Dynamic Pricing with “BOGO” Promotion

in Revenue Management

Kyoung-Kuk Kim∗, Sunggyun Park†

Korea Advanced Institute of Science and Technology

Chi-Guhn Lee‡

University of Toronto

Mar 2016

Abstract

We consider a dynamic pricing problem when a seller, facing uncertain demands, sells a singleproduct in a finite horizon. The seller actively adopts dynamic pricing and quantity discount schemes.The proposed model is based on the assumption that each customer has random reservation prices andthe purchase size depends on the posted price and discount. We particularly focus on the widely adoptedpromotional schemes “buy one get one free” and “50% off” and study the optimal strategic choicesof the seller. Analytical results together with numerical experiments are presented to help us obtainmanagerial insights. Additional numerical results for a generalized model are provided so as to examinethe effectiveness of promotional schemes.

KEYWORDS: Revenue Management; Price Promotion; Reservation Price;Copula

1 Introduction

Pricing is an important factor that determines retailers’ profitability. Among many success stories, the airlineindustry is regarded as one prominent example in which pricing optimization techniques have successfullyresulted in increased revenues (Phillips, 2005). Other examples include electricity pricing, hotels and rentalcars, etc. According to Sullivan (2005), companies that employ price optimization techniques were able toraise their gross margins ranging from one percent to three percent, and in some cases up to ten percent.Elmaghraby and Keskinocak (2003) point out the availability of customer data along with decision supporttools for analyzing such data as well as new technologies that make price changes easy as the drivers forthe development of dynamic pricing strategies. Firms in e-commerce, in particular, actively adopt dynamicpricing and experience real growth. For example, E-Bay useda dynamic pricing strategy that sold more than20 billion dollars worth of goods in 2005 (Sahay, 2007) and Amazon made 3 million daily price changesthrough the month of November 2013, outperforming other retailers (Berthiaume, 2014).

∗Industrial and Systems Engineering, E-mail:kkim3128@kaist.ac.kr†Corresponding author, Industrial and Systems Engineering, E-mail:sunggyun@kaist.ac.kr‡Mechanical and Industrial Engineering, E-mail:cglee@mie.utoronto.ca

1

And yet there are many popular strategies employed by retailers other than dynamic pricing. One suchexample is the phrases such as “buy one get one free” (“BOGOF”for short) that we easily encounter ineveryday life. This particularly assumes that demands occur in multi-units or batches and retailers needto consider optimal menus (combination of batch sizes, prices, and discounts) to customers, consideringcustomers’ willingness-to-pay. That certainly raises theproblem complexity considerably. Sometimes itis referred to asdynamic nonlinear pricingbecause different unit prices can be set depending on variousquantities (Levin and Nediak, 2014), or simplyquantity discountsbecause often more discounts are offeredfor larger purchases. Although such approaches have been dealt in the literature, it is still limited in terms ofdynamic consideration of pricing and other promotional strategies together. The present work was motivatedby this research gap.

Particularly interesting and prevalent promotional strategies are “BOGOF” or “50% off.” Even if itis certainly the latter that is more attractive to customersif both are offered at the same unit price, it isnot so clear which is the best for a seller as there is a possibility of raising a higher revenue by sellingtwo units through the former. Customer responses to those schemes, therefore, are the most importantfactor in determining effective promotional strategies and have been the target of active academic studies.For example, Sinha and Smith (2000) compare “BOGOF” and “50%off” empirically to understand thedifference in customers’ perceived transaction values under different schemes. By extending this work,Li et al. (2007) divide dairy foods into four types accordingto stock-up characteristics and consumptionlevels and run a survey about customer preferences. To mention other examples, Jayaraman et al. (2013)investigate consumer’s satisfaction and repurchase intension from “BOGOF” in Malaysia. Salvi (2013)studies the effectiveness of “BOGOF” compared to other strategies in Indian apparel retail industry. In thispaper, we aim to understand these popular strategies via quantitative models, which are scarce in the revenuemanagement literature, and derive insights to support the seller’s decision making.

Our contributions can be further specified as follows. First, we develop a dynamic programming for-mulation in order to find the seller’s optimal choices at eachsales epoch which depend on the profitabilitiesof “BOGOF” and “50% off.” It is dynamic pricing because the seller can dynamically adjust the price,say betweenp andp/2, but there is an additional option to provide “BOGOF” without altering the postedprice. Second, the common reservation price approach is extended to handle the case where there are tworandom willingness-to-pay, sayR1 for one unit andR2 for two units. Based on the analysis of the proposedmodel, some managerial insights can be obtained. Lastly, wecompare the proposed approach with existingdynamic (nonlinear) pricing strategies in an extended model setting. To sum up, our model formulates theseller’s problem as a stochastic dynamic program and offersa theoretical background to the above men-tioned empirical results. Thomas and Chrystal (2013) give another theoretical contribution by explainingschemes such as “BOGOF” and other quantity discount methodsthrough their relative utility pricing model,however our model considers this problem in a dynamic environment.

The rest of this paper is organized as follows. Relevant literatures are reviewed in the next section. Sec-tion 3 describes the setting and important modeling features. In Section 4, we analyze the optimal strategicchoices of the seller by comparing promotional schemes “BOGOF” and “50% off” with “no promotion”case. We further conduct numerical experiments in Section 5for a more complete understanding. In thefollowing section, we also run several numerical experiments for a generalized model for dynamic pricingand dynamic discounting, and examine the effectiveness of promotional schemes. Section 7 concludes.

2

2 Literature Review

Since the study of dynamic pricing for a single product and a single demand class (Gallego and van Ryzin,1994), the literature has flourished in many different modeling frameworks. One obvious extension is thecase of multiproduct with multiple customer classes as donein Gallego and van Ryzin (1997). Another workin a multiproduct setting that is somewhat relevant is Maglaras and Meissner (2006) where multiple productsrequest one unit of a single common resource. A relatively recent development is for the case where multiplecustomer classes exist for a single product. In Ding et al. (2006), the authors investigate dynamic pricingfor multiple customer classes where each customer requiresone unit of the product. However, there havebeen only a limited number of studies on multi-unit demands.Examples include Elmaghraby et al. (2008)and Wang et al. (2013). In the former, the authors look at optimal markdown designs with preannouncedprices, assuming that all participants have complete information on customer demands. On the other hand,the latter formulates a stochastic dynamic programming andstudy optimal prices for a monopolist who sellsmultiple identical, imperishable items over an infinite horizon.

There are a couple of the most recent and most relevant works which we review in more details. Onesuch paper is Levin and Nediak (2014) and they analyze optimal dynamic nonlinear pricing, includingdiscounts and premiums. More specifically, the authors investigate the pricing policy of a monopolisticfirm where customers can purchase multiple units based on their willingness-to-pay. Their model assumesseveral customer pools and attaches a pre-determined number of items to purchasing customers who arefrom the same pool. This means that customers do not make a choice between batches of different size,which is indeed pointed out in Levin and Nediak (2014). The proposed model in the current paper insteadassumes that every customer decides the number of items (up to two) to purchase after observing the postedprice and discount.

The other related paper is Lu et al. (2014) where the authors deal with a similar problem but they focuson a dual-pricing strategy which offers a unit selling priceand a quantity discount on a batch of fixed size.While presenting a joint analysis on quantity-based price differentiation and inventory control, a marketdemand is determined by a single random variable and the total market share is a deterministic function ofposted prices. An additional random variable is then introduced to derive an imperfect correlation betweenthe demands for unit-sales and for quantity-sales. In this paper, we instead consider that the demands fordifferent batch sizes are determined by their own randomness. We later show that those random variableshave some natural constraints. Furthermore, the dependence structure between reservation prices is quitegeneral, modeled via copula functions. As one consequence,the approach adopted in this paper is extendableto other promotional strategies which concern multiple products such as bundling.

Since one can consider “BOGOF” as bundling two units of the target product, there is a certain similar-ity between such promotional schemes andproduct bundling, which is also a type of nonlinear pricing formultiple products. In previous works, the focus has been on the profitability of bundling where customerbehaviors are often modeled via reservation prices. As early as in 60’s, Stigler (1963) shows that bundlingcan be profitable when reservation prices are negatively correlated. Using the same setting, Adams andYellen (1976) demonstrate the profitability of mixed bundling, a strategy that allows both single selling andbundling. Additional insights are obtained in Schmalensee(1984) who assumes that reservation prices fol-low a bivariate normal distribution in the Adams-Yellen framework. This normality assumption is weakenedin Long (1984). More recently, a general dependence structure for willingness-to-pay via copula functions

3

has been proposed in, for example, Chen and Riordan (2013) whose contribution lies in identifying suitableconditions for the profitability of bundling under different dependence structures of reservation prices. Laterin the paper, we remark on a necessary and sufficient condition for the profitability of discounting in ourproposed model. Also there are many discussios on bundling from the operations management point ofview. Since they are not essential in this paper, we simply refer the interested reader to Banciu et al. (2010),Bitran and Ferrer (2007), or Hitt and Chen (2005) and references therein for further information.

3 The Model

3.1 Problem formulation

Let us consider a seller who offers a single product to customers over a finite horizon. We assume thatthe inventory is fixed and possibly perishable. To maximize the revenue from the existing inventory, theseller chooses to offer discounts for anyone that purchasesan additional item. Among many possibilities,we focus on specific promotional strategies in this paper, namely “no promotion” (n), “BOGOF” (b), and“50% off” (f ). Possible prices are thenp or p/2. Although this consideration seems restrictive, they arewidely observed in real practices and our choice allows us toderive some analytical properties of optimalpromotional strategies and offer insights.

Nevertheless, a general setting can be given as follows. This generalized version is implemented andtested in a later section for numerical experiments. Discount decisions are fully dynamic and they arecombined with dynamic pricing as well. Specifically, the action space for pricep and discountq is given by

A ⊂ P×Q, P = {p1, . . . , pk} ⊂ R+, Q = {q1, . . . , qm} ⊂ [0, 1]

for some integersk,m. Selecting(p, q) ∈ A means that the first item is offered at the price ofp and thesecond atpq. Hence, for example, ifq = 1, then there is no discount; ifq = 0, then the additional item isfree or in other words, it is effectively the strategy “BOGOF.” Therefore, the three strategiesn, b, f for fixedunit pricep can also be written as(p, 1), (p, 0), (p/2, 1), respectively.

Variablet is time to maturity and it decreases fromt = T to t = 0, the end of the sales season. Theinventory level is denoted bys in {0, . . . , S} whereS is the initial inventory. Customers are assumed toarrive with probabilityλ at each time step. Note that this discrete time model can be understood as anapproximation to a continuous time model where customers arrivals follow a Poisson process with rateλby settingλ = λ∆t within a time interval of size∆t. Likewise, our approach can be easily extended tononhomogeneous Poisson arrivals. We also assume that customers are homogeneous in the sense that theirreservation prices are independent and identically distributed.

The most important feature in the model is how customer purchase probabilities are determined. Reser-vation price approach has been quite popular in the literature. In particular, some recent papers addresscustomer purchasing behaviors of bundled products by correlated reservation prices (Chen and Riordan,2013). However, there is a scarce source of information whenit comes to the multi-item and single productcase. One approach is to model multi-class customers who quote pre-determined batch sizes with indepen-dent Poisson arrival processes as in Levin and Nediak (2014). We do not assume separate customer poolsfor different batch sizes, but restrict the purchase amountto two for tractability.

4

Throughout the paper, we denote the purchase probabilitiesof each customer at timet by πi,t whereiis the number of items to purchase andi = 0, 1, 2. Wherever we need to specify a particular menu, we useπai,t for a ∈ A. Whenπa

i,t is independent oft, we simply writeπai . The main idea in computing purchase

probabilities is that the differences between reservationprices and offered prices determine customer choice.In more detail,πi,t’s are given as follows:

π1,t = P

(

R1,t > p,R1,t − p > R2,t − p(1 + q))

,

π2,t = P

(

R2,t > p(1 + q), R2,t − p(1 + q) > R1,t − p)

whereR1,t, R2,t stand for the reservation prices of homogeneous customers for a single item and two items,respectively.

A little thought reveals that modelingR1,t, R2,t as correlated random variables is not enough. This isbecause they have the intuitive constraint:R1,t ≤ R2,t ≤ 2R1,t. For this reason, we seek for an alternativebut equivalent expression that is better suited for our purpose, that is,

R1,t = γT−t(αX + βY ),

R2,t = γT−t(αX + 2βY ).

Consider a linear transformationx 7→

(

2 −1

−1 2

)

x, and this maps the region{(x1, x2)|0 ≤ x1 ≤ x2 ≤

2x2} to the first quadrant. Therefore, if we define(X ′, Y ′) = (2R1,t −R2,t, R2,t −R1,t), then(X ′, Y ′) is abivariate random vector with the first quadrant as its state space. Hence, any bivariate model for reservationprices can be expressed as above by setting(X ′, Y ′) = γT−t(αX, βY ). We consider(X,Y ) as the basicproduct features that induce purchase probabilities with “factor loading” parametersα, β, and the timedepreciation factorγ.

In our discrete-time model, the optimality equation is easily found to be

Vt(s) = max(p,q)∈A

{

λπ1,t(p+ Vt−1(s− 1)) + λπ2,t(p+ pq + Vt−1(s− 2)) (1)

+λπ0,tVt−1(s) + (1− λ)Vt−1(s)}

for s ≥ 2. Whens = 1, the retailer can control the price but not discount, thus

Vt(1) = maxp∈P

{

λπ1,t′p+ λ(1− π1,t

′)Vt−1(1) + (1− λ)Vt−1(1)}

whereπ1,t′ = P (R1,t > p). We setVt(0) ≡ 0 for any t. Sometimes a modified version becomes handy.If we set∆tVt(s) = Vt(s) − Vt−1(s) and∆iVt(s) = Vt(s) − Vt(s − i) for i = 1, 2, then the optimalityequation is re-written as

∆tVt(s) = max(p,q)∈A

{

λπ1,t(p−∆1Vt−1(s)) + λπ2,t(p+ pq −∆2Vt−1(s))}

.

Also, whens = 1, we get

∆tVt(1) = maxp∈P

{

λπ1,t′(p−∆1Vt−1(1))

}

.

Finally, we assume that the salvage value of the leftover items is zero, that is,V0(s) ≡ 0 for anys, meaningthat the product is not costly to dispose at the end. This is a mild assumption that can be easily relaxed.

5

Remark 1 It is a simple matter to check that the optimal value functionVt(s) is monotone in boths andt.Intuitively, when the inventory level is ats+ 1, any strategy that the seller can implement with inventorys

is applicable to a subset ofs units. The extra one unit is the source of additional revenue. It is similar in thecase of the remaining time. One additional sales period provides a chance for a higher revenue. This alsocan be proved mathematically by induction.

3.2 Interpretation of model ingredients

Previously, we introduced a 2-dimensional random vector(X,Y ) as a convenient modeling scheme for thecorrelated reservation prices(R1,t, R2,t) with R1,t ≤ R2,t ≤ 2R1,t. In this subsection, we argue that thismethod allows us to view(X,Y ) as hidden product characteristics. Additionally, factor loading parametersα andβ control the level of effects ofX andY on the product. This bridges the gap between quantitativemodeling and experimental observations of purchasing behaviors of customers regarding price promotions.In fact, there are more than a million hits on Google with “BOGOF” as of Feb. 28, 2014; nevertheless, therehas not been much attempts to explain it via quantitative modeling. However, one rare example is Thomasand Chrystal (2013) where the authors explain why ”BOGOF” are so widespread using their relative utilitypricing model.

According to Miracle (1965), product characteristics can be divided into as many as 9 categories accord-ing to which one can analyze the effects of different promotional strategies. More relevant to our context,Sinha and Smith (2000) focus on two features, namely consumption level and durability of a product. Thelevel of consumption can be understood as a measure of the amount or the frequency of purchases. Thehigher the consumption level is, the more a customer is willing to pay for an additional item. This can beunderstood in our model as a high ratio ofR2,t/R1,t. And we notice that it is effectively controlled by theratio c := β/α. On the other hand, the durability of a product is explicitlymodeled by the parameterγ.One can easily imagine that the reservation prices would deteriorate quickly as the end of the sales horizonapproaches for highly perishable products.

The interesting experiments conducted in Li et al. (2007) and Sinha and Smith (2000) test how customersrespond to the promotional strategies “BOGOF” and “50% discount.” In principle, customers are better offby being offered 50% discounts for all purchased items as that provides more flexibility than the otherpromotion. However, it is not always so to the retailers. In the experiments by Li et al. (2007), customerpreferences are compared across four product categories based on consumption level and durability. Forexample, one can consider powdered milk, fresh milk, powdered cheese, and yogurt. The main messagesfrom the experiments are, first, preference differences decrease between two promotions as the consumptionlevel increases, second, there is no statistically significant indication that the durability factor affects thedifference of promotional preferences of customers. However, the preference difference is slightly moresensitive with respect to the durability factor (or stock-up level according to Li et al. (2007)) at a highconsumption level than at a low consumption level.

Figure 1 illustrates the preference differences of customers with varying parameters. Here, preferencedifference is defined as the difference between purchase probabilitiesπ1,t + π2,t of “BOGOF” and “50%discount.” In the left panel, we observe that this difference decreases asc increases. This is consistent withthe first experimental observation described in the previous paragraph. In the right panel,γ refers to the

6

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

0.25

c=β/α

Pre

fere

nce d

iffe

ren

ce

(a)

0.8 0.85 0.9 0.95 1

0

0.05

0.1

0.15

0.2

0.25

γ

Pre

fere

nce d

iffe

ren

ce

c=3.5 (High consumption level)

c=0.5 (Low consumption level)

(b)

Figure 1: Preference differences between “BOGOF” versus “50% off”: X,Y are bivariate normal withmean 5, variance 1, correlation−0.7, andα+ β = 4.

durability factor so that it indicates that the products aremore durable asγ increases. Then, for each fixedconsumption level, we observe that the preference difference does not vary much compared to the differenceinduced byc at least for the parameter values considered. This is consistent with the second experimentalobservation. In the figure,X andY are assumed to be bivariate normal.

4 Analysis of Promotional Strategies

Recall that we consider three policiesA = {b, f, n}. We aim to understand the seller’s optimal choices atdifferent combinations of the inventory levels and the remaining timet. It is assumed thatγ = 1 in thissection. We start by analyzing the corresponding purchase probabilities and marginal revenues.

4.1 Purchase probabilities and marginal revenues

Sinceγ = 1, the purchase probabilities are independent oft. It is then readily checkable thatπb1 = 0 and

πb2 = P

(

αX + 2βY > p)

,

πf1 = P

(

αX + βY > p/2 > βY)

,

πf2 = P

(

2βY > p)

,

πn1 = P

(

αX + βY > p > βY)

,

πn2 = P

(

βY > p)

.

For eacha = (p, q) ∈ A, let us denote the associated total purchase probability and the marginal expectedrevenue byΠa = πa

1 + πa2 andΛa = pπa

1 + pqπa2 . These are important quantities that are directly related to

customers’ purchasing behavior. As an illustration, Figure 2 shows howΠb,Πf ,Πn differ from each other.Obviously,Πf ≥ Πb ≥ Πn. The magnitude of such a difference depends on the probability distribution of(αX, βY ) over the shaded regions.

7

(a) Πf− Πb (b) Πb

− Πn (c) Πf− Πn

Figure 2: The differences between purchase probabilities are the probabilities of(αX, βY ) belonging to theshaded regions.

In addition, simple but interesting behaviors can be shown by varying the model inputs such asα, β andp. For instance, we note thatR2/R1 = 1 + cY/(X + cY ) with c = β/α. It is clear thatlimc↓0 R2/R1 =

1 and limc↑∞R2/R1 = 2. In other words, the higher (the smaller)c is, the more (the less) customersvalue the second item on top of the first one. Such behaviors are found to depend on stock-up possibilitiesor consumption levels in the literature. In our model, we obtain the following asymptotic results. Theexpectations ofX,Y are denoted byµX andµY .

Lemma 1 Suppose thatγ = 1 and the expected value ofR1 is fixed atθ. For c = β/α, we have

1. asc increases,

limc↑∞

Πf

Πb= 1, lim

c↑∞

Πb

Πn= lim

c↑∞

Πf

Πn=

P (Y > 0.5p µY /θ)

P (Y > pµY /θ);

limc↑∞

Λf

Λb= 1, lim

c↑∞

Λb

Λn= lim

c↑∞

Λf

Λn=

P (Y > 0.5p µY /θ)

2P (Y > pµY /θ);

2. asc decreases,

limc↓0

Πb

Πn= 1, lim

c↓0

Πf

Πb= lim

c↓0

Πf

Πn=

P (X > 0.5p µX/θ)

P (X > pµX/θ);

limc↓0

Λb

Λn= 1, lim

c↓0

Λf

Λb= lim

c↓0

Λf

Λn=

P (X > 0.5p µX/θ)

2P (X > pµX/θ).

The proof is omitted because it is straightforward from the observation thatα → 0 andβ → θ/µY whenc increases to infinity andθ is fixed. The case of decreasingc can be handled in a similar fashion.

Albeit simple, the above observations shed some light on relative characteristics of the strategies. If weregard purchase probabilities as one measure of product attractiveness, then “50% off” and “BOGOF” areclose in that measure and dominant over “no promotion” at high level ofc whereas “50% off” is better thanthe others at low level ofc. Hence, the parameterc should not be too small for “BOGOF” to be attractive tocustomers compared to price discounts.

As shown in the next subsection, it is marginal revenueΛ rather thanΠ that affects optimal strategicchoices for the seller. Therefore, it is desirable to have a good understanding of the properties of marginalrevenues. From Lemma 1, it still holds that “50% off” and “BOGOF” are close in their marginal revenues at

8

high c, but they are not necessarily better than “no promotion.” The critical quantity in the first statement ofthe lemma is of form0.5FY (t)/FY (2t) whereFY is the complementary cumulative distribution function ofY . Thus, the role of the probability distribution ofY becomes notable as illustrated in the examples below.

Example 1 Suppose thatY is a normal random variable truncated at0 to ensure its nonnegativity. If theunit pricep is sufficiently high, then the marginal revenue of “50% off” or “BOGOF” is greater than that of“no promotion” at highc. This can be seen in the limit

limt↑∞

0.5FY (t)

FY (2t)= lim

t↑∞

exp(

−(t− µY )/2σ2Y

)

4 exp(

−(2t− µY )2/2σ2Y

) = ∞

whereσ2Y is the variance ofY .

Example 2 An intermediate behavior can be obtained by assuming thatY is regularly varying. IfY hasindexρ ≥ 0, then by definition of regularly varying functions

limt↑∞

FY (at)

FY (t)= a−ρ ⇒ lim

t↑∞

0.5FY (t)

FY (2t)= 2ρ−1.

Hence, the relative profitability of “50% off” or “BOGOF” depends onρ in this case.

The first example shows thatΛb andΛf exceedΛn if c, p are sufficiently large andY is normallydistributed. This is relevant to the case where we havemin{Λb,Λf} > Λn in the next subsection. WhenYis regularly varying, we have a similar behavior ifρ > 1; however, its magnitude is smaller. We note that asimilar argument applies to the lowc case as well by considering the variableX.

4.2 Properties of optimal strategies

Let us now consider the seller’s problem of choosing an optimal strategy inA = {b, f, n} to maximize thesales revenue. The optimality equation is given in Section 3. When the inventorys = 1, the seller cannotimplement strategyb. Thus, we assume that the posted price is simplyp/2. This leads to the optimalityequation

Vt(1) = λπ′1p

2+ (1− λπ′

1)Vt−1(1) ⇒ Vt(1) =p

2

{

1− (1− λπ′1)

t}

where the purchase probability isπ′1 = P (R1 > p/2), which is easily seen to beΠf .

The next proposition partially characterizes optimal promotional strategies that the seller takes over theselling horizon.

Proposition 1 WithA = {b, f, n} andγ = 1, the following statements hold:

1. if Λn > Λf , then it is never optimal to choose strategyf ;

2. if Λn > Λb, then it is never optimal to choose strategyb.

Consequently, ifΛn > max{Λf ,Λb}, then it is always optimal to choose strategyn.

9

100 80 60 40 20 02

6

10

14

18

Remaining time

Inven

tory

No promotion

(a) Λf > Λn

100 80 60 40 20 02

6

10

14

18

Remaining time

Inven

tory

(b) Λb > Λn

Figure 3: Optimal strategies over the sales horizon: (a)A = {f, n}, (b)A = {b, n}.

Proof: It is helpful to explicitly write maximands in the optimality equations, sayV at (s) for a ∈ A:

V ft (s) = λπf

1 (p/2 + Vt−1(s− 1)) + λπf2 (p+ Vt−1(s− 2)) + (1− λπf

1 − λπf2 )Vt−1(s),

V bt (s) = λπb

2(p+ Vt−1(s− 2)) + (1− λπb2)Vt−1(s),

V nt (s) = λπn

1 (p+ Vt−1(s− 1)) + λπn2 (2p + Vt−1(s− 2)) + (1− λπn

1 − λπn2 )Vt−1(s).

For the first case, we note that the condition implies thatn or b is optimal att = 1 for any inventorylevel. For induction, we also note that

V nt (s)− V f

t (s) = λp(

πn1 + 2πn

2 − 0.5πf1 − πf

2

)

+ λ(

πf1 + πf

2 − πn1 − πn

2

){

Vt−1(s)− Vt−1(s− 1)}

+λ(

πf2 − πn

2

){

Vt−1(s− 1)− Vt−1(s− 2)}

.

It is obvious that the value functionVt(s) is increasing ins at any timet. Collecting the conditions inΛn > Λf , Πf ≥ Πn, andπf

2 ≥ πn2 , we can conclude that strategyf is not optimal at timet. This completes

the induction step.

For the second case, we have similarly as above

V nt (s)− V b

t (s) = λp(

πn1 + 2πn

2 − πb2

)

+ λ(

πb2 − πn

1 − πn2

){

Vt−1(s)− Vt−1(s− 1)}

+λ(

πb2 − πn

2

){

Vt−1(s − 1)− Vt−1(s− 2)}

.

Then, the same argument withΛn > Λb, Πb ≥ Πn, andπb2 ≥ πn

2 yields the result.

The last statement is a trivial consequence of these two observations.

The above result permits us to consider the following remaining alternatives:

• A = {a, n} andΛa > Λn wherea is eitherb or f ;

• A = {b, f, n} andmin{Λb,Λf} > Λn.

10

The first case witha = f resembles the classical dynamic pricing problem as the seller sets the price ateitherp or p/2, but the important difference is that each customer is allowed to buy up to two items and thusany typical analytical approaches in the literature do not apply. However, the intuition behind this one andthe classical model is the same. Whent is close to the end of the horizon, it is optimal to offer a discount(or “BOGOF”); otherwise, “no promotion” is optimal if no pressure is on the seller to enhance the sales.

Figure 3 illustrates typical outcomes in the first case. Part(a) shows the optimal strategies among{f, n}.SinceΛf > Λn, it is always optimal to take “50% off” att = 1. As the time increases, there are moreopportunities to raise revenues by setting the price high atp. Therefore, “no promotion” becomes optimal ifthe remaining time is sufficiently large. The inventory level plays a role as well because higher revenues canbe achieved from selling more items to customers by taking the price promotionf . Quite similar behaviorsare observed for the caseA = {b, n} in part (b).

Unlike the first case, the second one has a new type of trade-off between “50% off” versus “BOGOF”that is different from the conventional dynamic pricing. Wesee that the insights in the previous paragraphare still true in this situation because it is best for the seller to set the price high as long as a full sale ishighly probable. However, it is interesting how the strategies f andb play as the time and the inventorylevel vary. To focus on this interplay and trade-off and to avoid unnecessary complications that do not affectmanagerial insights, we analyze the case ofA = {f, b} in the rest of this section.

When there are only two choices “50% off” or “BOGOF,” a few words can be said from our intuition.On the one hand, it is the former that is at least as good as as the latter from the customer’s viewpoint (buyinga second item is at the customer’s discretion). On the other hand, although the seller makesp from eitherstrategy by selling two items, strategyb offers the seller another option to raise revenue especially when itis desirable to reduce the inventory at a faster rate becauseπb

2 ≥ πf2 .

The proposition below characterizes the optimal strategicchoices of the seller in the(s, t) space. Thefirst statement confirms the previous reasoning by showing that the region in which strategyb is optimalbecomes larger and larger ast approaches the sales horizon ands is greater. However, such an opportunitydisappears if “50% off” has a higher expected revenue than “BOGOF.”

Proposition 2 WithA = {b, f} andγ = 1, the following statements hold:

1. if Λb > Λf and if strategyb is optimal fort = τ, τ − 1, . . . , 1 and ats = ς − 1, ς − 2 with ς ≥ 4, thenit is optimal to choose strategyb for t = τ + 1, . . . , 1 and ats = ς;

2. if Λf > Λb, then it is always optimal to choose strategyf .

Its proof is based on induction and the initiation step is proved in the next lemma. Proofs of both claimsare deferred to the appendix.

Lemma 2 Suppose thatA = {b, f}, γ = 1, ands = 2. If Λb > Λf , then there exists a finiteτ ≥ 1 suchthat it is optimal to take strategyf for t = T, T − 1, . . . , τ + 1 and to switch to strategyb from t = τ tot = 1. If Λf > Λb, then it is always optimal to take strategyf .

An interesting implication of two propositions is that it ispossible to deduce the retailer’s optimal actionsast gets close to the end of sales horizon.

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(a) δ = 1.1

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(b) δ = 1.9

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100 80 60 40 20 0

18

14

10

6

2

delt

a

1

1.2

1.4

1.6

1.8

2

(c) sensitivity

Figure 4: Optimal strategies over the sales horizon:A = {b, f} whereδ = πf1/(π

b2 − πf

2 ).

Corollary 1 Suppose that the same policy is optimal at the inventory level 2 and 3 fort = 1, 2, . . . , τ . Then,the policy is still optimal in the region

{

(s, t)|s = 2k, t ≤ τ + k − 1 or s = 2k + 1, t ≤ τ + k − 1 wherek = 1, 2, . . .}

.

Proof: We first note that ifΛb = Λf , then all the arguments above are still valid. Next, consider the caseΛb > Λf . Whenk = 1, the statement is true by assumption. Suppose it is true up tok ≤ k0. Then, at theinventory levels2k0 and2k0+1 the policy is optimal up tot = τ +k0−1. This implies that at the inventorylevel 2(k0 + 1), it is optimal up tot = τ + k0 by Proposition 2. We can apply the same reasoning for theinventory levels2k0 + 1 and2(k0 + 1). Then the same is true at level2(k0 + 1) + 1 up tot = τ + k0. Theinduction step is done. The other caseΛb < Λf is similar hence the proof is omitted.

We note that regardless of the condition on the purchase probabilities the same policy is applied for alls at t = 1. Hence,τ ≥ 1. An immediate consequence of this fact is that a single policy dominates in theregion

∞⋃

k=1

{

(s, t)|s = 2k or s = 2k + 1, t = 1, 2, . . . , k}

.

In order to illustrate our findings, we present numerically found optimal strategies in Figure 4. Lightlyshaded regions in the first two panels show the regions in which “BOGOF” is more profitable than “50%off.” Here,δ isπf

1 /(πb2−πf

2 ) and this turns out to be a convenient measure to compare two strategies. Basedon extensive numerical experiments, it is observed that optimal strategies with the sameδ value exhibitqualitatively similar behaviors. For instance,2πb

2 is greater thanπf1 +2πf

2 if and only if δ ∈ (1, 2); if δ > 2,then “50% off” is always optimal. And whenδ ∈ (1, 2) so thatΛb > Λf , the shapes of optimal strategieslook similar to those in Figure 4. On the other hand, two strategies offer similar marginal revenues asδ

converges to 2, and the region for “50% off” is expanded asδ increases to 2. This behavior is illustrated inpart (c) which shows the values ofδ at which the optimal strategy at a given(s, t) changes from “BOGOF”to “50% off.” As expected, the seller is quick in changing optimal strategies in the lower left corner of theregion as the relative benefit ofb decreases.

Remark 2 One last comment is that the region found above can be enlarged by imposing a little morestringent condition on the purchase probabilities. For instance, suppose2πb

2 > πf1 + 3πf

2 . Then, it can be

12

shown that strategyb is optimal fort = 1, . . . , τ + 1 if it is so for t = 1, . . . , τ at s = 2. Then, a similarargument as Corollary 1 yields a larger set.

5 Model Implementation and Computational Results

5.1 Modeling dependencies

The dynamic programming (1) in the generalized model is solved by backward recursion with boundaryconditionsV0(s) = Vt(0) = 0 for all s and t. The most important factor that affects optimal strategicchoices is the purchase probabilitiesπa

1,t andπa2,t for a = (p, q) ∈ A. A widely used and flexible approach

to modeling correlated random variables is the method of copulas.

A copula is a multivariate distribution function of random variables whose supports are the unit interval[0, 1]. It has been a very useful tool as it provides a way of handlingthe dependence structure between ran-dom variables via Sklar’s Theorem. It states that for a givenjoint multivariate distribution and correspondingmarginal distributions, there is a copula function, sayC, that relates the joint and marginal distributions. In2-dimensional case, for given random variablesX,Y , their joint distribution can be written as

P

(

X ≤ x, Y ≤ y)

= C(

FX(x), FY (y))

,

whereFX , FY are the marginal distribution functions ofX andY , respectively, andC is the joint distribu-tion function of two correlated uniform random variablesU = FX(X) andV = FY (Y ).

In our experiments, we assume that the marginal distributions ofX,Y are normal distributions withsuitably chosen means and variances so thatP(X < 0 or Y < 0) is close to zero. Since this makes eachRi normally distributed and a normal distribution is light-tailed, this models the situation in which thereservation prices quickly diminish as the price hikes. We however note that copulas are flexible enough tohandle much more general cases. The adoption of copulas enables us to change the wayRi’s depend oneach other and to test the effects of such dependence structure on the top of the usual Pearson correlationcoefficient.

It is known that any copula functionC has upper and lower bounds which are called Frechet-Hoeffdingbounds. The following examples illustrate how one can compute πi,t’s usingC and examine the extremecases of those bounds.

Remark 3 Suppose thatX,Y have probability densitiesfX , fY , respectively. We further assume thatC isdifferentiable almost everywhere andα = β = γ = 1 for simplicity. Then, fora = (p, q) ∈ A, it can beshown that

πa1 = P

(

X + Y > p,X + Y − p > X + 2Y − p(1 + q))

=

∫ pq

0P

(

X > p− y|Y = y)

fX(y)dy

= FY (pq)−

∫ pq

0C2

(

FX(p− y), FY (y))

fY (y)dy,

13

πa2 = FY (pq)−

∫ p(1+q)/2

pqC2

(

FX(p(1 + q)− 2y), FY (y))

fY (y)dy

whereC2 is the derivative ofC with respect to the second argument. Here we used the fact that P(U ≤

u|V = v) = C2(u, v).

Example 3 In the remark above, we additionally imposeC(u, v) = min{u, v} which is the Frechet-Hoeffding upper bound. In this case,X,Y become comonotonic random variables. Then,C2(u, v) =

1{u>v}. For strategyb = (p, 0) andf = (p/2, 1), it is readily found that

Πb = FY (y∗),

Πf = FY (y∗∗)

wherey∗ > y∗∗ are points in(0, p/2) such thatFX(p − 2y) = FY (y) andFX(p/2 − y) = FY (y),respectively. One can computeπn

i ’s in a similar way. This observation leads us to

Λb > Λf ⇔ P

(

y∗∗ < Y ≤ y∗)

< P

(

y∗ < Y ≤ p/2)

.

Example 4 In the other extreme case, we consider the Frechet-Hoeffding lower boundC(u, v) = (u+ v−

1)+. This choice makesX,Y counter-monotonic. SinceC2(u, v) = 1{u+v>1}, straightforward computa-tions yield

Πb = P(Y ∈ D∗),

Πf = P(Y ∈ D∗∗)

whereD∗ ⊂ D∗∗ are the complements of the sets{y ∈ [0, p/2]|FX (p − 2y) + FY (y) > 1} and{y ∈

[0, p/2]|FX (p/2− y) + FY (y) > 1}, respectively. Then, it is not difficult to see that

Λb > Λf ⇔ P

(

Y ∈ D∗∗\D∗

)

< P

(

Y ∈ D∗ ∩ [0, p/2]

)

.

5.2 Numerical results for optimal strategies

Purchase probabilities and optimal strategies.Previously, we analyzed the optimal strategic choices of theseller by focusing on the trade-off between “BOGOF” and “50%off.” In this subsection we shall strengthenour understanding of optimal strategies through several numerical experiments. Let us continue to assumeγ = 1. The remaining cases that need further investigation aremin{Λb,Λf} > Λn. See Section 4.2 for adetailed description of optimal strategies in other cases.

Figure 5 shows the optimal promotional strategies over the sales horizon withT = 100 under differentpurchase probability settings but with the constraintΛb > Λf > Λn. Part (a) of Figure 5 is similar to the

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(a) (πn1 , π

n2 ) = (0, 0), δ = 1.99

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(b) (πn1 , π

n2 ) = (0.05, 0.05), δ = 1.99

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(c) (πn1 , π

n2 ) = (0.1, 0.1), δ = 1.99

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(d) (πf1 , π

f2 ) = (0.25, 0.15), δ = 1.99

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(e) (πf1 , π

f2 ) = (0.1, 0.3), δ = 1.6

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(f) (πf1 , π

f2 ) = (0.3, 0.1), δ = 1.2

Figure 5: Optimal strategies over the sales horizon whereΛb > Λf > Λn: For upper row,(πf1 , π

f2 ) =

(0.15, 0.15) and for lower row,(πn1 , π

n2 ) = (0.05, 0.025).

high delta case of Figure 4(b) as the purchase probabilitiesfor “no promotion” are set to zero. However, theregion for strategyn grows asΛn increases. This coincides with our intuition that the seller chooses a highprice if the sales horizon is sufficiently long.

A more interesting behavior is shown in the lower row of Figure 5. In this case, the probabilityΠf

is fixed at0.4 but we vary the composition ofπfi ’s and the parameterδ. Panels (d) and (f) are similar to

Figure 4, except that some regions are taken by strategyn, where the region of strategyf is larger whenδis close to 2. In part (e), we observe that “50% off” appears insome small areas when the inventory levels is an odd integer. This can be understood as the effect of the subtle trade-off between “BOGOF” and“50% off”; for instance, even if strategyb is optimal at(s, t) = (2, t0), strategyf could be beneficial at(s, t) = (3, t0) due to its flexibility. However, such an advantage quickly disappears ast decreases orsincreases.

Regarding the case ofΛf > Λb > Λn, see Figure 6. In those three panels, “BOGOF” is not present.This is not at all surprising because, according to Proposition 2, strategyf dominates strategyb if Λf > Λb.This implies that the region for strategyb would be taken over by strategyf if we addf toA in Figure 3(b).Lastly, we also see that the region for strategyf diminishes asπf

i ’s increase. This is because the seller canpostpone price promotion because largerπf

i ’s mean better sales opportunities once the promotion starts.

Effects of model parameters.We next study how the optimal strategies depend on some important modelparameters: pricep, “consumption level”c, and time depreciation factorγ. First, as shown in Figure 7,strategyb becomes more attractive asc increases at a high price. This is because the reservation priceR2 islikely to have larger values whereasE[R1] = αµX +βµY is fixed at20. HereX,Y are normally distributed

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(a) (πf1 , π

f2 ) = (0.2, 0.2)

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(b) (πf1 , π

f2 ) = (0.3, 0.3)

100 80 60 40 20 02

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10

14

18

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tory

(c) (πf1 , π

f2 ) = (0.4, 0.4)

Figure 6: Optimal strategies over the sales horizon whereΛf > Λb > Λn, (πn1 , π

n2 ) = (0.1, 0.05) and

δ = 5.

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(a) price= 35, c = 0.01

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(b) price= 35, c = 100

Figure 7: Optimal strategies with respect to model parameter c : X,Y ∼ N(5, 1) and Gaussian copula withρ = 0.5, α+ β = 4, γ = 1 andc = β/α.

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(a) γ = 1

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(b) γ = 0.99

Figure 8: Optimal strategies with respect to model parameter γ : X,Y ∼ N(5, 1) and Gaussian copula withρ = 0.5, α = β = 2 and price= 20.

with mean 5 and variance 1. A Gaussian copulaC is given by

C(u, v) = Φ2

(

Φ−1(u),Φ−1(v))

whereΦ(·), Φ2(·, ·) are the cumulative distribution functions of univariate and bivariate normal randomvariables (with correlationρ), respectively. In contrast, strategyf dominates for smallc values at a highprice, which is implied by Lemma 1. We note, however, that these behaviors are not seen at a small pricep

where purchase probabilities without any promotion are already large enough.

Second, Figure 8 compares optimal strategies when different values for the depreciation factorγ areused. Notice that the effect of a smallerγ is somewhat different from the effect of a largerp because theeffect of the former on the purchase probabilities escalates over time. As shown in the figure, “50% off”could enter the picture atγ = 0.99 although there is nof whenγ = 1. This is due to the quickly decreasingpurchase probabilities ast gets closer to zero. Most notably, the boundary between strategiesb andf is nownonlinear unlike previous figures.

Effects of dependence structure.The bivariate random vector(X,Y ) is the key ingredient of our mod-eling approach. And its distributional properties determine the reservation pricesRi’s and eventually thepurchase probabilities and marginal expected revenues. Tobetter understand the effect of the dependencestructure of(X,Y ), we first assume that(X,Y ) is bivariate normal and see the optimal strategies as thecorrelation coefficientρ varies in Figure 9.

In the upper row of the figure, scatter plots of sampled(R1, R2) values are given forρ = −0.99,−0.5

and 0. The two dotted lines represent the extreme cases ofR2 = 2R1 andR2 = R1, respectively. Thepanels in the lower row show that the optimal strategies can change drastically asρ changes (all otherparameters fixed) and this eventually leads to quite different optimal expected revenues as well. In thisparticular example,πb

i ’s are large enough and thus strategyf is nowhere optimal for anyρ ≥ 0. However,asρ becomes more negative, one can easily check that the variances ofRi’s become smaller and this yieldssmaller purchase probabilities becausep = 25 > 20 = E[R1]. This decreasingρ eventually makes strategyf optimal in some regions as shown in parts (a) and (b) of Figure9.

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70

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R2

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(a) ρ = −0.99

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70

R1

R2

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(b) ρ = −0.5

0 5 10 15 20 25 30 35 400

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40

50

60

70

R1

R2

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(c) ρ = 0

Figure 9: Distributions ofRi’s and optimal strategies with respect to model parameterρ : X,Y ∼ N(5, 1)

and Gaussian copula withα = β = 2 and price= 25

While the Pearson correlation coefficientρ tells us a lot about the behaviors of the reservation pricesand optimal choices, a more subtle dependence structure canbe considered. As a final remark in thissubsection, let us compare the optimal expected revenues iftwo other copula functions are incorporated;namely, Clayton and Gumbel:

C(u, v) =

max(

u−θclayton + v−θclayton − 1, 0)−1/θclayton

, θclayton ∈ [−1,∞)\{0};

exp

{

−[

(− lnu)θgumbel + (− ln v)θgumbel

]1/θgumbel

}

, θgumbel ∈ [1,∞).

They are known to exhibit different tail behaviors. We first match their correlation coefficient via the rela-tionships

ρ = sin(π

2τ)

, τ =2θclayton

1− θclayton, τ =

1

1− θgumbel

whereτ is Kendall’s tau. We refer the reader to Nelson (2006) for more information about copulas.

Then, optimal expected revenues are compared under three different parameter settings in Figure 10.More precisely, they-axis represents the ratio of the optimal expected revenuesfrom Clayton copula andGumbel copula and we note that a different dependence structure can yield up to 3% difference in additionto the effect of linear correlation.

18

0 0.5 1 1.5 2 2.5 31

1.001

1.002

1.003

1.004

1.005

1.006

1.007

c

RD

(a) price= 20, τ = 0.5

20 25 30 350.97

0.98

0.99

1

1.01

1.02

1.03

1.04

Price

RD

(b) c = 1, τ = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91

1.001

1.002

1.003

tau

RD

(c) price= 20, c = 1

Figure 10: Ratio of optimal expected revenues withRD = revenue(Clayton)/revenue(Gumbel) : X,Y ∼

N(5, 1) andγ = 1, c = β/α

6 Extended Model

So far we have restricted our attention to the caseA = {b, f, n}. In this section, we consider a generalizedversion of the proposed model by enlarging the set of admissible strategiesA = P × Q for some sets ofprices and discountsP andQ. Actually, the benefits of dynamic pricing or discounting are widely known.However, the interplay between pricing and discounting makes the model not amenable to mathematicalanalysis. Several experiments are conducted in order to quantify the benefits of dynamic nonlinear pricing,compared to other strategies such as static pricing, dynamic pricing, etc.

Remark 4 As mentioned in the introduction, Chen and Riordan (2013) derive conditions on the dependencestructure of reservation prices under which a small discounting is beneficial for the seller of product bun-dles. In the same spirit, we can also derive a condition on(X,Y ) such that the marginal expected revenueincreases with a small fraction of discounting.

Suppose thatX,Y have continuous distributionsFX , FY with densitiesfX , fY . We further assume thatC has second order derivatives and thatα = β = γ = 1. Then, there existsq < 1 such thatΛ(p,q) > Λ(p,1)

if and only ifp h(p) > 1

whereh(x) = fY (x)/FY (x) is the hazard rate function ofY . This implies that the seller can improve thesales via discounting (in addition to optimal pricing) at least near the end of the sales horizon if the abovecondition is satisfied.

Model setting and preliminary analysis.For our numerical experiments, we consider a bivariate normalfor (X,Y ) with ρ = 0.9 and set the inventory level at 20. The set of admissible strategies is given by

A = P×Q, P = {8, 9, . . . , 40}, Q = {1%, 2%, . . . , 100%}. (2)

Figure 11 shows optimal prices and discounts as functions oftime-to-maturity. As one can expect, opti-mal prices and discount rates tend to decrease as the time-to-maturity decreases, but with some fluctuatingbehaviors near the end of sales horizon. This finding can be explained by the relationship between priceand discount rate; often price reduction and smallq induce similar effects on customers’ purchase behaviors

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15

20

25

30

35

40

Pri

ce

Remaining time020406080100

0

0.25

0.5

0.75

1

Dis

co

un

t

Price

Discount

(a) c = 3.9

10

15

20

25

30

35

40

Pri

ce

Remaining time020406080100

0

0.25

0.5

0.75

1

Dis

co

unt

Price

Discount

(b) c = 0.1

Figure 11: Optimal price and discount rate in remaining timewith inventory level 20,X,Y ∼ N(5, 1) andGaussian copula withρ = 0.9, α+ β = 4, γ = 1 andc = β/α.

and that results in subtle differences for optimal decisions. Such dynamics are more visible for small time-to-maturities. Consequently, the monotonicity of price ordiscount rate breaks down. We further compareoptimal values according to the “consumption level”c. Optimal prices in both cases are formed at similarlevels while there is an evident difference in the movementsof discount rates. Notably, the discount rate ata low c is shaped lower overall. This coincides with our intuition that smallq should be offered to inducepurchases ifR2 is relatively low.

Effects of adaptive thinning.It is typical to assume different customer pools for different purchase sizes.If we assume Poisson arrivals for instance, then this means that such different customer arrivals are formedby fixed thinning probabilities. In contrast, the proposed model assumes that thinning happens adaptivelyaccording to the given menu(p, q). For a better understanding of this feature, we compare the optimalrevenues from the proposed model and the model analyzed in Levin and Nediak (2014).

To be specific, let us denote the arrival probabilities of customers per period for one item and two itemsby λ1 andλ2, respectively. For simplicity, we setγ = 1. Then, the optimality equation becomes

Vt(s) = max(p,q)∈A

{

λ1π1(p+ Vt−1(s− 1)) + λ2π2(p+ pq + Vt−1(s − 2)) + (1− λ1π1 − λ2π2)Vt−1(s)}

,

whereπ1 = P(

αX + βY > p)

andπ2 = P(

αX + 2βY > p + pq)

. For comparison, we apply strategiesfrom the above equation to our modeling framework and then compute the expected revenue at each decisionepoch(s, t). We repeat the same procedure for different combinations of(λ1, λ2) such thatλ1+λ2 = λ. Theaverage difference of revenues is given in Figure 12. According to this experiment, we observe that adaptivethinning makes a larger difference if the inventory size or remaining time is large, under the assumption thatcustomer’s purchase sizes are dependent on the posted menu.

Comparison with other sales strategies.We lastly examine how much improvements in profits can bebrought by the joint consideration of dynamic pricing and discounting. For that matter, we generate total 108different problem instances, and for each instance we identify optimal prices and discounts over a 100-periodhorizon under various pricing and discounting schemes. More specifically, the set{1, 2, 3} is considered forpossible values ofα, β, and the time depreciation factorγ is assumed to take a value in{0.7, 0.8, 0.9, 1}.Also, three dependence structures are incorporated, namely Clayton, Gumbel and Gaussian.

20

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tory

100 80 60 40 20 0

10

Exp

ecte

d R

even

ue D

iffe

ren

ce

50

100

150

200

250

300

350

Figure 12: Average differences of expected revenues resulting from adaptive thinning.

Table 1: Comparison of optimal revenues compared to the benchmark case of dynamic pricing with discountrates:P = {8,9, . . . ,40} andQ = {0%,1%, . . . ,100%}.

Case Average OG Max OG Min OG

Dynamic programming+ Discount rate{0,1}2.11% 2.96% 0.58%

P = {8,9, ...40}, Q = {0,1}

Dynamic programming+ No discounting8.72% 11.14% 1.94%

P = {8,9, ...40}, Q = {1}

Monotonic constraint+ Discount rate[0,1]0.04% 0.07% 0.02%

P = {8,9, ...40}, Q = {0%,1%, . . . ,100%}

Monotonic constraint+ Discount rate{0,1}5.95% 10.56% 1.94%

P = {8,9, ...40}, Q = {0,1}

Fixed Price+ No discounting36.09% 54% 27.39%

P = {20}, Q = {1}

There are six pricing and discounting schemes that we consider. The benchmark case is the case wherewe apply fully dynamic pricing and discounting together. The action space is set as in (2). Then, we comparethe resulting expected revenue with other cases where we apply fully dynamic pricing with “BOGOF” onlyor dynamic pricing without any discounting. See the first tworows in Table 1. Three columns in the tablereport the average optimality gap (OG), the maximum OG, and the minimum OG among 108 probleminstances, respectively. We notice that “BOGOF” induces 6%increase on average compared to the nodiscounting case but has 2% difference compared to the fullydynamic case. To be more realistic about pricechanges, we experimented with the constraint that prices are not allowed to increase in time. As shown inthe third and fourth rows of Table 1, such a constraint turns out not to affect expected revenue much, makingit a reasonable pricing and discounting strategy. The last row in the table reports the static pricing case, anda large difference in revenues up to 54% is observed.

21

7 Concluding Remarks

We studied a dynamic pricing problem of a seller who utilizesprice promotional schemes in a finite horizon.A specific focus was given to “buy one get one free” promotion and “50% off” discount promotion. Basedon the reservation price approach, we transformed the two-dimensional vector of reservation prices into avector(X,Y ) in the first quadrant for modeling convenience. This approach provides a straightforward yetflexible modeling framework for reservation prices with certain constraints. We then studied the optimalstrategic choices of the seller mathematically and numerically.

Important managerial insights are, first, “BOGOF” is profitable near the end of the sales horizon aslong as its marginal expected revenue exceeds those of otherschemes; otherwise, “no promotion” or “50%off” tend to dominate. Second, such a tendency is more visible for larger inventories and larger differencesin marginal expected revenues. Additional observations have been made through numerical experiments.Optimal strategic choices were found to depend on model parameters such asc, which was interpreted asconsumption level, the posted pricep, and the time depreciation factorγ. Another interesting observationwas made by studying the impact of the dependence structure of X andY . The first order effect wasexamined by varying their linear correlation. We additionally noticed that there is a more subtle, secondorder but non-negligible effect induced by different copula structures.

Extended numerical experiments provided a better understanding of the model. We first quantified theeffect of adaptive thinning. It was shown that higher revenues can be gained by incorporating the fact that acustomer’s purchase size is affected by a posted menu. Second, more general promotional schemes such as“buy one get one with discount” were analyzed as well. Total 108 test cases were generated and their corre-sponding dynamic programs were solved. Although dynamic pricing is well known to increase the revenue,the additional increase thanks to dynamic discounting was more than 8% on average. Actually dynamicimplementation of simply “buy one get one free” induced morethan 6% increase in revenue compared todynamic pricing with no discounting. This certainly confirms the popularity of “BOGO” strategies, but itseffectiveness depends on several factors such as the dependence structure of product features, the remainingsales horizon and the inventory level.

There still remains a large room for improvement. Despite the fact that the reservation price approachhas been widely accepted in the literature, empirical studies are required to make the proposed model imple-mentable in practice. This is one limitation of the present work. Understanding product characteristics andthe dependence structure of reservation prices would be particularly intriguing and challenging. Anotherlimitation is that we gave only partial analytical results.It is beyond the scope of the present work but itwould be quite interesting to analyze complete structural properties of optimal pricing and discounting aswell as the value function possibly in a more general setting. A different direction for further developmentcan be mentioned by noting that the current practices of dynamic pricing and promotions are diverse andwidespread. For example, it is quite common to observe discounted offers as well as “buy one get onefree” or “buy one get two free” for many different (but possibly highly correlated) products in the samestore. However, a systemic approach to such diversified products and promotional schemes has yet to come.Lastly, the current investigation can be extended to incorporate consumers’ strategic acts when the seller isknown to implement certain promotional strategies.

22

Acknowledgement

The authors would like to thank Prof. Tim Huh and three anonymous reviewers for their helpful comments. Kim’s

work was supported by the Basic Science Research Program through the National Research Foundation of Korea

funded by the Ministry of Education (NRF-2014R1A1A2054868).

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Appendix

Proof of Lemma 2: SupposeΛb > Λf . This assumption implies that strategyb is optimal att = 1 becauseV1(2) = max{λπb

2p, λ(πf1 p/2 + πf

2p)}. Assume that strategyb is optimal fort = 1, 2, . . . , τ . Then, theoptimality equation becomes

Vt(2) = λπb2p+ (1− λπb

2)Vt−1(2) ⇒ Vt(2)− p = (1− λπb2)(Vt−1(2)− p).

24

This yieldsVτ (2) = p−p(1−λπb2)

τ . The same equation also implies that strategyf is optimal att = τ +1

if and only if

πb2(p − Vτ (2)) < πf

1

(p

2− Vτ (2) + Vτ (1)

)

+ πf2 (p− Vτ (2))

⇔(

Πf − πb2

) Vτ (2)

p<

1

2πf1 + πf

2 − πb2 + πf

1

Vτ (1)

p

⇔1

2πf1ϕ

τ < (Πf − πb2)

(

1−1

pVτ (2)

)

(3)

whereϕ = 1− λΠf . With η = 1− λπb2, (3) is simplified to

πf1

2(Πf − πb2)

<

(

η

ϕ

)τ

.

Sinceη > ϕ, this inequality holds for a sufficiently largeτ .

Suppose that it is optimal to choose strategyf for t = τ +1, . . . , τ ′. The same policy would be optimalif (3) still holds att = τ ′ + 1 with τ replaced byτ ′. Direct computations from the optimality equation giveus the dynamics ofVt(2) so that we obtain

1

pVτ ′(2) =

ϕ

pVτ ′−1(2) + λΠf −

λπf1

2ϕτ ′−1.

To check (3), we proceed as follows:

(Πf − πb2)

(

1−1

pVτ ′(2)

)

−1

2πf1ϕ

τ ′

= (Πf − πb2)ϕ

(

1−1

pVτ ′−1(2)

)

+1

2πf1ϕ

τ ′−1(

λ(Πf − πb2)− ϕ

)

>1

2πf1ϕ

τ ′ +1

2πf1ϕ

τ ′−1(

λ(Πf − πb2)− ϕ

)

=1

2πf1ϕ

τ ′−1λ(Πf − πb2)

which is clearly positive. Inductively, we conclude that, for all t > τ , policy f is optimal.

Now supposeΛb < Λf . It is then optimal to take strategyf at t = 1. Assume that the same strategy isoptimal fort = 1, 2, . . . , τ . Then the optimality equation reads

Vt(2) = λπf1

(p

2+ Vt−1(1)

)

+ λπf2p+

(

1− λπf1 − λπf

2

)

Vt−1(2), t = 1, 2, . . . , τ

from which it follows that

Vt(2)− p =(

1− λΠf)

(Vt−1(2)− p) + λπf1

(

Vt−1(1)−p

2

)

=(

1− λΠf)

(Vt−1(2)− p)− λπf1

p

2

(

1− λΠf)t−1

.

This is again simplified toVt(2) = p

(

1− ϕt)

−p

2λπf

1 tϕt−1

25

for t = 1, 2, . . . , τ . Next, we check the optimality of strategyf , (3), att = τ + 1:

(3) ⇔1

2πf1ϕ

τ <(

Πf − πb2

)

(

ϕτ +1

2λπf

1 τϕτ−1

)

⇔1

2πf1ϕ <

(

Πf − πb2

)

(

ϕ+1

2λπf

1 τ

)

.

SinceΛf − Λb > 0, this inequality always holds. Hence, strategyf is optimal in this case.

Proof of Proposition 2: Part 1

The assumptionΛb > Λf implies that strategyb is optimal for any inventory level att = 1. We firstrecord the condition that one prefers strategyb to strategyf at timet and at the inventory levelx ≥ 2. Fromthe optimality equation, it is easy to see that

Vt(x) = max{

λπb2(p−∆2Vt−1(x)), λπ

f1

(p

2−∆1Vt−1(x)

)

+ λπf2 (p−∆2Vt−1(x))

}

+ Vt−1(x).

Then, strategyb is better than the other if and only if

πb2(p−∆2Vt−1(x)) > πf

1

(p

2−∆1Vt−1(x)

)

+ πf2 (p−∆2Vt−1(x))

⇔ (p−∆2Vt−1(x)) > δ(p

2−∆1Vt−1(x)

)

(4)

whereδ is defined asπf1/(π

b2 − πf

2 ). We note thatδ ∈ (1, 2).

Now suppose that strategyb is optimal fort = 1, 2, . . . , τ when the inventory level is ats − 2 ≥ 2 andthat the same is true for inventory levels− 1 ands. Then, the optimality equation is

Vt(s) = λπb2(p+ Vt−1(s− 2)) + (1− λπb

2)Vt−1(s), t = 1, 2, . . . , τ.

Re-writing this asVt(s)− p = (1− λπb2)(Vt−1(s)− p) + λπb

2Vt−1(s− 2) and usingV1(s) = λπb2p, we get

t−1∑

j=0

(1− λπb2)

j(Vt−j(s)− p) =t−1∑

j=0

(1− λπb2)

j+1(Vt−1−j − p) +t−1∑

j=0

(1− λπb2)

jλπb2Vt−1−j(s− 2)

⇔ Vt(s) = p− p(

1− λπb2

)t+

t−2∑

j=0

(

1− λπb2

)jλπb

2Vt−1−j(s− 2).

The same formula holds fors−1 and fors−2. Then, fort = τ we take the difference between two resultingequations froms ands− 2:

p−∆2Vτ (s) = p−

τ−2∑

j=0

(

1− λπb2

)jλπb

2∆2Vτ−1−j(s− 2)

= p(

1− λπb2

)τ−1+

τ−2∑

j=0

(

1− λπb2

)jλπb

2 (p−∆2Vτ−1−j(s− 2))

> p(

1− λπb2

)τ−1+

τ−2∑

j=0

(

1− λπb2

)jλπb

2δ(p

2−∆1Vτ−1−j(s− 2)

)

26

where the inequality follows from the assumption that (4) should hold fors − 2 and fort = 1, 2, . . . , τ . Ifwe do the same exercise fors ands− 1, then we obtain

p

2−∆1Vτ (s) =

p

2−

τ−2∑

j=0

(

1− λπb2

)jλπb

2∆1Vτ−1−j(s − 2)

=p

2

(

1− λπb2

)τ−1+

τ−2∑

j=0

(

1− λπb2

)jλπb

2

(p

2−∆1Vτ−1−j(s− 2)

)

.

Consequently, we see that

p−∆2Vτ (s) > p

(

1−δ

2

)

(

1− λπb2

)τ−1+ δ

(p

2−∆1Vτ (s)

)

.

The assumption thatδ < 2 implies (4) holds fort = τ + 1 and at the inventory levels. Thus strategyb isoptimal in that case.

Suppose that the optimality of strategyb is true only ats − 1 ands − 2 for t = 1, 2, . . . , τ . We caninductively show that strategyb is also optimal ats for t = 1, 2, . . . , τ + 1. Indeed, the fact that strategybis optimal ats for t = 1 and the above arguments yield that the retailer choosesb at s for t = 2. We can dobackward induction until we arrive att = τ + 1 for the levels.

Part 2

As in Part 1, the assumption onΛb andΛf is necessary and sufficient for the optimality of strategyf att = 1 for any inventory level greater than 1. It also implies thatδ > 2. We first consider the cases ≥ 4.Suppose that strategyf is optimal fort = 1, 2, . . . , τ at the inventory levelss − 2, s − 1, ands. Then, theoptimality equation reads

Vt(s) = λπf1

(p

2+ Vt−1(s − 1)

)

+ λπf2 (p+ Vt−1(s − 2)) + (1− λπf

1 − λπf2 )Vt−1(s)

for t = 1, 2, . . . , τ . With ϕ = 1− λΠf , we get

Vt(s)− p = ϕ (Vt−1(s)− p) + λπf1

(

Vt−1(s− 1)−p

2

)

+ λπf2Vt−1(s− 2).

It is then not difficult to obtain

Vt(s) = p(

1− ϕt)

+t−1∑

j=0

λπf1ϕ

j(

Vt−1−j(s− 1)−p

2

)

+t−2∑

j=0

λπf2ϕ

jVt−1−j(s− 2) (5)

for t = 1, 2, . . . , τ . The same formula holds at the inventory levelss − 1 ands − 2 from which we get theexpressions for∆iVt(s) in terms of∆iVt−1−j(s − 1) and∆iVt−1−j(s − 2). In a straightforward manner,the next equalities follow:

p−∆2Vτ (s) =

τ−2∑

j=0

{

λπf1ϕ

j (p−∆2Vτ−1−j(s− 1)) + λπf2ϕ

j (p−∆2Vτ−1−j(s− 2))}

+p− p

τ−2∑

j=0

ϕjλΠf ,

27

δ(p

2−∆1Vτ (s)

)

=

τ−2∑

j=0

{

λπf1ϕ

jδ(p

2−∆1Vτ−1−j(s− 1)

)

+ λπf2ϕ

jδ(p

2−∆2Vτ−1−j(s − 2)

)}

+δp

2− δ

p

2

τ−2∑

j=0

ϕjλΠf .

The optimality of strategyf for inventory levelss− 1 ands− 2 and fort = 1, 2, . . . , τ implies that theinequality (4) is reversed forx = s− 1, s− 2 andt = 1, 2, . . . , τ . This observation withδ > 2 allows us toconclude that

p−∆2Vτ (s) < δ(p

2−∆1Vτ (s)

)

.

Therefore, it is optimal for the retailer to choose strategyf at the levels for t = τ + 1.

Now suppose that strategyf is optimal ats− 1 ands− 2 for t = 1, 2, . . . , τ . Then, a similar argumentas in Part 1 is applicable and we can confirm the optimality off for t = 1, 2, . . . , τ + 1. Since we provedthe optimality off for everyt at s = 2 in Lemma 2, it is enough to show the same is true ats = 3. Thedesired result then follows from induction.

Let us fixs = 3 and suppose that policyf is optimal up tot. To simplify notation, we writex andy forλπf

1 andλπf2 , respectively. The equation (5) implies

Vt(3) = p(1− ϕt)−p

2

x

x+ y(1− ϕt) +

t−2∑

j=0

ϕj{

xVt−1−j(2) + yVt−1−j(1)}

= p(1− ϕt)x+ 2y

2(x + y)+ p

(

x+y

2

)

(

1− ϕt−1

1− ϕ− (t− 1)ϕt−1

)

−p

2x2

t(t− 1)

2ϕt−2

=3

2p− pϕt−2G(t;x, y)

where we usedVt(1) = (p/2)(1 − ϕt), Vt(2) = p(1 − ϕt) − (p/2)xtϕt−1 and introduced an auxiliaryfunction

G(t;x, y) =x+ 2y

2(x+ y)ϕ2 +

2x+ y

2(x+ y)ϕ+

(

x+y

2

)

(t− 1)ϕ +x2t(t− 1)

4.

One can easily verify that formula is correct fort = 1 as well. To prove the optimality of policyf at t+ 1,we show the inequality in (4) is reserved, which is equivalent to

G(t;x, y) −1

2ϕ2 < δ

(

G(t;x, y) − ϕ2 −1

2xtϕ

)

.

For t = 1, the left side is(1 − y/2)ϕ while the right side isδϕ/2. Sinceδ > 2, it is done. On the otherhand, we compare the derivatives of both sides with respect to t:

(

x+y

2

)

ϕ+ x2(

t

2−

1

4

)

< δ

{

(x

2+

y

2

)

ϕ+ x2(

t

2−

1

4

)}

where the inequality holds again due toδ > 2. Therefore, the condition for the optimality off holds fort+ 1. By induction,t can be arbitrarily large. The proof is complete.

28