};\boldsymbol{\sigma}^{*}_{\bar{S}})=J_{v}(\boldsymbol{\sigma}^{*};S)$.
Therefore, if $\det\left[J_{v}(\boldsymbol{\sigma}^{*};S)\right]\neq 0$, using
the Implicit Function Theorem, there is a product neighborhood $N_{p}\times
N_{s}\subset(\mathbb{R}_{0}^{+})^{2}\times(0,1)^{l}$, containing
$(\mathbf{p}^{*},\mathbf{s}^{*})$, and a unique $C^{2}$ function
$\phi:N_{p}\to N_{s}$ such that $\Delta
U(\mathbf{p},\phi(\mathbf{p}))=\mathbf{0}$. Furthermore, in that neighborhood

$\sum_{i=1}^{l}\dfrac{\partial\Delta
U}{\partial\phi_{i}(\mathbf{p})}(\mathbf{p},\phi(\mathbf{p}))\dfrac{\partial\phi_{i}}{\partial
p_{a}}(\mathbf{p})=\dfrac{\partial\Delta U}{\partial
p_{a}}(\mathbf{p},\phi(\mathbf{p})).$

As in the same neighborhood $\Delta u_{i}(\mathbf{p},\boldsymbol{\sigma})=0$,
we get $v_{i}(\boldsymbol{\sigma})=\Delta p$, and

$J_{v}(\phi(\mathbf{p});S)\dfrac{\partial\phi}{\partial
p_{a}}(\mathbf{p})=\mathbf{1}_{l\times 1}.$

Using the Cramer rule,

$\dfrac{\partial\phi_{i}}{\partial
p_{a}}(\mathbf{p})=\dfrac{\det\left[J^{(i)}_{v}(\phi(\mathbf{p});S)\right]}{\det\left[J_{v}(\phi(\mathbf{p});S)\right]},$

where $J^{(i)}_{\Delta v}(\phi(\mathbf{p});S)$ is obtained by replacing the
entries of column $i$ with $1$ in $J_{v}(\boldsymbol{\sigma}^{*};S)$.

Therefore, for the outcome $(\mathbf{p}^{*},\boldsymbol{\sigma})$, given a
neighborhood $N$ and a continuous local selection $q$, demand is

$D_{a}(p_{a};p_{b},q)=\sum_{i\in\mathcal{G}}m_{i}q_{i}(\mathbf{p})=\sum_{i\in
S}m_{i}\phi_{i}(\mathbf{p})+\underline{D_{a}}(\boldsymbol{\sigma}^{*}_{\bar{S}}),$

where $\underline{D_{a}}(\boldsymbol{\sigma}^{*}_{\bar{S}})$ is locally a
constant: the number of groups that choose $a$ (i.e., satisfy $(i)$). As such
in that neighborhood (because groups that don’t split do not change)

$D^{\prime}(p_{a};p_{b},q)=\dfrac{\partial D_{a}}{\partial
p_{a}}(\phi(\mathbf{p}))=\sum_{i}m_{i}\dfrac{\det\left[J^{(i)}_{v}(\phi(\mathbf{p});S)\right]}{\det\left[J_{v}(\phi(\mathbf{p});S)\right]}.$
(4)

Observe that
$\det\left[J^{(i)}_{v}(\phi(\mathbf{p});S)\right]=\sum_{i^{\prime}}C_{i^{\prime}i}(J_{v}(\phi(\mathbf{p});S))$
and the right-hand side of (4) is $K_{S}$.

Analogously, for the second derivative,

$D^{\prime\prime}_{a}=m_{i}\frac{\partial^{2}q_{i}}{\partial
p_{a}^{2}},~{}~{}i\in S,$

which is implicitly given by $v_{i}(q(p_{a},p_{b}^{*}))=\Delta p$. So for all
$i\in S$,

$\sum_{i^{\prime}\in S}\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\frac{\partial q_{i^{\prime}}}{\partial p_{a}}=\mathbf{1}.$

Therefore, for all $i$

$\sum_{i^{\prime}\in S}\left[\frac{\partial}{\partial
p_{a}}\left(\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\right)\frac{\partial q_{i^{\prime}}}{\partial
p_{a}}+\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\frac{\partial^{2}q_{i^{\prime}}}{\partial
p_{a}^{2}}\right]=\mathbf{0}.$

As

$\frac{\partial}{\partial p_{a}}\left(\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\right)=\sum_{j\in S}\frac{\partial^{2}v_{i}}{\partial
q_{i^{\prime}}\partial q_{j}}\frac{\partial q_{j}}{\partial p_{a}},$

we get

$\sum_{i^{\prime}\in S}\left[\left(\sum_{j\in
S}\frac{\partial^{2}v_{i}}{\partial q_{i^{\prime}}\partial
q_{j}}\frac{\partial q_{j}}{\partial p_{a}}\right)\frac{\partial
q_{i^{\prime}}}{\partial p_{a}}+\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\frac{\partial^{2}q_{i^{\prime}}}{\partial
p_{a}^{2}}\right]=\mathbf{0},$

and as such

$\sum_{i^{\prime}\in S}\frac{\partial v_{i}}{\partial
q_{i^{\prime}}}\frac{\partial^{2}q_{i^{\prime}}}{\partial
p_{a}^{2}}=-\sum_{i^{\prime}\in S}\left[\left(\sum_{j\in
S}\frac{\partial^{2}v_{i}}{\partial q_{i^{\prime}}\partial
q_{j}}\frac{\partial q_{j}}{\partial p_{a}}\right)\frac{\partial
q_{i^{\prime}}}{\partial p_{a}}\right].$

The system on the left can be written as

$J_{v}(\boldsymbol{\sigma};S)\nabla^{2}q,$

and the system on the right can be written as

$\mathbf{k}H_{v_{i}}\mathbf{k}^{T},$

which, using the Cramer rule as before, leads to $R_{S}$ (instead of replacing
a column by 1’s we use the above). ∎

## References

  * (1)
  * Amaldoss and Jain (2005) Amaldoss, W., Jain, S. 2005. Pricing of conspicuous goods: A competitive analysis of social effects. Journal of Marketing Research. 42 (1), 30–42.
  * Armstrong (2006) Armstrong, M. 2006. Competition in two-sided markets. The RAND Journal of Economics. 37 (3), 668–691.
  * Banerji and Dutta (2009) Banerji, A., Dutta, B. 2009. Local network externalities and market segmentation. International Journal of Industrial Organization. 27 (5), 605–614.
  * Becker (1991) Becker, G. S. 1991. A note on restaurant pricing and other examples of social influences on price. Journal of Political Economy. 99 (5), 1109–1116.
  * Caplin and Nalebuff (1991) Caplin, A., Nalebuff, B. 1991. Aggregation and imperfect competition: On the existence of equilibrium. Econometrica. 59 (1), 25.
  * Grilo et al. (2001) Grilo, I., Shy, O., Thisse, J.-F. 2001. Price competition when consumer behavior is characterized by conformity or vanity. Journal of Public Economics. 80 (3), 385–408.
  * Kaldor (1934) Kaldor, N. 1934. Mrs. robinson’s "economics of imperfect competition". Economica. 1 (3), 335.
  * Karni and Levin (1994) Karni, E., Levin, D. 1994. Social attributes and strategic equilibrium: A restaurant pricing game. Journal of Political Economy. 102 (4), 822–840.
  * Soeiro et al. (2015) Soeiro, R., Mousa, A., Pinto, A. A. 2015. Externality effects in the formation of societies. Journal of Dynamics and Games. 2 (3,4), 303–320.
  * Soeiro and Pinto (2021) Soeiro, R., Pinto, A. A. 2021. Negative network effects and asymmetric pure price equilibria. Portuguese Economic Journal.
  * Tolotti and Yepez (2020) Tolotti, M., Yepez, J. 2020. Hotelling-bertrand duopoly competition under firm-specific network effects. Journal of Economic Behavior and Organization. 176, 105–128.