stochastic nash equilibrium

8/9/2019 Stochastic Nash Equilibrium

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Competitive Revenue Management of Perishable

Assets with Multiple Predetermined Options

Ming Hu

Department of Industrial Engineering and Operations Research,

Columbia University, New York, NY 10027, USA

[email protected]

Abstract

We study continuous-time revenue management models under price or sales competition with

multiple capacity providers competing to sell their own fixed initial inventories of perishable items

over a finite sales horizon. We assume the available menu of decision variables is given and each

player can observe others remaining capacities. For the sales competition of substitutable products

or price competition of complementary products, we obtain a threshold-type optimal control policy

for each player to switch sales target or price in closed form that can be sustained as an exact Nash

equilibrium for the stochastic game. Such a policy can be constructed in a reasonable computation

effort.

Furthermore, we provide counterexamples for price competition of substitutable products and

sales competition of complementary products to show that Nash equilibrium in these types of

competition fails to have monotone threshold structure generally. As a variant of price compe-

tition of substitutable products, dynamic competition with a Multinomial Logit customer choice

model where each player sells finite differentiated products by making available offer sets, has Nash

equilibrium policy not necessarily nested by fare, in contrast to its monopolistic version.

Key words: Revenue Management; Nash Equilibrium; Monotonicity; Optimal Switching Time

History:

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1 Introduction

Oligopoly is arguably the most interesting market condition for studying revenue management

(RM) since it is the prevailing competitive situation in many RM industries. The two classic

models of oligopoly are the Cournot model (competition in quantities Cournot (1838)) and the

Bertrand model (competition in prices Bertrand (1883)). The empirical study of Brander and

Zhang (1990) shows that in the airline industry the Cournot model seems much more consistent

with the data than the Bertrand model, where the volume or output is aggregated over a quarter.

Wei and Hansen (2007) models airlines quantity decisions of both aircraft size and service frequency

under competition, based on empirically derived cost function and market share model. Thus it is

empirically suggested that airlines long-term competitive business strategy is better modeled by a

Cournot model.

For the short-term decision at the operational level that RM is particularly focused on, it is

difficult to say that firms practicing RM compete purely on price or quantity. On the one hand, as

capacity is fixed in a short term, price becomes the relevant decision variable. So price competition 1

might seem more appropriate. On the other hand, the main decision variables in quantity-based

RM are capacity allocations, which are quantity-based variables. This would suggest that quantity

competition models are more relevant for quantity-based RM.

The price of a product provided by one firm can affect sales of another product offered by its

competitor in two ways, substitutably or complementarily. Most RM literature on competition

considers substitutable effects between differentiated products offered by competing firms, where

a price increase (decrease) in a product positively (negatively) influences sales of other related

products. For example, a price increase of a flight offered by one airline can increase the sales of a

competing flight. Most RM models often ignore complementary effects, where a price increase (de-

crease) in a product negatively (positively) influences sales of other related products. For example,

lowering the price of a particular flight may increase the demand for car rentals at the destination.

In this paper, we consider an oligopolistic market where multiple capacity providers com-

pete to sell their own fixed initial inventories of perishable products over a finite sales horizon.

1We mention price (quantity) competition rather than Bertrand (Cournot) competition here in the RM setupbecause our problem is slightly different from what Bertrand (Cournot) competition means in economics literaturein the sense that our decision is made dynamically in a continuous time over a finite horizon rather than a staticaggregated decision.

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There are four types of competition as a combination of quantity/price competition with substi-

tutable/complementary products. First, quantity competition with substitutable products may be

more appropriate for the competition of hotels, cruise ships, and rental cars and other common

quantity-based RM industries with substitutable products. Firms fix the prices for the duration of

the sale of their inventories, and only change allocations to the products. This is how quantity based

revenue management is practiced in the airline industry. Such static, fixed, prices are preferred

when prices have to be advertised, or resources are sold based on reservations, or it is otherwise

costly to change prices. Second, the current airline market with fierce competition from low cost

carriers demands a price competition model with substitutable products in a market environment

with price transparency and low cost of changing price. In addition, price or quantity competition

models with complementary products can best capture the independent efforts of airline, car rental

and hotel industry to secure its own business by taking the most favorable price-based or quantity-

based actions according to other industries decisions on related products with complementary

effects.

Singh and Vives (1984) points out quantity (price) competition with substitutable products is

the dual of price (quantity) competition with complementary products. We extend the traditional

one-shot economic problem to a RM problem with decision making in a continuous time. We

assume the decision options for each player are finite and pre-determined. For the quantity-based

RM competition problem with substitutable products and the price-based RM competition problem

with complementary products, we provide an algorithm to construct the exact Nash equilibrium

for the stochastic game.

In order to derive the feedback type of Nash equilibrium, we need to assume that the joint

inventory levels are observable by all players at any time. This requirement used to be unrealistic

but is arguably implementable now. For example, in 2004, to help customers avoid getting stuck

in a seat they do not want, Orbitz started offering a seat map feature that lets travelers compare

flight options by seat availability when reviewing fares. Now more online travel agencies (e.g.,

Expedia, Travelocity) and major airlines (e.g., Northwest, American, Delta, United, Continental,

US Airways) have joined to offer such preview seat features from their websites. It is indeed possible

to use scraper programs to keep track of competitors inventory levels as well as prices in a real-time.

The assumption of fixed number of pricing options is not restrictive but actually is the current

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practice. As Gallego and van Ryzin (1994) notes, discrete price points become possible when there

exists an explicit or implicit consensus at the industry level, or arise if a firm wants to achieve

certain market segmentation. For example, airlines typically publish promotional and full fares

in advance and retailers usually consider discounts of 25%-50% off list prices. After all, current

practice in airline industry makes the ticket price round in currency denomination within some

reasonable range and in retailing business practice prefers to have 9(or 99) as the last digit(s)

of price.

Gallego and Hu (2006) provides a choice-based, multi-player, pricing game theoretic formulation

as a stochastic control problem to model an oligopolistic market with multiple capacity providers

competing to sell their own fixed perishable inventories over a finite sales horizon. The open-loop

(essentially closed-loop) Nash equilibrium in the corresponding differential game can be proved to

be an Nash equilibrium for the original stochastic game asymptotically in the sense that either

the initial capacity of each player or the sales horizon is sufficiently large. This fluid heuristic sug-

gested by the open-loop (closed-loop) equilibrium provides a good approximation to the stochastic

game when a large sales volume and a long selling horizon are able to smooth out the stochastic

fluctuations in sales over the horizon. Similarly for a quantity-based stochastic game, the fluid

heuristic provides good performance asymptotically. However, it is less likely to be the case for

small initial inventories and a short selling horizon. It is reported in the paper that for a particular

MultiNomial Logit (MNL) model the relative performance of the open-loop heuristic is worse than

10% below the Nash equilibrium when the number of capacities from any player is fewer than 15

and is nearly optimal for more than 37 items. The heuristic is also tested not to perform very well

for a short selling horizon. Given the competitive nature and large size of the market2, a 5% gap

is significant3 and is truly the goal for revenue management systems.

Several papers model the price competition with gross substitution effects. Talluri (2003) (also

see Talluri and van Ryzin (2004, 8.4.3.2)) studies a duopoly quantity-based RM competition with

substitution, where each player makes decisions in a discrete time and can effectively change the

prices by deciding on what subset they make available simultaneously. We believe this model is

2Nowadays the major U.S. airlines have annual domestic revenue about $2-$12 billion. (Source: Forbes)3By most estimates, the revenue gains from the use of quantity-based RM systems are 4-5%, roughly comparable

to many airlines total profitability in a good year (see Talluri and van Ryzin (2004, 1.2.2)). It is estimated thatnetwork RM techniques add 0.5% to 1% of additional revenue on top of the revenue gains from single-leg capacityallocation and overbooking controls (see Phillips (2005, 8.2)).

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essentially price competition in the sense of Bertrand competition though the decision variables

are making available offer sets. Lin and Sibdari (2006) develops an oligopolistic model to describe

dynamic price competitions in a discrete time between firms that sell substitutable products. It

has been shown in both papers that in general it is even hard to prove the existence of a Nash

equilibrium for the stochastic game not to mention its uniqueness and structure properties. In this

paper, we also provide examples to show that in a continuous-time price competition model for

gross substitutes, we do not expect to have price monotonicity generally that we used to see in the

monopolistic RM problem.

We provide a sufficient condition for the existence of Nash equilibrium under either price or

quantity competition for both gross substitutes and complements. Based on the characterization,

we design a constructive approach to computing the best-response strategies (exact4 Nash equilib-

rium) when the available price or quantity options for each player are predetermined, discrete and

finite. We need to have submodularity to have the algorithm works. Thus for quantity competition

with gross substitutes or for price competition with gross complements, we can obtain Nash equi-

librium in closed-form by providing a recursive expression. For the two types of competition with

submodular property, the construction not only can practically guarantee performance for small

sales volumes or a short selling horizon, but also is theoretically elegant. It is rarely seen examples

of Nash equilibrium solved for multi-player stochastic games in the operations research (OR) lit-

erature due to its stochastic nature. In a seminal work, Vieille (2000) establishes the existence of

equilibrium payoffs in general two-person nonzero-sum undiscounted stochastic games with finite

action and state sets. It is appealing to have concrete examples from classic OR setups to com-

plement the theoretic literature. The monopolistic dynamic pricing problem initialized by Gallego

and van Ryzin (1994) can potentially be extended to the game context to provide such a example

because the no-replenishment nature of such RM problems only allow the inventory level to go

down monotonically over the horizon rendering simple finite-step state changes. This paper follows

the technique introduced by Feng and Xiao (2000a) to construct a Nash equilibrium for the game

version of the RM problem with finite price options. For the other two types of competition with

supermodular property, i.e., the price competition with gross substitutes and quantity competition

with gross complements, we do provide examples that the construction sometime also works.

4We put exact here to differentiate the close-loop or feedback strategies from the open-loop Nash equilibrium.

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Under the setup of Gallego and van Ryzin (1994), there is a series of papers studying the

optimal strategy for the monopoly with a menu of available prices. If price is only allowed to

change monotonically, i.e., either the markup or markdown policy is implemented, optional sampling

theorem from martingale theory can be utilized to characterize the optimal solutions. Feng and

Gallego (1995) provide an optimal threshold-type switching time policy when the monopoly is only

allowed at most one price change, from a given initial price to another given price. Feng and Gallego

(2000) further characterize the optimal timing of price changes within a given menu of allowable

price paths that is time-dependent. Feng and Xiao (2000b) identify an optimal pricing policy in a

recursive form under the assumption that the allowable price set is discrete and finite and the price

changes are irreversible. If reversible price changes are allowed, Feng and Xiao (2000a) show that

optimal policies can be constructed in a closed form by directly deriving sufficient optimal conditions

from the associated Bellman equation. Feng and Xiao (2001) further apply the idea to an airline seat

inventory control problem with multiple origins, one hub and one destination. Whether reversible

price changes are allowed, we always have the following results about the optimal solutions: (1)

An exact solution can be derived in a recursive form; (2) At each inventory level there exists a

sequence of nested time thresholds that guide price changes; (3) The threshold time points shift

monotonically as inventory level changes. It turns out under unilateral concavity and submodularity

of the revenue rates, these properties are retained in the game context for each player. We also

show that a strong version of unilateral concavity and submodularity can guarantee the uniqueness

of the Nash equilibrium.

The algorithm to compute the close-loop Nash equilibrium under submodularity needs a reason-

able computation effort due to the discretization of continuous-time analytic formulae. However,

cumulative discretization errors are limited to a negligible size for short planning horizon and small

inventory levels. For long planning horizon or large inventory levels, fixed pricing policy is proved

to provide very good performance (see Gallego and Hu (2006)).

The rest of the paper is organized in the following order: 2 discusses quantity competition

with substitutable products. Concretely, 2.1 presents a mathematical formulation of the stochas-

tic game. 2.2 derives the sufficient and necessary condition for a Nash equilibrium in such a

stochastic game. 2.3 constructs the Nash equilibria recursively. 3 studies price competition with

complementary products as a dual problem. Numerical results and counterexamples are placed in

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4, followed by concluding remarks and future research questions.

2 Quantity Competition with Substitutable Products

2.1 The Stochastic Game

We consider an oligopolistic market of m differentiated substitutable products with a set S =

{1, 2, . . . , m} of competitors. At time zero, firm i S has inventory ni Z+ = N

{0} units of the

substitutable perishable asset and the same finite time t > 0 to sell them. We assume the salvage

value of the asset at time t, the end of selling horizon, is zero and that all other costs are sunk.

We assume a discrete sales target menu for each firm that contains finite options. For each

firm i S, there are K(i) N number of sales target options and we denote the set of them by

Qi = {i,1, i,2, . . . , i,K(i)}, with 0 i,K(i) < < i,2 < i,1 < . We denote the joint strategy

space by Q := mi=1Qi.

The market is assumed to be an imperfect market where demand is a function of the prices

across the industry and in a Cournot competition each firm tries to set price to meet sales target

chosen. At any time s, the current price vector p(s) = (p1(s), p2(s), . . . , pm(s)) is determined by

the vector of non-homogeneous Poisson demand intensity vector (s) = (1(s), 2(s), . . . , m(s))

with i(s) Qi through a mapping p() : Q Rm+ , p() = (p1(), p2(), . . . , pm()). Let

i = (1, . . . , i1, i+1, . . . , m), i S denote the demand intensity vector of the other m 1

firms who compete with firm i. We denote the range of the mapping p() by P. For any given i,

the revenue rate function for player i, i S is ri(i; i) := ipi(). We assume the mapping

p() is known and that it satisfies the following assumptions:

Assumption 1 (One-to-One Mapping). The mapping p() : Q Rm+ has a one-to-one reverse

mapping (p) : P Rm+ .

Assumption 2 (Concavity). ri(i; i) is an increasing and concave of i for any i.

Assumption 3 (Decreasing Differences). ri(i,l; i) ri(i,r; i) is decreasing in i for any i,l >

i,r.

Assumptions 1-3 are satisfied for the inverse function of most commonly used demand functions,

e.g. linear demand function and MultiNomial Logit(MNL) demand function.

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Example 1 (MNL Demand Function). The demand mapping is

i(p) = Mai exp(bipi)

a0 +m

j=1 aj exp(bjpj), i S,

where a0 0 is the no-purchase option value and ai, bi 0, i S. Its corresponding price

mapping is

pi() =1

biln

ai[M m

j=1 j]

a0i, i S.

It is not hard to check that Assumption 2 is satisfied for ri(i; i). Additionally since

ri()

j=

ibi

1

M m

j=1 j, j = i,

is decreasing in i, Assumption 3 holds.

Example 2 (Linear Demand Function). The demand mapping is

i(p) = ai bipi +j=i

cijpj , i S,

where ai, bi > 0, i S, cij 0, j = i, i S. We assume that the matrix B with Bii = bi and

Bij = cij for j = i, is nonsingular, then the price mapping is

pi() = ai bii j=i

cijj, i S.

It is easy to see for linear demand function, 2ri(i; i)/2i = bi < 0 and ri(i; i)/j =

ciji, j = i is decreasing in i.

The firms compete for the market by adjusting its own targeted sales level. At any time s [0, t],

firm i, i S applies its own non-anticipating sales level i(s) Qi. Let

Ii,j(s) =

1, if i,j Qi is effective at s,

0, otherwise.

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A non-anticipating policy ui for firm i is defined to be

ui =

(Ii,1(s), Ii,2(s), . . . , I i,m(s)) :

K(i)j=1

Ii,j(s) = 1, 0 s t

,

where the imposed constraint is to ensure that one and only one sales target for each firm is active

at any given time. Let Ni,j(s), i S, j = 1, 2, . . . , K (i) represent the accumulated number of

items sold up to time s at sales target i,j Qi. A demand for any firm i S is realized at sales

target i,j at time s if dNi,j(s) = 1. We denote by U the joint Markovian allowable sales policy

space: any joint allowable sales policy u = (u1, u2, . . . , um) U must satisfy that for i S,

K(i)

j=1

t

0

Ii,j(s)dNi,j(s) ni, a.s.

and the sales level i(s) targeted by firm i is a function of the elapsed time s, its own inventory

level as well as the capacity levels of all other firms at time s, i.e.,

i(s) = i (s, n1(s), . . . , nm(s)) ,

where ni(s) := niK(i)

j=1 Ni,j(s) is the remaining inventory of firm i S at time s [0, t]. In terms

of game theory, we analyze strategies in feedback form, or in other words, closed-loop strategies.

Given sales target policy u U, joint initial stock vector n5 = (n1, n2, . . . , nm) Zm+ and a

finite sales horizon t > 0, we denote the expected profit for any firm i S by

Ji(t,n,u) := E

K(i)j=1

t0

Ii,j(s)pi ((s)) 1{ni(s)>0} dNi,j(s)

.

The goal of each firm i S is to maximize its total expected profit over [0, t] in the competitive

market. We assume all firms have perfect information of inventory levels about each other. More

specifically, all firms completely observe the joint state vector (ni(s), i S) at any time s [0, t],

and act upon that information. A joint policy u = (u1, u2, . . . , u

m) U constitutes a Nash equi-

librium if, whenever any firm modifies its policy away from the equilibrium, its own payoff will not

5We will omit vectos sign above all vectors for simplicity of notation. Readers should be able to tell that from thecontext.

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increase. More precisely, u U is called a Nash equilibrium if Ji

t,n,ui, ui

Ji

t,n,ui , ui

,

i S for any (ui, ui) =

u1, . . . , ui1, ui, u

i+1, . . . , u

m

U. In other words, we are requiring

that, for any i S, the policy ui provides the optimal solution to the dynamic pricing problem for

firm i while all firms j = i use policy uj . Generally it is extremely difficult to solve such a stochas-

tic game for Nash equilibria as current research still stays at theoretically proving the existence

of approximate equilibria. However, for this particular stochastic game, we can solve for the Nash

equilibrium in closed form and compute it with a reasonable effort.

2.2 Sufficient Equilibrium Condition

2.2.1 The Monopolistic Problem

Before we present a sufficient condition to characterize the Nash equilibrium, we present its reduced

version in one-dimensional state space (m = 1), where there is no competition, to gain some insights.

For the single capacity holder in a monopolistic environment, we drop the subscript i that specifies

a player for the time being and denote the set of potential sales target by Q = {1, 2, . . . , K}

with K < < 2 < 1. By Assumption 2, all the sales level options are efficient in the sense

that Q forms a maximum concave envelope6. Here the price function is a one-variable mapping of

the demand intensity decision, i.e., pi = p(i). Since price is strictly decreasing in sales, we have

p1 < p2 < < pK. The notion ri := pii represents the expected revenue rate at i. We assume

ri > rj ifi > j (or pi < pj). Given a sales target policy u U, an initial stock n Z+ and a sales

horizon t > 0, we denote the expected revenue by J(t,n,u). The firms problem is to find a sales

policy restricting itself to the discrete set of sales levels to maximize the total expected revenue

generated over [0, t], denoted by J(t, n). The following Lemma is adapted from Feng and Xiao

(2000a, Lemma 2) and it is crucial to our proof of the optimality condition that characterizes the

Nash equilibrium.

Lemma 1. LetV(t, n) be a differentiable function for all give n 0. If V(t, n) satisfies:

V(t, n)

t+ i [V(t, n 1) V(t, n)] + ri = 0, (1)

6See Feng and Xiao (2000a, Lemma 3) for how to determine the envelope if given an arbitrary set of predeterminedsales targets(or prices).

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while i is the smallest integer l = 1, 2, . . . , K 1 such that

V(t, n) V(t, n 1) rl rl+1l l+1

, (2)

then

(i) an optimal sales target at (t, n) is i and V(t, n) is the value function of the monopolistic

model, i.e., V(t, n) = J(t, n).

(ii) if P is a strictly maximum concave envelope, i is the unique optimal sales target at (t, n).

Proof. An informal derivation by Principle of Optimality leads to the following Hamilton-Jacobi-

Bellman (HJB) equation that characterizes the optimal policy:

J(t, n)

t+ max

j=1,2,...,K{j(t) [J

(t, n 1) J(t, n) + pj]} = 0, (3)

where

j(t) =

j, if Ij(t) = 1,

0, otherwise.

A rigorous justification can be obtained by using Theorem II.1 in Bremaud (1980). Suppose i is

the smallest integer such that (2) holds. By the maximum concave envelope assumption of P (a

crucial condition7), we have

(i) j < i, j > i,

V(t, n) V(t, n 1) >ri1 rii1 i

rj rij i

.

(ii) j > i, j < i,

V(t, n) V(t, n 1) ri ri+1i i+1

ri rji j

.

By (1), V(t, n)/t = i[V(t, n) V(t, n 1)] ri, thus for j {1, 2, . . . , K },

V(t, n)

t+ j [V(t, n 1) V(t, n)] + rj = (i j)[V(t, n) V(t, n 1)] + rj ri 0.

7The proof of Lemma 2 in Feng and Xiao (2000a) does not make this assumption explicitly.

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Hence, V(t, n) satisfies HJB equation (3) and i achieves the optimum at (t, n). In addition, no

other j, j = i can achieve the optimum if r() is strictly concave in .

2.2.2 The Oligopolistic Problem

The multi-dimensional version of maximum concave envelope can be guaranteed by Assumption 2,

which implies that the marginal revenue rate

ri (i,l; i) ri (i,l+1; i)

i,l i,l+1

of player i to switch sales target is increasing in l for any given i. For an oligopolistic market

(m > 0), let n = (n1, n2, . . . , nm) Zm+ be the joint state vector and ei Rm+ , i S be the i

th

unit vector. We have the following extension of Lemma 1 to the multi-player game.

Theorem 1. LetVi(t, n), i S be differentiable functions for all given n Zm+ . If Vi(t, n), i S

satisfies:

Vi(t, n)

t+ i,j(i) [Vi(t, n ei) Vi(t, n)] + ri

i,j(i); k,j(k)

= 0, i S, (4)

while j(i): S N is a function mapping i to the smallest integer l = 1, 2, . . . , K (i) 1 such that

Vi(t, n) Vi(t, n ei) ri

i,l; k,j(k)

ri

i,l+1; k,j(k)

i,l i,l+1

, (5)

where k,j(k) :=

k,j(k), k S \ {i}

, then an equilibrium sales policy u at (t, n) is (i,j(i), i S)

and Vi(t, n), i S are value functions of the Nash equilibrium, i.e., Vi(t, n) = Ji(t,n,u), i S.

Proof. If there exists a set of differentiable functions Vi(t, n) satisfying the system of (4) and (5)

for any i S simultaneously, by Lemma 1, then Vi(t, n), i S is the value function of the best

response problem for firm i when the competitors strategy is k,j(k) at (t, n), which indicates that

Vi(t, n), i S are value functions of the Nash equilibrium.

Remark 1. Since the system of HJB equations is a necessary and sufficient criterion to characterize

value functions of a Nash equilibrium, then if

i,j(i)(t, n), i S

is a unique solution to achieve

the optimum in the system (4), the stochastic game has a unique Nash equilibrium.

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2.3 Construction of the Nash Equilibrium

The system of equations (4) potentially provides a dynamic programming scheme to construct the

value function Vi(t, n), i S explicitly with boundary conditions

Vi(0, n) = 0, i S, n Zm+ , (6)

Vi (t, (n1, . . . , ni1, 0, ni+1, . . . , nm)) = 0, nj Z+, j = i, i S, t R+. (7)

We first walk through the construction procedure for a duopoly game(m = 2) to ease the

illustration and gain some intuition, and then provide a generic algorithm to treat the general

multiple-player case(m > 2).

2.3.1 Solution Scheme for Duopoly (m = 2)

For (n1, n2) = (0, 0), we have V1(t, (0, 0)) = 0, V2(t, (0, 0)) = 0, t R+.

For (n1, n2) = (1, 0) when only firm 1 has capacity to sell, we have V2(t, (1, 0)) = 0 and note that

V1(t, (1, 0)) 0 when t 0, the inequality

V1(t, (1, 0)) V1(t, (0, 0)) r1(1,1) r1(1,2)

1,1 1,2

can hold for t 0. Hence, in the right neighborhood of t = 0, 1,1 is the unique optimal sales for

firm 1. This is consistent with our intuitions: firm 2 has run out of inventory and with little time

left, it is necessary for firm 1 to target sales at the highest level 1,1.

Solving for V1(t, (1, 0)) in HJB equation (4) at 1,1 with the boundary condition V1(0, (1, 0)) = 0

yields

V1(t, (1, 0)) = p1(1,1)

1 e1,1t

, t 0.

Since V1(t, (1, 0)), t 0 is strictly increasing and concave in t,

z1,1(1, 0) := sup

t 0 : V1(t, (1, 0))

r1(1,1) r1(1,2)

1,1 1,2

is well-defined with sales 1,1 optimal for t (0, z1,1(1, 0)]. As 1,2 is effective for (z1,1(1, 0), t] when

t z1,1(1, 0), we can solve V1(t, (1, 0)) from (4) with the boundary condition

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V1 (z1,1(1, 0), (1, 0)) = (r1(1,1) r1(1,2)) /(1,1 1,2):

V1(t, (1, 0)) = p1(1,2) + 1,1p1(1,1) p1(1,2)

1,1 1,2e1,2(tz1,1(1,0)), t z1,1(1, 0).

Since V1(t, (1, 0)), t z1,1(1, 0) is strictly increasing and concave in t,

z1,2(1, 0) := sup

t z1,1(1, 0) : V1(t, (1, 0))

r1(1,2) r1(1,3)

1,2 1,3

is well-defined with sales 1,2 optimal for t (z1,1(1, 0), z1,2(1, 0)].

The above procedure is repeated until z1,K(1)1(1, 0) is calculated by the following recursive

formula, i = 2, . . . , K (1) 1:

V1(t, (1, 0)) = p1(1,i) + 1,i1p1(1,i1) p1(1,i)

1,i1 1,ie1,i(tz1,i1(1,0)), z1,i1(1, 0) < t z1,i(1, 0),

z1,i(1, 0) := sup

t z1,i1(1, 0) : V1(t, (1, 0))

r1(1,i) r1(1,i+1)

1,i 1,i+1

.

Then the optimal policy is to apply sales target 1,i when t (z1,i1(1, 0), z1,i(1, 0)], i = 1, 2, . . . , K (1)

1 with the convention z1,0(1, 0) = 0 and apply sales target 1,K(1) when t > z1,K(1)1(1, 0).

For (n1, n2) = (0, 1), we have a similar result as (n1, n2) = (1, 0).

For (n1, n2) = (1, 1), note that V1(t, (1, 1)) 0, V2(t, (1, 1)) 0 when t 0 and V1(t, (0, 1)) = 0,V2(t, (1, 0)) = 0, the inequalities

V1(t, (1, 1)) V1(t, (0, 1)) r1(1,1, 2,1) r1(1,2, 2,1)

1,1 1,2,

V2(t, (1, 1)) V2(t, (1, 0)) r2(1,1, 2,1) r2(1,1, 2,2)

2,1 2,2

can hold for t 0. Hence, by Theorem 1, in the right neighborhood of t = 0, (1,1, 2,1) is the

equilibrium sales target. Solving for V1(t, (1, 1)), V2(t, (1, 1)) in HJB equation (4) at 1,1, 2,1 with

the boundary conditions V1(0, (1, 1)) = 0, V2(0, (1, 1)) = 0 respectively yields

V1(t, (1, 1)) = p1(1,1, 2,1)

1 e1,1t

, t 0,

V2(t, (1, 1)) = p2(1,1, 2,1)

1 e2,1t

, t 0.

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Since both V1(t, (1, 1)) and V2(t, (1, 1)), t 0 are strictly increasing and concave in t,

z1,1(1, 1) sup

t 0 : V1(t, (1, 1))

r1(1,1, 2,1) r1(1,2, 2,1)

1,1 1,2

,

z2,1(1, 1) sup

t 0 : V2(t, (1, 1))

r2(1,1, 2,1) r2(1,1, 2,2)

2,1 2,2

are well-defined with (1,1, 2,1) to be the equilibrium for t (0, z1,1(1, 1) z2,1(1, 1)].

(1) If z1,1(1, 1) < z2,1(1, 1), (1,2, 2,1) should be the equilibrium for (z1,1(1, 1), t] when t

z1,1(1, 1). Hence, we can solve V1(t, (1, 1)), V2(t, (1, 1)) in HJB equation (4) at (1,2, 2,1) with the

boundary conditions

V1(z1,1(1, 1), (1, 1)) =r1(1,1, 2,1) r1(1,2, 2,1)

1,1

1,2

,

V2(z1,1(1, 1), (1, 1)) = p2(1,1, 2,1)

1 e2,1z1,1(1,1)

r2(1,1, 2,1) r2(1,1, 2,2)

2,1 2,2,

yielding t z1,1(1, 1),

V1(t, (1, 1)) = p1(1,2, 2,1)[ez1,1(1,1) e1,2t] + V1(z1,1(1, 1), (1, 1))e

1,2(tz1,1(1,1)),

V2(t, (1, 1)) = p2(1,2, 2,1)[ez1,1(1,1) e2,1t] + V2(z1,1(1, 1), (1, 1))e

2,1(tz1,1(1,1)).

We can verify that both V1(t, (1, 1)) and V2(t, (1, 1)), t z1,1(1, 1) are strictly increasing and concave

in t, where justifications are provided in the next section. By the concavity Assumption 2,

r1(1,1, 2,1) r1(1,2, 2,1)

1,1 1,2

r1(1,2, 2,1) r1(1,3, 2,1)

1,2 1,3.

By the decreasing differences Assumption 3,

r2(1,1, 2,1) r2(1,1, 2,2)

2,1 2,2

r2(1,2, 2,1) r2(1,2, 2,2)

2,1 2,2 .

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Therefore,

z1,2(1, 1) sup

t z1,1(1, 1) : V1(t, (1, 1))

r1(1,2, 2,1) r1(1,3, 2,1)

1,2 1,3

,

z2,1

(1, 1) supt z1,1

(1, 1) : V2

(t, (1, 1)) r2(1,2, 2,1) r2(1,2, 2,2)

2,1 2,2

are well-defined.

Firm 1 switches from 1,2 to 1,3 right at z1,2(1, 1) ifz1,2(1, 1) < z2,1(1, 1). Firm 2 switches from

2,1 to 2,2 right at z2,1(1, 1) ifz1,2(1, 1) > z2,1(1, 1). Otherwise both firms switch at the same time

of z1,2(1, 1) = z2,1(1, 1). We can repeat the above procedure to extend V1(t, (1, 1)) and V2(t, (1, 1))

piecewisely along the time horizon until z1,K(1)1(1, 1) and z2,K(2)1(1, 1) are obtained. Note that

both V1(t, (1, 1)) and V2(t, (1, 1)) could have at most K(1) + K(2) segments.

(2) If z1,1(1, 1) z2,1(1, 1), we have the same way of construction. Note that whenever the

switching time coincides, both firms switch the sales target simultaneously.

For n = (n1, n2) > (1, 1), we can follow the same procedure as n = (1, 1) of constructing the

equilibrium value functions and the optimal switching times.

2.3.2 Generic Algorithm for Oligopoly (m 2)

We first describe a generic algorithm to construct value functions and optimal switching times

for the multiple-player nonzero-sum Cournot competition, and then provide proofs to justify the

well-definedness of each step. Having gained insights from the case of m = 2, it will not be hard to

understand how the following algorithm works.

Algorithm 1. Generic Algorithm for Oligopoly Quantity Competition with Substitutable Prod-

ucts: To Compute the Equilibrium Switching Times for n, n {z Zm+ :m

i=1 zi L}:

Parameter. m; L 1; K(i), i S; Qi = {i,1, i,2, . . . , i,K(i)}, i S; pi(), mj=1Qj ,

i S.

Step 0. Initialization.

SET l 1;

SET Vi (t, (n1, . . . , ni1, 0, ni+1, . . . , nm)) 0, nj Z+, j = i, i S, t [0, +).

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Step 1. Construction.

FOR any n {z Zm+ : l 1 m

i=1 zi l}, DO the following:

Step 1.1. Initialization of current effective sales index c(i) and value functions around t = 0 for n.

SET c(i) 1, i S; SET

Vi(t, n)

t0

i,c(i)[Vi(s, n ei) + pi(j,c(j), j S)]ei,c(i)(ts) ds, t [0, +), i S.

Step 1.2. Construction of value functions and switching times zi,j(n) for n.

WHILE Sa {i S : c(i) < K(i)} = , DO the following: SET

zi,c(i)(n) sup

t zi,c(i)1(n) : Vi(t, n) Vi(t, n ei)

ri(j,c(j), j S) ri(i,c(i)+1, j,c(j), j S \ {i})i,c(i) i,c(i)+1

, i Sa;

SET zc min{zi,c(i)(n), i Sa};

SET Sc {i Sa : zi,c(i)(n) = zc};

SET c(i) c(i) + 1, i Sc; SET

Vi(t, n)

tzc

i,c(i)[Vi(s, n ei) + pi(j,c(j), j S)]ei,c(i)(ts) ds + Vi(zc, n)e

i,c(i)(tzc),

t [zc, +), i S.

Step 2. SET l l + 1;

IF l > L, STOP; OTHERWISE, GOTO Step 1.

Remark 2. Lemma 3 and 4 proved next can guarantee that Vi(t, n)Vi(t, nei) is strictly increasing

in t, which together with Assumptions 2-3 can guarantee that zi,c(i)(n) is always well-defined.

Remark 3. The whole idea behind Algorithm 1 is to use dynamic programming approach to con-

struct value functions that satisfy the system of HJB equations. We denote the index of the current

effective sales level for player i by c(i). zc is the point of time when there is a sales level switch

by some player(s). Vi(t, n), i S is then adjusted to such a switch and used to seek the next

switching point. Assumptions 2-3 make sure the switching point will not be altered after being

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determined, essentially guaranteeing zi,j zi,j+1, j = 1, 2, . . . , K (i) 2, i S. For any n, there

are K(i) 1 switching points of time for any player i along the whole time horizon, and Vi(t, n)

i S is piece-wise continuous in t but could have at mostm

j=1 K(j) pieces.

Next we rigorously justify that Algorithm 1 is well-defined and indeed generates value functions

and sales strategies of a Nash equilibrium.

Lemma 2. 8 Iff(s) : R+ R+ is increasing and concave function ofs with f(z) a (f(z) > a)

and f(z) [f(z) a] for fixed > 0, z 0 and a R, then the function

F(t) :=

tz

f(s)e(ts) ds + ae(tz)

is also a (strictly) increasing and concave function of t for t z.

Proof. F(t) is a unique solution to equation F(t) = F(t) + f(t), F(z) = a. Since 0 f(z)

f(s) f(t) for 0 z s t,

F(t)

t=

tz

f(s)e(ts) ds ae(tz) + f(t) [f(t) a]e(tz) [f(z) a]e(tz) 0,

with the last inequality strictly holding if f(z) > a.

Note that F(t) is a unique solution to equation G(t) = G(t) f(t), G(z) = a f(z).

Since f(t) f(s) f(z) 0 for 0 z s t,

F(t)t

t

=

tz

f(s)e(ts) ds (a f(z))e(tz) f(t)

[f(z) a2 f(t)]e(tz) [f(z) a2 f(z)]e(tz) 0.

Remark 4. If f(s) is strictly concave in s a.s., F(t) is also strictly concave in t a.s..

Lemma 3. Vi(t, ei) is a strictly increasing, continuously differentiable and concave function of t.

Proof. This is the case ofl = 1 in Algorithm 1, which is reduced to a monopolistic problem. In view

of the case of (n1, n2) = (1, 0) in 2.3.1, the algorithm gives us Vi(t, ei) = pi(i,1)(1 exp(i,1t))

8A simple version of this Lemma in Feng and Xiao (2001, Lemma 3.2.) is not sufficient to guarantee our resultsin Lemma 4 for z > 0.

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for 0 t < zi,1(ei) where zi,1(ei) = sup {t 0 : Vi(t, ei) (ri(i,1) ri(i,2))/(i,1 i,2)} and for

j = 2, . . . , K (i) 1,

Vi(t, ei) = pi(i,j) + i,j1pi(i,j1) pi(i,j)

i,j1 i,j

ei,j(tzi,j1(ei)), zi,j1(ei) < t zi,j(ei),

zi,j(ei) := sup

t zi,j1(ei) : Vi(t, ei)

ri(i,j) ri(i,j+1)

i,j i,j+1

.

The sales target i,K(i)1(ei) is effective when t > zi,K(i)1(ei). It is easy to verify Vi(t, ei) is a

strictly increasing in t 0, continuously differentiable and concave at t = zi,j , j = 1, 2, . . . , K (i)1.

It suffices to show that the differentiability also holds at these switching times. As a matter of fact,

we have for j = 1, 2, . . . , K (i) 1,

limtzi,j(ei)

Vi(t, ei)t

= limtzi,j(ei)+

Vi(t, ei)t

= i,ji,j+1pi(i,j) pi(i,j+1)

i,j i,j+1 0.

Lemma 4. If Vi(t, n ei) is a strictly increasing, continuously differentiable and strictly concave

function of t, then Vi(t, n) constructed from Vi(t, n ei) in Step 1.1.-1.2. of Algorithm 1 satisfies

that

(i) Vi(t, n) is a strictly increasing, continuously differentiable and strictly concave function of t;

(ii) Vi(t, n) Vi(t, n ei) is strictly increasing in t.

Proof. Whenever some switching time zi0,c(i0)(n) reaches the minimum among all indices of set

Sa in Algorithm 1, player i0 switches price from pi0,c(i0) to pi0,c(i0)+1 and Vi(t, n) is updated for

t zi0,c(i0)(n), i Sa. Under Assumptions 2-3, no matter how the competitors switch sales target

lower as the time-to-go increases, the barrier for Vi0(t, n) Vi0(t, n ei0) to reach when switching

from i0,c(i0)+1 to i0,c(i0)+2 is always greater than the current barrier of switching point zi0,c(i0)(n).

This implies zi,j zi,j+1, j = 1, 2, . . . , K (i) 2, i S.

It is not hard to show the differentiability since at any switching time zi,c(i)(n) (zc for short in

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this proof), we have

limtzc

Vi(t, n)

t= lim

tzc+

Vi(t, n)

t

=

i,c(i)i,c(i)+1 pi(j,c(j), j S) pi(i,c(i)+1, j,c(j), j S \ {i})i,c(i) i,c(i)+1 0,

where j,c(j), j S \ {i} is understood as the effective sales level of player j at the time of zc

when player i switches sales target from i,c(i) to i,c(i)+1.

Taking derivative w.r.t. t on both sides of (4) yields

i,c(i)[Vi(t, n) Vi(t, n ei)]

t=

2Vi(t, n)

t2 0, (8)

which reveals that the (strict) increasing property of Vi(t, n) Vi(t, n ei) in t is equivalent to the

(strict) concavity of Vi(t, n) in t.

We use induction to show the strict increasing property and concavity of the value function

Vi(t, n) in t since it is constructed piecewisely and sequentially in Step 1.1.-1.2. of Algorithm 1.

(1) In Step 1.1., where c(i) = 1, i S, in view of Lemma 2, it suffices to verify f(0) a and

f(0) [f(0)a], where f(t) = i,1[Vi(t, nei)+pi(j,1, j S)], = i,1 and a = 0. It is easy to

verify both inequalities by noticing that f(0) = pi(j,1, j S)i,1 and f(0) = pi(j,1, j S)(i,1)2.

By Lemma 2 and Remark 4, Vi(t, n) constructed in Step 1.1. is strictly increasing and concave in

t.

(2) Suppose the value function Vi(t, n) constructed up to the point of time zc is strictly increasing

and concave. Note that Vi(t, n), t zc is a unique solution to

Vi(t, n)

t= i,c(i)Vi(t, n) + i,c(i)

Vi(t, n ei) + pi(j,c(j), j S)

,

with boundary condition Vi(zc, n) = Vi(zc, n), the left limit at t = zc of the value function

obtained from the previous stage. In view of Lemma 2, it suffices to verify f(zc) a and f(zc)

[f(zc) a], where f(t) = i,c(i)[Vi(t, n ei) +pi(j,c(j), j S)], = i,c(i) and a = Vi(zc, n). It is

easy to see that f(zc) a is equivalent to Vi(zc, n) Vi(zc, n ei) pi(j,c(j), j S), which can

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be shown by the definition of zc, since

Vi(zc, n) Vi(zc, n ei) ri(j,c(j), j S) ri(i,c(i)+1, j,c(j), j S \ {i})

i,c(i) i,c(i)+1 pi(j,c(j), j S).

It is also easy to see that f(zc) [f(zc) a] is equivalent to

Vi(t, n ei)

t

t=zc

i,c(i)[Vi(zc, n ei) Vi(zc, n) + pi(j,c(j), j S)] =Vi(t, n)

t

t=zc

,

which can be guaranteed by the concavity of Vi(t, n) up to t = zc and the differentiability at

t = zc.

Theorem 2. Vi(t, n), i S constructed in Algorithm 1 are indeed the value functions of a Nash

equilibrium and

(i) Vi(t, n) is a strictly increasing, continuously differentiable and strictly concave function of t;

(ii) Vi(t, n) Vi(t, n ei) is strictly increasing in t.

Proof. Vi(t, n), i S satisfy Theorem 1 and thus are the value functions of a Nash equilibrium. By

Lemma 3, the statements (i)-(ii) are true for n = ei. Lemma 4 then leads to the conclusion for any

n through induction on n.

Remark 5. In a Nash equilibrium, for player i at the joint inventory level n, sales target i,1 is

effective for 0 t < zi,1, sales target i,j is effective for zi,j1 t < zi,j , j = 2, . . . , K (i) 1 and

sales target i,K(i) is effective if t zi,K(i)1.

Theorem 3. Ifp() satisfies Assumptions 2 and3 strictly, Vi(t, n), i S constructed in Algorithm

1 are the value functions of a unique Nash equilibrium.

Proof. Ifp() satisfies a strong version of Assumptions 2 and 3, then for any i the marginal revenue

rateri(j,c(j), j S) ri(i,c(i)+1, j,c(j), j S \ {i})

i,c(i) i,c(i)+1

is strictly increasing between two consecutive evaluations in Step 1.2. no matter which player

switches sales level in the previous stage: a strong version of Assumption 2 guarantees that the

marginal revenue rate is strictly increasing when player i just switches his own sales target and

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a strong version of Assumption 3 guarantees that the marginal revenue rate is strictly increasing

when some player j, j = i just switches sales target. Hence, zi,j(n) constructed in Algorithm 1

does not coincide for various j {1, 2, . . . , K (i) 1}. The solution described in Remark 5 is a

unique solution to the system of HJB equations and in view of Remark 1, it must be a unique Nash

equilibrium.

3 Price Competition with Complementary Products

We consider an oligopolistic market of m differentiated complementary products with a set S =

{1, 2, . . . , m} of competitors. At time zero, firm i S has inventory ni Z+ = N

{0} units of

the complementary perishable asset and the same finite time t > 0 to sell them. We assume the

salvage value of the asset at time t, the end of selling horizon, is zero and that all other costs are

sunk.

We assume a discrete price menu for each firm that contains finite options. For each firm

i S, there are K(i) N number of price options and we denote the set of them by Pi =

{pi,1, pi,2, . . . , pi,K(i)}, with 0 pi,1 < pi,2 < < pi,K(i) < . We denote the joint strategy

space by P := mi=1Pi. The market is assumed to be an imperfect market where demand is a

function of the prices across the industry and in a Bertrand competition each firm tries to set

price to compete. At any time s, the current non-homogeneous Poisson demand intensity vector

(s) = (1(s), 2(s), . . . , m(s)) is determined by the price vector p(s) = (p1(s), p2(s), . . . , pm(s))

with pi(s) Pi through a mapping (p) : P Rm+ , p (p) = (1(p), 2(p), . . . , m(p)). Let

pi = (p1, . . . , pi1, pi+1, . . . , pm), i S denote the price vector of the other m 1 firms who

compete with firm i. We denote the range of the mapping (p) by Q. For any given pi, the

revenue rate function for player i, i S is ri(pi;pi) := pii(p). We assume the mapping (p) is

known and that it satisfies the following assumptions:

Assumption 4 (One-to-One Mapping). The mapping (p) : P Rm+ has a one-to-one reverse

mapping p() : Q Rm+ .

Assumption 5 (Concavity). ri(pi;pi) is an increasing and concave of pi for any pi.

Assumption 6 (Decreasing Differences). ri(pi,l;pi) ri(pi,r;pi) is decreasing in pi for any pi,l >

pi,r.

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24/29

where ni(s) := ni K(i)

j=1 Ni,j(s) is the remaining inventory of firm i S at time s [0, t].

Given pricing policy u U, joint initial stock vector n9 = (n1, n2, . . . , nm) Zm+ and a finite

sales horizon t > 0, we denote the expected profit for any firm i S by

Ji(t,n,u) := E

K(i)j=1

t0

Ii,j(s)pi,j (s) 1{ni(s)>0} dNi,j(s)

.

The goal of each firm i S is to maximize its total expected profit over [0, t] in the competitive mar-

ket. We assume all firms have perfect information of inventory levels about each other. More specif-

ically, all firms completely observe the joint state vector (ni(s), i S) at any time s [0, t], and act

upon that information. A joint policy u = (u1, u2, . . . , u

m) U constitutes a Nash equilibrium if

Jit,n,u

i, u

i J

it,n,u

i, u

i, i S for any (u

i, u

i) = u

1, . . . , u

i1, u

i, u

i+1, . . . , u

m U.

As a dual to the quantity competition with substitutable products, we have the following con-

structive algorithm to compute the Nash equilibrium for the stochastic game. We omit the proofs

of the existence and uniqueness due to its analogy to the quantity competition case.

Algorithm 2. Generic Algorithm for Oligopoly Pricing Competition with Complementary Prod-

ucts: To Compute the Equilibrium Switching Times for n, n {z Zm+ :m

i=1 zi L}:

Parameter. m; L 1; K(i), i S; Pi = {pi,1, pi,2, . . . , pi,K(i)}, i S; i(p), p mj=1Pj ,

i S.

Step 0. Initialization.

SET l 1;

SET Vi (t, (n1, . . . , ni1, 0, ni+1, . . . , nm)) 0, nj Z+, j = i, i S, t [0, +).

Step 1. Construction.

FOR any n {z Zm+ : l 1 m

i=1 zi l}, DO the following:

Step 1.1. Initialization of current effective price index c(i) and value functions around t = 0 for

n.

9We will omit vectos sign above all vectors for simplicity of notation. Readers should be able to tell that from thecontext.

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SET c(i) 1, i S; SET

Vi(t, n)

t0

i(pj,c(j), j S)[Vi(s, n ei) + pi,c(i)]ei(pj,c(j),jS)(ts) ds, t [0, +), i S.

Step 1.2. Construction of value functions and switching times zi,j(n) for n.

WHILE Sa {i S : c(i) < K(i)} = , DO the following: SET

zi,c(i)(n) sup

t zi,c(i)1(n) : Vi(t, n) Vi(t, n ei)

ri(pj,c(j), j S) ri(pi,c(i)+1, pj,c(j), j S \ {i})

pi,c(i) pi,c(i)+1

, i Sa;

SET zc min{zi,c(i)(n), i Sa};

SET Sc {i Sa : zi,c(i)(n) = zc};

SET c(i) c(i) + 1, i Sc; SET

Vi(t, n)

tzc

i(pj,c(j), j S)[Vi(s, n ei) + pi,c(i)]ei(pj,c(j),jS)(ts) ds + Vi(zc, n)e

i(pj,c(j),jS)(tzc),

t [zc, +), i S.

Step 2. SET l l + 1;

IF l > L, STOP; OTHERWISE, GOTO Step 1.

4 Numerical Results

Example 4 (Quantity Competition). Two airlines compete to sell tickets for the same route.

Airline 1 has sales target or booking limit options 1,1 = 4, 1,2 = 3 and 1,3 = 2 and airline 2 has

options 2,1 = 3, 2,2 = 2 and 2,3 = 1. The demand system is estimated to fit an MNL model:

1(p1,j , p2,l) = Ma1e

b1p1,j

a0 + a1eb1p1,j + a2eb2p2,l,

2(p1,j , p2,l) = Ma2e

b2p2,l

a0 + a1eb1p1,j + a2eb2p2,l,

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with the no-purchase value a0 = 1, a1 = 1, a2 = 1.1, b1 = 1/100, b2 = 1/90 and M = 400/12.

This model assumes that each player has + as a null price but does not necessarily have it as a

price option. By using Algorithm 1, we compute the equilibrium switching times, which are listed

in Table 4 and plotted in Figure 1.

05

10

0246810

0

0.5

1

1.5

2

2.5

3

3.5

4

n1

Equilibrium Switching Times for Airline 1

n2

z1,2

(n1,n

2)

z1,1

(n1,n

2)

t

Sales Sample Path

02

46

810 0

2 46 8

100

1

2

3

4

5

6

7


t

Sales Sample Path

z2,1

(n1,n

2)

z2,2

(n1,n

2)

n1

n2

Figure 1: Equilibrium Switching Times for Airline 1 and Airline 2

Example 5 (Price Competition). Two airlines compete to sell tickets for the same route. Airline

1 has price options p1,1 = $200, p1,2 = $300 and p1,3 = $400 and airline 2 has price options

p2,1 = $250, p2,2 = $300 and p2,3 = $360. The demand system is assumed to the same as in

Example 1. By using Algorithm algor:ber, we still succeed in computing the equilibrium switching

times for this case, which are listed in Table 5 and plotted in Figure 2. However, in general the

algorithm might not work.

0

5

10

02468100

2

4

6

8

10

12


t

n1n2

Sales Sampel Path

z1,2

(n1,n

2)

z1,1

(n1,n

2)

0 2 4 6 8 100

5

10

0

2

4

6

8

10

12

Equilibrium Switiching Times for Airline 2

t

n1

n2

Sales Sampel Path

z2,2

(n1,n

2)

z2,1

(n1,n

2)

Figure 2: Equilibrium Switching Times for Airline 1 and Airline 2

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n z1,1(n) z1,2(n) z2,1(n) z2,2(n)

(0,0) 0 0 0 0(1,0) 0.19 0.32 0 0(2,0) 0.47 0.72 0 0(3,0) 0.77 1.12 0 0

(4,0) 1.08 1.53 0 0(5,0) 1.38 1.93 0 0

(0,1) 0 0 0.34 0.64(1,1) 0.16 0.29 0.26 0.58(2,1) 0.45 0.72 0.29 0.61(3,1) 0.76 1.12 0.29 0.59(4,1) 1.07 1.52 0.29 0.59(5,1) 1.37 1.92 0.29 0.59

(0,2) 0 0 0.78 1.37(1,2) 0.16 0.28 0.45 1.19

(2,2) 0.42 0.67 0.59 1.28(3,2) 0.71 1.07 0.66 1.34(4,2) 1.02 1.52 0.66 1.32(5,2) 1.36 1.92 0.66 1.28

n z1,1(n) z1,2(n) z2,1(n) z2,2(n)

(0,3) 0 0 1.24 2.07(1,3) 0.16 0.28 0.64 1.83(2,3) 0.42 0.65 0.77 1.93(3,3) 0.69 1.05 0.91 2.00

(4,3) 0.96 1.45 1.01 2.03(5,3) 1.30 1.85 1.04 2.07

(0,4) 0 0 1.68 2.78(1,4) 0.16 0.28 0.82 2.49(2,4) 0.42 0.65 0.95 2.58(3,4) 0.69 1.01 1.10 2.66(4,4) 0.97 1.43 1.24 2.70(5,4) 1.24 1.83 1.34 2.73

(0,5) 0 0 2.13 3.48(1,5) 0.16 0.28 1.01 3.15

(2,5) 0.42 0.65 1.13 3.24(3,5) 0.69 1.02 1.27 3.32(4,5) 0.97 1.37 1.43 3.37(5,5) 1.25 1.81 1.56 3.41

Table 1: Optimal Switching Times for (n1, n2), 0 n1 5, 0 n2 5

5 Conclusion

For the sales competition of substitutable products or price competition of complementary products,

we obtain a threshold-type optimal control policy for each player to switch sales target or price in

closed form that can be sustained as an exact Nash equilibrium for the stochastic game. Such a

policy can be constructed in a reasonable computation effort.

For price competition of substitutable products and sales competition of complementary prod-

ucts, Nash equilibrium in these types of competition fails to have monotone threshold structure

generally. As a variant of price competition of substitutable products, dynamic competition with a

Multinomial Logit customer choice model where each player sells finite differentiated products by

making available offer sets, has Nash equilibrium policy (if exists) not necessarily nested by fare,

in contrast to its monopolistic version.

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n z1,1(n) z1,2(n) z2,1(n) z2,2(n)

(0,0) 0 0 0 0(1,0) 0.28 0.92 0 0(2,0) 0.68 1.94 0 0(3,0) 1.08 2.96 0 0

(4,0) 1.48 3.97 0 0(5,0) 1.89 4.97 0 0

(0,1) 0 0 0.59 1.08(1,1) 0.30 0.97 0.64 1.15(2,1) 0.72 2.02 0.67 1.18(3,1) 1.13 3.05 0.67 1.21(4,1) 1.55 4.08 0.67 1.22(5,1) 1.96 5.10 0.67 1.22

(0,2) 0 0 1.28 2.22(1,2) 0.30 0.98 1.35 2.30

(2,2) 0.72 2.05 1.39 2.36(3,2) 1.14 3.09 1.42 2.40(4,2) 1.57 4.12 1.44 2.43(5,2) 1.99 5.14 1.44 2.46

n z1,1(n) z1,2(n) z2,1(n) z2,2(n)

(0,3) 0 0 1.97 3.32(1,3) 0.30 0.98 2.05 3.42(2,3) 0.72 2.07 2.11 3.48(3,3) 1.14 3.12 2.14 3.54

(4,3) 1.58 4.16 2.17 3.59(5,3) 2.01 5.19 2.20 3.62

(0,4) 0 0 2.64 4.41(1,4) 0.30 0.98 2.73 4.53(2,4) 0.72 2.07 2.79 4.59(3,4) 1.14 3.14 2.85 4.65(4,4) 1.58 4.19 2.88 4.71(5,4) 2.01 5.23 2.91 4.76

(0,5) 0 0 3.31 5.49(1,5) 0.30 0.98 3.41 5.63

(2,5) 0.72 2.07 3.47 5.69(3,5) 1.14 3.15 3.53 5.75(4,5) 1.58 4.21 3.58 5.81(5,5) 2.01 5.25 3.61 5.87

Table 2: Optimal Switching Times for (n1, n2), 0 n1 5, 0 n2 5

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stochastic nash equilibrium

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