capacity allocation with traditional and internet channels

Capacity Allocation with Traditional and Internet Channels*

Yue Dai,1 Xiuli Chao,2 Shu-Cherng Fang,2,3 Henry L.W. Nuttle2

1 School of Management, Fudan University, Shanghai 200433, China

2 Industrial Engineering and Operations Research, North Carolina State University,Raleigh, North Carolina, 27695-7906

3 Mathematical Sciences and Industrial Engineering, Tsinghua University, Beijing, China

Received 1 May 2006; accepted 1 May 2006DOI 10.1002/nav.20168

Published online 14 July 2006 in Wiley InterScience (www.interscience.wiley.com).

Abstract: In this paper we study a capacity allocation problem for two firms, each of which has a local store and an online store.Customers may shift among the stores upon encountering a stockout. One question facing each firm is how to allocate its finitecapacity (i.e., inventory) between its local and online stores. One firm’s allocation affects the decision of the rival, thereby creatinga strategic interaction. We consider two scenarios of a single-product single-period model and derive corresponding existence andstability conditions for a Nash equilibrium. We then conduct sensitivity analysis of the equilibrium solution with respect to price andcost parameters. We also prove the existence of a Nash equilibrium for a generalized model in which each firm has multiple localstores and a single online store. Finally, we extend the results to a multi-period model in which each firm decides its total capacityand allocates this capacity between its local and online stores. A myopic solution is derived and shown to be a Nash equilibriumsolution of a corresponding “sequential game.” © 2006 Wiley Periodicals, Inc. Naval Research Logistics 53: 772–787, 2006

Keywords: capacity allocation; game theory; nash equilibrium; sequential game

1. INTRODUCTION

The Internet, as a relatively inexpensive electronicmedium, dramatically reduces the transaction costs andincreases information availability. By utilizing the Internet,in-stock items can be made available to more customers, andorders can be placed in real time. Internet-based electronicmarketplaces have become an integral part of the moderneconomy. In this study, we consider two competitive firms,each of which has a local store and an online store, namely,its web page. Customers can either visit the local store ororder from the web page. We assume that the local store andthe online store hold separate stocks. When a stockout occursat a local store customers may go to the online store belongingto the same firm or visit the local store of the other firm. How-ever, as commonly observed, when a stockout occurs at anonline store, customers usually will not visit the local store

Correspondence to: Yue Dai ([email protected])* This work was partially supported by the National Textile Cen-

ter of the U.S. Department of Commerce under Grant I01-S01. Thefirst author is supported in part by Chinese NSF under 70502009.The second author is supported in part by NSF under DMI-0196084and DMI-0200306.

belonging to the same firm, but they may go to the onlinestore of the other firm. One question facing each firm is howto allocate its finite capacity (stock) between its local andonline stores to maximize its profit as a whole.

Because customers may shift from one firm to anotherwhen stockout occurs, the capacity allocation of one firmaffects the decision of its rival, thereby creating a strategicinteraction. Therefore, in this paper game theory is used forthe analysis. We first consider a single-product single-periodmodel and assume that the capacity of each firm is given andknown. We study two scenarios of this capacity allocationgame. Remember that when a stockout occurs at a local storecustomers may go to the rival’s local store or go to the onlinestore belonging to the same firm. In Scenario 1, it is assumedthat those customers who have visited both the local and theonline stores of the same firm will leave the system. In Sce-nario 2, we assume that these customers with demand unmetby one firm may go to the rival’s online store before leavingthe system. For both scenarios, we present some existenceand stability conditions for a Nash equilibrium and conductsensitivity analysis of the equilibrium solution with respectto price and cost parameters. We further extend the existenceresult of a Nash equilibrium to a generalized model with

© 2006 Wiley Periodicals, Inc.

Dai et al.: Capacity Allocation 773

very mild assumptions on shifting and to the case when eachfirm has multiple local stores and single online store. Wethen study the multi-period model and show that a myopicsolution is a Nash equilibrium solution for a correspondingmulti-period game.

Our study is related to revenue management in the sensethat each firm has a finite capacity and decides its allocationto maximize the profit. Historically, revenue managementstarted as an activity that focuses on capacity allocationgiven exogenous demand. The problem of distributing seatinventory across multiple fare classes has been studied bymany researchers since 1972 (see [1, 11, 13]). More andmore researchers and practitioners have come to realize thatthe pricing decisions cannot be separated from traditionalcapacity-oriented yield management decisions [3, 4, 7–9].For a comprehensive and up-to-date overview of revenuemanagement we refer the readers to [14], which containsa bibliography of over 190 references. In contrast to mostpapers on revenue management, instead of focusing on max-imizing the revenue or profit of a single firm, this paperapplies game theory to analyze capacity strategies for mul-tiple competitive firms. We propose a single-period modeland a multi-period model and derive Nash equilibrium solu-tions to corresponding games. Note that, for convenience,in this paper we focus on the analysis of the case wheneach firm has one local store and one online store. For ageneralized model in which each firm has multiple localstores and a single online store, we derive the existence ofa Nash equilibrium. Potential applications of our analysismay also be found in scenarios in which firms have multipledemand classes, such as in the airline and hotel industries.For example, two airlines offer direct flights between thesame origin and destination with departures and arrivals atsimilar times, and each airline must allocate the seats ontheir flight to the high-fare class and the low-fare class.Another related scenario occurs in the PC and printer indus-try where the firms bundle the products they manufacturewith value-added packages, and the firms must decide onthe allocation of the basic units to each package classes tomaximize revenue [2].

This paper is also related to [19], which examines seatinventory problem with two fare classes for two competingairlines. Each airline chooses an optimal booking limit forthe lower fare class while taking into account any overflowof passengers from its rival. They show that under certainconditions this “revenue management game” has a Nash equi-librium, and in some special cases, the Nash equilibrium isunique. They also compare the total number of seats allo-cated to each fare class with, and without, competition. Ourmodel differs from that of Netessine and Shumsky in the fol-lowing aspects: (i) We consider the possible overflow fromone demand class to another within a firm, while Netessineand Shumsky’s model does not allow this. (ii) In the model

by Netessine and Shumsky, all unsatisfied demand at oneairline overflows to the rival. In our model, this overflowoccurs probabilistically. (iii) In addition to the single-periodmodel in which each firm allocates its capacity, we extend ourresults to a multi-period model in which each firm decides itstotal capacity and allocates this capacity simultaneously.

We conclude this section by briefly reviewing the appli-cation of game theory to supply chain management. Fromthe game theory perspective, the different parties in a supplychain are called players [15]. The profit function of a playeris referred to as its payoff function. A player’s best response isits best strategy given the strategies of all other players. Theconcept of “Nash equilibrium” is used to represent a solutionto a game. A set of strategies constitutes a Nash equilibriumif each player’s strategy maximizes its own payoff given thestrategies of other players. Parlar [20] is perhaps the firstauthor to treat an inventory problem using game theory. Heexamines a substitutable product problem as an extension ofthe classical newsvendor problem and proves the existenceof a unique Nash equilibrium. Dai et al. [5], [6] considera distribution system with one supplier and two retailers.When a stockout occurs at one retailer, customers may goto the other retailer. They study a single-period model inwhich the supplier may have infinite or finite capacity. Chap-ter 9 of Heyman and Sobel [10] is on sequential games,which are multi-person decision processes in which eachplayer makes a sequence of decisions. Each player’s deci-sion sequence influences the evolution of the process andaffects the time streams of rewards to all players. A sequen-tial game is said to have a myopic solution if its data canbe used to easily specify a single-period game such thatad infinitum repetition of a Nash equilibrium of the single-period game comprises an equilibrium for the sequentialgame.

The rest of this paper is organized as follows. In Section 2we consider the single-product single-period model assum-ing that the total capacity of each firm is given and known.We study two scenarios of this capacity allocation game andderive corresponding existence and stability conditions fora Nash equilibrium. In Section 3 we conduct the sensitiv-ity analysis of an equilibrium solution with respect to theprice and cost parameters. In Section 4, we present sev-eral extensions. We first analyze a generalized single-periodmodel with very mild assumptions about shifting and extendthe equilibrium-existence result to the case when each firmhas multiple local stores and a single online store. We thenextend the results to a multi-period model in which each firmmakes simultaneous decisions on its total capacity and theallocation of this capacity. A myopic solution is derived forthe corresponding sequential game. Section 5 reports on anumerical study, comparing the performance of the single-period model and the multi-period model. In Section 6, wemake concluding remarks.

Naval Research Logistics DOI 10.1002/nav

774 Naval Research Logistics, Vol. 53 (2006)

2. SINGLE-PERIOD MODELS

We consider two firms selling the same product. Each firmhas a local store and an online store. To simplify the exposi-tion, we start with a single-period model and assume that thetotal capacity of each firm is given and known. Customersmay switch among the stores upon encountering a stockoutas shown in Figure 1. For each firm, a challenging problemis to allocate its finite capacity between its local and onlinestores in order to maximize its total profit.

The initial demand at the local store of firm i will bereferred to as firm i’s local demand, while the demand at theonline store as firm i’s online demand. Following Figure 1,when a stockout occurs at a local store, customers may go tothe online store belonging to the same firm or visit the localstore of its rival firm. However, when a stockout occurs atan online store, customers may only go to the rival’s onlinestore. Therefore, all stores are faced with a random initialdemand as well as demand shifting from other stores. Thetotal demand at a store will be referred to as its effectivedemand. We use a nonspecific shortage penalty cost, whichcould, depending on context, include payment to unsatisfiedcustomers, loss of goodwill, and/or opportunity cost. How-ever, to simplify the analysis, we assume that a lost customermay cause a shortage penalty cost to all stores visited, whichcannot satisfy his/her demand.

In the sequel, we use the following notation:

sLi: the selling price at the local store of firm i = 1, 2;

sOi: the selling price at the online store of firm i = 1, 2;

hLi: the holding cost at the local store of firm i = 1, 2;

hOi: the holding cost at the online store of firm i = 1, 2;

pLi: the stockout penalty cost at the local store of firm

i = 1, 2;pOi

: the stockout penalty cost at the online store of firmi = 1, 2;

DLi: a continuous random variable, denoting firm i’s

stochastic local demand, i = 1, 2;DOi

: a continuous random variable, denoting firm i’sstochastic online demand, i = 1, 2;

aLiLj: the fraction of unsatisfied local customers at firm

i visiting the local store of (the other) firm j , i, j =1, 2, j �= i;

aLiOi: the fraction of unsatisfied local customers at firm

i visiting its online store, i = 1, 2;aOiOj

: the fraction of unsatisfied online customers atfirm i visiting the online store of (the other) firm j , i, j =1, 2, j �= i;

Ci : the total capacity of firm i = 1, 2;Bi : the capacity of firm i allocated to its online store,

i = 1, 2;RLi

: the effective demand at the local store of firmi = 1, 2;

ROi: the effective demand at the online store of firm

i = 1, 2;πi(B1, B2): the expected payoff function of firm i, i =

1, 2.

As do most, if not all, papers in traditional revenue man-agement, we assume that the local and online demands areexogenous and independent. The decision variable for firm i

is the capacity allocated to its online store, i.e., Bi . Note thatRLi

, ROi, and πi(B1, B2), i = 1, 2, depend on both B1 and B2.

For obvious reasons, we further assume that aLiLj+ aLiOi

≤1, i, j = 1, 2, j �= i.

In studying the interactive allocation strategies of the twofirms and showing the existence of a Nash equilibrium, weshall make use of the following definition of submodularityand related results introduced by Topkis [22], [23].

DEFINITION 1: A function f (x1, x2) is submodular in(x1, x2) if f (xsmall

1 , x2) − f (xlarge1 , x2) is nondecreasing in x2

Figure 1. Customer shifting.



Figure 2. Scenario 1.

for all xsmall1 ≤ x

large1 . Function f (x1, x2) is supermodular if

−f (x1, x2) is submodular. If a function is both supermodularand submodular, it is a valuation.

LEMMA 1: Let g(x1) be a nondecreasing function inx1 and m(x2) be a nonincreasing function in x2. Then thefunction min{g(x1), m(x2)} is submodular in (x1, x2).

LEMMA 2: Let function g(x1, x2) be monotone in both x1

and x2 and submodular in (x1, x2). Also let m(·) be a non-decreasing concave function. Then the composition functionm(g(x1, x2)) is submodular in (x1, x2).

We consider two scenarios for the single-period model.Remember that when a stockout occurs at a local store cus-tomers may go to the rival’s local store or go to the online storebelonging to the same firm. In Scenario 1, it is assumed thatthose customers who have visited both local and online storesof the same firm will leave the system, as shown in Figure 2.In Scenario 2, we assume that these customers with demandunmet at one firm will exhibit the same shifting behavior as

the online demand of this firm and may go to the rival’s onlinestore before leaving the system (Figure 3).

2.1. Scenario 1

In Scenario 1, based on the customer shifting structure inFigures 1 and 2, we can write down the effective demands atthe local and online stores of firm i:

RLi= DLi

+ aLj Li(DLj

− (Cj − Bj))+,

ROi= DOi

+ aOj Oi(DOj

−Bj)+ + aLiOi

(DLi− (Ci −Bi))

+,

where (x)+ = max{x, 0}.For firm i, the expected payoff function is

πi(B1, B2)

= E[sLi

min{RLi, Ci − Bi} − pLi

(RLi− (Ci − Bi))

+

− hLi(RLi

− (Ci − Bi))− + sOi

min{ROi, Bi}

− pOi(ROi

− Bi)+ − hOi

(ROi− Bi)

−]

Figure 3. Scenario 2.



= E[sLi

min{RLi, Ci − Bi} − pLi


− (pLi+ hLi

)(RLi− (Ci − Bi))

− + sOimin{ROi

, Bi}− (pOi

+ hOi)(ROi

− Bi)+ + hOi

(ROi− Bi)

], (1)

where (x)− = max{−x, 0}.

THEOREM 1: In our two-firm model, each payoff func-tion πi(B1, B2), i = 1, 2, is submodular in (B1, B2).

PROOF: First, we prove that π1(B1, B2) is submodular in(B1, B2) by showing that each term in (1) is submodularin (B1, B2) when i = 1. By using Lemma 1, we know thatmin{RL1 , C1 − B1} is submodular in (B1, B2). From Defi-nition 1, we know that (RL1 − (C1 − B1)) is a valuation.By using Lemma 2, we know that −(RL1 − (C1 − B1))

− issubmodular. To see the submodularity of min{RO1 , B1}, notethat

min{RO1 , B1} = min{DO1 + aO2O1(DO2 −B2)

+, B1 − aL1O1

× (DL1 − (C1 − B1))+} + aL1O1(DL1 − (C1 − B1))

+,

where min{DO1 +aO2O1(DO2 −B2)+, B1−aL1O1(DL1 −(C1−

B1))+} is submodular by Lemma 1 and aL1O1(DL1 − (C1 −

B1))+ is a valuation. By using Lemma 2 we can prove that

−(pO1 +hO1)(RO1 −B1)+ is submodular. The last term in (1),

hO1(RO1 −B1), is a valuation. Therefore, as the sum of severalsubmodular functions, π1(B1, B2) is submodular in (B1, B2).Similarly, we can prove that π2(B1, B2) is submodular in(B1, B2). �

By Definition 1, Theorem 1 shows that ∂πi (B1,B2)

∂Biis decreas-

ing in Bj . In other words, Theorem 1 implies that eachplayer’s best response is decreasing in its rival’s strategy. Thisresult goes with our intuition. Take firm 1’s best responsegiven firm 2’s decision as an example. If B2 increases, fewercustomers will switch from the online store of firm 2 to theonline store of firm 1, and so firm 1’s best response B1 willdecrease.

Theorem 3.1 of [23] asserts that if the strategy spaceis a complete lattice, the joint payoff function is upper-semicontinuous, and each player’s payoff function is super-modular (submodular), then there exists a pure strategyNash equilibrium. In our model, both players’ payoffs aresubmodular, so then each player’s best response is decreas-ing in its rival’s strategy. When the best responses exhibitthis monotonicity property, the players’ strategies are saidto be strategic complements, and the existence of a Nashequilibrium is easy to establish (see [12]).

THEOREM 2: For Scenario 1 of our two-firm model,there exists a Nash equilibrium (BNash

1 , BNash2 ), which can

be obtained by solving the following system of equations:

∂π1(B1,B2)

∂B1= 0,

∂π2(B1,B2)

∂B2= 0.

(2)

PROOF: The proof follows Theorem 3.1 of [23] and thesubmodularity of the payoff functions. �

To calculate the Nash equilibrium, we can apply a simplegradient algorithm, e.g., Newton’s method. Given the distri-butions of DLi

and DOi, at each iteration, we can apply Monte

Carlo simulation to simulate the distributions of the effectivedemand RLi

and ROiand then to evaluate the gradients given

in (2). Section 5 of this paper reports on a numerical study,comparing the performance of single-period and multi-periodmodels.

Now, we study the stability of a Nash equilibrium solution.The concept of stability of a Nash equilibrium was introducedby Moulin [16] using the concept of a Cournot tatonnement.A Cournot tatonnement is a sequence formed by the bestresponses of all players. For example, in our two-firm game,let ri(Bj ) be the best response function of firm i given thestrategy of firm j , Bj , then the Cournot tatonnement is thefollowing sequence

(B1, B2) → (r1(B2), r2(B1))

→ (r1(r2(B1)), r2(r1(B2))) → · · ·A Nash equilibrium is locally stable if the Cournot taton-nement starting within a neighborhood of this Nash equilib-rium converges to it.

LEMMA 3 (Moulin [16]): If | ∂2πi(B1,B2)

∂2Bi|Bi=BNash

i ,Bj =BNashj

>

| ∂2πi(B1,B2)

∂Bi∂Bj|Bi=BNash

i ,Bj =BNashj

, i, j = 1, 2, j �= i, then the Nashequilibrium is locally stable.

Taking derivatives using the Leibnitz formula in order tocheck Moulin’s condition for the expected payoff functionsin Scenario 1 is complicated. We follow the approach ofRudi [21] (see also [17–19]) and obtain the following results:

∂πi(B1, B2)

∂Bi

= hLi−hOi

−(sLi+pLi

+hLi) Pr(RLi

> Ci−Bi)

+ (sOi+ hOi

)aLiOiPr(DLi

> Ci − Bi)

+(sOi+pOi

+hOi)(1−aLiOi

) Pr(ROi> Bi , DLi

> Ci −Bi)

+ (sOi+ pOi

+ hOi) Pr(ROi

> Bi , DLi≤ Ci − Bi), (3)

∂2πi(B1, B2)

∂Bi∂Bj

= −(sLi+ pLi

+ hLi)aLj Li

fRLi|DLj

>Cj −Bj

× (Ci − Bi) Pr(DLj> Cj − Bj) − (sOi

+ pOi+ hOi

)



× (1 − aLiOi)aOj Oi

fROi|DOj

>Bj ,DLi>Ci−Bi

(Bi)

× Pr(DOj> Bj , DLi

> Ci − Bi)

− (sOi+ pOi

+ hOi)aOj Oi

fROi|DOj

>Bj ,DLi≤Ci−Bi

(Bi)

× Pr(DOj> Bj , DLi

≤ Ci − Bi) ≤ 0, (4)

and

∂2πi(B1, B2)

∂2Bi

= −(sLi+ pLi

+ hLi)fRLi

(Ci − Bi)

− (sOi+ pOi

+ hOi)(1 − aLiOi

)2fROi|DLi

>Ci−Bi(Bi)

×Pr(DLi> Ci −Bi)− (sOi

+pOi+hOi

)fROi|DLi

≤Ci−Bi(Bi)

× Pr(DLi≤ Ci − Bi) + (sOi

+ hOi)aLiOi

fDLi(Ci − Bi).

(5)

THEOREM 3: If aLiOi+aOj Oi

≤ 1 in our two-firm modelfor Scenario 1, a sufficient condition for a Nash equilibrium,(BNash

1 , BNash2 ), to be locally stable is

Pr(DLj≤ Cj − BNash

j ) >(sOi

+ hOi)aLiOi

sLi+ pLi

+ hLi

,

i, j = 1, 2, j �= i. (6)

PROOF: It is easy to see that

fRLi(Ci−Bi) = fRLi

|DLj>Cj −Bj

(Ci−Bi) Pr(DLj> Cj −Bj)

+ fRLi|DLj

≤Cj −Bj(Ci − Bi) Pr(DLj

≤ Cj − Bj)

= fRLi|DLj

>Cj −Bj(Ci − Bi) Pr(DLj

> Cj − Bj)

+ fDLi(Ci − Bi) Pr(DLj

≤ Cj − Bj),

where fRLi|DLj

>Cj −Bj(Ci −Bi) is the density function of RLi

condition on DLj> Cj − Bj . Other conditional density

functions are defined in the same way.Therefore, from (5), if

Pr(DLj≤ Cj − Bj) >

(sOi+ hOi

)aLiOi

sLi+ pLi

+ hLi

,

then∂2πi(B1, B2)

∂2Bi

≤ 0.

It follows that

∣∣∣∣∂2πi(B1, B2)

∂2Bi

∣∣∣∣ −∣∣∣∣∂2πi(B1, B2)

∂Bi∂Bj

∣∣∣∣ > (sOi+ pOi

+ hOi)(1 − aLiOi

)(1 − aLiOi− aOj Oi

)fROi|DLi

>Ci−Bi(Bi)

× Pr(DLi> Ci − Bi) + (sLi

+ pLi+ hLi

)aLj LifRLi

|DLj>Cj −Bj

(Ci − Bi) Pr(DLj> Cj − Bj)

+ (sLi+ pLi

+ hLi)fRLi

(Ci − Bi) − (sOi+ hOi

)aLiOifRLi

|DLj≤Cj −Bj

(Ci − Bi)

> (sOi+ pOi

+ hOi)(1 − aLiOi


)fROi|DLi

>Ci−Bi(Bi) Pr(DLi

> Ci − Bi)

+ fRLi|DLj

≤Cj −Bj(Ci − Bi)((sOi

+ hOi)aLiOi

− (sLi+ pLi

+ hLi) Pr(DLj

≤ Cj − Bj)).

It is easy to see that if aLiOi+aOj Oi

≤ 1 and Pr(DLj≤ Cj −

BNashj ) >

(sOi+hOi

)aLiOi

sLi+pLi

+hLi

, then | ∂2πi(B1,B2)

∂2Bi|Bi=BNash

i ,Bj =BNashj

>

| ∂2πi(B1,B2)

∂Bi∂Bj|Bi=BNash

i ,Bj =BNashj

. And the result follows fromLemma 3. �

REMARK 1: If aLiOi= 0 for all i, then

∣∣∣∣∂2πi(B1, B2)

∂2Bi

∣∣∣∣ >

∣∣∣∣∂2πi(B1, B2)

∂Bi∂Bj

∣∣∣∣always holds. Following Theorem 3 in Chapter 5 of Moulin[16], we know that a unique and globally stable Nash equili-brium exists. This special case has been studied by Netessineand Shumsky [19].

REMARK 2: It is easy to see that an instance of thetwo-firm model in Scenario 1 with higher sLi

, pLi, and hLi

and lower sOi, hOi

, and aLiOiis more likely to have a stable

Nash equilibrium solution.

2.2. Scenario 2

In Scenario 2 we allow customers who have encounteredstockouts at both the local and the online stores of one firm tovisit the online store of the rival firm, as shown in Figure 3.For this scenario, we can write the effective demands at thelocal and online stores of firm i as

RLi= DLi

+ aLj Li(DLj

− (Cj − Bj))+,

ROi= DOi

+ aOj Oi(DOj

+ aLj Oj(DLj

− (Cj − Bj))+

− Bj)+ + aLiOi

(DLi− (Ci − Bi))

+.



Analogous to Scenario 1, the expected payoff function offirm i is

πi(B1, B2) = E[sLi

min{RLi, Ci−Bi}−pLi

(RLi−(Ci−Bi))

− (pLi+ hLi

) (RLi− (Ci − Bi))

− + sOimin{ROi

, Bi}− (pOi

+ hOi) (ROi

− Bi)+ + hOi

(ROi− Bi)

],

which has the same representation as in Scenario 1, exceptthat the form of RO1 is different.

THEOREM 4: For Scenario 2 of our two-firm model,there exists a Nash equilibrium (BNash

1 , BNash2 ), which can

be obtained by solving the following system of equations:

∂π1(B1,B2)

∂B1= 0,

∂π2(B1,B2)

∂B2= 0.

PROOF: To prove the theorem, we show πi(B1, B2) is sub-modular in (B1, B2), and we need only to prove that the termswith ROi

are still submodular in (B1, B2). For firm 1, note that

min{RO1 , B1} = min{DO1 + aO2O1(DO2 + aL2O2

×(DL2 −(C2−B2))+−B2)

+, B1−aL1O1(DL1 −(C1−B1))+}

+ aL1O1(DL1 − (C1 − B1))+.

By using Lemma 1, we know that min{RO1 , B1} is submodu-lar. Meanwhile, since (RO1−B1) is a submodular function anddecreasing in both B1 and B2, by using Lemma 2, we knowthat −(pO1 + hO1)(RO1 − B1)

+ is submodular. Therefore,π1(B1, B2) is submodular in (B1, B2). Similarly, we can provethat π2(B1, B2) is submodular in (B1, B2). From Theorem 3.1of Topkis [23] and the submodularity of payoff functions, wecan prove the existence of a Nash equilibrium. �

THEOREM 5: In Scenario 2, if aLiOi+ aOj Oi

≤ 1, a suf-ficient condition for the Nash equilibrium, (BNash

1 , BNash2 ), to

be locally stable is

Pr(DLj≤ Cj − BNash

j ) >(sOi

+ hOi)aLiOi

sLi+ pLi

+ hLi

,

i, j = 1, 2, j �= i.

PROOF: Since ROiis not related to Bi , we know that

∂πi (B1,B2)

∂Biand ∂2πi(B1,B2)

∂2Biare the same as (3) and (5), respec-

tively. Analogous to Scenario 1, denoting DOj+aLj Oj

(DLj−

(Cj − Bj))+ by D

′Oj

, we obtain

∂2πi(B1, B2)

∂Bi∂Bj

= −(sLi+ pLi

+ hLi)aLj Li

fRLi|DLj

>Cj −Bj

×(Ci −Bi) Pr(DLj> Cj −Bj)−(sOi

+pOi+hOi

)(Y +Z),

where

Y = (1 − aLiOi)aOj Oi

fROi|DLi

>Ci−Bi ,D′Oj

>Bj ,DLj≤Cj −Bj

(Bi)

× Pr(DLi> Ci − Bi , D

′Oj

> Bj , DLj≤ Cj − Bj)

+ aOj Oi(1 − aLiOi

)2fROi|DLi

>Ci−Bi ,D′Oj

>Bj ,DLj>Cj −Bj

(Bi)


′Oj

> Bj , DLj> Cj − Bj),

and

Z = aOj OifROi

|DLi≤Ci−Bi ,D

′Oj

>Bj ,DLj≤Cj −Bj

(Bi)

× Pr(DLi≤ Ci − Bi , D

′Oj

> Bj , DLj≤ Cj − Bj)

+ (1 − aLiOi)aOj Oi

fROi|DLi

>Ci−Bi ,D′Oj

>Bj ,DLj>Cj −Bj

(Bi)


′Oj

> Bj , DLj> Cj − Bj).

We know that if Pr(DLj≤ Cj − Bj) >

(sOi+hOi

)aLiOi

sLi+pLi

+hLi

, then∂2πi(B1,B2)

∂2Bi≤ 0. After some calculations, analogous to Sce-

nario 1, we have

∣∣∣∣∂2πi(B1, B2)

∂2Bi

∣∣∣∣ −∣∣∣∣∂2πi(B1, B2)

∂Bi∂Bj

∣∣∣∣ > (sOi+ pOi

+ hOi)(1 − aLiOi


)fROi|DLi

>Ci−Bi(Bi)

× Pr(DLi> Ci − Bi) + (sLi

+ pLi+ hLi

)aLj LifRLi

|DLj>Cj −Bj

(Ci − Bi) Pr(DLj> Cj − Bj)

+ (sLi+ pLi

+ hLi)fRLi

(Ci − Bi) − (sOi+ hOi

)aLiOifRLi

|DLj≤Cj −Bj

(Ci − Bi)

> (sOi+ pOi

+ hOi)(1 − aLiOi


)fROi|DLi

>Ci−Bi(Bi) Pr(DLi

> Ci − Bi)

+ fRLi|DLj

≤Cj −Bj(Ci − Bi)((sOi

+ hOi)aLiOi

− (sLi+ pLi

+ hLi) Pr(DLj

≤ Cj − Bj)) > 0


This completes the proof. �

3. SENSITIVITY ANALYSIS

In this section, we analyze the sensitivity of an equilib-rium solution to the value of system parameters, such asselling price, holding cost, and stockout penalty cost. Basedon the above results, we know that in both scenarios theNash equilibrium is characterized by the following optimalityconditions:

G1(B1, B2) � ∂π1(B1,B2)

∂B1= 0,

G2(B1, B2) � ∂π2(B1,B2)

∂B2= 0,

where

∂πi(B1, B2)

∂Bi

= hLi−hOi

−(sLi+pLi

+hLi) Pr(RLi

> Ci−Bi)

+ (sOi+ hOi

)aLiOiPr(DLi

> Ci − Bi) + (sOi+ pOi

+ hOi)

×(1−aLiOi) Pr(ROi

> Bi , DLi> Ci−Bi)+(sOi

+pOi+hOi

)

× Pr(ROi> Bi , DLi

≤ Ci − Bi), i, j = 1, 2, j �= i.

Recall that the two scenarios differ in the form of ROi. In

this section, we discuss some properties that hold for bothscenarios.

THEOREM 6: Assume that the stability condition (6)holds at an equilibrium solution. The following results holdfor both Scenarios 1 and 2:

(i) Relative to selling price, BNashi is decreasing in sLi

and sOj, while it is increasing in sLj

and sOi.

(ii) Relative to holding cost, BNashi is increasing in hLi

and hOj, while it is decreasing in hLj

and hOi.

(iii) Relative to stockout penalty cost, BNashi is decreas-

ing in pLiand pOj

, while it is increasing in pLjand

pOi.

PROOF: We only show the relationship between BNashi

and selling prices sLiand sLj

. Other cases follow using thesame reasoning. By the implicit function theorem applied at(BNash

1 , BNash2 ), we have

∂B1

∂sL1

∂B1∂sL2

∂B2∂sL1

∂B2∂sL2

= −

[∂G1∂B1

∂G1∂B2

∂G2∂B1

∂G2∂B2

]−1 ∂G1

∂sL1

∂G1∂sL2

∂G2∂sL1

∂G2∂sL2

= 1(∂G1∂B1

∂G2∂B2

− ∂G1∂B2

∂G2∂B1

) − ∂G2

∂B2

∂G1∂sL1

∂G1∂B2

∂G2∂sL2

∂G2∂B1

∂G1∂sL1

− ∂G1∂B1

∂G2∂sL2

.

Recall from (4), ∂Gi

∂Bj= ∂2πi(B1,B2)

∂Bi∂Bj< 0. Under the stability

condition (6),

∂Gi

∂Bi

= ∂2πi(B1, B2)

∂2Bi

≤ 0,

and

∂G1

∂B1

∂G2

∂B2− ∂G1

∂B2

∂G2

∂B1=

∣∣∣∣∂2π1(B1, B2)

∂2B1

∣∣∣∣∣∣∣∣∂2π2(B1, B2)

∂2B2

∣∣∣∣−

∣∣∣∣∂2π1(B1, B2)

∂B1∂B2

∣∣∣∣∣∣∣∣∂2π2(B1, B2)

∂B1∂B2

∣∣∣∣ > 0.

Further, we have

∂Gi

∂sLi

= − Pr(RLi> Ci − Bi) ≤ 0, i = 1, 2.

Hence, we know ∂Bi

∂sLi

is nonnegative and ∂Bi

∂sLj

is non-positive

at the Nash equilibrium (BNash1 , BNash

2 ). In other words, BNashi

is decreasing in sLiand increasing in sLj

. �

The results of Theorem 6 go with our intuition. For exam-ple, the rationale behind (i) is as follows: If the selling priceof the local store at firm i, sLi

, increases, then at equilibriummore capacity should be allocated to that store, and thus lesscapacity is allocated to the online store. Consequently, moreonline demand is unsatisfied at firm i and shifts to the onlinestore of firm j . Therefore, at equilibrium, firm j allocatesmore capacity to its online store. Intuitively, a small changein sOi

causes an opposite effect on the equilibrium solution.

4. EXTENSIONS

We now present several extensions of the basic model. InSubsection 4.1, we generalize the model by relaxing the rout-ing rules of unsatisfied demand and analyze the existence ofthe Nash equilibrium. In Subsection 4.2, we consider the casewhen each firm has multiple local stores and a single onlinestore. In Subsection 4.3, we consider the case in which eachfirm must decide its total capacity and allocate this capacitybetween its local and online stores. Thus, each firm has twodecision variables to consider. Finally, in Subsection 4.4, weextend the single-period model to a multi-period model, inwhich each firm makes a sequence of decisions, and we derivea myopic solution for a corresponding sequential game.

4.1. A General Routing Model

In this subsection, we generalize the model by relaxingthe routing rules for unsatisfied demand. Different from thesections above, we now allow the shifting of unsatisfied



Figure 4. Model with general routing.

demand from the online store of firm i to its local store anddenote it by D′

OiLias shown in Figure 4. In addition, we drop

the assumption that a fixed percentage of unsatisfied demandwill shift. Instead, we now allow general shifting patterns ofunsatisfied demand. For example, as shown in Figure 4, weuse the notation D′

LiLjto denote the shifting of unsatisfied

demand from the local store of firm i to the local store offirm j . Similarly, D′

OiOjis used to denote the shifting from

the online store of firm i to that of firm j . The unsatisfieddemand shifting from the local store of firm i to its onlinestore is now denoted by D′

LiOi.

For this generalized model, the effective demands at thelocal and online stores of firm i can be written as

RLi= DLi

+ D′Lj Li

+ D′OiLi

,(7)

ROi= DOi

+ D′Oj Oi

+ D′LiOi

.

Recall that the payoff function of firm i is given in formula(1) in Subsection 2.1 and it can be written as

πi(B1, B2) = E[(sLi+pLi

+hLi) min{RLi

, Ci −Bi}−pLiRLi

− hLi(Ci − Bi) + (sOi

+ pOi+ hOi

) min{ROi, Bi}

− pOiROi

− hLiBi]. (8)

We now discuss the submodularity of this payoff function.Applying Lemma 1 in Section 2, we know that the payofffunction πi(B1, B2) is submodular in (B1, B2) if the followingtwo conditions are satisfied:

Condition 1: RLiis submodular in (B1, B2) and non-

decreasing in Bj , i �= j ;Condition 2: ROi

is submodular in (B1, B2) and non-increasing in Bj , i �= j .

Given the definition of RLiand ROi

in (7), we rewrite theconditions above as, again, for i �= j .

(A1): D′Lj Li

is nondecreasing in Bj and independent of Bi ;(A2): D′

OiLiis independent of Bj ;

(A3): D′Oj Oi

is nonincreasing in Bj and independent of Bi ;(A4): D′

LiOiis independent of Bj ;

With these assumptions and applying the lemmas and defi-nition given in Section 2, we obtain the following theorem.

THEOREM 7: If the assumptions (A1) through (A4) aresatisfied, then for our generalized model shown in Figure 4,there exists a Nash equilibrium.

PROOF: In payoff function (8), by Lemma 1, if assump-tions (A1) and (A2) are satisfied, min{RLi

, Ci − Bi} issubmodular and RLi

is a valuation. Therefore, the first line in(8), E[(sLi

+pLi+hLi

) min{RLi, Ci−Bi}−pLi

RLi−hLi

(Ci−Bi)], is submodular in (B1, B2). Similarly, we can prove thatif assumptions (A3) and (A4) are satisfied, the second line in(8), E[(sOi

+ pOi+ hOi

) min{ROi, Bi} − pOi

ROi− hLi

Bi],is submodular in (B1, B2). Hence, from Theorem 3.1 of Top-kis [23] and the submodularity of payoff function, we canprove the existence of a Nash equilibrium. �

Note that assumptions (A1) through (A4) are quite mildand they are very plausible. Take assumption (A1) as anexample. Recall that D′

Lj Liis the demand shifting from the

local store j to the local store i and (thus) it is nonincreas-ing in the capacity of local store j , Cj − Bj . In other words,D′

Lj Liis nondecreasing in Bj . Meanwhile, D′

Lj Liis obviously

related to the capacity of local store j but not that of localstore i. In other words, D′

Lj Liis independent of Bi . In general



D′Lj Li

is a function of the demand that cannot be satisfied bylocal store j , i.e., (Dj −Cj +Bj)

+, and it is always less thanor equal to (Dj − Cj + Bj)

+. It is reasonable to assume thatthis function is nondecreasing since the larger the demandthat cannot be satisfied at local store j , the larger the numberof customers that will switch. This will guarantee that D′

Lj Li

is nondecreasing in Bj . The other assumptions also go withour intuition.

We remark that Scenarios 1 and 2 in Section 2 are justspecial cases of this generalized model and they satisfy theassumptions (A1) through (A4). Therefore, the existence ofNash equilibrium is guaranteed, as proved in Section 2. Com-pared to the general model in this subsection, the specialshifting patterns in Scenarios 1 and 2 allow uniqueness resultsand sensitivity analysis.

4.2. A Model with Multiple Local Stores

In this subsection, we extend the original model to thecase in which each firm has multiple local stores and a singleonline store. In this case the unsatisfied customers at a localstore may overflow to the online store of the same firm or anyother local store (either of the same firm or of the other firm).We now introduce some new notation. Suppose that firm i hasmi local stores and one online store. Let Bis be the capacityallocated to local store s of firm i and Ci − ∑mi

s=1 Bis thecapacity allocated to the online store of firm i. We use aLisLjr

to represent the fraction of unsatisfied customers at local stores of firm i visiting local store r of firm j . Note that here i mayequal j . Let aLisOi

be the fraction of unsatisfied customers atlocal store s of firm i visiting its online store and aOiOj

thefraction of unsatisfied online customers at firm i visiting theonline store of (the other) firm j .

In the following we only analyze firm 1. The analysisfor firm 2 is similar. For firm 1, we can write the effectivedemands of local store r and the online store as

RL1r= DL1r

+2∑

i=1

mi∑s=1

aLisL1r(DLis

− Bis)+,

RO1 = DO1 +m1∑s=1

aL1sO1(DL1s− B1s)

+

+ aO2O1(DO2 − C2 +m2∑s=1

B2s)+,

We define revenue and cost notation at each store for thismodel similar to those in Section 2. The expected payofffunction of firm 1 is

π1((B11, . . . , B1m1), (B21, . . . , B2m2))

= E

[m1∑r=1

sL1rmin{RL1r

, B1r} −m1∑r=1

pL1r(RL1r

− B1r )+

−m1∑r=1

hL1r(B1r − RL1r

)+ + sO1 min

{RO1 , C1 −

m1∑r=1

B1r

}

−pO1

(RO1 − C1 +

m1∑r=1

B1r

)+− hO1

(C1−

m1∑r=1

B1r − RO1

)+].

(9)

It is noteworthy that, based on Topkis [22], Lemma 2 inSection 2 still holds when x1 and x2 are vectors. Applyingthis lemma, we obtain the following theorem.

THEOREM 8: When each firm has multiple local storesand a single online store, the expected payoff function offirm 1, π1((B11, . . . , B1m1), (B21, . . . , B2m2)), is a submodularfunction of (B11, . . . , B1m1), and it is submodular in (B1r , B2s)

for all r and s.

PROOF: First we consider the term∑m1

r=1 sL1rmin{RL1r

,B1r} in (9). We have

min{RL1r, B1r} = min

{DL1r

+2∑

i=1

mi∑s=1

aLisL1r(DLis

− Bis)+

− B1r , 0

}+ B1r .

Since DL1r+ ∑2

i=1

∑mi

s=1 aLisL1r(DLis

− Bis)+ − B1r is a

separable function that is decreasing in all its variables, itis a valuation. Therefore, DL1r

+ ∑2i=1

∑mi

s=1 aLisL1r(DLis

−Bis)

+ − B1r is a decreasing submodular function. It fol-lows from the concavity of min{x, 0} and Lemma 2 thatmin{RL1r

, B1r} is submodular in ((B11, . . . , B1m1), (B21, . . . ,B2m2)). Thus, it is submodular both in (B11, . . . , B1m1) and in(B1r , B2s) for all r and s.

We next consider terms − ∑m1r=1 pL1r

(RL1r− B1r )

+ and− ∑m1

r=1 hL1r(B1r −RL1r

)+ in the second line of (9). We have

−(RL1r− B1r )

+ = min{B1r − RL1r, 0}

and−(B1r − RL1r

)+ = min{RL1r− B1r , 0}.

The same argument as the one used above shows that boththese two are submodular functions. Note that one functionis decreasing in all variables and the other is increasing in allits variables, but Lemma 2 applies to both cases.

We now consider the term min{RO1 , C1 −∑m1s=1 B1s} in the

third line of (9). We have

min

{RO1 , C1 −

m1∑s=1

B1s

}

= min

{RO1 +

m1∑s=1

B1s , C1

}−

m1∑s=1

B1s .



Since − ∑m1s=1 B1s is submodular, it suffices to prove that

min{RO1 + ∑m1s=1 B1s , C1} is submodular in (B11, . . . B1m1)

and in (B1r , B2s) for all r and s. Note that for fixed(B21, . . . , B2m2), the function RO1 + ∑m1

s=1 B1s is a separablefunction of (B11, . . . , B1m1), thus, it is a submodular functionof (B11, . . . , B1m1). To prove that min{RO1 + ∑m1

s=1 B1s , C1}is submodular in (B1r , B2s), note that

min

{RO1 +

m1∑s=1

B1s , C1

}

= min

{DO1 +

m1∑s=1

(B1s + aL1sO1(DL1s− B1s)

+)

+aO2O1(DO2 − C2 +m2∑s=1

B2s)+, C1

}.

We know that the function

DO1 +m1∑s=1

(B1s + aL1sO1(DL1s− B1s)

+)

+ aO2O1

(DO2 − C2 +

m2∑s=1

B2s

)+

is clearly increasing in (B1r , B2s) and it is valuation in(B1r , B2s). Hence, it follows from Lemma 2 that min{RO1 +∑m1

s=1 B1s , C1} is submodular in (B1r , B2s).Similarly, we can show that −(RO1 − C1 + ∑m1

s=1 B1s)+

and −(C1 − ∑m1s=1 B1s − RO1)

+ are both submodular in(B11, . . . B1m1) and are also both submodular in (B1r , B2s)

for all r and s. This completes the submodularity proof. �

THEOREM 9: When each firm has multiple local storesand a single online store, a Nash equilibrium exists for thiscapacity allocation game.

PROOF: Following similar reasoning process as givenabove, we can easily prove that the payoff function offirm 2, π2((B11, . . . , B1m1), (B21, . . . , B2m2)), is a submod-ular function of (B21, . . . , B2m2), and it is submodular in(B1r , B2s) for all r and s. Therefore, for this capacity alloca-tion game, the payoff function of firm i, πi((B11, . . . , B1m1),(B21, . . . , B2m2)), is submodular in its own decision variables(Bi1, . . . , Bimi

), and it is also submodular in (B1r , B2s) for allr and s. According to Topkis [22], it is a submodular gameand a Nash equilibrium exists. �

REMARK 3: According to Topkis [22], we do not needπi((B11, . . . , B1m1), (B21, . . . , B2m2)) to be submodular in(Bj1, . . . , Bjmj

), i �= j for the problem to be a submodu-lar game. In fact, πi((B11, . . . , B1m1), (B21, . . . , B2m2)) is notsubmodular in (Bj1, . . . , Bjmj

), i �= j in this model.

4.3. Total Capacity Decision

We now go back to our original model, Scenarios 1 and 2.However, we now assume that each firm decides its totalcapacity and allocates this capacity between its local andonline stores. In other words, the total capacity at each firmis a decision variable as well. For the sake of brevity, weonly discuss Scenario 1. Scenario 2 follows the same logic.Recall that Ci is the total capacity of firm i, Bi is the capacityof firm i allocated to its online store, and (thus) Ci − Bi

is the capacity of firm i allocated to its local store. Sincethe total capacity Ci is a decision variable, we consider theprocurement cost and assume ui is the unit cost that firm i

pays for the capacity. Denoting the expected payoff functionof firm i by πi(B1, C1, B2, C2), for firm i, we have

πi(B1, C1, B2, C2) = E[sLiRLi

+ hLi(RLi

− (Ci − Bi))

−(sLi+pLi

+hLi)(RLi

−(Ci−Bi))++sOi

ROi+hOi

(ROi−Bi)

− (sOi+ pOi

+ hOi)(ROi

− Bi)+ − uiCi], (10)

which is almost the same as (1) except the −uiCi term. Thefirst derivative of π1(B1, C1, B2, C2) with respect to B1 is thesame as (3), and the first derivative with respect to C1 is

∂π1

∂C1= −hL1 − ui + (sL1 + pL1 + hL1) Pr(RL1 > C1 − B1)

− (sO1 +hO1)aL1O1 Pr(DL1 > C1 −B1)+ (sO1 +pO1 +hO1)

× aL1O1 Pr(RO1 > B1, DL1 > C1 − B1). (11)

With some simple calculations, we obtain the Hessian matrix, ∂2π1

∂2B1

∂2π1∂B1∂C1

∂2π1∂C1∂B1

∂2π1∂2C1

,

where

∂2π1

∂B1∂C1= (sL1 + pL1 + hL1)fRL1

(C1 − B1)

− (sO1 + hO1)aL1O1fDL1(C1 − B1)

− (sO1 + pO1 + hO1)aL1O1(1 − aL1O1)

× fRO1 |DL1 >C1−B1(B1) Pr(DL1 > C1 − B1),

∂2π1

∂2C1= −(sL1 + pL1 + hL1)fRL1

(C1 − B1)

+ (sO1 + hO1)aL1O1fDL1(C1 − B1)

− (sO1 + pO1 + hO1)a2L1O1

fRO1 |DL1 >C1−B1(B1)

× Pr(DL1 > C1 − B1),

and ∂2π1∂2B1

is the same as (5).



A sufficient condition for π1(B1, C1, B2, C2) to be jointlyconcave in (B1, C1) is that the Hessian matrix is negativesemidefinite, i.e.,

∂2π1

∂2B1

∂2π1

∂2C1−

(∂2π1

∂B1∂C1

)2

> 0.

Denoting

fRO1 |DL1 >C1−B1(B1) Pr(DL1 > C1 − B1),

which is positive, by M and

fRO1 |DL1 ≤C1−B1(B1) Pr(DL1 ≤ C1 − B1),

which is also positive, by N , we have

∂2π1

∂2B1

∂2π1

∂2C1−

(∂2π1

∂B1∂C1

)2

= N((sL1 + pL1 + hL1)fRL1(C1 − B1) − (sO1 + hO1)

×aL1O1fDL1(C1 −B1))+M((sL1 +pL1 +hL1)fRL1

(C1 −B1)

− (sO1 +hO1)aL1O1fDL1(C1 −B1)−2a2

L1O1(sL1 +pL1 +hL1)

× fRL1(C1 − B1)) + (sO1 + pO1 + hO1)a

2L1O1

MN .

Therefore, if

(2a2L1O1

− 1)(sL1 + pL1 + hL1)fRL1(C1 − B1)

+ (sO1 + hO1)aL1O1fDL1(C1 − B1) ≤ 0, (12)

then π1(B1, C1, B2, C2) is jointly concave in (B1, C1). Sym-metrically, we have the following sufficient condition for firm2 to have a concave payoff function:

(2a2L2O2

− 1)(sL2 + pL2 + hL2)fRL2(C2 − B2)

+ (sO2 + hO2)aL2O2fDL2(C2 − B2) ≤ 0, (13)

By using [24], we have the following theorem.

THEOREM 10: For Scenario 1 of our two-firm model withtwo decision variables, if conditions (12) and (13) are sat-isfied, then there exists a Nash equilibrium, which can beobtained by solving the following system of equations:

∂π1(·)∂B1

= 0,

∂π1(·)∂C1

= 0,

∂π2(·)∂B2

= 0,

∂π2(·)∂C2

= 0.

PROOF: If (12) and (13) are satisfied, then πi(B1, C1,B2, C2) is jointly concave in (Bi , Ci), i, j = 1, 2, j �= i.Then, as stated in [24], the equilibrium solution of our two-firm model with two decision variables can be obtained bysolving the system of four equations consisting of the twofirst-order conditions for each of the two firms. �

Now we show that conditions (12) and (13) are valid;namely, they can be expected to be satisfied in a real setting.For example, note that (12) is equivalent to

aL1O1 ≤ −V +√

V 2 + 1

2,

where V = (sO1 +hO1 )fDL1(C1−B1)

4(sL1 +pL1 +hL1 )fRL1(C1−B1)

≥ 0.

Since −V +√

V 2 + 12 is decreasing in V , we denote

(sO1 +hO1 )

4(sL1 +pL1 +hL1 )max

0≤y≤C1

fDL1(y)

fRL1(y)

by V . Thus, if

aL1O1 ≤ −V +√

V2 + 1

2, (14)

then π1(B1, C1, B2, C2) is jointly concave in (B1, C1). Notethat the right-hand side of (14) is decreasing in the sellingprice at the online store of firm 1, sO1 , and increasing in theselling price at the local store of firm 1, sL1 . This goes withour intuition, since aL1O1 is the fraction of unsatisfied localcustomers at firm 1 visiting its online store.

4.4. Multi-Period Model

Now we consider a multi-period model in which eachplayer makes a sequence of decisions. At the beginning ofeach period t (t = 1, 2, . . .), firm i has initial on-hand inven-tories xt

Liand xt

Oiat its local and online stores, respectively.

It makes decisions Cti and Bt

i , where Cti is its total stock,

Bti is the allocation to its online store, and (thus) Ct

i − Bti

is the allocation to its local store, as shown in Figure 5. Weassume that the inventory replenishment is instantaneous, soBt

i and Cti − Bt

i , respectively, are the actual inventory avail-able at the online and local stores of firm i in period t . Thelocal demand, online demand, and effective demands are alsoperiod-related, and we denote them by Dt

Li, Dt

Oi, Rt

Li, and

RtOi

, respectively. We assume that DtLi

and DtOi

are randomvariables with independent identical distributions. The result-ing inventory levels of firm i at the beginning of period t + 1are

xt+1Li

= (Cti − Bt

i − RtLi

)+

andxt+1

Oi= (Bt

i − RtOi

)+.

Let βi < 1 be the discount factor per period for firm i.Therefore, firm i’s expected total profit is



Figure 5. The initial stock and order of firm i at the beginning of period t .

�i = E

[ ∞∑t=1

(βi)t−1[sLi

min{RtLi

, Cti − Bt

i } − pLi(Rt

Li− (Ct

i − Bti ))

+ − hLi(Rt

Li− (Ct

i − Bti ))

−

+ sOimin{Rt

Oi, Bt

i } − pOi(Rt

Oi− Bt

i )+ − hOi

(RtOi

− Bti )

− − ui(Cti − Bt

i − xtLi

) − ui(Bti − xt

Oi)]

]

= E

[ ∞∑t=2

(βi)t−1[sLi

min{RtLi

, Cti − Bt

i } − pLi(Rt

Li− (Ct

i − Bti ))

+ − hLi(Rt

Li− (Ct

i − Bti ))

− + sOimin{Rt

Oi, Bt

i }

− pOi(Rt

Oi− Bt

i )+ − hOi

(RtOi

− Bti )

− − ui(Cti − Bt

i − (Ct−1i − Bt−1

i − Rt−1Li

)+) − ui(Bti − (Bt−1

i − Rt−1Oi

)+)]+ sLi

min{R1Li

, C1i − B1

i } − pLi(R1

Li− (C1

i − B1i ))

+ − hLi(R1

Li− (C1

i − B1i ))

− + sOimin{R1

Oi, B1

i }

− pOi(R1

Oi− B1

i )+ − hOi

(R1Oi

− B1i )

− − ui(C1i − x1

Li− x1

Oi)

]

= ui(x1Li

+ x1Oi

) + E

[ ∞∑t=1

(βi)t−1[sLi

min{RtLi

, Cti − Bt

i } − pLi(Rt

Li− (Ct

i − Bti ))

+ − hLi(Rt

Li− (Ct

i − Bti ))

−

+ sOimin{Rt

Oi, Bt

i } − pOi(Rt

Oi− Bt

i )+ − hOi

(RtOi

− Bti )

− − uiCti + βiui((C

ti − Bt

i − RtLi

)+ + (Bti − Rt

Oi)+)]

]

= ui(x1Li

+ x1Oi

) + E

[ ∞∑t=1

(βi)t−1[(sLi

− βiui)RtLi

+ hLi(Rt

Li− (Ct

i − Bti )) − (sLi

− βiui + pLi+ hLi

)(RtLi

− (Cti − Bt

i ))+

+ (sOi− βiui)R

tOi

+ hOi(Rt

Oi− Bt

i ) − (sOi− βiui + pOi

+ hOi)(Rt

Oi− Bt

i )+ − (ui − βiui)C

ti ]

].

Denoting sLi−βiui by s ′

Li, sOi

−βiui by s ′Oi

, and ui −βiui

by u′i we have

�i = ui(xiLi

+ xiOi

) +∞∑t=1

(βi)t−1Wt

i (Bt1, Ct

1, Bt2, Ct

2), (15)

where

Wti (B

t1, Ct

1, Bt2, Ct

2) = E[s ′Li

RtLi

+ hLi(Rt

Li− (Ct

i − Bti ))

− (s ′Li

+ pLi+ hLi

)(RtLi

− (Cti − Bt

i ))+ + s ′

OiRt

Oi

+ hOi(Rt

Oi− Bt

i ) − (s ′Oi

+ pOi+ hOi

)(RtOi

− Bti )

+ − u′iC

ti ],

(16)

which is very close to (10).

We now apply the theory of sequential games developedin [10] to analyze this multi-period model. In general, asequential game is difficult to solve. Hence, we turn to theconcept of a “myopic solution” to simplify the analysis.A sequential game is said to have a myopic solution if its datacan be used easily to specify a single-period game such that adinfinitum repetition of a Nash equilibrium of the single-periodgame comprises an equilibrium for the sequential game.

We refer to the single-period game, in which each firm haspayoff function (16) without superscript t , as game �. Forour sequential game characterized by (16), from Section 9–4of Heyman and Sobel, we know that if the following threeconditions are satisfied, then a myopic solution exists and this



sequential game can be simplified into single-period game �.Those sufficient conditions are

(B1) the demands in all periods are random variables withindependent identical distribution;(B2) the discount factor per period for firm i is less than 1,i.e., βi < 1;(B3) the equilibrium solution of the game � is a feasiblesolution to the multi-period game in each period t .

For our model, conditions (B1) and (B2) are satisfiedautomatically. Condition (B3) is equivalent to:

xt+1Li

≤ (Cti − Bt

i ) = (Ci − Bi),

xt+1Oi

≤ Bti = Bi , i = 1, 2,

where (B1, C1, B2, C2) is an equilibrium solution of thegame �. Since (B1, C1, B2, C2) is an equilibrium solutionof the game �, we know that (B1

1 , C11 , B1

2 , C12) ≤ (B1, C1,

B2, C2). Further, we have

xt+1Li

= (Cti − Bt

i − RtLi

)+ = (Ci − Bi − RtLi

)+ ≤ Ci − Bi ,

xt+1Oi

= (Bti − Rt

Oi)+ = (Bi − Rt

Oi)+ ≤ Bi , i = 1, 2.

Hence, condition (B3) is also satisfied.Therefore, we have the following theorem.

THEOREM 11: For our multi-period model characterizedby (16), a myopic solution exists that simplifies this corre-sponding sequential game into the single-period game �, inwhich firm i makes decisions on Bi and Ci and has payofffunction (16) without superscript t , i.e.,

Wi(B1, C1, B2, C2) = E[s ′Li

RLi+ hLi


− (s ′Li

+ pLi+ hLi

)(RLi− (Ci − Bi))

+ + s ′Oi

ROi

+ hOi(ROi

− Bi) − (s ′Oi

+ pOi+ hOi

)(ROi− Bi)

+ − u′iCi].(17)

The task of finding a Nash equilibrium solution for thegame � is almost the same as that discussed in Subsec-tion 4.3. The myopic equilibrium can be found by solvingthe following system of equations:

∂W1(·)∂B1

= 0,

∂W1(·)∂C1

= 0,

∂W2(·)∂B2

= 0,

∂W2(·)∂C2

= 0.

Note that the payoff function (17) is very close to the payofffunction (10) of the single-period game in Subsection 4.3.Compared to (10), (17) has three different parameters: theselling price of local store s ′

Lirather than sLi

, the selling priceof online store s ′

Oirather than sOi

, and the unit purchasing costu′

i rather than ui . Recall that s ′Li

= sLi− βiui , s ′

Oi= sOi

−βiui , and u′

i = ui − βiui . Therefore, in the simplified modelfor the sequential game �, both the selling price and the unitpurchasing cost are deducted by the same amount βiui , and(thus) the unit margin remains the same. In addition, fromformulas (15) and (16), we can see that, when the discountfactor βi equals zero, the game � reduces to the single-periodgame analyzed in Subsection 4.3 with initial inventory x1

Li+

x1Oi

. In other words, the single-period game in Subsection 4.3is a special case of the sequential game � when the discountfactor βi equals zero and there is no initial inventory. Thisgoes with our intuition.

5. NUMERICAL EXPERIMENTS

In this section, we compare the results of the single-periodmodel in Subsection 4.3 with those for the multi-period modelof Subsection 4.4. Compared with equation (10), in the sim-plified payoff function of game �, i.e., equation (17), boththe selling price and the unit purchasing cost are reduced.Therefore, it is not obvious whether the firms stock more orless at equilibrium.

Following the method by Netessine and Shumsky [19],equilibria are found by Newton’s method. At each iteration,given the values of Ci and Bi and the distributions of DLi

and DLj, we apply Monte Carlo methods to simulate the

distributions of the effective demands RLiand RLi

and use theresults to evaluate the gradients, e.g., equations (3) and (11).

In Section 3, we analyzed the sensitivity of an equilibriumsolution with respect to cost parameters, such as selling price,holding cost, and stockout penalty cost. In this section, ourgoal is to analyze and compare the effects of demand varianceand of the discount factor βi on the equilibrium results forsingle-period model and the myopic solution for the multi-period model. To avoid the effects of cost parameters, weconsider two symmetric firms with the same cost parametersas follows: sLi

= 8, sOi= 6, pLi

= 4, pOi= 4, hLi

= 1,hOi

= 0.5, ui = 2. The following demand shifting parame-ters are used: aL1L2 = aL2L1 = 0.5, aL1O1 = aL2O2 = 0.05,aO1O2 = 0.3, aO2O1 = 0.4. Demands are normally distributedwith the following parameters. The means of DL1 and DO1

are set to be 500. The means of DL2 and DO2 are set to be 100and their standard deviations are set to be 10. We first gen-erate scenarios by considering a different value of demandvariance at firm 1. The standard deviations of DL1 and DO1

are set to be equal and vary from 1 to 200. For each scenario,we have computed the total capacity, capacity allocation, andpayoff at equilibrium for each firm.



Figure 6. Payoff of firm 1 in single and multi-period models whenβi = 0.2. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

Figures 6 and 7 exhibit the equilibrium payoff and totalcapacity of firm 1 in single and multi-period models whenβi = 0.2, i = 1, 2, respectively. We observe that firm 1 isbetter off and stocks more in the multi-period model than inthe single-period model under the current parameter settings.As the standard deviation of firm 1’s demand increases from1 to 200, firm 1’s equilibrium payoff goes down even thoughit stocks more. This goes with our intuition. It is noteworthythat as the demand variance increases, compared to the single-period model, the multi-period model tends to perform betterand better. In other words, the multi-period model has theadvantage of handling demand turbulence. Figure 8 exhibitsthe equilibrium payoff of firm 2 in the single and multi-period

Figure 7. Total capacity of firm 1 in single- and multi-period mod-els when βi = 0.2. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

Figure 8. Payoff of firm 2 in single- and multi-period models whenβi = 0.2. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

models with βi = 0.2. We observe that firm 2’s payoff isaffected by the demand variance at firm 1. This is due tothe shifting of unsatisfied demand. It is interesting that firm2’s payoff seems to be concave w.r.t. the standard deviationof firm 1’s demand. In other words, as firm 1’s rival, firm 2prefers a “proper" demand variation at firm 1.

To analyze the effect of discount factor βi on the myopicsolution of multi-period model, we have computed (firm 1’spayoff for) scenarios as βi increases from 0.1 to 0.9. Theseresults along with that for the single-period model are dis-played in Figure 9. Figure 9 shows that the payoff of firm1 is increasing in βi under the current parameter settings.In addition, it demonstrates (as mentioned in Subsection 4.4)that numerically the single-period model is just a special caseof the multi-period model when the discount factor βi equalszero.

Figure 9. Payoff of firm 1 in single- and multi-period models whenβi = 0.1, . . . , 0.9.



6. CONCLUDING REMARKS

In this study, we have considered the capacity allocationproblem for two firms selling the same product. Each firm hasa local store and an online store. When a stockout occurs at alocal store customers may go to the online store belonging tothe same firm or visit the rival’s local store. However, when astockout occurs at an online store, customers usually will notvisit the local store belonging to the same firm, but insteadthey may go to the rival’s online store. One question facingeach firm is how to allocate its finite capacity between itslocal and online stores to maximize its profit as a whole.

Because customers may shift from one firm to the otherwhen a stockout occurs, one firm’s allocation affects thedecision of the rival, thereby creating a strategic interaction.In this paper game theory has been used for the analysis.We first considered a single-product single-period modeland assumed that the total capacity of each firm is givenand known. We studied two scenarios for this model andderived corresponding existence and stability conditions fora Nash equilibrium. We also conducted sensitivity analy-sis of the equilibrium solution with respect to price andcost parameters and analyzed a generalized model withvery mild shift assumptions. We then studied the case inwhich each firm decides its total capacity and allocates thiscapacity between its local and online stores simultaneouslyand derived existence conditions for a Nash equilibrium.Finally, we extended the results to a multi-period model andshowed that a myopic solution is a Nash equilibrium for thecorresponding sequential game.

In our study, we have assumed that the demands at localand online stores are exogenous. A potential avenue of futureresearch would be to study the case in which the demands areaffected by the quality of service (such as the percentage ofdemand satisfied) or by the selling prices. It would also beinteresting to analyze the case when the fraction of customershifting, such as aLiLj

, aLiOi, aOiOj

, is related to these factors.Another avenue of future research would be to study potentialcollaboration between the two firms, such as using side pay-ments to make both firms better off by sharing a single webpage. We could also study the behavior of a monopolist whoowns both firms (a centralized model) as in [19] and comparethe results with the behavior of two firms in competition (adecentralized model).

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