Omega 92 (2020) 102148

Contents lists available at ScienceDirect

Omega

journal homepage: www.elsevier.com/locate/omega

A review of non-cooperative newsvendor games with horizontal

inventory interactions

�

Lena Silbermayr

Department of Information Systems and Operations, WU Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Wien, Austria

a r t i c l e i n f o

Article history:

Received 23 November 2018

Accepted 15 October 2019

Available online 17 October 2019

Keywords:

Newsvendor

Non-cooperative simultaneous game

Transshipments

Substitution

Horizontal coordination

a b s t r a c t

There are numerous applications of game theory in the analysis of supply chains where multiple actors

interact with each other in order to reach their own objectives. In this paper we review the use of non-

cooperative game theory in inventory management within the newsvendor framework describing a single

period inventory control model with the focus on horizontal interactions among multiple independent

newsvendors. We develop a framework for identifying these types of horizontal interactions including,

for example, the models with the possibility of inventory sharing via transshipments, and situations with

substitutable products sold by multiple newsvendors. Based on this framework, we discuss and relate the

results of prior research and identify future research opportunities.

© 2019 The Author. Published by Elsevier Ltd.

This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

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. Introduction

The newsvendor model is a single-period inventory problem,

here the decision maker places an inventory order in advance

f the selling season before demand is realized. As demand is not

nown in advance the amount of stock to be kept becomes a chal-

enging decision. Excess inventory results in high inventory related

osts while shortages results in low service levels and lost rev-

nues.

Whenever there are multiple newsvendors (retailers) selling

ame or related products to customers then any horizontal interac-

ion between the retailers has to be taken into consideration when

aking inventory decisions. In practice, several forms and varying

egrees of horizontal interactions can occur. For example, inven-

ory aggregation or risk pooling through inventory transshipments,

nables retailers to pool their demand risks while increasing prof-

ts and service levels. That is, a retailer might use the option to

ransship excess inventory to a retailer facing a shortage to ben-

fit from reduced leftover inventory. The transshipment receiving

etailer, on the other hand, can benefit from increased sales. An-

ther possible form of horizontal interaction is product substitu-

ion where customers, in case of a stockout, find a substitute for

he product at another retailer.

In many real situations retailers in the same supply chain eche-

on that interact in such ways are independent ( decentralized ) play-

rs. They act independently by optimizing their own objectives.

� This manuscript was processed by Associate Editor AE F. Zhang.

E-mail address: lena.silbermayr@wu.ac.at

ttps://doi.org/10.1016/j.omega.2019.102148

305-0483/© 2019 The Author. Published by Elsevier Ltd. This is an open access article un

his constitutes a non-cooperative static game where retailers in

dvance of the selling season make their decisions simultaneously

aking any inventory interaction into consideration.

Previous works surveying the application of non-cooperative

ewsvendor games are, for example, Cachon [10] who focuses on

ertical coordination between supplier and buyer (newsvendor)

r [11,15] who provide tutorials on non-cooperative newsvendor

ames with horizontal interactions where two newsvendors com-

ete on product availability. There are also reviews on inventory

ransshipments. Paterson et al. [58] review inventory models with

ateral transshipments in general, in centralized and decentralized

upply chains. Huang [38] considers transshipments and substitu-

ion between retailers focusing on supply chain relationships be-

ween the manufacturer and the retailers.

This paper fills a gap in the literature by providing a review on

uantitative models for multiple independent newsvendors with

ny type of horizontal inventory interactions. In the last years, es-

ecially due to new business models and advanced information

echnologies several papers dealing with the application of non-

ooperative newsvendor games focusing on several aspects leading

o horizontal inventory interactions have appeared. What is still

issing is a structured classification of the different aspects and

pplications that enable horizontal inventory interactions in a sup-

ly chain with multiple independent retailers in order to provide

n overview on the main findings in this field and to discuss future

esearch opportunities.

To summarize, the contributions of this paper are as follows:

• We describe and classify the specific settings where game the-

ory applications to horizontal inventory interactions arise.

der the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

2 L. Silbermayr / Omega 92 (2020) 102148

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• We discuss the modelling approaches and conditions required

for the existence and uniqueness of Nash equilibrium under

each setting. • We address horizontal coordination and discuss possible coor-

dination mechanisms that can be applied for horizontal supply

chain relationships. • We provide an up-to-date literature review summarizing the

important findings in that field. • We identify gaps and discuss important future research direc-

tions.

The reminder of this paper is organized as follows. In

Section 2 we introduce the classical newsvendor model and the

basic concepts of non-cooperative game theory and discuss possi-

ble interactions between decentralized newsvendors that may re-

sult in inventory interactions leading to the review structure and

methodology used in this paper. In Sections 3 and 4 the different

models are analyzed using non-cooperative game theory. In each

section we introduce some basic concepts of the non-cooperative

games and then proceed to review and discuss the related litera-

ture in the respective field. Section 5 discusses miscellaneous as-

pects with combinations or comparisons of different models dis-

cussed in Sections 3 and 4 . Finally, Section 6 concludes the paper

and discusses directions for future research.

2. Preliminaries

We first introduce the classical newsvendor model, discuss pos-

sible interactions among multiple decentralized newsvendors in

a non-cooperative fashion and provide a brief overview on basic

game theoretic concepts in this context. Then, the structure of the

literature review and the review methodology is presented.

2.1. The classical newsvendor model

Consider a single period inventory model with n retailers i =1 , 2 , . . . , n . The retailers act to maximize their own profits. They

face random demands D i with marginal distribution function F D i and demands can be correlated. The product is sold at a selling

price r i per unit and purchased at c i per unit.

In the classical newsvendor setting, if demand of retailer i, D i ,

turns out to be larger than the stocking quantity Q i , unsatisfied

demand is assumed to be immediately lost and a penalty cost of p i is incurred per unsatisfied demand. On the other hand, if demand

turns out to be smaller than the stocking quantity, leftovers are

salvaged at a value of s i per unit where r i > c i > s i . All parameters as

well as marginal and joint distributions of demands are common

knowledge. 1

In this case, the expected profit of a retailer i is

�NV i (Q i ) = E(r i min (D i , Q i ) + s i (Q i − D i )

+ − p i (D i − Q i ) + − c i Q i ) ,

(1)

where we define (X ) + = max (X, 0) . The expected profit function is

concave and the optimal order quantity satisfies

F D i (Q i ) =

r i − c i + p i r i − s i + p i

, (2)

where the right-hand side of (2) is the newsvendor’s critical ratio.

However, there are several settings which cause interaction

among the retailers when there are shortages and leftovers at dif-

ferent retailers. For example, the retailers with a shortage can pro-

cure goods from the retailers with leftovers to satisfy some of their

customer demand, which describes a system with lateral transship-

ments; or customers whose demand could not be satisfied might

1 For games with incomplete informations we refer to e.g. [77] .

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earch for a retailer with leftover inventory and make a purchase

n this substitute location. Because of these possible interactions,

newsvendor has to consider the decisions of other newsvendors

hich gives rise to a game theoretic approach. The common char-

cteristic of the different settings is that the unsatisfied demand of

ne retailer can be satisfied by another retailer. Therefore the risks

elated to shortages and leftovers differ from the classical newsven-

or setting.

.2. Basic concepts in non-cooperative game theory

We give a short overview of the basic concepts in non-

ooperative static games with complete information , i.e. all players

re in possession of all information in the game. Since our focus

s on single-period newsvendor games, only one-shot games (i.e.

ames with only one play-through) are discussed and we will di-

ectly link the basic concepts to this setting. For further details we

efer to e.g. [11,15,25] .

A game has a set of rational decision makers, called players , de-

oted by retailer i = 1 , . . . , n, a set of strategies available to each

etailer denoted by stocking quantities Q i , and payoffs received by

ach player given by the expected profits �i . In a non-cooperative

tatic game the strategies (stocking quantities) are chosen simulta-

eously and the players are unable to make binding commitments

efore choosing their strategies.

The rational outcome of such a game is the Nash equilib-

ium which is characterized by solving the system of n best re-

ponse functions of all players. The best response function of

layer i given vector of fixed strategies of the other players, Q −i

ives player i ’s response that maximizes its payoff (i.e. Q

∗i ( Q −i ) =

ax Q i �i (Q i , Q −i ) , hence, it is typically defined by the players’ first

rder conditions. A Nash equilibrium Q

∗i ( Q

∗−i

) is considered a sta-

le outcome as no player has incentive to deviate from his strategy

hoice.

The existence of a Nash equilibrium is guaranteed when the

layers’ payoffs �i are concave with respect to Q i . The unique-

ess of an equilibrium can be proven e.g. through contraction map-

ing argument showing that the best response mapping is a con-

raction, which then implies that the mapping has a unique fixed

oint. In other words, one has to verify that no column sum or no

ow sum of the matrix of derivatives of the best response functions

xceeds one [11] .

In general decentralized decision making results in supply chain

nefficiency, since the Nash equilibrium usually is not equal to the

ystem optimal solution where a centralized decision maker maxi-

izes the total payoff of all players.

.3. Review structure and methodology

We group different settings which give rise to interaction

mong decentralized newsvendors in two categories: 1) designed

nteraction and 2) customer driven interaction . The first category in-

ludes settings where the newsvendors design operations in such a

ay that the supply chain design enables risk pooling via inventory

haring. Lateral transshipments among retailers (i.e. virtual pool-

ng), and physical centralization of stocks at a central warehouse

i.e. physical pooling) belong to this first category. Customer driven

nteraction, on the other hand, stems from the perceived substi-

utability and complementarity of different products and/or sellers.

f the customers are willing to substitute products and/or sellers, a

hortage at one retailer causes higher demand at other retailers.

n the other hand, if products are complementary, shortage at one

etailer hinders sales of another complementary retailer.

One of the main differences among the two categories is

hat for the designed interaction there is a necessity to develop

orizontal contracts so that the horizontal interaction through

L. Silbermayr / Omega 92 (2020) 102148 3

Table 1

Organization of literature review including section numbers.

Designed interaction Transshipments Two retailers

( Section 3 ) ( Section 3.1 ) ( Section 3.1.1 )

n > 2 retailers

( Section 3.1.2 )

Physical centralization

( Section 3.2 )

Customer driven interaction Substitution

( Section 4 ) ( Section 4.1 )

Complementarity

( Section 4.2 )

Combination/Comparison of designed

and customer driven interaction

( Section 5 )

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nventory pooling can take place. If the benefits of the designed

ystem are not appropriately distributed, the players might decide

ot to participate in the game at all. Then, their decisions can be

roperly modelled using the classical newsvendor setting. On the

ther hand, the customer driven interaction is an external factor

hich should not be ignored. Accordingly, the literature on the first

ategory (designed interaction) puts some focus on horizontal co-

rdination and contracting while the literature on the second cat-

gory (customer driven interaction) does not deal with horizontal

ontracting.

The structure of our literature review is shown in Table 1 . In the

ase of a designed interaction through transshipments we explic-

tly discuss a two-retailer and n > 2 retailer problem, since the gen-

ral model with n > 2 retailers includes an additional complexity

f how to allocate residual stocks to stocked-out retailers. For the

hysical centralization example the conclusions are similar, hence,

hey will not be repeatedly discussed in such a detail. In the fi-

al Section 5 papers dealing with a combination or comparison of

esigned and customer driven interactions will be discussed.

The inclusion criteria for this review are articles studying

orizontal inventory interactions among multiple decentralized

ewsvendors that make non-cooperative stocking decisions. Exclu-

ion criteria applied for this review are articles focusing on purely

ooperative operational decisions and information asymmetries or

ncomplete information. Since we put our focus on inventory-

elated implications of horizontal interactions we also exclude ar-

icles with price competition.

. Designed interaction

In this section we consider non-cooperative risk pooling games

esulting from the sellers designed operations that enable inven-

ory sharing. For independent retailers selling identical products

he most commonly studied setting in this category is the lateral

ransshipments system which is the focus of Section 3.1 . Another

ossible setting to exploit the risk pooling benefit is the physical

onsolidation of inventories at a central location (see Section 3.2 ).

he detailed classification of the literature discussed in this section

s shown in Table 3 .

.1. Transshipments

First, we discuss virtual pooling through transshipments, i.e.

esidual inventory at one retailer is shipped to another retailer that

aces a stockout.

.1.1. Two retailers

Transshipments from i to j (throughout the paper, when using

his indexing, we assume i � = j ) incurs a transshipment cost c ij per

nit. Since i and j are independent a transshipment price r ij per

nit is charged by retailer i and paid by j . If there is still unsatisfied

emand at i after the transshipments are realized, then a shortage

enalty cost of p i is incurred.

We assume that c i < c j + c ji , s j > s i + c ji , and r i + p i < r j + p j + ji . These conditions guarantee that it is not beneficial to always

urchase and/or salvage through the other retailer, and to sell to

he other retailer instead of own customers. To ensure mutually

rofitable transshipments we need to assume that the unit trans-

hipment price r i j ∈ [ s i + c i j , r j + p j ] , where s i + c i j < r j + p j (see

36,61] ), then neither retailer is worse off by performing transship-

ents. We will call the mutually beneficial transshipment system

s bidirectional setting.

For given order quantities Q i and Q j , transshipments from i to

are T i j = min ((D j − Q j ) + , (Q i − D i )

+ ) . Sales at retailer i are S i =in (D i , Q i ) + T ji , leftovers at retailer i are L i = (Q i − D i − T i j )

+ , and

nsatisfied demand at retailer i is P i = (D i − Q i − T ji ) + .

When the two retailers make their ordering decisions locally, in

decentralized manner, then the expected profit of retailer i is

i (Q i , Q j ) = E(r i S i + (r i j − c i j ) T i j − c i Q i − r ji T ji + s i L i − p i P i ) . (3)

A special case of the problem where transshipments are only

ossible in one direction has been studied in some recent papers

3,32,63] . Under the so-called unidirectional transshipments set-

ing, we assume that transshipments are only allowed from retailer

to retailer j , i.e. T ji = 0 . Unidirectional transshipments can be ap-

licable, for example, in omni-channel systems with online and

ffline stores where in-store customers in contrast to online cus-

omers are not willing to wait a regular shipment time for product

elivery [63] . Other reasons might be that certain transshipment

outes are impossible, due to e.g. high cost, unavailable transport

r different proximities, limiting inventory pooling [4,68,69] .

quilibrium analysis

For the bidirectional setting, following [37] , we define D

e i

= D i +(D j − Q j )

+ as the effective demand for retailer i . It includes the ini-

ial demand at retailer i and all the secondary demand, i.e. the un-

atisfied demand at other retailers which can be potentially satis-

ed by retailer i through transshipment (or substitution discussed

n Section 4.1 ). Similarly, D

n i

= D i − (Q j − D j ) + is the net demand

t retailer i , which is the initial demand minus the part which can

e potentially satisfied by retailer j [3] . Note that D

e i

≥ D i ≥ D

n i

and,

ence, F D e i (Q i ) ≤ F D i (Q i ) ≤ F D n

i (Q i ) .

Hu et al. [36] show that the expected profit under decentralized

ecision making of retailer i , Eq. (3) , is concave in Q i . The first or-

er conditions characterizing the optimal order quantities Q

∗i (Q j )

or i = 1 , 2 are

(r i − c i + p i ) − (r ji − r i j + c i j ) F D i (Q i ) − (r i − r ji + p i ) F D n i (Q i )

− (r i j − c i j − s i ) F D e i (Q i ) = 0 . (4)

he existence of the Nash equilibrium is guaranteed by the con-

avity of the expected profit functions, and the uniqueness of the

4 L. Silbermayr / Omega 92 (2020) 102148

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equilibrium is shown by a contraction mapping argument (see e.g.

[11] for further details). The proof follows along the lines of the

proof of Proposition 1 in [61] (see also [3] ). Implicit differentiation

of Eq. (4) results in

∂Q i

∂Q j

= −(r i j − s i − c i j ) g D e

i (Q i ) + (r i + p i − r ji ) g D n

i (Q i )

(r i + p i − s i ) f D i (Q i ) + (r i j − s i − c i j ) f D e i (Q i ) + (r i + p i − r ji ) f D n

i (Q i )

.

(5)

For D

e i , define f D e

i (Q i )

. = ∂F D e i (Q i ) /∂Q i and g D e

i (Q i )

. = ∂F D e i (Q i ) /∂Q j ,

and similarly for D

n i . From the definitions of the effective demand

and the net demand, one can derive F D e i (Q i ) ≤ F D i (Q i ) ≤ F D n

i (Q i ) ,

which leads to f D e i (Q i ) ≥ g D e

i (Q i ) and f D n

i (Q i ) ≥ g D n

i (Q i ) , respec-

tively. Combining these properties, one can conclude that −1 ≤∂Q i ∂Q j

≤ 0 . That is, the slopes of the best response functions are less

than one in absolute terms, which is sufficient condition for the

uniqueness of the Nash equilibrium [11] .

Note that the analysis regarding existence and uniqueness of

the Nash equilibrium for the unidirectional transshipment setting

is similar.

Horizontal coordination

In a system with decentralized retailers that maximize their

own profits, there are incentive problems that prevent coordina-

tion. However, there is literature that deals with horizontal coor-

dination discussing mechanisms or contracts that can be designed

in order to reach system optimal solutions in which a single deci-

sion maker acts so as to maximize the total profit of the system. If

the ordering decisions are centrally made then there is no need to

charge a transshipment price and the total expected profit is

�C (Q i , Q j ) = E

( 2 ∑

i =1

(r i S i − c i j T i j − c i Q i + s i L i − p i P i ) ). (6)

The total expected profit given in Eq. (6) is concave [61] and the

first order conditions characterizing the optimal order quantities

Q

∗C i

for i = 1 , 2 are

(r i − c i + p i ) − (c i j − r j + c ji + s j − p j ) F D i (Q i ) −(r i + p i − c ji − s j )

×F D n i (Q i ) − (r j + p j − c i j − s i ) F D e

i (Q i ) = 0 . (7)

As discussed before, players, without coordination, only focus

on their own objectives which leads to an inefficiency in the sup-

ply chain (see difference between Eqs. (4) and (7) ), i.e. they can-

not achieve the system optimal solution of a centralized decision

maker. Literature on supply chain coordination seeks to find sim-

ple mechanisms (contracts) that provide incentives for the players

to coordinate the decentralized system and achieve the system op-

timum.

Rudi et al. [61] study the simple transshipment price contract

discussed above that from the practical perspective is easy to exe-

cute, i.e. a retailer facing a stockout has to pay a predetermined

unit transshipment price in order to receive leftovers from the

other retailer. Rudi et al. [61] show that the decentralized sys-

tem can be coordinated by appropriately set transshipment prices

equating Eqs. (4) and (7) and solving for the transshipment price

r ij .

Hu et al. [36] provide examples which show that such coordi-

nating prices may not exist in several cases considering a more

general model than [61] including uncertain capacity. Especially

with increasing asymmetries in the economic parameters for the

two retailers, coordination of bidirectional transshipments may

not be possible by varying the transshipment prices. For exam-

ple, Arikan and Silbermayr [3] numerically show that coordinating

transshipment prices might only exist for situations with low de-

mand correlation between the retailers when they differ in their

shortage cost. That is, coordination is only possible in cases where

ransshipments are very beneficial. A sufficient condition that co-

rdinating transshipment prices exist is that the retailers are sym-

etric [36] .

Unlike [36] that assume that retailers subject to uncertain ca-

acity are supplied by two independent suppliers, [44] consider

two-retailer transshipment game supplied by a single supplier

ith uncertain capacity. Hence their model includes competition

or supplier capacity and inventory sharing through transship-

ents. They show that retailers increase their orders when ca-

acity is allocated proportional to the orders (i.e. rationing game)

nd that coordinating transshipment prices exist in a more limited

ange due to the capacity uncertainty.

Shao et al. [64] examine transshipment incentives under the

ransshipment price contract for a fully decentralized two-echelon

upply chain including also the manufacturer’s decision that dis-

ributes the product through two identical retailers. They show

hat if the manufacturer has control, it will set a high transship-

ent price in order to increase its profit. If, however, the retailers

ave control they prefer low transshipment prices and as a result

he manufacturer might prefer dealing with centralized retailers.

Feng et al. [23] consider the same transshipment price contract,

ut under partial backordering. That is, the transshipment quantity

orm i to j is T i j = min (δ(D j − Q j ) + , (Q i − D i )

+ ) , with δ being the

raction of backordered demands. However, they do not discuss the

oordinating transshipment prices under this setting.

Hezarkhani and Kubiak [34] find a coordinating contract with

n implicit pricing mechanism where transshipment prices depend

n the retailers’ inventory decisions. Thereby, they split up the

ontract into two phases. In the first phase inventory dependent

ransshipment prices are set and then in a second phase - after

eciding on inventories but before demand realization - the ne-

otiated transshipment prices are fixed. The contract has the de-

irable property of flexibility as it allows to arbitrarily divide the

otal expected profit between the two retailers depending on the

argaining power of the retailers. This flexibility is not provided by

he coordinating transshipment price contract since it leads to a

ingle split-up of the total expected profit.

The unidirectional transshipment system ( T ji = 0 ) cannot be co-

rdinated at all with an unit transshipment price contract (see [3] ).

his is similar to the setting under a wholesale price contract in

two echelon system with a supplier and a buyer (e.g. [10] ) or

o the bidirectional transshipment setting of [36] with asymmetric

arameters between two retailers. Limiting transshipments in only

ne direction causes extreme asymmetry in the system. Arikan and

ilbermayr [3] study a number of simple contracts that coordinate

he unidirectional transshipment setting. These contracts include a

ombination of transshipment price r ij , leftover subsidy τ L per unit

f leftover at the transshipment giver i paid by the transshipment

eceiving retailer j and shortage subsidy τ S per unit of shortage in j

aid by i . Contract types including ( r ij , τL ) or ( r ij , τ

S ) can not guar-

ntee that the contract is beneficial for both parties. This problem

s shown to be overcome by designing a contract with three terms

r ij , τL , τ S ) which can achieve coordination such that both parties

re better off compared to a no-transshipment setting. Such a con-

ract again has the desirable property of flexibility allowing an ar-

itrary division of the total expected profit.

.1.2. n > 2 Retailers

So far we have focused on the case of two retailers. Analyzing a

ransshipment system with more than two decentralized players is

nown to be a nontrivial task [3,40,61,64] . In the case of two retail-

rs it is clear that any leftovers of one retailer are transshipped to

he other retailer facing a stockout. In the n > 2 retailer case, how-

ver, if there is more than one retailer facing a stockout any left-

vers have to be allocated according to a specific allocation rule. In

L. Silbermayr / Omega 92 (2020) 102148 5

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non-cooperative environment, finding a proper allocation rule is

hallenging.

For simplicity and tractability [3,40,64] extend their analyti-

al tractable two-retailer models by numerically investigating a

roportional allocation rule and assume n > 2 symmetric retailers.

hat is, a retailer facing a stockout (leftover) receives transship-

ents (transships excess inventory) proportional to its excess de-

and (inventory), if total excess demand (inventory) of all retail-

rs is higher than their total excess inventory (demand). [3] show

hat, for both the bidirectional and unidirectional setting, most of

he results in the two retailer case also hold for the case with

> 2 newsvendors (this is also concluded by Huang and Soši ́c

40] and Shao et al. [64] for the bidirectional transshipment set-

ing). For example, the transshipment price contract can coordi-

ate the symmetric bidirectional multi-retailer transshipment sys-

em. However, similar to the two-retailer setting, the system with

retailers might not be coordinated via simple unit transship-

ent prices as soon as certain asymmetries are included in the

ystem. In practice, however, there may be certain asymmetries

n the supply chain. Reasons can be differences in retailers’ size

nd/or proximity to transportation hubs or differences in the re-

ailers’ shortage cost [4,43] . Ben-Zvi and Gerchak [5] provide an

xample where a hospital and adjacent pharmacy share inventory

f medical equipment or materials, where a shortage of an item

eans something different at the two locations.

Hanany et al. [29] develop a transshipment pricing mechanism

ith a transshipment fund, that unlike the unit transshipment

rice contract discussed in [61] and others for the case of 2 retail-

rs, always coordinates the general n -retailer transshipment prob-

em in a fully non-cooperative setting. Retailers make initial pay-

ents to the fund (i.e. a third party financial entity that contracts

ith the retailers on transshipment payments) and after demand

ealization, transshipped residuals and payments from the fund to

he retailers dependent on the retailers’ announcements about ex-

ess supply or demand and are specified according to a predeter-

ined rule. This mechanism creates a large set of feasible trans-

hipment payments that coordinate the n -retailer transshipment

roblem.

Another stream of literature discusses two-stage non-

ooperative/cooperative games, so-called ’biform games’ [8] ,

ith non-cooperative inventory decisions in the first stage and

ooperative shipping decisions in the second stage termed ’coope-

ition’ (for a review on cooperative game theory and inventory

anagement see e.g. [24,50,51] ). The main difference to the liter-

ture discussed before is that although the allocation mechanism

s as-well defined before demand realization the transshipment

rices are set after demand realization (ex post) and are agreed on

ooperatively. Anupindi et al. [2] analyze a very general framework

ith n decentralized newsvendors that share all residual invento-

ies through such a two-stage non-cooperative/cooperative game.

hey show that the core of the game is non-empty and that there

xists an allocation mechanism in the form of a fractional alloca-

ion rule with a side payment scheme that achieves a coordinated

olution for inventory deployment and allocation. The authors,

owever, point out the main difficulty in such allocation problems.

hat is, while the inventory decision can be done without any

greement with the other players in the game, for the allocation

ecision, in contrast, the players must find a mutual consent.

uang and Soši ́c [40] compare the performance of the ex ante

ransshipment price method of [61] and others against the ex

ost allocation rule of [2] showing that neither allocation method

ominates the other.

Granot and Soši ́c [28] extend the model of [2] including the

etailers’ decision of how much residual inventory to share with

he other players. They present a three-stage model, where in the

rst stage retailers make inventory decision before demand real-

zation, in the second stage (after demand realization) they decide

n how much residual stock to share and in the third stage, in-

entories are transshipped and profits allocated according to an al-

ocation rule. They discuss the impact of different allocation rules

n the retailers willingness to share residual inventory with oth-

rs in the second stage and if coordination can be achieved. Yan

nd Zhao [79] develop a mechanism to coordinate n retailers that

ill completely share their residuals by involving a third party

e.g. a manufacturer) who subsidizes the transshipment profit

llocation.

Huang and Soši ́c [39] compare the single-shot transshipment

ame of n newsvendors with a game that is repeated infinitely

any times. They show that it could be a subgame-perfect Nash

quilibrium to share all the residuals at the second stage in the

epeated game if the discount factor is large enough.

.2. Physical centralization of inventories

Physical aggregation is another common practice of inventory

ooling. Instead of having local inventories for each retailer, inven-

ories are stored at a central storage facility (warehouse). Usually

he storage cost at the centralized facility is lower than the total

torage cost at the retailers. However, physical centralization also

ncreases response time and transportation cost to customers [17] .

The majority of literature dealing with the physical cen-

ralization of inventories assume either a central control (e.g.

21,14,27,19] ) or a cooperative newsvendor game (e.g. [31,30] ). This

ay be caused by the fact that the physical sharing of a storage

ocation and consequently a joint stock and ownership can be crit-

cal for independent retailers. Further, one again needs a proper

llocation rule for assigning the joint stock to the retailers.

In practice, however, there do exist situations where the ware-

ouse is the only source for replenishment in case of a stockout.

or independent retailers this is especially appropriate when orga-

izational structures, like franchising arrangements, inhibit trans-

hipments among retailers [75] .

Although in respect to the design of the network the non-

ooperative game with physical centralization of inventories is dif-

erent from the game discussed in Section 3.1 , mathematically they

re related to each other regarding existence and uniqueness of the

ash equilibrium (see [66] ). Hence, in this section we briefly dis-

uss main differences and findings in literature.

In terms of modelling the main differences compared to the

etting discussed in Section 3.1 are that i) there is a single per unit

urchase cost c at the central storage location for all retailers, ii)

he final allocation of stocks to the retailers or customers may in-

lude additional transportation cost due to the centralization and

ii) there might be a compensation cost for using the other firms

ontribution to the common stock [66] . Now, in a non-cooperative

nvironment each retailer i decides on his contribution Q i to the

oint stock based on uncertain demand D i [5] . The retailers make

heir decisions simultaneously. After demand is realized each re-

ailer i first receives the minimum of the order Q i and the demand

i . Then, residual stock is allocated to retailers facing a shortage

ccording to a given allocation rule.

Consider the model of [5] with two independent firms differ-

ng in their shortage cost. If a retailer faces a stockout situation,

hen he will receive an allocation proportional to its contribution,

.e. Q i / (Q i + Q j ) . Ben-Zvi and Gerchak [5] provide first order con-

itions of such a model and prove existence and uniqueness of a

ash equilibrium. Gerchak [26] analyze the consequences of such a

on-cooperative game with a modified scheme that is beneficial to

ll parties relative to a no-pooling situation. Ben-Zvi and Gerchak

5] and Gerchak [26] assume that it is costless for retailer i to use

esidual stock from j ’s contribution to the joint stock Q j .

6 L. Silbermayr / Omega 92 (2020) 102148

Table 2

Overview for literature on horizontal contracts between retailers.

Horizontal Inventory No. Flexible Paper

contract type interaction retailers profit div.

Transshipment price Transsh. 2 no [36,61,40,64,3,44]

(ex ante)

Compensation cost for Physical cent. 2 no [66]

using centralized stock

Implicit transsh. price Transsh. 2 yes [34]

depend. on inventory dec.

Combining transsh. price, Unid.-transsh. 2 yes [3]

leftover & shortage subsidy

Transshipment fund Transsh. n yes [29]

arranged by third party

Ex post allocation rule Transsh. n yes [2,28,40,39,79]

(biform game) (Physical cent.)

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t

t

4

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t

j

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p

Netessine and Rudi [53] study the practice of drop-shipping for

multiple decentralized newsvendors and a wholesaler as a non-

cooperative game. They analyze a combined strategy where retail-

ers use own inventories and drop-shipping stocked at a central lo-

cation owned by the wholesaler. The difference to the literature

discussed before is that centralized inventory decision is done by

another independent player at an upstream echelon and not by

the independent retailers. They discuss structural properties of the

equilibrium solution and show that such a combined strategy often

benefits retailers as well as wholesaler.

Silbermayr and Gerchak [66] address horizontal coordination in

a model with two independent retailers that might partially pool

their inventories. Each retailer decides on the quantity stocked at

its local storage facility and on its contribution to the joint ware-

house. They assume a non-negative compensation cost for using

the other firms contribution to the pool. Transshipments between

the retailers are prohibitively expensive. They show that, similar to

the transshipment setting with a transshipment price contract, a

system with physical centralization of inventories can be coordi-

nated if retailers appropriately set a compensation cost for using

the other retailers’ residual stock at the warehouse.

The general framework of [2] with n newsvendors constitut-

ing a biform game with an ex post inventory allocation rule also

considers physical centralization at one or several jointly owned

warehouses in addition to transshipments between retailers. They

introduce the notion of ’claims’ that establish ownership for each

unit of inventory in the system, regardless of its locations. After

demand is realized the claim holder owns the right to determine

how its purchased units are to be used. This reduces the inven-

tory decision (at the first stage of the biform game) to a non-

cooperative game in spite of the presence of common inventory

in central warehouses with joint ownership. That is, with ’claims’

the ex post allocation rule allows for horizontal coordination also

under physical centralization of inventories.

A summary of literature on horizontal contracts that coordinate

a system of independent retailers that physically or virtually pool

their inventories is given in Table 2 . We report whether the con-

tract has been studied analytically for the 2 or n retailer case and

whether it has the flexibility to arbitrarily divide the total profit

between the retailers.

4. Customer driven interaction

In this section we consider non-cooperative newsvendor games

that arise from distinct customer driven interactions, whereby the

sales of one product affects the sales of another product. First,

there is product substitution where an unsatisfied customer is will-

ing to buy a substitute product at another retailer, i.e. a system

with competition between the newsvendors (Section 4.1). Then,

there is product complementarity where an unsatisfied customers

f one product hinders the sales of a complementary product (Sec-

ion 4.2). The detailed classification of the literature discussed in

his section is shown in Table 4 .

.1. Substitution

The general setting is based on n competing retailers selling

ubstitutable products. As in the classical newsvendor model, at

he beginning of the selling period each newsvendor decides about

wn stocking quantity in advance of demand realization. If the de-

and for i ’th newsvendor turns out to be larger than its stock-

ng quantity, D i > Q i , then a customer might be willing to search

nd buy the product at the other retailers. If the customer can-

ot find the product at retailer j he can continue searching in

ther retail locations (e.g. [49] ) or he stops and his demand re-

ains unsatisfied. In the following we discuss the latter setting

here the search is over after one attempt. The search is gener-

lly modelled such that a proportion a ij of customers demanding

product at newsvendor i will take product at newsvendor j as

ubstitute.

Modelling the proportion of the unsatisfied demand of i switch-

ng to j by 0 < a ij ≤ 1 is general enough to model partial substitu-

ion, full substitution and one-way substitution of demand. For an

verview we refer to [70] . Under partial substitution only a fraction

f demands will switch to a competitor, i.e. 0 < a ij < 1. Full substitu-

ion means that all customers of i are willing to accept the product

when retailer i is out of stock, i.e. a i j = 1 . Finally, one-way sub-

titution expresses a system where, for example, customer i will

witch to j with probability a ij in case of a stockout but not vice

ersa, i.e. 0 < a ij ≤ 1 and a ji = 0 . This can be the case if the product

f j has a higher quality than the product of i . One might addition-

lly need an adjustment cost whenever the lower quality product

ill be substituted by the higher quality product [20] . Note that

ne-way substitution can also be related to unidirectional trans-

hipments discussed in Section 3.1 .

One of the earliest works under this setting is [56] which con-

iders only two competing newsvendors, but highlights that in

ractice substitution often takes place between different products

old by independent retailers. Wang and Parlar [74] extend the

ork of [56] to three competing newsvendors. Li and Ha [46] ex-

end the two retailer case selling substitutable products including

eactive capacity to fill uncertain demand in addition to initial in-

entory. They show that additional reactive capacity has a positive

ompetitive effect on a retailer but a negative effect on the com-

etitor. Wu et al. [76] explore the two competing retailer case with

symmetric bargaining power of the retailers where the weak re-

ailer is capital-constrained and takes a trade credit with a whole-

ale price dictated by the manufacturer. The trade credit can be

sed by the manufacturer as a strategic response to the bargaining

ower of the other retailer.

L. Silbermayr / Omega 92 (2020) 102148 7

Table 3

Overview for the literature on designed horizontal interactions.

No. dec. Non-ident. Transsh. Physical Comp. to Coord. Incl. Specifics

newsv. newsv. Pooling Central

con.

Mechanism Upstream dec.

Anupindi et al. [2] n √ √ √ √

B

Rudi et al. [61] 2 √ √ √

T

Granot and Soši ́c [28] n √ √ √

B

Netessine and Rudi [53] n √ √ √

Drop-shipping

Hu et al. [36] 2 √ √ √

T Uncertain supplier capacity

Hanany et al. [29] n √ √ √

F

Hezarkhani and Kubiak [34] 2 √ √ √

I

Huang and Soši ́c [40] n √ √ √

T, B

Huang and Soši ́c [39] n √ √ √

B Repeated game

Shao et al. [64] 2 √ √

T √

Ben-Zvi and Gerchak [5] 2 √ √ √

Yan and Zhao [79] n √ √ √

B

Gerchak [26] 2 √

Lee and Park [44] 2 √ √ √

T Uncertain supplier capacity

(rationing game)

Arikan and Silbermayr [3] 2 √ √ √

T + S+L Unidirectional transshipments

Feng et al. [23] 2 √ √ √

T Partial backordering

Silbermayr and Gerchak [66] 2 √ √ √

C Partial pooling

Coordination mechanism: T: transshipment price (ex ante), B: biform (non-cooperative/cooperative) game, I: implicit pricing, F: transshipment fund, S: shortage subsidy, L:

leftover subsidy, C: compensation cost

Table 4

Overview for the literature on customer driven interactions and on combination/comparison of designed and customer driven interaction.

No. dec. Non-ident. Substitutes Complements Comp. to Incl. Specifics

newsv. newsv. Central con. Upstream dec.

Parlar [56] 2 √ √ √

Wang and Parlar [74] 3 √ √ √

Lippman and McCardle [47] n √

Splitting rule for industry

demand allocation

Mahajan and Van Ryzin [49] n √ √ √

Dynamic consumer choice

Cachon [10] n √ √ √

Industry demand allocated

propor. to inventory

Netessine and Rudi [52] n √ √ √

Netessine and Zhang [54] n √ √ √ √ √

Complements vs.

substitutes

Li and Ha [46] 2 √ √

Reactive capacity

Huang et al. [37] n √ √ √

Zhang et al. [81] 2 √ √ √

Qi et al. [59] n √ √ √ √

Wu et al. [76] 2 √ √ √ √

Dominant and weak

(capital-contained) retailer

Anupindi and Bassok [1] 2 √ √ √

Physical pooling vs.

decentralized stocks

Zou et al. [84] 2 √ √

Substitution vs.

transshipments

Çömez et al. [18] 2 √ √

Transshipment request

after each customer arrival

Chen et al. [12] 2 √ √ √ √

Transshipments vs.

substitution (incl. fast-ship

option)

Li and Li [45] 2 √ √

Transshipments vs.

substitution

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Mahajan and Van Ryzin [49] , Netessine and Rudi [52] and

uang et al. [37] , for example, study the problem for any number

f newsvendors. Lippman and McCardle [47] study n competing

ewsvendors with a probabilistic aggregate industry demand that

s allocated across firms by a predefined rule (e.g. deterministic or

andom splitting). Cachon [10] studies supplier retailer coordina-

ion with n competing retailers where stochastic industry demand

s divided proportional to the retailers’ inventory levels.

For the general model with n retailer facing random demands

i , i = 1 , . . . , n and substitution, the effective demand is D

e i

= D i +

i � = j a ji (D j − Q j ) + while the net demand does not play a role in

his customer search setting.

There are slight differences in the literature with respect to

odelling the total shortage cost. A limiting case would be a

ewsvendor with emergency orders. Some (e.g. [52] ) do not con-

ider any shortage costs. Parlar [56] , Wang and Parlar [74] and Qi

t al. [59] , for example, assume that the shortage cost is only in-

urred for the unsatisfied demand from initial customers, i.e. re-

ailer i incurs a total shortage cost of p i (D i − Q i ) + . On the other

and, Lippman and McCardle [47] and Huang et al. [37] assume a

hortage cost for all the unsatisfied demand both from initial and

econdary customers, i.e. retailer i incurs a total shortage cost of

p i (D

e i − Q i )

+ . In the latter case, the expected profit of retailer i is

i = E(r i min (D

e i , Q i ) − c i Q i + s i (Q i − D

e i )

+ − p i (D

e i − Q i )

+ ) . (8)

f the shortage cost is only incurred for the unsatisfied demand

rom initial customers then the last term in Eq. (8) has to be re-

laced by p (D − Q ) + .

i i i

8 L. Silbermayr / Omega 92 (2020) 102148

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Equilibrium analysis

From the modelling perspective the advantage of considering

shortage cost p i (D

e i − Q i )

+ is the fact that expected profit �i only

includes D

e i

as the random component while the other one has

both D

e i

and D i . As a result, the first model gives a more compact

formulation of the optimality condition. The optimal order quan-

tity Q

∗i

satisfies

P r(D

e i < Q

∗i ) = F D e

i (Q

∗i ) =

r i − c i + p i r i − s i + p i

. (9)

Hence, the critical ratio of retailer i under substitution is equal to

the critical ratio of the classical newsvendor in Eq. (2) , but the de-

mand distribution is different.

�i is concave in Q i and the slope of the best response function

is

∂Q i

∂Q j

= −a ji g D e

i (Q i )

f D e i (Q i )

. (10)

Since f D e i (Q i ) ≥ g D e

i (Q i ) , the slope is non-positive and smaller than

one in absolute terms.

On the other hand, under the model with shortage cost p i (D i −Q i )

+ the optimality condition is

r i (1 − F D e i (Q

∗i )) − p i (1 − F D i (Q

∗i )) + s i F D e

i (Q

∗i ) − c i = 0 (11)

and the slope of the best response function is

∂Q i

∂Q j

= −a ji (r i − s i ) g D e

i (Q i )

(r i − s i ) f D e i (Q i ) + p i f D i (Q i )

. (12)

Similarly, the slope is non-positive and smaller than one in abso-

lute terms.

Concavity of �i guarantees the existence of a Nash equilibrium

under both models. The uniqueness of the equilibrium is again

proven using the contraction mapping argument. In order to show

that the mapping is a contraction it is sufficient to show that no

column or row sum of matrix J exceeds one, where J is the Jaco-

bian of the best response mapping. Under both models, | ∂Q i ∂Q j

| < a ji ,

and the following holds for all columns i :

∑

j=1 ,.,n

∣∣∣∣ ∂Q i

∂Q j

∣∣∣∣ <

∑

j=1 ,.,n

a ji . (13)

Note that the same holds for the row sums.

Therefore, if either ∑

j=1 ,.,n a ji < 1 for all i or ∑

i =1 ,.,n a ji < 1 for

all j , then there exists a unique Nash equilibrium.

Horizontal coordination

In the centralized case, when all products/locations are man-

aged by a single decision maker, the total expected profit with

shortage cost p i (D

e i − Q i )

+ is

�C = E

(

n ∑

i =1

r i min (D

e i , Q i ) − c i Q i + s i (Q i − D

e i )

+ − p i (D

e i − Q i )

+

)

.

(14)

Eq. (14) is concave for n = 2 (see [57] ) and n = 3 and partial substi-

tution (see [22] ). Netessine and Rudi [52] , however, show that the

profit function might not be concave and not even quasiconcave in

a setting with n > 2 retailers and full substitution considering the

analytically tractable deterministic analog of Eq. (14) and numeri-

cal experiments. Hence, the first order condition does not guaran-

tee the global optimum in such settings. Their numerical experi-

ments, however, indicate that for reasonable demands where the

coefficient of variation is more than 0.1 the objective function is

concave in Q for i = 1 , . . . n .

i

The first order necessary optimality conditions for i = 1 , . . . , n

re given by

P r(D

e i < Q i ) − P r(D i < Q i < D

e i ) +

∑

i � = j

r j − s j + p j

r i − s i + p i a i j

P r(D

e j < Q j , D i > Q i ) −

r i − c i + p i r i − s i + p i

= 0 . (15)

If the other approach with shortage cost p i (D i − Q i ) + is taken

he first order conditions are given by

r(D

e i < Q i ) −P r(D i < Q i < D

e i ) +

∑

i � = j

r j − s j

r i − s i a i j P r(D

e j < Q j , D i > Q i )

+

p i r i − s i

P r(D i < Q i ) −r i − c i + p i

r i − s i = 0 . (16)

Comparing the first order conditions under centralized con-

rol ( Eqs. (15) and (16) ) with those under decentralized control

Eqs. (9) and (11) ), respectively, it can be concluded that the sys-

em is suboptimal in the competitive newsvendor case with prod-

ct/demand substitution (e.g. [37,52] ). Compared to the centralized

ase, in competition each retailer receives less expected profit. That

s, competition among retailers introduces challenges for supply

hain coordination [10] .

Lippman and McCardle [47] find that competition makes the re-

ailers order more in anticipation of overflow demand from the

ompetitors who may stockout. This effect is also noticed by e.g.

10,49] . Netessine and Rudi [52] show that although competi-

ion compared to centralized control usually leads to overstocking,

here are also some cases with the counterintuitive situation that

ompetition leads to understocking. Qi et al. [59] who consider a

wo-echelon framework with one manufacturer and two retailers

how that the total stock of two competing newsvendors is de-

reasing in the (endogenous) wholesale price and that competition

ay result in lower total supply chain efficiency compared to a

entralized system.

As already discussed in Section 2 the customer driven inter-

ction is an external factor and no literature on horizontal con-

ract design exist in this setting. Coordination under substitution

s only addressed through vertical contracts with manufacturer-

etailer competition, as for example in [7] who studies a multiple-

hannel distribution system in which a manufacturer sells its prod-

ct through an independent retailer and its wholly-owned channel

ith customer spill-over. While simple vertical contracts fail to co-

rdinate such a system, they show that a penalty contract and a

wo-part compensation-commission contract under which the re-

ailer earns revenues only as commissions on his channel sales that

xceed a flexible target can coordinate the supply chain.

.2. Complementarity

Unlike product substitution where an unsatisfied customer

hooses to buy a substitute product which benefits the seller of

he substitute, the existence of cross-selling or complementar-

ty may produce opposite effects [81] . Complementarity implies

hat customers are willing to purchase some related products to-

ether with the original and if one of the products is out of stock

he related product is no longer demanded. Especially for elec-

ronic products sold by independent companies the complemen-

arity may be an issue. Netessine and Zhang [54] give a few exam-

les such as, e.g., music players whose sales are impacted by the

vailability of compatible rerecorded discs.

The topic has been mainly studied in areas such as e.g. cus-

omer behavior or marketing. Within the newsvendor framework,

owever, it has been rarely studied. In particular, we only came

cross [54,81] that deal with cross-selling/complementarity in non-

ooperative newsvendor problems.

L. Silbermayr / Omega 92 (2020) 102148 9

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Let a ij be the decreased demand for item i caused by a stock

ut situation of item j , then the expected profit of a newsvendor

ith cross-selling is obtained by replacing the effective demand D

e i

n the expected profit functions defined in Section 4.1 by the net

emand D

n i

= D i −∑

i � = j a ji (D j − Q j ) + . Hence, cross-selling is com-

lementary to substitution where a newsvendor has to consider

he effective demand when making the inventory decision. Note

hat [54] describe the net demand for i as D

n i

= D i −∑

i � = j a ji (D j − j )

+ +

∑

i � = j a ji D j , where a ij is the fraction of customers willing to

urchase from retailer i will also purchase from retailer j but only

f retailer i has the product on stock.

Similar to the substitution setting, the optimal order quantity

∗i (Q j ) under complementarity and shortage cost p i (D

n i

− Q i ) + sat-

sfies

r(D

n i < Q

∗i ) = F D n

i (Q

∗i ) =

r i − c i + p i r i − s i + p i

. (17)

he critical ratio of retailer i under complementarity again is equal

o the critical ratio of the classical newsvendor in Eq. (2) , but the

emand D i is replaced by the net demand D

n i .

Again a unique Nash equilibrium for the game exists if

j=1 ,.,n a ji < 1 for all i or ∑

i =1 ,.,n a ji < 1 for all j [81] .

Assuming a centralized decision maker under complementarity

unique solution is guaranteed [54,81] , while under substitution

his is not the case (see discussion in Section 4.1 ). Comparing the

rst order conditions of the centralized and decentralized system

t can again be concluded that the system with independent play-

rs is not coordinated. Netessine and Zhang [54] and Zhang et al.

81] show that decentralization in such a setting leads to under-

tocking. Netessine and Zhang [54] also discuss the intuitive result

hat the expected profits in the decentralized case are lower than

n the centralized case, while the numerical results of [81] show

hat in some cases the opposite is true, i.e. more sales do not mean

igher profits under complementarity.

Netessine and Zhang [54] also discuss the implication of com-

lementarity on a two-echelon supply chain including the man-

facturer’s decision. They conclude that competition on comple-

ents induces both retailers and the wholesaler to coordinate the

upply chain.

. Combination or comparison of designed and customer

riven interactions

There exist some work that deal with the i) combination of or

i) comparison between the different settings of non-cooperative

ames with horizontal inventory interactions. Anupindi and Bas-

ok [1] study a decentralized two-echelon supply chain with one

anufacturer and two retailers and partial substitution. They dis-

uss the impact of the retailers designed operations, in particular

entralization of stocks at a warehouse versus decentralized stocks

t the retail level, on the supply chain when a fraction α of cus-

omers who do not find the good at their retailer attempt to buy

he good at the other retailer. In the system with centralization of

tocks they assume a single decision maker (central control). This

ssumption implies that centralization of stocks is always more

eneficial for the retailers. However, looking at the total supply

hain profit including the manufacturers profit they show that it

ay not always increase upon centralization of stocks by retail-

rs as for certain α the expected sales in the decentralized system

ith substitution is larger. Çömez et al. [18] study decentralized re-

ailers with customer over-spills and transshipments. They assume

hat transshipments do not occur after all demands are realized,

ut a transshipment request is sent after each customer arrival at

stock-out retailer. The other retailer has the flexibility to reject or

ccept each request individually. They show that the optimal trans-

hipment policy is characterized by the inventory holdback levels.

Zou et al. [84] compare two alternative scenarios of a system

ith two retailers and a common manufacturer: one with trans-

hipments and another without transshipments but substitution.

hey discuss the impact of the customer switching rate and the

ransshipment price on the benefit of transshipments compared

o a scenario with substitution. Chen et al. [12] also compare the

wo scenarios but they additionally include a customer’s willing-

ess to wait in case the product is not available and the possibil-

ty to arrange a fast-shipment directly to the customer from the

upplier arranged by the stocked out retailer. They find that when

ast-ship participation rate in the substitution scenario is high,

he supplier tends to prefer the retailers to transship, while the

etailers prefer substitution. Li and Li [45] study a decentralized

upply chain with one manufacturer and two symmetric retail-

rs comparing transshipments with substitution, i.e. without trans-

hipments a fraction of unsatisfied customers buy the product at

he competing retailer. They show that as long as the transship-

ent price is properly chosen the retailers will always prefer trans-

hipments independent of the proportion of customers switching

o the competitor. They also study how the power structure of

he supply chain members affects the transshipment decision. If

he manufacturer has power to control transshipment prices he

ill choose transshipments over substitution and set a high trans-

hipment price which increases the retailers inventories. If the

anufacturer can only decide whether to transship or not, then

e will avoid transshipment if the customer search probability is

igh.

. Summary and future research directions

.1. Summary

In practice retailers are often independent players in the sup-

ly chain. Due to risk pooling or customer interactions between

ndependent retailers the inventory decision of one retailer may

ffect the decision of another. Hence, non-cooperative game the-

ry is a tool that can be applied in this context. This paper pro-

ides an overview of the specific settings where non-cooperative

tatic game theory with horizontal inventory interactions among

ewsvendors can be applied and discusses findings in the lit-

rature. Thereby, we distinguish between inventory interactions

hat are caused i) through the design of the retail-networks,

.e. local storage with inventory sharing through transshipments

r inventory sharing through physical centralization of invento-

ies and ii) through the customer driven interactions whereby

etailers sell substitutable or complementary products. For each

pecific setting we discuss the conditions required for the ex-

stence and uniqueness of a Nash equilibrium in such a non-

ooperative game. Further, we analyze the differences and rela-

ions between the different settings and discuss the main find-

ngs in the literature. We also compare the decentralized systems

o a centrally controlled systems in order to emphasize the im-

act of horizontal coordination and horizontal contracting in such

ames.

To summarize, in this survey we review the contributions to

ate for the practical settings of horizontal inventory interactions

etween independent newsvendors and also give a comparison

ith a system of a single decision maker. From the managerial

erspective, the reviewed mathematical models and their solutions

rovide important insights for inventory managers into the hori-

ontal interactions in decentralized retailer networks. The findings

rovide guidance for both practice and future research how retail-

rs considering the impact of these interactions can improve sup-

ly chain performance. The presented models can serve as building

locks for topics that have not been addressed so far and deserve

urther investigation.

10 L. Silbermayr / Omega 92 (2020) 102148

6.2. Future research directions

The findings of this review of the literature on non-cooperative

newsvendor games with horizontal inventory interactions open

various promising future research directions.

1. Combining vertical and horizontal relationships in the supply

chain: Although some work already discusses the vertical re-

lationships, e.g. with a supplier or a manufacturer, in ad-

dition to the horizontal relationships we have focused on,

there is still a lack of fully understanding how the verti-

cal relations impact the supply chain performance under in-

ventory pooling and competition, respectively. There is also

potential in comparing all the individual settings discussed

here in order to get a better understanding under which par-

ticular scenario a focus should either be put on the designed

interaction or the customer driven interaction in a system

with decentralized retailers.

2. Horizontal contracting: There is also a lot of potential in

searching for more horizontal contracts that coordinate the

supply chain, since most of the contracts that have been ad-

dressed so far are either not always coordinating or they are

very complex and difficult to implement in practice. One has

to find simple contracts that are flexible in allocating the to-

tal profits arbitrarily between the independent newsvendors.

Consider the simple ex ante transshipment price contract

that might not coordinate the decentralized transshipment

system as soon as certain asymmetries among retailers arise

[36] . This is due to the implied unbalanced risk sharing be-

tween the retailers whenever they are asymmetric. For ex-

ample, the retailers could add another payment, e.g. on left-

overs or stockouts, in addition to the transshipment price in

order to balance the risk between them (see also [3] ). This

would also lead to more flexibility as the introduction of

another price also allows to arbitrarily divide the total ex-

pected profit between the retailers.

3. Omni-channel supply chains and off-price retailers: The set-

tings reviewed here are also relevant in today’s omni-

channel supply chains that include online-shops and/or dif-

ferent costumer segments where the non-cooperative game-

theoretic framework can be applied (see e.g. [63] and [82] ).

Online and offline retailers need to compete in new and in-

novative ways [9] . In addition to e-commerce, off-price re-

tailers such as T.J. Maxx and Ross have seen their market

share increase greatly in the last three decades [41] . Regu-

lar retailers are transshipping leftover inventory to off-price

retailers offering the retailers a useful sale channel. Off-price

retailers offer these products at considerable price discounts,

which puts them into a unique competitive position [33] .

It would be interesting to model such settings as a non-

cooperative newsvendor game. To summarize, using the ex-

isting concepts reviewed here and apply them to the nowa-

days challenging supply chain structures of omni-channel

retailing and/or the presence of off-price retailers in order

to provide decision support is a promising research field.

4. New technologies and data driven processes: Recent technol-

ogy advances are creating new business models presenting

new challenges to retailers. For example, smart phones en-

able tracking of customers and also remove barriers from re-

tailers such as geography and customer ignorance [9] . Large

amounts of valuable data from e.g. social, mobile or lo-

cal channels provide potential for better understanding cus-

tomer interactions and effectively control inventory levels.

However, they are also creating a new competitive environ-

ment for retailers. Hence, retailers that are able to analyze

these new data can gain competitive advantage in the new

business environment [80] . It would be interesting to inte-

grate big data from internal and external sources into the

existing models to generate new business models for re-

tailers in competitive environments. Choi et al. [16] discuss

how different types of big data techniques can be applied in

modern operations management. From the inventory related

side new technologies such as e.g. 3D printing, also known

as additive manufacturing, with its built to order fashion

also lead to new challenges [13] . Retailers will have to re-

think their traditional approaches with this new technology;

however, it could also lead to new ways of horizontal col-

laboration between independent retailers if they are willing

to share these new technological resources or raw materials.

5. Sustainability of horizontal inventory interaction: When look-

ing at the designed operations, inventory pooling is known

to be a best practice from the economic perspective, how-

ever its consequence on the environment and the impact on

the product carbon footprint of the product should also be

taken into consideration [67] . It is well-known that inven-

tory pooling leads to higher expected profits than no pool-

ing. However, the consequences on the environmental sus-

tainability of such horizontal interactions for independent

players has not been analyzed yet.

6. Rules for inventory allocations: Especially in networks with

multiple (more than two) retailers, understanding how to

design and negotiate proper allocation rules for assigning

residual inventory or demand to the individual retailers has

to be advanced. While there is some work on allocation

mechanisms, the relevant factors that influence the perfor-

mance of the mechanism, and the problem of putting the

right incentives to the retailers in order to engage in hori-

zontal interactions deserve future exploration. This is espe-

cially relevant in the case of physical pooling, where there

are additional issues in sharing joint warehouses in a decen-

tralized manner. Empirical investigations could help in or-

der to better understand the main challenges and extend the

theory in such settings.

7. Newsvendors with alternative optimization objectives: Previous

work on the newsvendor problem has mainly focused on

analytical approaches assuming risk neutral expected profit

maximizing decision makers [72] . The findings of this review

of the literature are all based on maximizing the expected

profits. There exist some work in the setting with indepen-

dent risk-averse or loss-averse retailers selling substitutable

products (e.g. [48,65,73,78] ). Future research could focus on

using alternative risk preferences rather than risk neutrality

to describe risk behavior under both designed and customer

driven inventory interactions. This would change the (equi-

librium) inventory quantities under decentralized and cen-

tralized supply chains depending on the retailers’ risk atti-

tudes. However, still very little is known about actual risk

attitudes of the newsvendors in a decentralized setting.

8. Behavioral newsvendors: The success of the existing opera-

tions management tools reviewed in this paper and the ac-

curacy of its theory rely heavily on understanding human

behaviour [6] . There is a great potential in studying the

human factor in an environment with independent play-

ers that optimize their own objectives in order to under-

stand decision makers’ attitudes and extend the standard

theory on non-cooperative newsvendor games. Ovchinnikov

et al. [55] is the first study that analyzes the human fac-

tor through an laboratory experiment for the setting with

two newsvendors selling substitutable products. Their ex-

perimental result shows that on the aggregate level the

participants did not respond to the other players actions.

This was also shown in [42,60,71,83] . The use of human

L. Silbermayr / Omega 92 (2020) 102148 11

A

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f

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[

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experiments can help to relate the empirical findings to the

analytical tools discussed here and extend them by includ-

ing behavioral factors. Since studying the behavioral aspects

in decentralized supply chains is just at its beginning it

would be very interesting to further explore the human fac-

tor whenever horizontal interaction in a decentralized en-

vironment appear in order to integrate the experimental

observations into the existing tools. Thereby, one has to ana-

lyze the aggregate and also the individual level outcomes of

the experiments. Note that, for example, a newsvendor with

per unit penalty cost discussed here has been defined in the

seminal work of [62] as a newsvendor with stockout-averse

preferences. Ho et al. [35] and Kirshner and Ovchinnikov

[42] , for example, show by means of experiments that the

psychological aversion to leftovers is greater than the disutil-

ity for stockouts. Therefore, it would be interesting to study

this further by extending the centralized multi-retailer in-

ventory model of [35] to a setting with decentralized retail-

ers. This is an important research direction and might lead

to some additional models and tools that can assist in de-

signing and negotiating such interactions in practice.

cknowledgments

The author would like to thank Emel Arikan, Yigal Gerchak,

erner Jammernegg, the editor and the referees for their valuable

omments and suggestions.

upplementary material

Supplementary material associated with this article can be

ound, in the online version, at doi: 10.1016/j.omega.2019.102148

eferences

[1] Anupindi R , Bassok Y . Centralization of stocks: retailers vs. manufacturer.Manag Sci 1999;45(2):178–91 .

[2] Anupindi R , Bassok Y , Zemel E . A general framework for the study of decen-tralized distribution systems. Manuf Serv Oper Manag 2001;3(4):349–68 .

[3] Arikan E , Silbermayr L . Risk pooling via unidirectional inventory transship-

ments in a decentralized supply chain. Int J Prod Res 2018;56(17):5593–610 . [4] Axsäter S . Evaluation of unidirectional lateral transshipments and substitutions

in inventory systems. Eur J Oper Res 2003;149(2):438–47 . [5] Ben-Zvi N , Gerchak Y . Inventory centralization in a newsvendor setting when

shortage costs differ: Priorities and costs allocation. In: Choi, T-M, editor.Handbook of Newsvendor Problems. Springer; 2012. p. 263–76 .

[6] Bendoly E , Donohue K , Schultz KL . Behavior in operations management:

assessing recent findings and revisiting old assumptions. J Oper Manag2006;24(6):737–52 .

[7] Boyaci T . Competitive stocking and coordination in a multiple-channel distri-bution system. IIE Trans 2005;37(5):407–27 .

[8] Brandenburger A , Stuart H . Biform games. Manag Sci 2007;53(4):537–49 . [9] Brynjolfsson E , Hu YJ , Rahman MS . Competing in the age of omnichannel re-

tailing. MIT Sloan Management Review; 2013 .

[10] Cachon GP . Supply chain coordination with contracts. Handbook Oper ResManag Sci 2003;11:227–339 .

[11] Cachon GP , Netessine S . Game theory in supply chain analysis. Tutor Oper Res20 06:20 0–33 .

[12] Chen H-W , Gupta D , Gurnani H . Balancing inventory and stockout risk in retailsupply chains using fast-ship. Prod Oper Manag 2016;25(12):2103–15 .

[13] Chen L , Cui Y , Lee HL . Retailing with 3d printing. working paper 2017:1–56 .

[14] Chen MS , Lin CT . Effects of centralization on expected costs in a multi-locationnewsboy problem. J Oper Res Soc 1989;40:597–602 .

[15] Chinchuluun A , Karakitsiou A , Mavrommati A . Game theory models and theirapplications in inventory management and supply chain. In: A Chinchuluun,

Pardalos P, editors. Pareto Optimality, Game Theory And Equilibria. Springer;2008. p. 833–65 .

[16] Choi T-M , Wallace SW , Wang Y . Big data analytics in operations management.Prod Oper Manag 2018;27(10):1868–83 .

[17] Chopra S , Meindl P . Supply chain management. Strategy, planning & operation,

6. Springer; 2016 . [18] Çömez N , Stecke KE , Çakanyıldırım M . In-season transshipments among com-

petitive retailers. Manuf Serv Oper Manag 2012;14(2):290–300 . [19] Corbett C , Rajaram K . A generalization of the inventory pooling effect to non-

normal dependent demand. Manuf Ser Oper Manag 2006;8(4):351–8 .

20] Deflem Y , Van Nieuwenhuyse I . Managing inventories with one-way substitu-tion: a newsvendor analysis. Eur J Oper Res 2013;228(3):484–93 .

[21] Eppen G . Effects of centralization on expected costs in a multi-location news-boy problem. Manag Sci 1979;25(5):498–501 .

22] Ernst R , Kouvelis P . The effects of selling packaged goods on inventory deci-sions. Manag Sci 1999;45(8):1142–55 .

23] Feng P , Wu F , Fung RY , Jia T . Evaluation of two transshipment policies ina two-location decentralized inventory system under partial backordering.

Transp Res Part E 2018;118:207–24 .

[24] Fiestras-Janeiro MG , García-Jurado I , Meca A , Mosquera MA . Cooperative gametheory and inventory management. Eur J Oper Res 2011;210(3):459–66 .

25] Friedman JW . Game theory with applications to economics. Oxford UniversityPress, USA; 1990 .

26] Gerchak Y . Allocating pooled inventory according to contributions and entitle-ments. Decis Making Manuf Serv 2016;9(1):51–4 .

[27] Gerchak Y , He Q-M . On the relation between the benefits of risk pooling and

the variability of demand. IIE Trans 2003;35(11):1027–31 . 28] Granot D , Soši ́c G . A three-stage model for a decentralized distribution system

of retailers. Oper Res 2003;51(5):771–84 . 29] Hanany E , Tzur M , Levran A . The transshipment fund mechanism: coordinating

the decentralized multilocation transshipment problem. Naval Res Logist (NRL)2010;57(4):342–53 .

30] Hartman BC , Dror M . Allocation of gains from inventory centralization in

newsvendor environments. IIE Trans 2005;37(2):93–107 . [31] Hartman BC , Dror M , Shaked M . Cores of inventory centralization games. Game

Econ Behav 20 0 0;31(1):26–49 . 32] He Y , Zhang P , Yao Y . Unidirectional transshipment policies in a dual-channel

supply chain. Econ Modell 2014;40:259–68 . [33] Hess RL , Ring L . Off-price versus price-off: use of discriminant analysis to iden-

tify competitive differences across retail formats. Int J Retail Distribut Manag

2014;42(10):902–28 . 34] Hezarkhani B , Kubiak W . A coordinating contract for transshipment in a two–

company supply chain. Eur J Oper Res 2010;207(1):232–7 . [35] Ho T-H , Lim N , Cui TH . Reference dependence in multilocation newsvendor

models: a structural analysis. Manag Sci 2010;56(11):1891–910 . 36] Hu X , Duenyas I , Kapuscinski R . Existence of coordinating transshipment prices

in a two-location inventory model. Manag Sci 2007;53(8):1289–302 .

[37] Huang D , Zhou H , Zhao Q-H . A competitive multiple-product newsboy problemwith partial product substitution. Omega 2011;39(3):302–12 .

38] Huang X . A review on policies and supply chain relationships under inventorytransshipment. Bull Stat Oper Res 2013;29(1):21–42 .

39] Huang X , Soši ́c G . Repeated newsvendor game with transshipments under dualallocations. Eur J Oper Res 2010;204(2):274–84 .

40] Huang X , Soši ́c G . Cooperation between multiple newsvendors with ware-

houses. Manuf Ser Oper Manag 2010;12(2):299–318 . [41] Khouja M , Liu X , Zhou J . To sell or not to sell to an off-price retailer in the

presence of strategic consumers. Omega 2018:1–17 . 42] Kirshner SN , Ovchinnikov A . Heterogeneity of reference effects in the compet-

itive newsvendor problem. Manuf Serv Oper Manag 2018 . 43] Kranenburg A , Van Houtum G-J . A new partial pooling structure for spare parts

networks. Eur J Oper Res 2009;199(3):908–21 . 44] Lee C , Park KS . Inventory and transshipment decisions in the rationing game

under capacity uncertainty. Omega 2016;65:82–97 .

45] Li M , Li T . Consumer search, transshipment, and bargaining power in a supplychain. Int J Prod Res 2018;56(10):3423–38 .

46] Li Q , Ha AY . Reactive capacity and inventory competition under demand sub-stitution. IIE Trans 2008;40(8):707–17 .

[47] Lippman SA , McCardle KF . The competitive newsboy. Oper Res1997;45(1):54–65 .

48] Liu W , Song S , Wu C . Impact of loss aversion on the newsvendor game with

product substitution. Int J Prod Econ 2013;141(1):352–9 . 49] Mahajan S , Van Ryzin G . Inventory competition under dynamic consumer

choice. Oper Res 2001;49(5):646–57 . 50] Montrucchio L , Norde H , Özen U , Scarsini M , Slikker M . Cooperative newsven-

dor games: A review. In: Choi, T-M, editor. Handbook of Newsvendor Problems.Springer; 2012. p. 137–62 .

[51] Nagarajan M , Soši ́c G . Game-theoretic analysis of cooperation among supply

chain agents: review and extensions. Eur J Oper Res 2008;187(3):719–45 . 52] Netessine S , Rudi N . Centralized and competitive inventory models with de-

mand substitution. Oper Res 2003;51(2):329–35 . 53] Netessine S , Rudi N . Supply chain choice on the internet. Manag Sci

2006;52(6):844–64 . 54] Netessine S , Zhang F . Positive vs. negative externalities in inventory man-

agement: implications for supply chain design. Manuf Serv Oper Manag

2005;7(1):58–73 . 55] Ovchinnikov A , Moritz B , Quiroga BF . How to compete against a behavioral

newsvendor. Prod Oper Manag 2015;24(11):1783–93 . 56] Parlar M . Game theoretic analysis of the substitutable product inventory prob-

lem with random demands. Naval Res Logist (NRL) 1988;35(3):397–409 . [57] Parlar M , Goyal S . Optimal ordering decisions for two substitutable products

with stochastic demand. OPSEARCH 1984;21(1):1–15 .

58] Paterson C , Kiesmüller G , Teunter R , Glazebrook K . Inventory models with lat-eral transshipments: a review. Eur J Oper Res 2011;210(2):125–36 .

59] Qi Y , Ni W , Shi K . Game theoretic analysis of one manufacturer tworetailer supply chain with customer market search. Int J Prod Econ

2015;164:57–64 .

12 L. Silbermayr / Omega 92 (2020) 102148

[60] Quiroga BF , Moritz B , Ovchinnikov A . Behavioral ordering, competition andprofits: an experimental investigation. Prod Oper Manag 2019 .

[61] Rudi N , Kapur S , Pyke DF . A two-location inventory model with transshipmentand local decision making. Manag Sci 2001;47(12):1668–80 .

[62] Schweitzer ME , Cachon GP . Decision bias in the newsvendor problemwith a known demand distribution: experimental evidence. Manag Sci

20 0 0;46(3):404–20 . [63] Seifert RW , Thonemann UW , Sieke MA . Integrating direct and indi-

rect sales channels under decentralized decision-making. Int J Prod Econ

2006;103(1):209–29 . [64] Shao J , Krishnan H , McCormick ST . Incentives for transshipment in a supply

chain with decentralized retailers. Manuf Serv Oper Manag 2011;13(3):361–372 .

[65] Shen Y , Xie J , Li T . The risk-averse newsvendor game with competition on de-mand. J Ind Manag Optim 2016;12(3):931–47 .

[66] Silbermayr L , Gerchak Y . Partial pooling by independent firms with allocation

according to contribution to pool. Int J Prod Econ 2019;218:375–85 . [67] Silbermayr L , Jammernegg W , Kischka P . Inventory pooling with environmental

constraints using copulas. Eur J Oper Res 2017;263(2):479–92 . [68] Smirnov D , Gerchak Y . Inventory sharing via circular unidirectional chaining.

Eur J Oper Res 2014;237(2):474–86 . [69] Smirnov D , Gerchak Y . Inventory sharing via circular bidirectional chaining. Int

J Prod Econ 2016;179:141–52 .

[70] Turken N , Tan Y , Vakharia AJ , Wang L , Wang R , Yenipazarli A . The multi--product newsvendor problem: Review, extensions, and directions for future

research. In: Choi, T-M, editor. Handbook of Newsvendor Problems. Springer;2012. p. 3–39 .

[71] Villa S , Castañeda JA . Transshipments in supply chains: a behavioral investiga-tion. Eur J Oper Res 2018;269(2):715–29 .

[72] Wang C , Webster S , Zhang S . Newsvendor models with alternative riskpreferences within expected utility theory and prospect theory frameworks.

In: Choi, T-M, editor. Handbook of Newsvendor Problems. Springer; 2012.p. 177–96 .

[73] Wang CX . The loss-averse newsvendor game. Int J Prod Econ2010;124(2):448–52 .

[74] Wang Q , Parlar M . A three-person game theory model arising in stochasticinventory control theory. Eur J Oper Res 1994;76(1):83–97 .

[75] Wee KE , Dada M . Optimal policies for transshipping inventory in a retail net-

work. Manag Sci 2005;51(10):1519–33 . [76] Wu D , Zhang B , Baron O . A trade credit model with asymmetric competing

retailers. Prod Oper Manag 2019;28(1):206–31 . [77] Wu H , Parlar M . Games with incomplete information: a simplified exposition

with inventory management applications. Int J Prod Econ 2011;133(2):562–77 .[78] Wu M , Zhu SX , Teunter RH . A risk-averse competitive newsvendor problem

under the cvar criterion. Int J Prod Econ 2014;156:13–23 .

[79] Yan X , Zhao H . Inventory sharing and coordination among n independent re-tailers. Eur J Oper Res 2015;243(2):576–87 .

[80] Zhan Y , Tan KH . An analytic infrastructure for harvesting big data to enhancesupply chain performance. Eur J Oper Res 2018 .

[81] Zhang R-Q , Zhang L-K , Zhou W-H , Saigal R , Wang H-W . The multi-itemnewsvendor model with cross-selling and the solution when demand is jointly

normally distributed. Eur J Oper Res 2014;236(1):147–59 .

[82] Zhao F , Wu D , Liang L , Dolgui A . Lateral inventory transshipment problem inonline-to-offline supply chain. Int J Prod Res 2016;54(7):1951–63 .

[83] Zhao Y , Zhao X . How a competing environment influences newsvendor order-ing decisions. Int J Prod Res 2016;54(1):204–14 .

[84] Zou L , Dresner M , Windle R . A two-location inventory model with transship-ments in a competitive environment. Int J Prod Econ 2010;125(2):235–50 .