# Subgame perfect Nash equilibrium for dynamic pricing competition with finite
planning horizon

Niloofar Fadavi<EMAIL_ADDRESS>Department of Industrial Engineering, Sharif
University of Technology

###### Abstract

Having fixed capacities, homogeneous products and price sensitive customer
purchase decision are primary distinguishing characteristics of numerous
revenue management systems. Even with two or three rivals, competition is
still highly fierce. This paper studies sub-game perfect Nash equilibrium of a
price competition in an oligopoly market with perishable assets. Sellers each
has one unit of a good that cannot be replenished, and they compete in setting
prices to sell their good over a finite sales horizon. Each period, buyers
desire one unit of the good and the number of buyers coming to the market in
each period is random. All sellers’ prices are accessible for buyers, and
search is costless. Using stochastic dynamic programming methods, the best
response of sellers can be obtained from a one-shot price competition game
regarding remained periods and the current-time demand structure. Assuming a
binary demand model, we demonstrate that the duopoly model has a unique Nash
equilibrium and the oligopoly model does not reveal price dispersion with
respect to a particular metric. We illustrate that, when considering a
generalized demand model, the duopoly model has a unique mixed strategy Nash
equilibrium while the oligopoly model has a unique symmetric mixed strategy
Nash equilibrium.
Keywords: price dispersion,mixed strategy, sub-game perfect, Nash equilibrium,
dynamic programming, stochastic programming, duopoly, oligopoly.

##  1 Introduction

The goal of revenue management is to manage the sale of a limited quantity of
a resource (such as hotel rooms, airline seats, advertising slots, etc.) to a
potential set of customers (the market).Airlines are one example of a
perishable capacity provider who competes by pricing fixed capacities for sale
over a limited sales horizon. Expedia and other online travel agencies gather
information and display real-time flight prices from rival airlines [12]. This
makes it possible for customers to compare the quality and price of products
and choose the best one. In a rapidly-changing technological world, company
managers have become increasingly intelligent, and revenue management has
become a key factor in their decision-making process. This is because in a
competitive environment, it is important for companies to have a good
understanding of their competitors’ potential actions in order to make
suitable decisions. For example, companies may fail to sell their products
When they set a price too high because they might make decision Regardless of
their competitors’ likely behavior. For this reason, there has been
considerable recent interest and research activity to understand revenue
management competition. In this paper, we consider an oligopoly stochastic
model, in which sellers compete by setting prices in each period. This model,
which is an advanced version of Bertrand-Edgworth competition aims to
determine the managers’s consideration and the pricing behavior in this
uncertain situation.

The remainder of the paper is structured out as follows: We review the related
literature in section 2. In section 3, we describe the problem and provide
equilibrium properties for different demand settings. Theoretical evidence for
the declared properties is presented in section 4. The results of the
numerical experiments are presented in 5. Finlay, the research conclusions and
future research opportunities are discussed in 6.

##  2 Related Literature

We have divided the related works into two categories in this section. The
discussion of dynamic competition models is followed by a brief introduction
of static competition models.

###  2.1 Static Models

The two classic models of oligopoly are the Cournot model in which firms
compete on quantities [7] and the Bertrand model in which firms compete on
prices [4]. The Cournot model starts with companies deciding how much to
produce, and the market then sets the prices. According to the Bertrand model
of price competition, consumers only shop at businesses who have the lowest
prices because they make similar goods. At the beginning of the period, prices
are set by competing businesses. The Bertrand model also makes the assumption
that each firm can meet the demand for its product as it sees it. In this
model, each firm sets its price at its marginal cost, and there is only one
possible equilibrium. In his study, Edgworth demonstrated that, when capacity
constraints are taken into account, there is no pure strategy Nash equilibrium
in the Bertrand model [11]. This study focuses on the Bertrand-Edgeworth
model, which takes into account Bertrand competition with capacity
constraints. In this case, clients pick the company with the lowest price, and
when two companies have comparable prices, they make a random selection.
Customers arrive at the market in a random order, which introduces demand
uncertainty. Although there isn’t necessarily a pure-strategy equilibrium in
this situation, there are mixed-strategy equilibria [1], [8], [13]. Bertrand-
Edgeworth appears to be a good place to start if we want to learn more about
revenue management competition. The majority of the time, revenue management
is concerned with situations where a set capacity needs to be sold quickly
(like the quantity of hotel rooms or airplane capacity), and businesses often
compete on pricing [15]. The absence of a pure-strategy equilibrium makes it
challenging to study and forecast potential market circumstances. The absence
of a pure-strategy equilibrium makes it challenging to study and forecast
potential market circumstances. The model (Bertrand-Edgeworth) is
straightforward, however it doesn’t give precise information about the state
of the market. Therefore, it would be useless to design the model for more
complex cases like several client segments. As a result, some of the other
pertinent publications were concentrating on mixed-strategy Nash equilibrium
at the same time as these fundamental models were being developed.

###  2.2 Dynamic Models

Many researchers have taken into consideration dynamic models of competition
after the emergence of numerous static models. In this part, we focus on the
most significant ones. Chamberlin [6] was of the first researchers to point
out that repeated interaction between competitors in an oligopoly market, can
facilitate collusion. When businesses interact frequently enough, they can
threaten price wars in order to deter rivals from engaging in collusive
pricing. Dudy’s paper is one of the earliest papers on dynamic competitions
with capacity constraints [10]. In spite of its nonexistence in Bertrand-
Edgworth competition, Dody proved that subgame perfect equilibrium is unique
for dynamic competition. Proving equilibrium’s uniqueness was a significant
achievement. This is because non-uniquness of equlibrium is problematic for
economic applications. Without a unique result, we do not know which
equilibrium is exactly considered by each competitor, and we cannot
legitimately perform comparative static exercises [16]. Dudy established that
the sub-game perfect equilibrium is unique for dynamic competition while not
being in Bertrand-Edgworth competition. The proof of the uniqueness of
equilibrium was a great accomplishment. This is due to the fact that non-
uniquness of equilibrium presents difficulties in economic applications.
Without an unique result, we are unable to compare static exercises amongst
competitors because we cannot determine precisely which equilibrium each
competitor is taking into account. In Dudy’s model, companies in each period
set a price to maximize their undiscounted profits in a duopoly variant of
Bertrand-Edgworth competition with capacity constraint. They assumed that the
demand structure is deterministic in their model.
Biglaiser and Vettas [5], investigate a dynamic price game in which businesses
have limited capacities, and customers are strategic. They discover that there
is no pure-strategy subgame perfect equilibrium, and that seller’s market
shares are highly likely to be asymmetric.
The key contribution of [2] is exploring the influence of search costs on the
equilibrium in a dynamic price model. They demonstrated that when the search
cost is not zero, duopolistic prices are greater than those of monopolists.
Mantin [14] analyzed a multiperiod duopoly pricing game where a homogeneous
perishable good is sold to consumers who visit one of the retailers in each
period.
Perakis and Sood in [18], studied an open-loop equilibrium. Prices are
supposed to be set at the start of the horizon and cannot change until the end
of the horizon, therefore they do not need to be subgame perfect. They used
variational inequality tools and performed comparative statics on equilibrium
prices as a function of demand- capacity ratios.
Dynamic price dispersion in perfect competition was derived in Deneckere and
Peck [9]. They developed a dynamic version of Prescott’s hotel model. Their
model presents a price dispersion for each period, and updates the demand
function regarding the last periods revealed demand. In each period, some
firms post a low price and sell for sure, while others post a higher price and
can only sell during that period if demand is high. Firms update their
knowledge and make decisions for future periods based on their revised demand
function when demand is revealed. In their research, no discount factor was
taken into account.
Talluri and Martínez de Albéniz in [15], studied perfect competition of a
homogeneous product when demand function is a Bernoulli random variable, and
derived a closed-form solution to the equilibrium price paths. They showed a
structural property of the equilibrium policy such that the seller with the
lower equilibrium reservation value sells a unit at a price equal to the
competitor’s equilibrium reservation value.
[12] analyzed a deterministic demand function in the first section of their
work, and then proved that under certain special conditions, the first
section’s results can be utilized as an approximation of the circumstance
where the demand function is random.
[19] considered an oligopolistic competition in which each firm has one unit
to sell, and products are not perishable. In his model, each firm tries to
maximize its discounted expected profit. They discovered the equilibrium price
dispersion, and they proved it is unique when demand follows Bernoulli
dispersion.
Although there are a huge number of researches done in the last decade, their
concentration is mostly on the customer behavior modeling and its enhancement.
On the other hand, there is an obvious lack of attention to competition
modeling. This is because when we add a constraint to these models, it becomes
difficult to prove the existence of subgame perfect equilibrium in a
stochastic revenue management game. Even the simplest model, Bertrand-Edgworth
model, does not have a unique pure Nash equilibrium, so it seems difficult to
develop these models. Also most of the related studies are in static models
and infinite games, and some of the studies considered special situation to
find pure Nash equilibrium with no regard to mixed strategies’ existence. Our
main contribution is that we find equilibrium for a finite long run horizon,
and we show that there is a unique mixed strategy Nash equilibrium in the
duopoly model with the generalized demand. Furthermore, we show that the
symmetric mixed strategy Nash equilibrium is unique for oligopoly model. In
order to be able to examine how various factors affect the state of the market
and sellers’ behavior in equilibrium, our goal is to create a mathematical
model for the given problem. Although it is difficult to meet all of the
model’s assumptions, this model can be used as a benchmark to understand how
the various parameters affect the market’s behavior and price dispersion’s
features. For this aim, we make use of the mathematical model presented in
[19].

##  3 Methodology

In this section, we examine a dynamic Bertrand-Edgworth competition. We first
discuss Bernoulli demand distribution before moving on to general demand
distribution. For each demand setting, We examine equilibrium for both doupoly
and oligopoly competitions.

Let’s first discuss the fundamentals of Nash equilibrium before delving into
the specifics of the equilibrium study. Think of a strategic game with $N$
players. The set of pure strategies for competitor $i$ is $A_{i}$, then a
mixed strategy is a randomization on set $A_{i}$ and is denoted by $F$. The
mixed strategies of the present competitors are assumed to be independent.
Finally, we have a new game with mixed strategies rather than the previous
simple pure strategies. This game is called mixed extension of the original
game. Now, we have the following definition from [17]:

######  Definition 3.1.

"A mixed strategy Nash equilibrium of a strategic game is a Nash equilibrium
of its mixed extension."

The other useful concept is subgame perfect Nash equilibrium (SPNE). This is a
variation of Nash equilibrium applied to dynamic games. A set of strategies
demonstrate a SPNE, if each point of this dynamic game, which is a subgame,
results in a Nash equilibrium [17].

###  3.1 The Model and Notation

We consider a market that includes $N$ Competing firms, each selling one unit
of a homogeneous perishable item. Time is considered as a discreet parameter
and is indexed by $t=1,2,\ldots.$ in an infinite horizon. Each firm should set
a price before the demand is revealed in every period. Customers only buy one
unit if the price is less than their reservation price, which is the same for
all of them. Once a customer arrive to the market in period $T$, they ask for
one unit of the product and leave the market if they fail to buy in this
period. When two businesses set their prices equally, they have an equal
chance of selling their goods. Each firm’s purpose is to maximize the present
discounted value of its profit. To discuss the variants of this problem, let
us first introduce certain notations:

  * •

$T$ : number of time units left until the product expires.

  * •

$\bar{p}$ : reservation Price.

  * •

$V(\bar{p},\mathrm{~{}N},\mathrm{~{}T})$ : the expected firm’s option value
when there are $N$ firms in the market and $T$ periods remained.

  * •

$P_{N,T}^{*}$ : the minimum price each firm tend to consider when there are
$N$ firms in the market and $T$ periods remained.

  * •

$\delta$ : discount factor.

Calculating option values is a crucial factor to consider when developing a
pricing strategy for a company. This option value is equal to the expected
profit they make if they do nothing and keep the price at the reservation
price till the planning horizon is over [19]. Because they have this option,
they have a lower bound for the profit of their pricing strategy they adopt.
So, from definition of option value, we have the following equation:

$\displaystyle\mathrm{V}(\bar{p},\mathrm{~{}T},\mathrm{~{}N})=0$ (1)

for $N>T$.

In the next sections, we explore Nash equilibrium for the duopoly model, then
create and find the equilibrium for the oligopoly market.

####  3.1.1 Binary Demand

In this section, we study the market when the demand distribution follows
Bernoulli distribution. We first discuss some properties of option value that
apply to both duopoly and oligopoly markets. Following that, we discuss the
equilibrium for each case individually. Let us first introduce some additional
notations:

  * •

$q$ : probability of zero demand.

  * •

$\Gamma(N,q,T)$ : a game in which $\mathrm{N}$ firms are present in the
market, and planning horizon is $T$, and demand parameter is equal to $q$.

We can perform the computations for our case using the same reasoning as [19].
For this market, closed-form of the option value can be calculated. Based on
Bayes’ theorem, we have:

$\displaystyle\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T})=q\delta
V(\bar{p},N,T-1)+(1-q)\delta V(\bar{p},N-1,T-1).$ (2)

Similarly, $\mathrm{V}(\bar{p},1,\mathrm{~{}T})$ can be computed as follows :

$\displaystyle V(\bar{p},1,\mathrm{~{}T})=q\delta
V(\bar{p},1,T-1)+(1-q)\bar{p}=$ (3)
$\displaystyle(1-q)\bar{p}+q\delta(1-q)\bar{p}+\ldots+(q\delta)^{T-1}(1-q)\bar{p}=$
$\displaystyle(1-q)\bar{p}(1-(q\delta)^{T})/(1-q\delta).$

According to (2) and (3), we can calculate
$\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T})$ according to following
equation:

$\displaystyle\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T})=\sum_{i=1}^{T-N+1}[(q\delta)^{i-1}(1-q)\delta
V(\bar{p},N-1,T-i)],$ (4)

when $N\leq T$. Because firms have an option value, there is a lower bound for
the price they are willing to accept in a given period. To figure out what
this price is, we need to find a price $P_{N,T}^{*}$ where the expected profit
from posting this price is equal to the current option value. As a result, it
fulfills the following equation:

$\displaystyle\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T})=q\delta\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T}-1)+(1-q)P_{N,T}^{*}$
(5)

for $N\leq T$. In conclusion, having $V(\bar{p},N,T)$, we can calculate
$P_{N,T}^{*}$ as:

$\displaystyle
P_{N,T}^{*}=\frac{\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T})-q\delta\mathrm{V}(\bar{p},\mathrm{~{}N},\mathrm{~{}T}-1)}{1-q}=\delta\mathrm{V}(\bar{p},\mathrm{~{}N}-1,\mathrm{~{}T}-1)$
(6)

for $N\leq T$. Equation (6) has an intuitive explanation as well. This idea
which is mentioned in [19] for nonperishables assets works here as well. When
the demand distribution is Bernoulli, in each period at most only one buyer
comes to the market. Thus, $P_{N,T}^{*}$ is the price at which a seller is
indifferent between selling or letting others sell when there is a one unit
demand. The profit in the first case is $P_{N,T}^{*}$, and the profit in the
second case is $\delta\mathrm{V}(\bar{p},\mathrm{~{}N}-1,\mathrm{~{}T}-1)$.
Therefore,
$P_{N,T}^{*}=\delta\mathrm{V}(\bar{p},\mathrm{~{}N}-1,\mathrm{~{}T}-1)$.

####  3.1.2 Duopoly Competition with Binary Demand

In this section we examine the equilibrium for duopoly markets. To examine the
mixed-strategy equilibrium, we model the sellers’ strategy as a random
variable. Here, we introduce some notations:

  * •

$F_{iT}(p)$ : the probability that seller $i$ sets a price less than or equal
to $p$ in period $T$,

  * •

$F_{-iT}(p)$ : the joint probability that all sellers except $i$ set prices
less than or equal to $p$ in period $T$,

  * •

$F_{iT}(p_{-})=\lim_{p\rightarrow p^{-}}F_{iT}(p)$,

  * •

$I_{iT}=\sup\\{p\mid F_{iT}(p)=0\\}$,

  * •

$U_{iT}=\inf\\{p\mid F_{iT}(p)=1\\}$.

Lemma 3.1 and 3.2 demonstrate the relationship between the minimum acceptable
price $P_{N,T}^{*}$ and the bounds associated with the adopted random pricing
strategy. These two together help to attain the equilibrium properties in
proposition 3.3.

######  Lemma 3.1.

The following equation holds in equilibrium:

$\displaystyle I_{1T}=I_{2T}\geq P_{2,T}^{*}$ (7)

######  Lemma 3.2.

The following equation holds in equilibrium:

$\displaystyle U_{1T}=U_{2T}=P_{2,T}^{*}.$ (8)

######  Proposition 3.3.

In the selling game $\Gamma(2,q,T)$, each seller adopts the pricing strategy
as $\mathrm{p}=P_{2,T}^{*}$ for all $T\geq 2$, and the expected profit for
each seller is exactly $\delta V(\bar{p},1,T)$.

As a result, when there are two firms competing on price to sell their
product, they both offer an identical price equal to $P_{2,T}^{*}$. The reason
for this is that no one intends to sell for less than their option value, and
if they let the other person sell, their expected profit is the same because
of equation (6).

####  3.1.3 Oligopoly Competition with Binary Demand

In this section, we examine the equilibrium when there are more than two firms
in the market. First, we introduce some notations for this part:

  * •

$F_{iNT}(p)$ : the probability that seller $i$ sets a price less than or equal
to $p$ in period $T$ in a market containing $N$ sellers,

  * •

$F_{-iNT}(p)$ : the probability at least one seller except seller $i$ set a
price less than or equal to $p$ in period $T$ in a market containing $N$
sellers,

  * •

$F_{iNT}(p_{-})=\lim_{p\rightarrow p^{-}}F_{iNT}(p)$,

  * •

$\gamma_{NT}(p)$ : the number of firms offering a price equal to $p$,

  * •

$I_{iNT}=\sup\\{p\mid F_{iNT}(p)=0\\}$,

  * •

$U_{iNT}=\inf\\{p\mid F_{iNT}(p)=1\\}$.

Now, we can discuss the equilibrium properties of this oligopoly market.

######  Proposition 3.4.

When there are N firms in the market and the demand structure is binary, the
three following conditions hold in equilibrium:
1\. $I_{iNT}\geq P_{N,T}^{*}$.
2\. $\gamma_{NT}\left(p_{N,T}^{*}\right)\geq 2$.
3\. The expected profit for each firm is $V(\bar{P},N,T)$.

Based on the proposition, when the demand structure is binary, offering a
price more than the reservation price is possible, and the price dispersion is
not predictable. However, based on the measure presented in [3], and the
corresponding case discussed in [19], if we only consider marginal prices,
this oligopoly model does not show any price dispersion.

####  3.1.4 General Demand Model

In this section, we have a generalization of the cases that we discussed
before. As a result, we investigate a general demand structure and investigate
the duopoly and oligopoly models’ equilibria. We start by introducing a few
extra notations:

  * •

$q_{i}$: the probability that the demand equals to $i$.

  * •

$F(p)$ : the identical assumed strategy chosen by all of the firms.

  * •

$F_{iNT}(p)$ : the probability that seller $i$ sets a price less than or equal
to $p$ in period $T$ in a market containing $N$ sellers.

  * •

$F_{-iNT}(p)$ : the probability that at least one seller except $i$ set a
price less than or equal to $p$ in period $T$ in a market containing $N$
sellers.

  * •

$F_{iNT}\left(p_{-}\right)=\lim_{p\rightarrow p^{-}}F_{\text{iNT }}(p)$.

  * •

$I_{iT}=\sup\left\\{p\mid F_{iT}(p)=0\right\\}$.

  * •

$U_{iT}=\inf\left\\{p\mid F_{iT}(p)=1\right\\}$.

Now, similar to the binary demand section, we first calculate the option value
and reservation price. The option value can be calculated from the following
equation:

$\displaystyle V(\bar{p},N,T)=\sum_{i=0}^{N-1}q_{i}\delta
V(\bar{p},N-i,T-1)+\sum_{i=N}^{+\infty}q_{i}\bar{p}.$ (9)

The first component of (9) is the expected profit we get if the demand is less
than the number of players, and the second one represents the expected profit
of the cases where demand is more than the number of players. In addition,
since $P_{N,T}^{*}$ is the least price that sellers are willing to offer, it
satisfies the following equations:

$\displaystyle V(\bar{p},\mathrm{~{}N},\mathrm{~{}T})=q_{0}\delta
V(\bar{p},\mathrm{~{}N},\mathrm{~{}T}-1)+\left(1-q_{0}\right)P_{N,T}^{*}.$
(10)

So, we have:

$\displaystyle
P_{N,T}^{*}=\frac{V(\bar{p},\mathrm{~{}N},\mathrm{~{}T})-q_{0}\delta
V(\bar{p},\mathrm{~{}N},\mathrm{~{}T}-1)}{1-q_{0}}=\frac{\sum_{i=1}^{N-1}q_{i}\delta
V\left(\bar{p},\mathrm{~{}N}-\mathrm{i}_{1},\mathrm{~{}T}-1\right)+\sum_{i=N}^{+\infty}q_{t}\bar{p}}{1-q_{0}}$
(11)

Equation (11) has an intuitive explanation similar to the one mentioned by
[19] for the perishable assets. It means that if demand is not zero,
$P_{N,T}^{*}$ is equal to the expected option value.
The last parameter that can be calculated regardless of the demand
distribution is the monopolist’s option value, and it satisfies the following
equation:

$\displaystyle V(\bar{p},1,T)=q_{0}\delta
V(\bar{p},1,T-1)+\left(1-q_{0}\right)\bar{p}.$ (12)

Using the equation (12), a closed-form of $V(\bar{p},1,T)$ can be derived as
follows:

$\displaystyle
V(\bar{p},1,T)=\left(1-q_{0}\right)\bar{p}+q_{0}\delta\left(1-q_{0}\right)\bar{p}+\ldots+q_{0}\delta^{T-1}\left(1-q_{0}\right)\bar{p}$
(13)
$\displaystyle=\left(1-q_{0}\right)\bar{p}\frac{1-\left(q_{0}\delta\right)^{T}}{1-q_{0}\delta}.$

In the following sections, we first analyse the equilibrium for duopoly and
then for oligopoly market.

####  3.1.5 Duopoly Competition with General Demand Function

To examine the properties of equilibrium, we need to first calculate the
option value. From the option value we obtained for monopolist, the option
value for a firm in a duopoly market can be calculated from the following
equation:

$\displaystyle V(\bar{p},2,1)=\left(1-q_{0}-q_{1}\right)\bar{p},$ (14)

and we have:

$\displaystyle V(\bar{p},2,T)=\sum_{l=0}^{1}q_{i}\delta
V(\bar{p},2-\mathrm{i},\mathrm{T}-1)+\sum_{i=2}^{+\infty}q_{i}\bar{p}$ (15)

for $T>2$. The right-hand side of equation (15) can be calculated using the
option value of $T=1$ and the monopolist option value in (13). The following
proposition introduces the symmetric mixed strategy equilibrium for the
duopoly case. Similar result exists and is proved in [19] for not perishable
assets. The proof of this proposition is stated in section 4.

######  Proposition 3.5.

Considering $q_{1}>0$ and $\sum_{l=2}^{+\infty}q_{l}>0$, the mixed strategy
$W_{T}(p)$ represents a Nash equilibrium, and its associated expected profit
is $V(\bar{p},N,T)$.

$\displaystyle W_{T}(p)=\left\\{\begin{array}[]{cc}0&p\leq P_{N,T}^{*}\\\
\frac{V(\bar{p},2,T)-q_{0}\delta
V(\bar{p},2,T-1)-\sum_{i=1}^{+\infty}q_{i}p}{q_{1}(\delta
V(\bar{p},1,T-1)-p)}&P_{N,T}^{*}<p<\bar{p}\\\
1&p\geq\bar{p}\end{array}\right.$ (19)

In proposition 3.5, we show that $W_{T}(p)$ can result in an equilibrium.
Another characteristic we discovered for the duopoly case is the equilibrium’s
uniqueness. To show that this is a unique equilibrium, we take advantage of
the expected profit in equilibrium and the equilibrium strategy’s. The
evidence we need to demonstrate the uniqueness in 3.8 can be found in the
following lemmas.

######  Lemma 3.6.

In equilibrium the following equations hold.

$\displaystyle\checkmark I_{1T}=I_{2T}=P_{2,T}^{*}$ (20)
$\displaystyle\checkmark U_{1T}=U_{2T}=\bar{P}^{*}$ (21)

######  Lemma 3.7.

The equilibrium strategy is symmetric.

######  Proposition 3.8.

$W_{T}(p)$ is the unique mixed-strategy equilibrium.

####  3.1.6 Oligopoly Competition with General Demand Function

We define the function $Z_{K,N}(x)$ according to the following equation:

$\displaystyle Z_{K,N}(x)=\sum_{i=0}^{K}\left(\begin{array}[]{c}N-1\\\
i\end{array}\right)(1-x)^{N-l-1}x^{l}.$ (24)

If all of the firms choose an identical strategy $F(p)$, the probability of an
observation in which at most $k$ firms offer a price less than or equal to
$\mathrm{p}$ is $Z_{K,N}(F(p))$.
Same as duopoly case, we can again calculate the option value as follows:

$\displaystyle V(\bar{p},N,1)=\sum_{l=N}^{+\infty}q_{l}\bar{p}.$ (25)

Thus, having $V(\bar{p},N,1)$ and monopolist option value, the values of
$P_{N,T}^{*}$ and $V(\bar{p},N,1)$ can be calculated. Having the equilibrium
for oligopoly cases in [19] for not perishable assets, the following statement
proposes a symmetric mixed strategy equilibrium.

######  Proposition 3.9.

With the conditions $\sum_{i=2}^{+\infty}q_{i}>0$, when every firm choose
$W_{N,T}(p)$ as its strategy, this condition results in a unique symmetric
equilibrium and their expected profit in equilibrium is $V(\bar{P},N,T)$:

$\displaystyle W_{N,T}(p)=\left\\{\begin{array}[]{cc}0&p\leq P_{N,T}^{*}\\\
G^{-1}(p)&P_{N,T}^{*}<p<\bar{p}\\\ 1&p\geq\bar{p}\end{array}\right.$ (29)
$\displaystyle
G(x)=\frac{\sum_{l=1}^{+\infty}q_{i}\bar{p}+\sum_{l=1}^{n-1}q_{i}z_{i-1}(x)\delta
V(\bar{P},N-i,T-1)}{\sum_{l=1}^{+\infty}q_{i}+\sum_{i=1}^{n-1}q_{i}Z_{i-1}(x)}$
(30)

As it was mentioned before, it is important to examine uniquness of
equilibrium. Therefore, the following lemma is employed in proposition 3.11 to
demonstrate the uniqueness of equilibrium under some particular circumstances.
For this aim, we need to provide a situation in which asymmetric equilibrium
is not possible because we showed that the only possible symmetric strategy is
$W_{N,T}(p)$.

######  Lemma 3.10.

With the conditions $\sum_{i=2}^{+\infty}q_{l}>0,W_{T}(p)$ is a right
continuous increasing function at $p\in$ $\left(P_{N,T}^{*},\bar{p}\right)$.

######  Proposition 3.11.

If the conditions $\sum_{l=2}^{+\infty}q_{i}>0$ and (31) are satisfied, the
equilibrium should be symmetric.

$\displaystyle p_{N,T}^{*}\leq I_{1NT}=l_{2NT}=\cdots=l_{NNT}$ (31)

##  4 Appendix

For the sake of proof following, we first define a few notations.

  1. 1.

Duopoly competition with binary demand: The equilibrium expected value of the
profit of firm $i$ is denoted by $\bar{V}_{i}^{2}(q,T)$. In addition the
expected value of the profit of firm $i$ when firm $j$’s strategy $F_{jT}$ is
given, is indicated by $V_{i}^{2}(p;q,T)$ and is calculated as follows:

$\displaystyle V_{i}^{2}(p;q,T)$
$\displaystyle=q\delta\bar{V}_{i}^{2}(q,T-1)+(1-q)[F_{jT}(p_{-})V(\bar{P},1,T-1)$
(32) $\displaystyle+(F_{jT}(p)-F_{jT}(p_{-}))\frac{1}{2}(\delta
V(\bar{P},1,T-1)+p)$ (33) $\displaystyle+(1-F_{jT}(p))p]$

  2. 2.

Oligopoly competition with binary demand: The equilibrium expected value of
the profit of firm $i$ is denoted by $\bar{V}_{i}^{N}(q,T)$. The expected
value of the profit of firm $i$ when we are given $F_{-iNT}$ is given, is
indicated by $V_{i}^{N}(p;q,T)$ and is calculated as follows:

$\displaystyle V_{i}^{N}(p;q,T)$
$\displaystyle=q\delta\bar{V}_{i}^{N}(q,T-1)+(1-q)\left[F_{-iNT}(p_{-}\right)\delta
V(\bar{P},N-1,T-1)$ (34)
$\displaystyle+(F_{-iNT}(p)-F_{-iNT}(p_{-}))(\frac{m}{m+1}\delta\bar{V}_{i}^{N-1}(q,T-1)+\frac{1}{m+1}p)$
(35) $\displaystyle+(1-F_{-iNT}(p))p].$

Parameter $m$ can be calculated using the joint distribution of all $N-1$
firms. For out purpose, it is enough to know that it is a scalar in $[1,N-1]$.

  3. 3.

Duopoly competition with general demand: The expected value of the profit of
firm $i$ when firm $j$’s strategy $F_{jT}$ is given, is indicated by
$W_{i}^{2}(p;q,T)$ and is calculated as follows:

$\displaystyle W_{i}^{2}(p;q,T)$
$\displaystyle=q_{0}\delta\bar{W}_{i}^{2}(q,T-1)+q_{1}[F_{jT}(p_{-})\delta
V(\bar{P}_{,},1,T-1)$ (36)
$\displaystyle+(F_{jT}(p)-F_{jT}(p_{-}))\frac{1}{2}(V(\bar{P},1,T-1)+p)+(1-F_{jT}(p))p]$
(37) $\displaystyle+\sum_{i=2}^{+\infty}q_{i}p,$ (38)

Here, we introduce a crucial theory that is applied throughout the proofs from
[17].

######  Theorem 4.1.

If we have game with infinite strategy set, In this case,
$(F_{1},\ldots,F_{N})$ is a mixed strategy Nash equilibrium of if and only if
for every player $i$ no action in its strategy set yields, given
$\\{F_{1},\ldots,F_{N}\\}\setminus F_{i}$ , a payoff to player $i$ that
exceeds his equilibrium payoff, and the set of actions that yield, given
$\\{F_{1},\ldots,F_{N}\\}\setminus F_{i}$, a payoff less than his equilibrium
payoff has probability zero.

Let’s now discuss the proofs offered for the equilibrium properties. Since
several equilibrium characteristics are comparable in both the finite and
infinite cases, we might frequently mimic the infinite case proofs provided in
[19] to show the same characteristics in our context.

A.1 Proof of lemma 3.1.

###### Proof.

Let us use proof by contradiction. First, we assume $I_{1T}<I_{2T}$.
Considering this assumption, firm $1$ can increase its profit by assigning the
probability of the interval $\left(I_{1T},I_{2T}\right)$ to a point in the
left neighborhood of $I_{2T}$. The reason is that they are sure that all of
these prices would be sold with probability one if there exists any demands,
thus they try to choose the maximum of them. So, this is a contradiction
arising from our false assumption, and we have $I_{1T}=I_{2T}$ in equilibrium.
In addition, if we assume $I_{1T}=I_{2T}<P_{2,T}$ in equilibrium, there will
be two possible states, and we demonstrate that neither of them can lead to
equilibrium. Therefore, we prove that the equation $I_{1T}=I_{2T}<P_{2,T}$
cannot stand in equilibrium.

  1. 1.

Assume $F_{2T}(I_{2T})<1$. Because cumulative distribution function is right-
continuous, we have:

$\displaystyle\exists\eta\in\left(l_{2T},P_{2,T}^{*}\right)\mid\forall
p\in[I_{2T},\eta)\rightarrow F_{2T}(I_{2T})\leq F_{2T}(p)<1.$ (39)

Now, considering firm $2$’s strategy, firm $1$ achieves smaller profit for
$p\in\left[I_{2T},\eta\right)$ than deciding not to sell in this period. The
following equation demonstrates this. Utilising equation (6) and
$p<P_{2,T}^{*}=\delta V(\bar{P},1,T-1)$, we have:

$\displaystyle
V_{1}^{2}(p;q,T)<q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)P_{2,T}^{*}.$ (40)

The inequality (40) is obtained by substituting $p$ by its upper bound $\delta
V(\bar{P},1,T-1)$ in (32), and the right-hand side demonstrates the expected
profit of firm $1$, when they allow the other seller to sell. As a result,
based on theorem 4.1, it is not possible to have a pure strategy with better
result than the equilibrium’s one. So, firm $1$ never offers
$p\in[I_{2T},\eta)$, and this is a contradiction. So, state 1 is not possible
in equilibrium.

  2. 2.

Assume $F_{2T}\left(I_{2T}\right)=1$ in equilibrium. The, firm $2$’s expected
profit associated with $p=I_{2T}$ is absolutely less than its corresponding
value for postponing the selling. This is shown in the following inequality:

$\displaystyle
V_{2}^{2}(p;q,T)<q\delta\bar{V}_{2}^{2}(q,T-1)+(1-q)P_{2,T}^{*}.$ (41)

The right-hand side of (41) is written based on the assumption
$I_{2T}<p_{2,T}^{*}$ and by substituting $p$ in (32) with its upper bound
$P_{2,T}^{*}=\delta V(\bar{P},1,T-1)$. Because firm $2$ has the option to
increase its profit by choosing not to sell, this state cannot be an
equilibrium according to the definition of a Nash equilibrium.

∎

A.2 Proof of lemma 3.2.

###### Proof.

First, we assume $U_{1T}\neq U_{2T}$ and $U_{1T}<U_{2T}$. Then, from lemma
3.1, we have $p_{2,T}^{*}\leq U_{1T}<U_{2T}$. If $U_{1T}<U_{2T}$ in
equilibrium, there will be four possible states, and we demonstrate that none
of them can lead to equilibrium. In other words, we prove that the equation
$U_{1T}\neq U_{2T}$ is not possible in equilibrium.

  1. 1.

$P_{2,T}^{*}<U_{1T}<U_{2T}$: Considering
$p_{1}\in\left(P_{2,T}^{*},U_{1T}\right)$ and $p_{2}\in(U_{1T},U_{2T})$ the
following equation holds:

$\displaystyle V_{2}^{2}(p_{1};q,T)>q\delta\bar{V}_{2}^{2}(q,T-1)+(1-q)\delta
V(\bar{P},1,T-1)=V_{2}^{2}(p_{2};q,T).$ (42)

The inequality is obtained from substitution of $p_{1}$ with its lower bound
$P_{2,T}^{*}=\delta V(\bar{P},1,T-1)$ in the function $V_{2}^{2}(p_{1};q,T)$.
Therefore, firm $2$’s expected profit for $(U_{1T},U_{2T})$ is less than the
equilibrium profit, and based on theorem 4.1 this interval should be assigned
zero probability. This is in contradiction with our assumption
$P_{2,T}^{*}<U_{1T}<U_{2T}$.

  2. 2.

$P_{2,T}^{*}=U_{1T}<U_{2T}$: From lemma 3.1, we can conclude
$P_{2,T}^{*}=I_{1T}=U_{1T}=I_{2T}$. For $p_{1}\in(P_{2,T}^{*},U_{2T})$, the
following inequality holds:

$\displaystyle V_{1}^{2}(p_{1};q,T)>q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)\delta
V(\bar{P},1,T-1)=V_{1}^{2}(P_{2,T}^{*};q,T).$ (43)

We have (43) by substituting $p_{1}$ with its lower bound $P_{2,T}^{*}=\delta
V(\bar{P},1,T-1)$ in (32). Therefore, firm $1$’s profit for
$p_{1}\in\left(p_{2,T}^{*},U_{2T}\right)$ is greater than its profit for
$p_{2,T}^{*}$. Thus, it is in contradiction with the definition of Nash
equilibrium, and this state cannot be an equilibrium.

As a result of rejection of state 1 and 2, we can conclude that the equation
$U_{1T}=U_{2T}$ should stand in equilibrium.

  3. 3.

Assume $U_{1T}=U_{2T}>I_{1T}=I_{2T}=P_{2,T}^{*}$. Because cumulative
distribution function is right-continues, we have:

$\displaystyle\exists\eta\in(P_{2,T}^{*},U_{2T})\mid\forall
p\in[I_{2T},\eta]\rightarrow F_{2T}(p)<1.$ (44)

So, for $p=\eta$, we have:

$\displaystyle
V_{1}^{2}(p;q,T)>q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)+(1-q)\delta
V(\bar{P},1,T-1).$ (45)

The inequality results from $p>P_{2,T}^{*}=\delta V(\bar{P},1,T-1)$, and the
right-hand side is equal to the expected profit associated with $P_{2,T}^{*}$.
In addition, for the points in a small right neighbourhood of $P_{2,T}^{*}$,
we have:

$\displaystyle\lim_{p\rightarrow
P_{2,T}^{*}+}V_{1}^{2}(p;q,T)=q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)\delta
V\left(\bar{P},1,T-1\right),$ (46)

which is obtained from the fact that $p\rightarrow P_{2,T}^{*}=\delta
V\left(\bar{P},1,T-1\right)$ and its substitution in $V_{1}^{2}(p;q,T)$. Based
on limit definition, there exists $\epsilon>0$ as the expected profit of
$P\in[p_{2,T}^{*},P_{2,T}^{*}+\epsilon]$ is less than the expected profit of
$p=\eta$. This means that there is a pure strategy with the expected profit
greater than the equilibrium one, and based on theorem 4.1 this state cannot
be an equilibrium. In other words based on Nash equilibrium definition, firm
$1$ never offers a price in this interval, and we have:

$\displaystyle I_{1T}=\sup\left\\{p\mid F_{1T}(p)=0\right\\}>p_{2,T}^{*},$
(47)

and this is in contradiction with our initial assumption regarding the lower
bound $I_{1T}$.

  4. 4.

Assume $U_{1T}=U_{2T}\geq I_{1T}=I_{2T}>P_{2,T}^{*}$. If the probability firm
$2$ assigned to $U_{2T}$ is not positive, for each $\epsilon>0$, there exists
$\eta\in(p_{2,T}^{*},U_{2T})$ as:

$\displaystyle\forall p\in(\eta,U_{2T}]\rightarrow F_{2T}(p)>1-\epsilon.$ (48)

When we consider $\epsilon=\frac{I_{1T}-P_{2T}^{*}}{\frac{1}{2}\left(\delta
V(\bar{P},1,T-1)+3U_{2T}\right)}$, the expected profit of firm $1$ for
$\mathrm{p}\in(\eta,U_{2T}]$ satisfies the following inequality:

$\displaystyle V_{1}^{2}(p;q,T)$ $\displaystyle<q\delta\bar{V}_{1}^{2}(q,T-1)$
(49) $\displaystyle+(1-q)[\delta V(\bar{P},1,T-1)+\frac{1}{2}\epsilon(\delta
V(\bar{P},1,T-1))+\frac{3}{2}U_{2T})]$
$\displaystyle=q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)I_{1T}.$

The inequality is obtained by substituting $1-F_{2T}(p)$,
$F_{2T}(p)-F_{2T}(p_{-})$ and $p$ with their upper bounds, $1$, $\epsilon$ and
$U_{2T}$ in $V_{1}^{2}(p;q,T)$, the following inequality is derived: The last
expression is obtained by substitution of $\epsilon$. On the other hand, in a
left neighbourhood of $I_{1T}$, the expected profit is:

$\displaystyle\lim_{p\rightarrow
I_{1T}^{-}}V_{1}^{2}(p;q,T)=q\delta\bar{V}_{1}^{2}(q,T-1)+(1-q)I_{1T}$ (50)

So, in a left neighbourhood of $I_{1T}$, the expected profit is more than the
expected profit of $p\in(\eta,U_{2T}]$. Thus, based on theorem 4.1, this state
is not possible in equilibrium. So, we should have
$F_{2T}\left(U_{2T}\right)-F_{2T}\left(U_{2T-}\right)>0$. In this case, since
we have $U_{1T}>P_{2,T}^{*}=\delta V(\bar{P},1,T-1)$, the value of
$V_{1}^{2}(p;q,T)$ in a small left neighborhood of $U_{1T}$ is greater than
its value in $U_{1T}$. Accordingly, based on theorem 4.1, the probability
given to the points with expected value below the equilibrium expected value
is zero, and based on this, the probability firm $1$ assigns to $U_{1T}$ is
zero. However, we proved that the probability assigned to the upper bound
should not be zero. As a result, we showed that state 4 is not possible to
maintain equilibrium.

∎

A.3 Proof of proposition 3.3

###### Proof.

Considering lemma 3.1 and 3.2, for a duopoly competition with binary demand
and $T\geq 2$, the necessary condition for equilibrium is to have a pure
strategy $I_{1T}=$ $I_{2T}=U_{1T}=U_{2T}$. Since we have $P_{2,T}^{*}=\delta
V(\bar{P},1,T-1)$, it can be shown that this strategy is a Nash equilibrium.
When $T=1$, this is a simple Bertrand model in which every firms offers zero
in equilibrium. ∎

A.4 Proof of proposition 3.4

###### Proof.

The proof of this proposition is similar to the proof of proposition 3.3.
These conditions are proved to be true for $N=2$. So, we can use inductive
reasoning. Assume that the three sections of the proposition are valid for
$\Gamma(N-1,q,T)$. Let us first demonstrate $I_{iNT}\geq P_{N,T}^{*}$. Without
loss of generality, let us first consider the following equation:

$\displaystyle I_{1NT}=\min_{1\leq i\leq N}I_{iNT}.$

Now, we use proof by contradiction. For this aim, we assume
$I_{1NT}<P_{N,T}^{*}$, and we examine the two resulting possibilities.

  1. 1.

If $F_{-1NT}\left(I_{1NT}\right)<1$, from right-continuity property of
cumulative distribution function, we have:

$\displaystyle\exists\eta\in\left(I_{1NT},p_{N,T}^{*}\right)\mid\forall
p\in[I_{1NT},\eta]\rightarrow F_{-1NT}(p)<1.$ (51)

Now, consider $p\in[I_{1NT},\eta]$. Since we have $p<\delta
V(\bar{P},N-1,T-1)=P_{N,T}^{*}$, if we substitute $p$ in (34) with the upper
bound $\delta V(\bar{P},N-1,T-1)$, we have the following inequality:

$\displaystyle V_{1}^{N}(p;q,T)<q\delta\bar{V}_{i}^{N}(q,T-1)+(1-q)\delta
V(\bar{P},N-1,T-1).$ (52)

The right-hand side of (52) is firm $1$’s expected profit if they decide to
postpone their selling. So, based on theorem 4.1, the probability assigned to
the points with the expected profit less than the equilibrium expected profit
must be zero in equilibrium, and as a result it is not possible that firm $1$
post a price in $[I_{1NT},\eta]$ in equilibrium. This is a contradiction, and
this state is rejected.

  2. 2.

Assume $F_{-1NT}(I_{1NT})=1$. Because we assumed that $I_{1NT}$ is the least
lower bound, there is at least one firm, other than $1$, that posts a price
equal $I_{1NT}$ with probability one. As a result, there exists firm $i\neq
1$, such that $F_{iTN}(I_{1NT})-F_{iTN}(I_{1NT}^{-})>0$. Since we have
$I_{1TN}<p_{N,T}^{*}=\delta V(\bar{P},N-1,T-1)$, and $I_{iNT}=I_{1NT}$ is the
least lower bound, if we substitute $I_{1NT}$ with its upper bound $\delta
V(\bar{P},N-1,T-1)$ in (34), we have:

$\displaystyle V_{i}^{N}(p;q,T)<q\delta\bar{V}_{i}^{N}(q,T-1)+(1-q)\delta
V(\bar{P},N-1,T-1).$ (53)

The right-hand side is the expected value of firm $i$’s profit when they
postpone selling. So, firm $i$ should not assign positive probability to
$I_{iNT}$, and this is a contradiction.

Now, the first part of proposition is proved. We again use proof by
contradiction to prove the second part
$\gamma_{NT}\left(p_{N,T}^{*}\right)\geq 2$. We first assume:

$\displaystyle p_{N,T}^{*}\leq I_{1NT}<I_{2NT}\leq\cdots\leq I_{NNT}$ (54)

Considering this assumption, we have:

$\displaystyle\exists\eta\in\left(I_{1NT},I_{2NT}\right)\mid F_{1NT}(\eta)>0.$
(55)

We only need to calculate firm $1$’s expected profit for $p\in[I_{1NT},\eta]$.
Since firm $1$ offers the lowest price in the market, it can sell for sure if
demand is not zero. So, we have:

$\displaystyle V_{i}^{N}(p;q,T)=q\delta\bar{V}_{i}^{N}(q,T-1)+(1-q)p.$ (56)

The right-hand side of (56) is bounded from above with its value for $p=\eta$.
As a result, offering a price equal to $p\in\left[I_{1TN},\eta\right)$ is not
possible in equilibrium, and it is in contradiction with our assumption. As a
result, the following relation stands:

$\displaystyle P_{N,T}^{*}\leq I_{1NT}=I_{2NT}\leq\cdots\leq I_{NNT}.$ (57)

To prove the second part, we should also show $P_{N,T}^{*}<I_{1NT}=I_{2NT}$ is
not possible in equilibrium. We use proof by contradiction. Let us assume that
$P_{N,T}^{*}<I_{1NT}=I_{2NT}$ in equilibrium. Now, we examine and reject the
two possible resulting states.

  1. 1.

Assume $F_{-1NT}(I_{1NT})>0$. Then, Since $I_{1NT}$ is greater than $\delta
V(\bar{P},N-1,T-1)$, firm $1$’s expected profit satisfies the following
equation:

$\displaystyle
V_{1}^{N}(I_{1NT};q,T)<q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)I_{1NT}.$ (58)

We also have the following inequality and equality for the limit of
$V_{1}^{N}(p;q,T)$ when $p$ approaches to $I_{1NT}$ from its right and left
neighbourhoods, respectively:

$\displaystyle\lim_{p\rightarrow
J_{1NT}+}V_{1}^{N}(p;q,T)<q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)I_{1NT},$ (59)
$\displaystyle\lim_{p\rightarrow
I_{1NT}-}V_{1}^{N}(p;q,T)=q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)I_{1NT}.$ (60)

Based on the limit definition, we can find $\epsilon_{1}$ and $\epsilon_{2}$
as the expected profit of $p\in[I_{1NT},I_{1NT}+\epsilon_{1})$ is less than
the expected profit of $p\in\left(I_{1NT}-\varepsilon_{2},I_{1NT}\right)$. So,
based on theorem 4.1, firm $1$ never offers a price from
$[I_{1NT},I_{1NT}+\epsilon_{1})$ in equilibrium, and we can reject the
assumption $F_{-1}\left(I_{1NT}\right)>0$.

  2. 2.

Assume $F_{-1NT}\left(I_{1NT}\right)=0$. In this situation, every firm has the
option to sell at $I_{1NT}$ when there is a positive demand. So, since
$I_{1NT}>V_{1}^{N}(p;q,T)=P_{N,T}^{*}$, the expected profit of firm $i$,
$V_{1}^{N}(p;q,T)$, for $p=I_{1NT}$ is strictly greater than its corresponding
value for selling postponement. Thus, there is no firm tend to let others
sell, and the following expression holds:

$\displaystyle U_{1NT}=U_{2NT}=$ $\displaystyle\cdots=U_{NNT}.$ (61)

In addition, the probability assigned to the upper bound is zero for all
firms. In order to show this we examine two different possibilities. First, if
firm $i$ assigns a positive probability to $U_{iNT}$, and there exists at
least one another firm $j$ who assigns zero probability to $U_{jNT}$, firm
$i$’s expected profit for $U_{iNT}$ is equal to the expected profit of
postponement. So, from theorem 4.1 the probability firm $i$ assigns to
$U_{iNT}$ should be zero. This is a contradiction.
Secondly, If all of the firms assign positive probability to the upper bound,
since $\delta V(\bar{P},N-1,T-1)<U_{iNT}$, the expected profit of firm $i$ at
$U_{iNT}$ is less than its corresponding value for a small left neighborhood
of $U_{iNT}$. This is because in the left neighborhood, it has more chance to
sell. So, it has to assign zero probability to $U_{iNT}$, and this is a
contradiction. So, we showed that the upper bounds are equal, and the
probability assigned to them is zero. As a result, the expected value of
profit for firm $1$ satisfies the following equation:

$\displaystyle\lim_{p\rightarrow
U_{1NT}{}^{-}}V_{1}^{N}(p;q,T)=q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)\delta
V(\bar{P},N-1,T-1).$ (62)

Since we have $\delta V(\bar{P},N-1,T-1)<I_{1NT}$, we also have:

$\displaystyle\lim_{p\rightarrow
U_{1NT}{}^{-}}V_{1}^{N}(p;q,T)<q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)I_{1NT}.$
(63)

The right-hand side is the expected profit of firm $1$ for $p=I_{1NT}$. So,
the probability assigned to a left neighborhood of $U_{1NT}$ must be zero, and
this is a contradiction with upper bound definition. So, we have the following
result is established:

$\displaystyle I_{1NT}=I_{2NT}=P_{N,T}^{*}$ (64)

Applying proof by contradiction, we show that at least two firms assign
probability one to $P_{N,T}^{*}$. Assume that non of the firms assign
probability one to $P_{N,T}^{*}$. First, we have that the limit of the
expected profit of firm $1$ when $p$ approaches to $I_{1NT}=P_{N,T}^{*}$ from
its right neighborhood is equal to its corresponding value at $P_{N,T}^{*}$.
In other words, we have:

$\displaystyle\lim_{p\rightarrow
I_{1NT}^{+}}V_{1}^{N}(p;q,T)=q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)P_{N,T}^{*}.$
(65)

Secondly, we can find $\eta$ such that:

$\displaystyle\exists\eta>p_{N,T}^{*}\mid 0<F_{-1NT}(\eta)<1.$ (66)

Now, because $P_{N,T}^{*}=\delta V(\bar{P},N-1,T-1)<\eta$, firm $1$’s expected
profit satisfies the following inequality:

$\displaystyle
V_{1}^{N}(\eta;q,T)>q\delta\bar{V}_{1}^{N}(q,T-1)+(1-q)P_{N,T}^{*}.$ (67)

As a result of (66) and (67) and based on the limit definition, parameter
$\epsilon$ and interval $\left[I_{1NT},I_{1NT}+\epsilon\right)$ can be found
such that for each price in this interval the expected profit of firm $1$ is
less than the expected profit of $\eta$. So, this state is not possible in
equilibrium. Therefore, we should have $F_{-i}(P_{N,T}^{*})=1$. Since we
assume that the mixed strategy randomizations are independent in definition of
mixed strategy [17], There should be a firm $j$ that offers $I_{1NT}$ with
probability one. With the same reasoning for firm $j$, there should be another
firm $i$ that posts $I_{1NT}$ with probability one, and the result is
established. Therefore, the second part of equation is proved.
We can infer from the preceding sections that the expected profit is equal to
$V(\bar{P},N,T)$. Let us first prove it for $T=N$. We have
$P_{N,N-1}^{*}=V(\bar{P},N,N-1)=0$, and based on the previous sections two
firms offer zero price. So, we have:

$\displaystyle V_{i}^{N}(P_{N,T}^{*};q,N)=q\delta 0+(1-q)\delta
V(\bar{P},N-1,T-1)=V(\bar{P},N,T).$ (68)

So, the induction reasoning can be applied. Note that there are two induction
assumptions we are making to prove the third part of proposition. First, we
start with the assumption we made regarding a game with $N-1$ firms, and that
is applied in all the three sections’ proofs. In the meantime, we made another
assumption regarding a game with the planning horizon $T-1$ to prove the third
section of the proposition. As a result, we now assume that the third
statement is valid for $T-1$, and we calculate the expected profit for
$\Gamma(N,T,q)$ as follows:

$\displaystyle V_{i}^{N}(P_{N,T}^{*};q,T)=q\delta V(\bar{P},N,T-1)+(1-q)\delta
V(\bar{P},N-1,T-1)$ (69)

Thus, the result is established. ∎

A.5 Proof of proposition 3.5

###### Proof.

Let us prove this by induction. First, we must establish that $W_{1}(p)$ leads
to equilibrium, and the equilibrium profit equals $V(\bar{p},2,1)$.
Substituting $V(\bar{p},1,0)=0$ in the given formula for $W_{1}(p)$, we have:

$\displaystyle W_{1}(p)=\left\\{\begin{array}[]{cc}0&p\leq P_{N,T}^{*}\\\
\frac{\sum_{i=1}^{+\infty}q_{i}p-V(\bar{p},2,1)}{q_{1}p}&P_{N,T}^{*}<p<\bar{p}\\\
1&p\geq\bar{p}\end{array}\right.$ (73)

Then, if we substitute $F_{2T}(p)=w_{1}(p)$ in (36), we have:

$\displaystyle W_{1}^{2}(p;q,1)$
$\displaystyle=q_{1}p[\frac{V(\bar{p},2,1)+q_{1}(p)-\sum_{i=1}^{+\infty}q_{i}p}{q_{1}(p)}]+\sum_{i=2}^{+\infty}q_{i}p=V(\bar{p},2,1)$
(74)

for $p\in(P_{2,1}^{*},\bar{p}))$, and $W_{1}^{2}(p;q,1)=0$ for other values of
$p$. The expected value of profit for $p\in(P_{2,1}^{*},\bar{p}))$ is similar.
In conclusion, no pure strategy has an expected profit greater than the
equilibrium expected profit, and there is no probability ascribed to any pure
strategy that has an expected profit that is less than the equilibrium profit.
As a result, the necessary and sufficient conditions of equilibrium stated in
4.1 are satisfied.
To use inductive reasoning, we assume that the conditions are satisfied for
$T-1$. substituting $W_{T}(p)$ instead of $F_{2T}(p)$ in $W_{1}^{2}(p;q,T)$,
we have:

$\displaystyle W_{1}^{2}(p;q,T)$ $\displaystyle=q_{0}\delta
V(\bar{p},2,T-1)+q_{1}[\frac{V(\bar{p},2,T)-q_{0}\delta
V(\bar{p},2,T-1)-\sum_{i=1}^{+\infty}q_{i}p}{q_{1}(\delta
V(\bar{p},1,T-1)-p)}\delta V(\bar{p},1,T-1)$ (75)
$\displaystyle+(\frac{q_{1}(\delta
V(\bar{p},1,T-1)-p)-V(\bar{p},2,T)+q_{0}\delta
V(\bar{p},2,T-1)+\sum_{i=1}^{+\infty}q_{i}p}{q_{1}(\delta
V(\bar{p},1,T-1)-p)})p]$ (76)
$\displaystyle+\sum_{i=2}^{+\infty}q_{i}p=V(\bar{p},2,T)$

for $p\in(P_{2,T}^{*},\bar{p})$. In addition, based on the formulation of
$W_{T}(p)$, we have $W_{1}^{2}(p;q,T)=0$ for $p\notin(P_{2,T}^{*},\bar{p})$.
As a result, the necessary and sufficient of equilibrium in 4.1 are satisfied,
and the result is established. ∎

A.6 Proof of lemma 3.6. Let’s start by demonstrating that $I_{1T}=I_{2T}$. To
employ proof by contradiction, assume $I_{1T}<I_{2T}$. According to this
assumption, there exists $\eta>I_{1T}$ such that $F_{1T}(\eta_{-})>0$. In this
case, since firm $2$ is absent, firm $1$ acts like a monopolist. Therefore,
the expected profit of $p\in[I_{1T},\eta)$ is smaller than the expected value
resulting from offering $\eta$. This interval should not have a positive
probability in equilibrium according to the theorem 4.1, which is a
contradiction. So, we have $I_{1T}=$ $I_{2T}$ in equilibrium. Now, we need to
show $I_{1T}=I_{2T}=P_{2T}$. We again use proof by contradiction. If we assume
$I_{1T}=I_{2T}<P_{2T}$, there will be two possible states. Here, we examine
both of them and demonstrate why it is impossible for them to establish an
equilibrium.

  1. 1.

$F_{2T}\left(I_{2T}\right)<1$: Because cumulative distribution function is
right-continues, we have:

$\displaystyle\exists\eta\in(I_{2T},P_{2,T}^{*})\mid\forall
p\in[I_{2T},\eta)\rightarrow F_{2T}\left(I_{2T}\right)\leq F_{2T}(p)<1.$ (77)

Since when $\sum_{1=2}^{+\infty}q_{i}>0$, we have $P_{2T}^{*}>\delta
V(\bar{P},1,T-1)$, we can substitute $P_{2T}^{*}$ as an upper bound for both
$p$ and $\delta V(\bar{P},1,T-1)$ in $W_{1}^{2}(p;q,T)$. As a result of this
substitution, we obtain the following inequality:

$\displaystyle
W_{1}^{2}(p;q,T)<q_{0}\delta\bar{W}_{1}^{2}(q,T-1)+q_{1}P_{2T}^{*}+\sum_{i=2}^{+\infty}q_{i}P_{2T}^{*}.$
(78)

for $p\in[I_{2T},\eta)$. From (9) and (10), we have:

$\displaystyle(1-q_{0})P_{N,T}^{*}=q_{1}\delta
V(\bar{p},1,T-1)+\sum_{i=2}^{+\infty}q_{i}\bar{p}.$ (79)

Now, we substitute $(1-q_{0})P_{N,T}^{*}$ in (78):

$\displaystyle W_{1}^{2}(p;q,T)<q_{0}\delta\bar{W}_{1}^{2}(q,T-1)+q_{1}\delta
V(\bar{P},1,T-1)+\sum_{i=2}^{+\infty}q_{i}\bar{P}.$ (80)

The right-hand side of (78) demonstrates the expected value associated with
offering the maximum price $\bar{P}.$ So, the expected benefit of
$p\in[I_{2T},\eta)$ is less than its value for $p=\bar{P}$. So, we found a
pure strategy with expected profit greater than the equilibrium expected
profit, and this stat cannot establish an equilibrium based on theorem 4.1.

  2. 2.

Assume $F_{2T}(I_{2T})=1$. The intuition is comparable to the prior section.
Because $P_{2,T}^{*}>\delta V(\bar{P},1,T-1)$ and by (9), firm $2$ achieves
less profit for $p=I_{2T}$ than $p=\bar{P}$:

$\displaystyle
W_{2}^{2}(p;q,T)<q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}P_{2T}^{*}+\sum_{i=2}^{+\infty}q_{i}P_{2T}^{*}$
(81)

So, refusing these two states, we have $I_{1T}=I_{2T}\geq P_{2T}^{*}$ in
equilibrium.

Now, we only need to show $p_{N,T}^{*}<I_{1T}=I_{2T}$ is not possible to
establish an equilibrium. We again employ proof by contradiction. Below, we
examine the two possible states arising from our assumption
$p_{N,T}^{*}<I_{1T}=I_{2T}$.

  1. 1.

Assume $F_{2T}(I_{2T})>0$. Then, because we have $I_{1T}>p_{N,T}^{*}$ and
$p_{N,T}^{*}>\delta V(\bar{P},1,T-1)$, substituting $I_{1T}$ as an upper bound
instead of $\delta V(\bar{P},1,T-1)$ in $W_{1}^{2}(I_{1T};q,T)$, we have:

$\displaystyle
W_{1}^{2}(I_{1T};q,T)<q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}I_{1T}+\sum_{i=2}^{+\infty}q_{i}I_{1T}.$
(82)

By same intuition, when $p$ approaches to $I_{1T}$ from its right
neighbourhood, we have the following inequality:

$\displaystyle\lim_{P\rightarrow
I_{1T^{+}}}W_{1}^{2}(q;T)<q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}I_{1T}+\sum_{i=2}^{+\infty}q_{i}I_{1T}$
(83)

On the other hand, when $p$ approaches to $I_{1T}$ from its left
neighbourhood, the limit of expected profit is calculated as follows:

$\displaystyle\lim_{P\rightarrow I_{1T^{-}}}W_{1}^{2}(q;T)=q_{0}\delta
V_{1}(T-1)+q_{1}I_{1T}+\sum_{i=2}^{+\infty}q_{i}I_{1T}.$ (84)

Based on limit definition, there are $\epsilon_{1}>0$ and $\epsilon_{2}>0$
such that for $p\in[I_{1T},I_{1T}+\epsilon_{1})$ the expected profit is less
than the expected profit for $p\in(I_{1T}-\varepsilon_{2},I_{1T})$. According
to theorem 4.1, firm $1$ never assign a positive probability to this interval
in equilibrium, and this is a contradiction.

  2. 2.

Assume $F_{2T}\left(I_{2T}\right)=0$. as a result of the prior state being
rejected, we also have $F_{1T}\left(I_{1T}\right)=0$. In this situation, both
firms have an option to sell for $I_{1T}$. Based on our assumption
$p_{N,T}^{*}<I_{1T}=I_{2T}$, and the relationship we have in (79), the
expected profit of postponing the sell is less than the existing option of
offering $I_{1T}=I_{2T}$. Therefore, probability assigned to postponement is
zero, and the following equation holds:

$\displaystyle U_{1T}=U_{2T}\leq\bar{P}.$ (85)

In addition, it can be shown that the probability assigned to upper bounds
cannot be positive. This is due to the fact that if only one of them gives the
upper bound a positive probability, that would be the same as giving the
postponement a positive probability, which we have shown is impossible in
equilibrium. Furthermore, it can be demonstrated that the expected profit of
the upper bounds is lower than the expected profit associated with the prices
in a left neighborhood of the upper bound, making it impossible for both of
them to simultaneously assign a positive probability to the upper bound. The
reason is that $V(\bar{P},1,T-1)<p_{N,T}^{*}<I_{1T}=I_{2T}\leq U_{1T}=U_{2T}$.
Therefore, when $p$ approaches to $U_{1T}$ from its left neighbourhood, the
limit of firm $1$’s expected profit’s can be obtained as:

$\displaystyle\lim_{P\rightarrow
U_{1T^{-}}}W_{1}^{2}(p;q,T)=q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}\delta
V(\bar{P},1,T-1)+\sum_{i=2}^{+\infty}q_{i}U_{1T}.$ (86)

Considering (9), (10) and our assumption $p_{N,T}^{*}<I_{1T}=I_{2T}$, firm
$1$’s expected profit for $I_{1T}$ is more than for a right neighborhood of
$U_{1T}$:

$\displaystyle\lim_{p\rightarrow U_{1T^{-}}}W_{1}^{2}(p;q,T)$
$\displaystyle\leq q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+(1-q_{0})P_{N,T}^{*}$
(87) $\displaystyle<q_{0}\bar{W}_{2}^{2}(q,T-1)+(1-q_{0})I_{1T}.$ (88)

Based on limit definition, there exists $\epsilon_{1}>0$ such that for
$p\in\left(U_{1T}-\varepsilon_{1},U_{1T}\right)$, the expected profit is less
than the expected profit for $I_{1T}$. So, firm $1$ never assign a positive
probability to this interval in equilibrium, and it is a contradiction. Thus,
the following equation is proved:

$\displaystyle l_{1T}=I_{2T}=P_{2,T}^{*}.$ (89)

Now, we examine the second part of lemma 3.6. First, assume $U_{1T}\neq
U_{2T}$ and $U_{1T}<U_{2T}$. According to the previous section, we have
$p_{2,T}^{*}\leq U_{1T}<U_{2T}$. So, we have:

$\displaystyle\exists\eta\in\left(U_{1T},U_{2T}\right)\mid 0<F_{2T}(\eta)<1.$
(90)

Firm $2$’s expected profit can be calculated from the following equation for
$p\in(U_{1T},U_{2T}]$:

$\displaystyle\begin{aligned} W_{2}^{2}(p;q,T)=q_{0}\delta
V_{1}(T-1)+q_{1}\delta\bar{W}_{2}^{2}(q,T-1)+\sum_{l=2}^{+\infty}q_{i}p\\\
<q_{0}\delta V_{1}(T-1)+q_{1}&\delta
V(\bar{P},1,T-1)+\sum_{l=2}^{+\infty}q_{l}U_{2T}\end{aligned}$ (91)

The right-hand side of (91) is the expected profit at $U_{2T}$. As a result,
since firm $1$ is absent in $p\in(U_{1T},U_{2T}]$, firm $2$ assigns no
probability to $(U_{1T},U_{2T})$. In addition, because we know $U_{2T}$ is
greater than $U_{1T}$, firm $2$ has to assigns positive probability to the
upper bound $U_{2T}$. In addition, it is not possible to have equilibrium if
firm $2$ assigns a positive probability to $U_{1T}$. To see this, we use proof
by contradiction. Thus, assuming $F_{2T}(U_{1T})-F_{2T}(U_{1T-})>0$, firm
$1$’s expected profit for $U_{1T}$ is less than its value for a right
neighborhood of $U_{1T}$. This is due the fact that we have
$V(\bar{P},1,T-1)<p_{N,T}^{*}<U_{1T}$. So, firm $1$ assigns zero probability
to $U_{1T}$. Considering all we’ve mentioned, firm $2$’s expected profit of
$U_{1T}$ would be less than its corresponding value for $U_{2T}$. So, we
showed that firm $2$ assigns zero probability to $U_{1T}$. Let’s examine at
the situation this time from the standpoint of firm $1$. Because firm $2$ is
absent in $(U_{1T},U_{2T})$, we have:

$\displaystyle W_{1}^{2}(U_{1T};q,T)=\lim_{P\rightarrow
U_{1T^{-}}}W_{1}^{2}(p;q,T)=q_{0}\delta\bar{W}_{1}^{2}(q,T-1)+q_{1}[F_{2T}(U_{1T})\delta
V(\bar{P},1,T-1)$ (92)
$\displaystyle+(1-F_{2T}(U_{1T}))U_{1T}]+\sum_{i=2}^{+\infty}q_{i}U_{1T}<q_{0}\delta\bar{W}_{1}^{2}(q,T-1)+q_{1}[F_{2T}(U_{1T})\delta
V(\bar{P},1,T-1)$
$\displaystyle+(1-F_{2T}(U_{1T}))\eta]+\sum_{i=2}^{+\infty}q_{i}\eta.$

when $\eta\in(U_{1T},U_{2T})$. The right-hand side of (93) demonstrates the
expected profit of $\eta$. Thus, we showed that there is a pure strategy for
firm $1$ for which the expected profit is greater than the expected profit of
the prices in a left neighborhood of $U_{1T}$. Based on theorem 4.1, firm $1$
should assign zero probability to an interval $(U_{1T}-\epsilon,U_{1T}]$, when
$\epsilon$ is small positive value. This is a contradiction, and we have
$U_{1T}=U_{2T}$ in equilibrium.
In order to prove that $U_{1T}=U_{2T}<\bar{P}$ is not feasible, we employ
proof by contradiction. We assume $U_{1T}=U_{2T}<\bar{P}$ in equilibrium.
According to this assumption, there are two probable states that we examine
here.

  1. 1.

Assume $1-F_{1T}(U_{1T-})>0$. The limit of the expected profit of firm $2$,
when $p$ approaches to $U_{1T}$ from its left neighbourhood is calculated as
follows:

$\displaystyle\lim_{P\rightarrow
U_{1T^{-}}}W_{2}^{2}(p;q,T)=q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}[F_{1T}(U_{1T_{-}})\delta
V(\bar{P},1,T-1)$ (93)
$\displaystyle+(1-F_{1T}(U_{1T_{-}}))U_{1T}]+\sum_{i=2}^{+\infty}q_{i}U_{1T}.$

We also can calculate the expected value of profit at $U_{1T}$ as follows:

$\displaystyle W_{2}^{2}(U_{1T};q,T)$
$\displaystyle=q_{0}\delta\bar{W}_{2}^{2}(q,T-1)+q_{1}[F_{1T}(U_{1T_{-}})\delta
V(\bar{P},1,T-1)$ (94)
$\displaystyle+(1-F_{1T}(U_{1T_{-}}))\frac{1}{2}(V(\bar{P},1,T-1)+U_{1T})+\sum_{i=2}^{+\infty}q_{i}U_{1T}.$

Comparing the right-hand side of (93) and (94), we have:

$\displaystyle\lim_{P\rightarrow
U_{2T}-}W_{2}^{2}(p;q,T)>W_{2}^{2}(U_{2T};q,T).$ (95)

Relationship (95) holds because $\delta V(\bar{P},1,T-1)<P_{2,T}^{*}$ and
$P_{2,T}^{*}\leq U_{1T}$, and $W_{2}^{2}(p;q,T)$ in the left neighbourhood of
$U_{1T}$ assigns less probability to $\delta V(\bar{P},1,T-1)$ than in
$U_{1T}$. Therefore, firm $2$ would not give a positive probability to
$U_{2T}$. As a result, firm $1$ gains less profit at $p=U_{1T}$ than
$p=\bar{P}$, because both $p=U_{1T}$ and $p=\bar{P}$ can be effective only
when there are more than two units of demand in the market. This is in
contradiction with equilibrium conditions stated in theorem 4.1. So, this
state is not possible in equilibrium.

  2. 2.

If $F_{iT}\left(U_{iT}\right)-F_{iT}\left(U_{iT-}\right)=0$ for $i=1$ and $2$,
it can be proved that equilibrium is not possible. When $p$ approaches to
$U_{1T}$ from its left neighbourhood, based on limit definition, there is
$\epsilon>0$ as for $p\in(U_{1T}-\epsilon,U_{1T}]$ the expected profit is less
than the expected profit at $p=\bar{P}$, and this is in contradiction with our
assumption.

So, we showed that $U_{1T}=U_{2T}=U\geq\bar{P}$. Also, it is not possible to
offer a price greater than $\bar{P}$ in equilibrium. This is because offering
this price will not result in sale when there is extra demand, and it results
in a profit less than when they offer $\bar{P}$. Thus, the result is
established, and we have $U_{1T}=U_{2T}=\bar{P}$.

A.7 Proof of lemma 3.7

If the domain of two firms’ strategies is $[P_{2,T}^{*},\bar{P}]$, both firms
assign zero probability to $P_{2,T}^{*}$. To show this, we can employ proof by
contradiction. Assume firm $2$ assigns a positive probability to
$P_{2,T}^{*}$. Then, we have:

$\displaystyle\lim_{p\rightarrow
P_{2,T}^{*}-}W_{2}^{2}(p;q,T)=q_{0}\delta\bar{W}_{i}^{2}(q,T-1)+q_{1}P_{2,T}^{*}+\sum_{i=2}^{+\infty}q_{i}P_{2,T}^{*},$
(96) $\displaystyle\lim_{\begin{subarray}{c}p\rightarrow
P_{2,T}^{*}+\end{subarray}}W_{2}^{2}(p;q,T)=W_{2}^{2}(P_{2,T}^{*};q,T)<q_{0}\delta\bar{W}_{i}^{2}(q,T-1)+q_{1}P_{2,T}^{*}+\sum_{T=2}^{+\infty}q_{i}P_{2T}^{*}.$
(97)

We have (97), due to the fact that $P_{2,T}^{*}>\delta V(\bar{P},1,T-1)$. As a
result, based on limit definition, there are $\epsilon_{1}>0$ and
$\epsilon_{2}>0$ such that for
$p\in[P_{2,T}^{*},P_{2,T}^{*}+\varepsilon_{1})$, the expected profit is less
than the expected profit for
$p\in\left(P_{2,T}^{*}-\epsilon_{2},P_{2T}^{*}\right)$. This is a
contradiction with our assumption regarding $P_{2,T}^{*}$ being the lower
bound. So, both firms assign a probability equal to zero to $P_{2,T}^{*}$.
Based on theorem 4.1, the expected profit of a set of pure strategies cannot
be grater than the equilibrium expected profit. We also have that the expected
profit of a set of pure strategies with positive assigned probability should
not be less than the equilibrium expected profit. Furthermore, we know that
there $\forall\epsilon>0$, we have $F_{1T}(P_{2T}^{*}+\epsilon)>0$. For a
small $\epsilon$, we assume the expected profit of the set of points in
$(P_{2T}^{*},P_{2T}^{*}+\epsilon)$ is equal to a common value $W$. Since we
have:

$\displaystyle W=\lim_{\begin{subarray}{c}p\rightarrow
P_{2,T}^{*}+\end{subarray}}W_{2}^{2}(p;q,T)=q_{0}\delta\bar{W}_{i}^{2}(q,T-1)+q_{1}P_{2,T}^{*}+\sum_{T=2}^{+\infty}q_{i}P_{2T}^{*},$
(98)

and the initial condition $\bar{W}_{i}^{2}(q,1)=V(\bar{P},2,1)=0$, we can
conclude from (10), that the expected profit in equilibrium is
$V(\bar{P},2,T)$, which is similar for both firms.
To complete the proof, we again use proof by contradiction. Let us assume that
the equilibrium strategy is not symmetric, then we have:

$\displaystyle\exists p\in(P_{2T}^{*},\bar{P})\rightarrow
F_{2T}(p)>F_{1T}(p).$ (99)

Because cumulative distribution function is right-continues, for any small
parameters $\epsilon_{1},\epsilon_{2}>0$ there exists $\eta$ such that:

$\displaystyle F_{iT}(\eta)-F_{iT}\left(\eta^{-}\right)=0$ (100)
$\displaystyle F_{iT}(\eta)-F_{iT}(p)=e_{i}<\epsilon_{i}.$ (101)

Then, firm $1$’s and firm $2$’s expected profit at $p=\eta$ can be calculated
from the following equations:

$\displaystyle W_{1}^{2}(\eta;q,T)=q_{0}\delta\bar{W}_{1}^{2}(q,T-1)$
$\displaystyle+q_{1}[(F_{2T}(p)+e_{2})\delta V(\bar{P},1,T-1)$ (102)
$\displaystyle+(1-\left(F_{2T}(p)+e_{2}))\eta\right]+\sum_{i=2}^{+\infty}q_{i}\eta$
$\displaystyle W_{2}^{2}(\eta;q,T)=q_{0}\delta\bar{W}_{2}^{2}(q,T-1)$
$\displaystyle+q_{1}[(F_{1T}(p)+e_{1})\delta V(\bar{P},1,T-1)$ (103)
$\displaystyle+(1-(F_{1T}(p)+e_{1}))\eta]+\sum_{i=2}^{+\infty}q_{i}\eta.$

Considering negligible $\epsilon_{1},\epsilon_{2}$, firm $1$’s expected profit
is greater than firm $2$’s in $\eta$. This is in contradiction with our
previous result of similar equilibrium profit. So, the assumption of different
equilibrium distribution function cannot be true, and the result is
established.

A.8 Proof of proposition 3.8. This can be shown regarding the result of
proposition 3.5 and lemma 3.8.

A.9 Proof of proposition 3.9. To show that is a mixed-strategy equilibrium, we
have to show that it satisfies the necessary and sufficient conditions in
theorem 4.1. In addition, we need to show that it is a cumulative distribution
function, so it should be a monotonically increasing function on $[0,1]$. We
have the following equation according to the definition:

$\displaystyle V(\bar{P},N,0)=0.$ (104)

In addition, if we assume that all the firms have a common mixed-strategy
$F(p)$, the expected profit of firm $1$ can be calculated as follows:

$\displaystyle
W_{1}^{N}(p;q,1)=q_{0}\delta\bar{W_{1}^{N}(q,0)}+\sum_{i=1}^{N-1}q_{i}Z_{i-1,N}(F(p))p$
$\displaystyle+\sum_{i=N}^{+\infty}q_{i}p$ (105)

Now, we obtain the equilibrium strategy which results in
$W_{1}^{N}(p;q,1)=V(\bar{P},N,1)$. Thus, by substituting
$W_{1}^{N}(p;q,1)=V(\bar{P},N,1)$ and $\bar{W_{1}^{N}(q,0)}=V(\bar{P},N,0)=0$
in (105), we can conclude the following equation:

$\displaystyle\sum_{i=1}^{N-1}q_{i}Z_{i-1,N}(F(p))p$
$\displaystyle+\sum_{l=N}^{+\infty}q_{i}p=\sum_{i=n}^{+\infty}q_{i}\bar{p}.$
(106)

Thus, if we solve $\eqref{eq:5.672}$ for $F(p)$, we have:

$\displaystyle F(p)=\left\\{\begin{array}[]{cc}0&p\leq P_{N,T}^{*}\\\
G^{-1}(p)&P_{N,T}^{*}<p<\bar{p}\\\ 1&p\geq\bar{p}\end{array}\right.$ (110)
$\displaystyle
G(x)=\frac{\sum_{i=n}^{+\infty}q_{i}\bar{p}}{\sum_{i=n}^{+\infty}q_{i}+\sum_{i=1}^{n-1}q_{i}Z_{i-1}(x)}.$
(111)

Since we have $W_{N,1}(p)=F(p)$ in $\eqref{eq:5.671}$, all three equilibrium
conditions are satisfied. Because the assumption of inductive reasoning is
proved, we can assume that we have the equilibrium profit for $T-1$. As a
result, $W_{1}^{N}(p;q,T)$ can be calculated as follows:

$\displaystyle
W_{1}^{N}(p;q,T)=q_{0}V(\bar{P},N,T-1)+\sum_{i=1}^{N-1}q_{i}[Z_{i-1,N}(F(p))p+(1-Z_{i-1,N}(F(p))$
(112) $\displaystyle\delta
V(\bar{p},N-i,T-1)]+\sum_{i=N}^{+\infty}q_{i}p=\sum_{i=0}^{N-1}q_{i}\delta
V(\bar{p},N-i,T-1)+\sum_{i=N}^{+\infty}q_{i}\bar{p}.$

If we solve (112) for $F(p)$ , we have $F(p)=W_{N,T}(p)$. So, the equilibrium
conditions are satisfied, and the result is established.

A.10 Proof of lemma 3.10. To show that $W_{N,T}(p)$ is a function, we first
should show that its inverse function is an injective function. Now we show
that its inverse function is monotonous and increasing at
$p\in\left(P_{N,T}^{*}\bar{p}\right)$. The denominator and numerator of the
function is positive and monotonous, so because $Z_{K,N}(x)$ is a continues
and strictly decreasing this function is monotonous. The derivative of these
functions can be calculated from the following relations:

$\displaystyle\frac{dG(x)}{dx}=\sum_{i=1}^{n-1}q_{i}^{2}(\delta
V(\bar{p},N-i,T-1)-\bar{P})\frac{dZ_{i-1}(x)}{dx}$ (113) $\displaystyle\forall
x\in(0,1)\rightarrow\frac{dZ_{k}(x)}{dx}=-(k+1)\left(\begin{array}[]{l}n-1\\\
k+1\end{array}\right)(1-x)^{n-2-k}x^{k}<0$ (116)

So, $W_{N,T}(p)$ is an increasing monotonous function at
$p\in\left(P_{N,T}^{*},\bar{p}\right)$.
A.11 Proof of proposition 3.11.
Assuming (31) , the probability every firm assigns to lower bound is zero.
This can be shown by employing a proof by contradiction. If a firm $2$ assigns
positive probability to this price, firm $1$’s expected profit when $p$
approaches to $P_{N,T}^{*}$ from its left neighbourhood is more than its
expected profit at $P_{N,T}^{*}$ and more than its expected profit when $p$
approaches to $P_{N,T}^{*}$ from its right-neighbourhood. This is in
contradiction with equilibrium conditions. Here, we have the limit of the
expected profit of firm $1$ as follows:

$\displaystyle\lim_{P\rightarrow
p_{N,T}^{*}-}W_{1}^{N}(p;q,T)=\delta\bar{W}_{1}^{N}(q,T-1)+\sum_{i=1}^{\infty}q_{t}p_{N,T}^{*}.$
(117)

Since $\sum_{l=2}^{+\infty}q_{i}>0$, we have $V(\bar{P},N-i,T-1)<P_{N,T}^{*}$.
As a result, when at least one of other $N-1$ firms assign a positive
probability to $P_{N,T}^{*}$, we have the following equation at
$p=P_{N,T}^{*}$:

$\displaystyle\lim_{p\rightarrow
P_{N,T}^{*}+}W_{1}^{N}(p;q,T)=W_{1}^{N}(p;q,T)<\lim_{P\rightarrow
p_{N,T}^{*}-}W_{1}^{N}(p;q,T)$ (118)

Based on (117) and (118) and limit definition, we can find two intervals at
the left and right of $p_{N,T}^{*}$, as the expected profit of the points in
the left interval is greater than the expected profit of the points in the
right interval containing $p_{N,T}^{*}$. So it is in contradiction with
equilibrium definition and firm $1$ and firm $2$ assign zero probability to
$p_{N,T}^{*}$ and the expected profit is $V(\bar{P},N,T)$. (It can be shown
with same reasoning as the one we employed for duopoly)
If we assume that the equilibrium strategy is asymmetric, we can find $p$ as
the following equation holds:

$\displaystyle F_{-1NT}(p)>F_{-2NT}(p).$ (119)

Because cumulative distribution function is right-continues, $\epsilon$ and
$\eta$ can be found as for $i=1,2$ the equations (120) and (121) hold:

$\displaystyle F_{-iNT}\left(\eta^{-}\right)-F_{-iNT}(\eta)=0,$ (120)
$\displaystyle F_{-iNT}(\eta)-F_{-iNT}(p)=e_{i}<\epsilon_{i}.$ (121)

The existence of $p$ can be shown from the fact that we assume the random
strategies are independent, so the we have
$F_{1NT}(p)=1-\prod_{i=2}^{N}(1-F_{iNT}(p))$. So, different values for
$F_{iNT}$ and $F_{jNT}$ would result in different values for $F_{-iNT}$ and
$F_{-jNT}$. Based on limit definition, $\epsilon_{i}$ can be chosen as small
as we wish, and firm $1$ and firm $2$ ’s expected profit can be obtained from
the following equations:

$\displaystyle W_{1}^{N}(p;q,T)=\delta\bar{W}_{1}^{N}(q,T-1)$ (122)
$\displaystyle+\sum_{i=1}^{N-1}q_{i}\left[\left(F_{-1NT}(p)+e_{1}\right)\delta
V(\bar{P},N-i,T-1)\right.$ (123)
$\displaystyle\left.+\left(1-\left(F_{-1NT}(p)+e_{1}\right)\right)\eta\right]+\sum_{i=2}^{+\infty}q_{i}\eta$
$\displaystyle W_{2}^{N}(p;q,T)==q_{0}\delta\bar{W}_{2}^{N}(q,T-1)$ (124)
$\displaystyle+\sum_{i=1}^{N-1}q_{i}\left[\left(F_{-2}(p)+e_{2}\right)\delta
V(\bar{P},N-i,T-1)\right.$ (125)
$\displaystyle\left.+\left(1-\left(F_{-2}(p)+e_{2}\right)\right)\eta\right]+\sum_{i=2}^{+\infty}q_{i}\eta$

for $p\in(p,p+\min(\epsilon_{1},\epsilon_{2}))$ Considering a small enough
$\epsilon_{i}$, firm $1$’s expected profit is less than firm $2$’s and it is
in contradiction with our assumption of identical expected profit. Thus, the
assumption of different equilibrium distribution function is not true.

##  5 Numerical Results

In this chapter, the results from the previous chapters are analyzed, along
with the impact of various parameters. The derived distribution functions are
used to study the equilibrium strategies. Additionally, numerical examples and
graphs are presented to help the reader grasp the results. First, as part of
the model evaluation and validation process, large values of the planning
horizon are considered in the obtained models, and the outcomes are compared
with the results of [19]. In these experiments, we consider $\delta=0.9$ and
$\bar{P}=40$.
Here, in figure 1, we have obtained the equilibrium price of duopoly
competition from our model for $T=2\ldots 1000$ and the associated result from
[19]. The blue lines indicate the result of [19] and the red lines show the
result of this research. The four figure are showing the results corresponding
to $q=0.2,0.4,0.6,0.8$, respectively.

Figure 1: Duopoly model with binary demand

Same experiment is conducted for oligopoly model with $N=3$, and the effective
prices in equilibrium are compared in figure 2.

Figure 2: Opligopoly model with binary demand

As we showed in the preceding sections, when $\sum_{i=2}{+\inf}q_{i}>0$, then
we have a mixed-strategy equilibrium. In figure 3 and 4, we have the
equilibrium distribution of our model for $N=2$ and $N=3$, respectively. The
result of our model is colored in blue for $T=2\ldots 1000$, and the
equilibrium distribution of [19] is denoted by red color. We considered that
the demand follows a poison distribution with expected value of $0.5$.

Figure 3: Duopoly model with general demand Figure 4: Oligopoly model with
general demand

##  6 Conclusions

Four limited capacity pricing models are described in this study, and
equilibrium strategies are found. The equilibrium strategies are symmetric
except for the duopoly case with binary demand. This study show that up to
According to efficient price criteria, the price dispersion is zero while
there is no chance of an excess of demand, but when that possibility arises,
the market will not have a single price and will only have a predictable price
pattern.

The equilibrium strategies in this research can be used by vendors of limited
capacity perishable commodities to enhance their profits. Sellers are likely
to have lower market prices that generate the option value in cases of low
potential demand values, and they hardly select and offer high prices that
turn into a form of risk. The numerical data-based results indicate that as
the planning horizon is increased, the results of the model with a restricted
planning horizon tend to the results of the model presented by [19]. Thus,
when the planning horizon is large, the model in [19] can be a good
approximation. Each vendor can only sell one product, according to the models
described in this study. Future studies can examine generalizing the model and
investigating equilibrium conditions for larger capacities. Such models would
be more compatible with real world conditions. This research did not prove the
uniqueness of the equilibrium strategy in the oligopoly model with general
demand. In future study, requirements for rejecting asymmetric strategies will
be taken into consideration. Also the option is a mental image in the minds of
sellers before it is a mathematical formula Which is equated here with the
profits from becoming a monopolist. The option might be different and the same
analysis might be used with a slight adjustment depending on the circumstances
in other markets.

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