%Paper: ewp-io/9309001
%From: Neil Gandal <GANDAL@VM.TAU.AC.IL>
%Date: Mon, 13 Sep 93 13:57:19 IST




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\pagestyle{myheading}\markright{NEVER WORKS}
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\begin{titlepage} \vspace*{0.3in}


\begin{center}
{\Large\bf  Experimentation and Learning with Network Effects}  \\
\vspace{.4in} Arthur Fishman \\ Tel Aviv University \\ and \\ Neil
Gandal\\ Tel Aviv University \\


\vspace{0.2in}\today \\
\vspace{.2in}{\bf Abstract}\\
\end{center}
%\marginpar{draft=dur17.tex}
This paper considers learning in an imperfectly competitive
setting.  By allowing an opponent a ``head start," unsuccessful
unilateral experimentation may jeopardize future sales and profits.
We show that even in the absence of spillover and signalling
effects, competition can inhibit the scope of learning, relative to
a monopoly.

\vspace{4 mm}

\noindent {\bf JEL Classification Numbers:} D83

\vspace{0.1in}\noindent
\end{titlepage}

\baselineskip=0.2in
\section{Introduction}
\hspace*{6 mm}

Consider a seller who is imperfectly informed about market demand.
She can improve her information by experimenting with prices.
However, experimentation is costly: charging a high price generates
valuable information, but will reduce current sales and profits if
demand is actually low.  Therefore the seller's optimal learning
policy must balance the costs of experimentation against the
potential gains.  In a seminal paper, Rothschild (1974) examined
the optimal dynamic learning strategy of a monopoly seller.  He
showed that the cost of experimentation will typically inhibit
complete learning, even though through sufficiently extensive
experimentation, the true state of demand can be learned with
arbitrarily high probability.  More recent studies of the optimal
learning problem include McLennan (1984), Easley and Kiefer
(1988), Creane (1989), Mirman, Samuelson, and Urbano (1989) and
Aghion, Bolton, Harris and Jullien (1991).  All of these papers
considered a monopoly seller, i.e., a single optimizing decision
maker.

In this paper, we consider equilibrium learning in an imperfectly
competitive setting.  In our setting, learning involves in
addition to the costs considered in the existing literature, a
novel cost.  By allowing an opponent a ``head start," unsuccessful
unilateral experimentation involves not only the conventional loss
of current profits, but may jeopardize future sales and profits.
In settings where initial market shares are important for
generating future profits, unilateral experimentation at the
market's inception may lead to the loss of current and {\bf future}
sales.  Settings in which this effect may be important include
markets characterized by network externalities, switching costs or
lock-in, and learning by doing.

In a competitive setting, there may be additional factors impinging
upon the decision to experiment.  For example, Rob (1991) has shown
that spillover effects may reduce the amount of learning.
Similarly signalling effects might also distort the learning
process.  For example, a firm might charge low prices to convince
a potential entrant that demand is low, further inhibiting
learning.  Aghion, Espinosa, and Jullien (1990) and Mirman,
Samuelson, and Schlee (1991) explore the incentives for strategic
manipulation of information in duopoly settings.

In this paper, we present a simple model with two decision makers
that allows us to isolate a ``market share" effect,
i.e., the framework has been constructed so that spillover and
signalling effects are absent.  In this setting, we show that there
is a collectively suboptimal amount of learning which is a direct
consequence of competition.  In other words, there exists a set of
parameters for which a monopoly seller would experiment and learn
about demand, but in the competitive setting, the unique subgame
perfect equilibrium is for neither firm to experiment:  We term
this lack of lack of learning ``the strategic Rothschild effect."
The message of the paper is thus that competition may exacerbate
the Rothschild effect.\footnote{Smith (1991) finds that the
Rothschild effect is mitigated by the presence of multiple decision
makers.  His result is due to the absence of any strategic
interaction among agents: In his setting the payoff of
each agent is independent of the actions of other agents.  Our goal
is precisely to address the inhibitory strategic effects on
equilibrium learning associated with multiple {\bf competing}
agents.}

\section{An Experimentation Model}
\hspace*{6 mm} The market opens in two periods.  In each period, a
new consumer cohort enters the market.  The size of each cohort is
normalized to one.  Consumers are homogeneous and demand one unit,
at the most.
Consumers knows their
reservation values, but the firms are ex ante uncertain whether
demand is high or low: each firm assigns the prior $q$ ($1-q)$ that
the common consumer reservation value is $v_h$ ($v_l)$, where
$v_h>v_l$.

There are two firms, each of whom can supply an unlimited amount of
the product at zero cost.  The firms are risk neutral, and there is
no discounting.

Consumers must make actual contact with a seller in order to learn
its price and it is assumed to be too costly to contact two firms
in the same period.  This grants the firms market power to set
prices.\footnote{Actually, in the framework considered here, even
a small search cost will be sufficient to grant firms full market
power with respect to pricing.  See Diamond (1971).}  Therefore
first period consumers choose a firm at random, and buy from that
firm if the price does not exceed the reservation
value.\footnote{Consumers are not dynamically strategic, i.e., they
do not delay purchasing in the hope of realizing a lower future
price.  This is a common assumption in the network literature.  See
Katz and Shapiro (1986) for example.}
Thus, in the presence of complete
information about demand, firms would have full monopoly power.
As discussed in the following paragraph, the only competitive
``liability" arises from the strategic inhibition to learn about
demand, i.e., to experiment in prices in the markets' formative
stage.

In the second period, there is a network effect based
on the firms' realized first period market shares.  We make the
simplifying assumption that the network effect is so important that
no second period consumer is willing to purchase from the firm with
a smaller installed base at any positive price.\footnote{Our
results remain qualitatively unchanged as long as there is some
network effect.}  Thus if a firm realizes a smaller market share in
the first period, it faces zero demand in the second period.  If
the firms achieve equal first period market shares, second period
demand for each firm is identical to that of the first period.

At the beginning of the second period, firms simultaneously set
prices.  We assume that they do so before learning which firm has
achieved the larger first period market share or learning their
rival's first period price.  This assumption eliminates the
presence of any learning externalities, i.e., it eliminates the
possibility of free-riding on learning achieved by one's rival.
Thus the case for experimentation is as compelling as possible.

The firms are initially imperfectly informed about market demand.
Clearly this information is of value to them.  Specifically, under
conditions of complete information about demand, each firm would
charge the true reservation price in each period.  This is because,
by assumption, each first period consumer accepts any price not
exceeding its reservation price.  Thus by charging the actual
reservation price, a firm is assured of not suffering any adverse
networks effects in the future.  Conversely, it cannot obtain any
network advantage by charging less than this price.  Therefore, in
the absence of any incomplete information about demand, each firm
would enjoy complete monopoly power over its segment of the market.



Given incomplete information, a firm can become perfectly
informed by charging the high price in the first period: acceptance
of price $v_h$ perfectly reveals that demand is high, while its
rejection reveals that demand is low.  This represents the
potential benefits from experimentation.

Experimentation is costly, however.  In our framework, this cost
has two components.  The first is the standard one, the potential
loss of sales in the first period.  The second and novel component,
which our model is constructed to address in the
starkest possible way, is a {\bf strategic} cost.  The source of
this cost is the loss of future sales via an adverse network
effect, which occurs if unilateral experimentation causes that firm
to realize a smaller first period market share.

A natural benchmark which is useful to assess the
significance of the second component, is comparison with monopoly
learning in the same context.  A monopoly also faces the first cost
associated with experimentation.  However, the strategic cost is
absent because there is no rival who can capture its future sales.
Because there is no scope for any {\bf direct} price competition in
our framework, the absence of a network effect would effectively
transform each firm into a monopoly.

In the following section, we derive conditions under which learning
is the monopolist's optimal strategy.  This serves a benchmark with
respect to which we subsequently identify the effect of competition
on the optimal learning strategy.
\section{Optimal Monopoly Learning}
\hspace*{6 mm} If
the monopolist chooses to experiment in the first period, i.e., if
it charges $v_h$, it becomes perfectly informed.  If the consumer
does not purchase in the first period, the monopoly concludes that
the reservation price is $v_l$ and charges this price in the second
period.  Otherwise $v_h$ is charged in the second period.   From
the perspective of the first period,  the expected second period
profits are $qv_h+(1-q)v_l$.  Thus the expected total profits
from experimenting (charging price $v_h$ in
the first period) are $qv_h + (qv_h+(1-q)v_l)$,  where $qv_h$ are
the expected first period profits.
Therefore a monopolist will experiment if and only if $qv_h +
(qv_h+(1-q)v_l) \geq 2v_l$, where $2v_l$ are the total profits
earned by the monopolist if it does not experiment in the first
period and hence charges $v_l$ in both periods.\footnote{If the
monopolist has no incentive to experiment
in the first period, it will not experiment in the second period
either.}  Rearranging terms in the above equation, the monopolist
will experiment if and only if
\begin{equation}
q \geq {v_l \over (2v_h-v_l)} \equiv q^m.
\label{mon2}
\end{equation}
%\marginpar{mon2}

If $q<q^m$, the monopolist gains no new information in the first
period and consequently continues to charge the low price in the
second period.  Even if demand is actually high, the monopolist
will persist in its belief that demand is low if the initial belief
in high demand is below $q^m$.
This effect has been termed the persistence of error by Rothschild.
In the following section we how strategic considerations may
exacerbate this effect.

\section{Duopoly Analysis}

We now turn to the analysis of the duopolists' equilibrium learning
strategies.  The solution concept is subgame perfect equilibrium.
We solve the extensive form game described earlier through
backwards induction, beginning with the second period.  Let
$(p_i,p_j)$ be the first period prices chosen by firms i and j
respectively.  Since first period consumers will accept any price
up to their true reservation value, no firm will ever charge a
price less than $v_l$ or a price between $v_l$ and $v_h$.  Thus the
four possible subgames are (I) $(v_l,v_h)$,
(II) $(v_l,v_l)$, (III) $(v_h,v_h)$, and (IV) $(v_h,v_l).$


\subsection{Second Period}
\begin{center}
\vspace*{3 mm}
{\bf Case I $(v_l,v_h)$}
\vspace*{3 mm}
\end{center}
First suppose that firm $i$ charges $v_l$ in the second period.
With probability $q$ demand is high and both firms made first
period sales and begin the second period with equal market shares.
In this case, firm i has no network advantage, so its second period
profits are just $v_l$.  With probability $1-q$ demand is low, and
firm $j$, having charged $v_h$, begins period two with a smaller
(no) market share.  In this case firm i enjoys a network advantage,
and earns profits of $2v_l.$  Thus firm $i$'s expected second
period profits from charging $v_l$ in that period are

\begin{equation}
\p^I_l=qv_l+(1-q)2v_l.
\label{case1l}
\end{equation}

If firm i charges $v_h$ in the second period, it cannot enjoy a
network advantage under any circumstances.  This is because if
demand is high, both firms begin the second period with equal
market shares.   If demand is low, firm $i$ makes no
sales.  Thus not having acquired any new information about demand,
firm i's expected second period profits from charging
$v_h$ in that period are

\begin{equation}
\p^I_h=qv_h.
\label{case1h}
\end{equation}

Comparing equations (\ref{case1l}) and (\ref{case1h}), firm i will
optimally charge $v_l$ if

\begin{equation}
q < {2v_l \over (v_h+v_l)}.
\label{case1}
\end{equation}

\begin{center}

\vspace*{3 mm}
{\bf Case II $(v_l,v_l)$}
\vspace*{3 mm}

\end{center}


In this case, there is no possibility of a network advantage
because the firms have equal first period market shares regardless
of the true state of demand.  Thus firm $i$'s second period profits
are just $v_l$ if its price is $v_l$ and are $qv_h$ if its price is
$v_h$.  Therefore firm i will optimally charge $v_l$ in the second
period if

\begin{equation}
q < {v_l \over v_h}.
\label{case2}
\end{equation}

\begin{center}
\vspace*{3 mm}
{\bf Case III $(v_h,v_h)$}
\vspace*{3 mm}

\end{center}


Again there is no scope for network effects: Whether the true state
of demand is high or low,
both firms begin the second period with equal market shares.
Having experimented in the first period, firm i is perfectly
informed about the true state of demand before setting its price,
i.e., its price is $v_h$ if demand was learned to be high or $v_l$
if demand turned out to be low.  Thus second period ex ante
expected profits,  (as perceived at the beginning of the first
period) are

\begin{equation}
\p^{III}=qv_h + (1-q)v_l.
\label{case3}
\end{equation}

\begin{center}

\vspace*{3 mm}
{\bf Case IV $(v_h,v_l)$}
\vspace*{3 mm}

\end{center}

As in the preceding case, firm $i$ is fully informed about the true
state of demand before setting its second period price.  If demand
has been learned to be high, its optimal second period price is
$v_h$, while if demand has been learned to be low, firm $i$ is
unable to make a sale at any price, because of its rivals' network
advantage.  Thus its second period ex ante profits (as perceived at
the beginning of the first period) are

\begin{equation}
\p^{IV}=qv_h.
\label{case4}
\end{equation}
\subsection{Equilibrium}

Based on the above analysis of the second period, we  can state the
following proposition.


\begin{prop}
If $q<v_l/v_h$, the unique subgame perfect equilibrium is that no
firm experiments, i.e, each firm charges $v_l$ in each period.
\label{main.p}
\end{prop}
{\em Proof.}
We prove the proposition by showing that it is a dominant strategy
to charge $v_l$ in the first period.

We will first show that $v_l$ is a best response to $v_l$. If firm
$i$ charges $v_l$ in the first period, its second period profits
are $v_l$ under the condition of the proposition, since from case
II, firm $i$ will charge $v_l$ in the second period.  Thus, its
expected total profits from charging $v_l$ are $2v_l.$

If firm $i$ charges $v_h$ in the first period, by case IV, its
expected second period profits are $qv_h$.  Since its first period
expected profits from charging $v_h$ are $qv_h$, its total profits
from charging $v_h$ in the first period are $2qv_h$.  Thus $v_l$ is
a best response to $v_l$ if $2v_l>2qv_h$, which is the case for the
q under consideration.

We will now show that $v_l$ is also a best response to $v_h$.  From
the analysis of case I, if firm $i$ charges $v_l$ in the first
period, it will also charge $v_l$ in the second period if $q< {2v_l
\over (v_h+v_l)}$, which is implied by the assumption that
$q<v_l/v_h$.  Its expected second period profits from charging
$v_l$ in the first period are $2v_l-qv_l$.
Thus its total expected profits from charging $v_l$ in the first
period are $v_l+2v_l-qv_l=2v_l+(1-q)v_l.$

If firm $i$ charges $v_h$ in the first period, from the analysis of
case III, its expected second period profits are $qv_h+(1-q)v_l$.
Thus its total expected profits from charging $v_h$ in the first
period are $qv_h+qv_h+(1-q)v_l=2qv_h+(1-q)v_l.$  Comparing these
profits, for $q<v_l/v_h$, $v_l$ is a best response to $v_h$.
\hfill {\em Q.E.D.}

\vspace{4 mm}

We can now state our main result.

\begin{prop}
Suppose that ${ v_l \over (2v_h-v_l)} \leq q < {v_l \over v_h}$.
Then the monopoly always experiments and learns the true state of
demand, while neither of the duopolists experiment.
\label{main2.p}
\end{prop}
%\marginpar{main2.p}
{\em Proof.}
This result follows directly from Proposition (\ref{main.p}) and
equation (\ref{mon2}).
\hfill {\em Q.E.D.}

Thus for the relevant parameters, as specified in Proposition
(\ref{main2.p}), the monopolist learns the true state of demand
with probability one, while the duopolists will persist in
``error," i.e., they will charge $v_l$ with probability one, even
if the true state of demand is high.  Thus we have demonstrated the
existence of the strategic Rothschild effect: Experimentation is
less likely to occur in its presence.
\vspace{4 mm}


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\end{thebibliography}

\end{document}

