Stochastic Dynamic Pricing with Strategic Customers

and Reference Price Effects

Rainer Schlosser

Hasso Plattner Institute, University of Potsdam, Potsdam, Germany

Keywords:

Dynamic Pricing, Strategic Customers, Price Anticipations, Waiting Customers, Reference Prices.

Abstract:

In many markets, customers strategically time their purchase by anticipating future prices and by checking

offer prices multiple times. Typically, customers base their decisions on historical reference prices and their

individual willingness-to-pay, which can change over time. For sellers it is challenging to derive successful

pricing strategies as customers’ reservation prices cannot be observed. In this paper, we present a stochastic

dynamic ﬁnite horizon framework to determine optimized price adjustments for selling perishable goods in the

presence of strategic customers, which (i) recur and (ii) anticipate future prices based on reference prices. We

analyze the impact of different strategic behaviors on expected proﬁts and the evolution of sales. Compared to

myopic settings, we ﬁnd that recurring customers lead to higher average prices, delayed sales, and increased

proﬁts. The presence of forward-looking customers has opposing but less intense effects.

1 INTRODUCTION

In electronic markets, it has become easy to adjust and

to observe offer prices. Customers repeatedly check

prices in order to strategically time their purchase de-

cisions. Occasionally, they even try to anticipate fu-

ture prices based on historical reference prices.

Applications can be found in a variety of contexts,

particularly in the case of perishable products or when

the sales horizon is limited. Prominent examples are,

for instance, the sale of airline tickets, event tickets,

accommodation services, fashion goods, and seasonal

products.

To derive effective dynamic pricing strategies in

the presence of strategic customers is an important

problem in revenue management. Practical relevance

is high, but the problem appears challenging. The

challenge is (i) to account for returning and forward-

looking customers, (ii) to include reference prices,

and (iii) to derive pricing decisions with acceptable

computation times.

In this paper, we study pricing strategies that take

strategic customer behavior into account when updat-

ing offer prices. Existing dynamic pricing techniques

cannot handle such scenarios efﬁciently and, hence,

force practitioners to limit the scope of their strate-

gies, e.g., by ignoring strategic behavior or by using

simple heuristics. Naturally, this limits the potential

quality of pricing strategies.

The inefﬁciency of existing techniques to account

for strategic customers stems from several factors that

reﬂect the challenges behind dynamic pricing: (i) the

solution space of strategies is enormous, (ii) a cus-

tomer’s individual willingness-to-pay is not observ-

able, and (iii) customers track the market to strategi-

cally time their buying decision.

1.1 Related Work

Selling products is a classical application of rev-

enue management theory, cf., e.g., Talluri, van Ryzin

(2004), Phillips (2005), and Yeoman, McMahon-

Beattie (2011). An excellent overview about recent

literature in the ﬁeld of dynamic pricing is given by

Chen, Chen (2015).

The analysis of the impact of strategic consumer

behavior has been studied since Coase (1972). Sur-

veys about strategic customer behavior in revenue

management are proposed by Su (2007) and Goen-

sch et al. (2013). Further, the recent survey article by

Wei, Zhang (2018) provides a very detailed overview

of publications studying strategic constumers in oper-

ations management.

Wei, Zhang (2018) distinguish three streams of

counteracting strategic customer behavior: (i) pric-

ing, (ii) inventory, and (iii) information. To account

for strategic customers via pricing aims to minimize

a customer’s incentive to wait for future price drops.

Schlosser, R.

Stochastic Dynamic Pricing with Strategic Customers and Reference Price Effects.

DOI: 10.5220/0007519401790188

In Proceedings of the 8th International Conference on Operations Research and Enterprise Systems (ICORES 2019), pages 179-188

ISBN: 978-989-758-352-0

179

This includes strategies such as ﬁxed price strategies,

pre-announced increasing prices (price commitment

strategy), or reimbursements for decreasing prices

(price-matching strategy).

The second stream seeks to mitigate strategic

waiting behavior by limiting the product availability

in order to address a customer’s concern of not being

able to purchase the product in the future (cf. run-

out). Similarly, sellers can also counteract strategic

consumer behavior by strategically announcing (or

hiding) partial inventory information to highlight the

product’s scarcity.

As typically observed in practice, in our model,

we allow for free price adjustments. On average, how-

ever, this leads to comparatively stable price paths.

These reference prices can be estimated by strate-

gic customers, cf. Wu et al. (2015). Further mod-

els focusing on reference price effects are studied by

Popescu, Wu (2007), Wu, Wu (2015), or Chenavaz,

Paraschiv (2018).

Finally, the key question is how (i) reference

prices, (ii) consumer’s price anticipations, and (iii) a

ﬁrm’s pricing strategy affect each other. Further, the

mutual dependencies will have to be determined by

buyers and sellers based on their partially observable

asymmetric (market) data. In addition, the complex

interplay of their mutual beliefs is further complicated

when multiple seller compete for the same market, cf.,

e.g., Levin et al. (2009), Liu, Zhang (2013).

1.2 Contribution

In the literature, for simplicity, mostly so-called my-

opic customers are considered. They simply arrive

and decide; they do not return to check prices again

and they do not anticipate future prices.

In our model, we consider the following two

sources of strategic customer behavior. First, we al-

low that customers return with a certain probability

in case they refuse to buy, i.e., if their willingness-

to-pay (WTP) does not exceed the current offer price.

To reﬂect planning uncertainty over time, we model a

customer’s future WTP as a random variable. Second,

we allow a certain share of customers to anticipate fu-

ture offer prices (based on the current offer price and

predetermined reference prices) as well as their indi-

vidual future WTP in order to check whether there is

an incentive to delay their purchase decision - even if

their current WTP exceeds the offer price. This allows

customers to optimize their consumer surplus.

We study a ﬁnite horizon model with limited ini-

tial inventory (i.e., products cannot be reproduced or

reordered). While in the literature demand is often

assumed to be of a special highly stylized functional

form, we allow for fairly general demand deﬁnitions.

In our model, demand is characterized by randomized

evolutions of individual WTP, which are not observ-

able for the seller. To this end, demand is allowed to

generally depend on time, the current offer price, and

reference prices.

The main contributions of this paper are the

following: We (i) present a demand model which

is based on individual reservation prices, reference

prices, and expected consumer surpluses, (ii) we com-

pute optimized feedback pricing strategies, (iii) we

study the impact of different strategic behaviors com-

pared to myopic settings, and (iv) we propose a Hid-

den Markov version of our model.

This paper is organized as follows. In Section 2,

we describe our model setup and deﬁne strategic cus-

tomer behaviors. In Section 3, we present our solution

approach and illustrate its results using different nu-

merical examples. In Section 4, we study a relaxed

version of our model with partially observable states.

Conclusions are summarized in the ﬁnal Section 5.

2 MODEL DESCRIPTION

We consider the situation in which a ﬁrm wants to sell

a ﬁnite number of goods (e.g., airline tickets, event

tickets, accommodation services) over a certain time

frame. We assume a monopoly situation. Further, a

certain ratio of customers acts strategically, i.e., they

(i) repeatedly track prices and wait for acceptable of-

fers and (ii) they are forward-looking, i.e., they com-

pare their current consumer surplus with expected fu-

ture consumer surpluses.

We assume that the time horizon T is ﬁnite. We

assume that products cannot be reproduced or re-

ordered. If a sale takes place, shipping costs c have

to be paid, c ≥ 0. A sale of one item at price a, a ≥ 0,

leads to a proﬁt of a −c. Discounting is also included

in the model. For the length of one period, we use the

discount factor δ, 0 ≤ δ ≤ 1.

2.1 Individual Buying Behavior

We consider a discrete time model with T periods.

We assume that consumers have individual (random)

reservation prices denoted by R

for periods (t,t + 1),

t = 0,1,...,T − 1. The reservation prices particularly

account for a customer’s planning uncertainty to be

able to beneﬁt from the product (e.g., an airline ticket

or an event ticket in time T ). In this context, the av-

erage planning uncertainty typically decreases over

time as the time horizon T gets closer.

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180

Further, customers may check current (ticket)

prices at multiple points in time. While some cus-

tomers start to check prices early (cf. economy class)

other customers start to check prices shortly before

the end of the sales period (cf. business class). More-

over, the evolutions of reservation prices of each in-

dividual customer are different. Figure 1 illustrates

individual reservation prices R

and average reserva-

tion prices (denoted by

) over time. We observe

that, on average, the willingness-to-pay is increasing

over time. The individual paths resemble, e.g., ran-

dom (sudden) changes of a customer’s planning un-

certainty. In other use-cases (e.g., the sale of fashion

goods) reservation prices may also tend to decrease.

0 10 20 30 40 50

100

200

300

400

Figure 1: Examples of individual reservation prices R

and

average reservation prices

(smooth blue curve) over time.

We assume that forward-looking customers com-

pare their current individual consumer surplus with

their expected future surplus of the next period. Given

a current offer price a

and an individual reservation

price R

at time t, the current consumer surplus is

:= R

− a

. The expected future surplus CS

t+1

the next period depends on the (expected) offer price

t+1

and the (expected) individual reservation price

t+1

of the consumer at time t + 1. Finally, a cus-

tomer purchases a product at time t only if condition

(i) is satisﬁed: The current consumer surplus is posi-

tive, i.e., t = 0, 1, ...,T − 1,

≥ 0 (1)

Further, for some consumer, also a second condition

(ii) has to be satisﬁed: The expected future surplus

E(CS

t+1

) does not exceed CS

plus a certain risk pre-

mium ε (e.g., mirroring a customer’s risk aversion

about a product’s future availability), so that there is

no incentive to wait for a higher consumer surplus),

t = 0, 1,...,T − 1, ε ≥ 0,

− a

≥ E(R

t+1

) − E(a

t+1

) − ε (2)

Consumers are assumed to be able to estimate fu-

ture prices E(a

t+1

), e.g., based on average historical

reference prices. In our model, we assume a known

predetermined path of reference prices denoted by,

t = 0, 1,...,T − 1,

(re f )

(3)

Based on a current offer price a

and the reference

price a

(re f )

t+1

, cf. (3), consumers can estimate expected

future prices, e.g., by, t = 0, 1,...,T − 2,

E(a

t+1

) ≈ (a

(re f )

t+1

+ a

)/2

We assume that a certain share of the consumers

are forward-looking and consider condition (2). This

share is denoted by, t = 0, 1,...,T − 1,

∈ [0,1] (4)

Finally, given the distribution of the expected evolu-

tion of a random customer’s reservation price R

, from

(1) - (3), we obtain the average purchase probabil-

ity of a random customer arriving in period (t,t + 1),

t = 0, 1,...,T , a ≥ 0, ε ≥ 0,

(buy)

(a) := (1 − γ

) · P (R

> a)

+γ

· P

> a and

− a ≥ E(R

t+1

) −

(re f )

t+1

− ε

(5)

The purchase probabilities (5) are characterized

by (i) the consumers’ mixture of reservation prices,

(ii) the share of forward-looking customers, cf. (4),

and (iii) the reference prices, cf. (3).

Note, we do not assume that reservation prices are

observable for the seller. As in practice, we only as-

sume arriving customers and realized sales to be ob-

servable fro sellers. The probabilities (5) can be esti-

mated by the conversion rate of interested and buying

customers for different offer prices at different time t,

cf., e.g., Schlosser, Boissier (2018).

2.2 Waiting Customers

Typically, customers tend to track the market and ob-

serve prices over time. In the literature, however, cus-

tomers are often assumed to be myopic, i.e., they ran-

domly occur, observe offers, and decide whether to

purchase or not. In case of no purchase they do not

further track the offer.

In reality, many customers are not myopic. In

our model, we consider recurring customers. In case

an interested customer does not purchase, we assume

that he/she checks the next period’s offer with a cer-

tain probability denoted by, t = 0, 1,...,T − 1,

∈ [0,1] (6)

Stochastic Dynamic Pricing with Strategic Customers and Reference Price Effects

181

Note, this nontrivially affects the arrival process

of potential customers. In the following, we distin-

guish between initially arriving (new) customers and

waiting/recurring (old) customers.

Arriving new customers are modelled as fol-

lows. We assume arbitrary given probabilities, t =

0,1,...,T − 1, j = 0,1,...,

(new)

( j ) (7)

that in period (t,t +1) exactly j new customers arrive,

where

∑

j≥0

(new)

( j) = 1 for all t = 0, 1, ...,T − 1.

For the time being, we assume that the number of

waiting customers can be effectively determined by

the selling ﬁrm as new arriving customers and old re-

curring customers can be observed (cf. cookies, etc.).

The random number of customers that did not pur-

chase in period (t − 1,t) and recur in the next period

(t,t + 1) are denoted by K

, t = 0,1,...,T − 1. A list

of variables and parameters is given in the Appendix,

cf. Table 5.

2.3 Problem Formulation

In our model, we use sales probabilities that depend

on (i) the number of arriving new customers j, (ii) the

number recurring waiting customers k, and (iii) the

offer price a. The individual purchase decisions, cf.

(5), are based on individual (expectations of future)

reservation prices and predetermined reference prices.

The random inventory level of the seller at time t

is denoted by X

, t = 0, 1, ..., T . The end of sale is the

random time τ, when all of the seller’s items are sold,

that is τ := min

t=0,...,T

{

t : X

= 0

}

∧T . As long as the

seller has items left to sell, for each period (t,t + 1),

a price a

has to be chosen. By A we denote the set

of admissible prices. For all remaining t ≥ τ the ﬁrm

cannot sell further items and we let a

:= 0.

We call strategies (a

)

admissible if they belong

to the class of Markovian feedback policies; i.e., pric-

ing decisions a

≥ 0 will depend on (i) time t, (ii) the

current inventory level X

, and (iii) the current number

of waiting customers K

Depending on the chosen pricing strategy (a

)

the random accumulated proﬁt from time/period t on

(discounted on time t) amounts to, t = 0, 1, ..., T ,

T −1

∑

s=t

s−t

· (a

) − c) · (X

s+1

− X

) (8)

The objective is to determine a (Markovian) feed-

back pricing policy that maximizes the expected total

discounted proﬁts, t = 0,1,...,T ,

E(G

= n,K

= k) (9)

conditioned on the current state at time t (cf. inven-

tory level n and waiting consumers k). An optimized

policy will balance expected short-term and long-term

proﬁts by accounting for the evolution of the inven-

tory level and the number of waiting customers.

3 COMPUTATION OF OPTIMAL

PRICING STRATEGIES WITH

OBSERVABLE STATES

In this section, we want to derive optimal feedback

pricing strategies that incorporate the strategic cus-

tomer behavior described in Section 2.

3.1 State Transition Probabilities

The state of the system to be controlled over time is

described by time t, the current inventory level n, and

the current number of waiting customers k. The tran-

sition dynamics can be described as follows. Given

an inventory level n at time t and a demand for i items

during the period (t,t + 1), we obtain the new inven-

tory level X

t+1

:= max(n −i, 0).

Given k waiting customers at time t and j new ar-

riving customers, cf. (7), we have k + j interested

customers in period (t,t + 1). Assuming i buying

customers, i = 0,...,k + j, cf. (1) - (5), we obtain

k + j − i customers that did not purchase an item. If

m of them plan to recur in (t + 1,t + 2) the new state,

i.e., the number of waiting customers at time t + 1 is

m, m = 0,...,k + j − i.

Assuming k waiting customers and j new cus-

tomers, the probability that exactly i items can be

sold at period (t,t + 1) is binomial distributed, i =

0,...,k + j, t = 0, 1,...,T − 1, cf. (5),

(demand)

(i|a,k, j)



k + j



· p

(buy)

(a)



1 − p

(buy)

(a)



k+ j−i

(10)

Assuming k waiting customers at time t, j new

customers, and i customers that want to buy, the prob-

ability that m of the remaining k + j −i customers re-

turn in period (t +1,t +2) is also binomial distributed,

m = 0,...,k + j − i, t = 0, 1, ...,T − 1, cf. (6),

(wait)

(m|k, j,i)



k + j − i



· η

· (1 − η

)

k+ j−i−m

(11)

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

182

3.2 Solution Approach

The problem of ﬁnding the best pricing strategy can

be solved using dynamic programming techniques. In

this context, the so-called value function describes the

best expected discounted future proﬁts E(G

|n,k) for

all possible states n and k at time t, cf. (9).

If either all items are sold or the time is up, no

future proﬁts can be made, i.e., the natural boundary

conditions for the value functions V are given by, t =

0,1,...,T − 1, n = 0,1,...,N, k = 0,1,...,

(0,k) = 0 and V

(n,k) = 0 (12)

For the remaining states, the value function is

determined by the Bellman equation, n = 0,1,...,N,

k = 0,1,...,M, t = 0,1,..., T − 1, cf. (7), (10), (11),

(n,k) = max

a∈A











∑

j=0,1,...,J

i=0,1,..., j+k

m=0,1,..., j+k−i

(new)

( j )

·p

(demand)

a, j , k ) · p

(wait)

j, k,i )

·((a − c) · min(i,n) + δ ·V

t+1

(max(n − i, 0), m))

}

(13)

Note, to obtain a bounded number of potential

events, in (13) we use a maximum number of new cus-

tomers J. To guarantee a limited state space, we use

a maximum number M of waiting customers. Both

bounds have to be chosen sufﬁciently large such that

the optimal solution is not conﬁned.

The nonlinear system of equations (13) can be

solved recursively. The associated optimal pric-

ing policy denoted by a

∗

(n,k), n = 0, 1, ...,N, k =

0,1,...,M, t = 0,1,...,T − 1, is determined by the arg

max of (13), i.e.,

∗

(n,k) = argmax

a∈A











∑

j=0,1,...,J

i=0,1,..., j+k

m=0,1,..., j+k−i

(new)

( j)

·p

(demand)

a, j , k ) · p

(wait)

j, k,i )

·((a − c) · min(i,n) + δ ·V

t+1

(max(n − i, 0), m))

}

(14)

If a

∗

(n,k) is not unique, we choose the smallest one.

3.3 Numerical Example

To illustrate our solution approach, we consider the

following numerical example.

Example 3.1. We let T = 50, N = 20, δ = 1, c =

10, J = 5, M = 8, a ∈ A := {10,20, ..., 500}, ε :=

U(0, 20), and γ

= η

= 0.5. Further, we use:

(i) Arrival of new customers, cf. (7), j = 0,1,...,

(new)

( j) =





· u

· (1 − u

)

J− j

where u

= 1 −e

−

t/T −0.6

, 0 ≤ t < T .

(ii) Individual reservation prices, cf. (5), 0 ≤ t < T ,



(min)

,t = 0

i f U (0,1) < 0.75 then R

t−1

else H

,t > 0

where L

(min)

:= U(0, 200) (initial reservation price),

(max)

:= U (100,800) (upper reservation price), and

:= L

(min)

+ D

·U(0.6,1.4) (random updates) with

:= E



(max)

− L

(min)



· (t/T )

(average increase).

By U(·,·), we denote Uniform distributions.

(iii) Reference prices, cf. (3), 0 ≤ t < T ,

(re f )

:= 150 +200 · (t/T )

(iv) We use 10 000 random realization of R

, cf.

(ii), to determine the average reservation prices

E(R

) and the conditional expectations E(R

t+1

which are the basis to derive p

(buy)

(a), a ∈ A, cf. (5).

Figure 2 depicts purchase probabilities p

(buy)

(a)

in the setting of Example 3.1 for different periods t

and prices a. Note, the seller cannot infer individual

reservation prices from p

(buy)

0 100 200 300 400 500

0.2

0.4

0.6

0.8

1.0

(buy)

(a)

t=10

t=45

Figure 2: Purchase probabilities p

(buy)

(a), cf. (5), for dif-

ferent prices a ≤ 500 and periods t = 10, 20,30,40,45; Ex-

ample 3.1.

Stochastic Dynamic Pricing with Strategic Customers and Reference Price Effects

183

Table 1 illustrates the value function V

for dif-

ferent inventory levels n and points in time t (for the

case that the number of waiting customers is k = 3),

cf. (13). We observe that expected future proﬁts are

(convex) decreasing in time and (concave) increasing

in the number of items left to sell.

Table 1: Expected proﬁts V

(n,3), Example 3.1.

n\t 0 10 20 30 40 45

1 350 350 350 350 356 356

2 654 654 654 655 655 637

5 1409 1409 1410 1410 1351 1265

10 2413 2413 2414 2383 2037 1475

15 3246 3239 3219 3035 2114 1476

20 3924 3883 3763 3230 2115 1476

Table 2 shows the expected proﬁts for different

inventory levels n and different numbers of waiting

customers k (for time t = 45). We observe that the

expected future proﬁts are increasing in the number

of waiting consumers k. The impact of k is higher the

larger the remaining inventory is.

Table 2: Expected proﬁts V

(n,k), Example 3.1.

n\k 0 1 2 3 4 5

1 288 318 339 356 369 380

2 492 552 599 637 667 693

5 744 953 1131 1265 1349 1408

10 762 1000 1238 1475 1711 1942

15 762 1000 1238 1476 1714 1952

20 762 1000 1238 1476 1714 1952

Table 3 illustrates optimal feedback prices a

∗

, cf.

(14), for different inventory levels n and points in time

t (for the case that the number of waiting customers

is k = 3), cf. Table 1. We observe that offer prices

are decreasing in n and increasing in time t. Further,

prices are (slightly) increasing in k. The impact of k

is higher if n is small and t is large.

Note, if n and t are small, optimal prices are cho-

sen such that the probability to sell is basically 0, cf.

Figure 2. This way, it is ensured that items are not

sold too early. Due to increasing demand (cf. Figure

1) items can be sold later at higher prices.

Table 3: Optimal prices a

∗

(n,3), Example 3.1.

n\t 0 10 20 30 40 45

1 200 220 280 370 410 420

2 200 220 280 330 370 380

5 200 220 250 280 280 290

10 200 200 220 230 250 290

15 180 180 190 200 250 290

20 160 160 170 190 250 290

Remark 3.1. (Properties of expected proﬁts)

(i) The expected proﬁts are increasing in the

inventory level.

(ii) If there is no discounting then the expected

proﬁts are increasing in time-to-go.

(iii) The expected proﬁts are increasing in the

number of waiting customers, especially if

time-to-go is small and the inventory is large.

Remark 3.2. (Properties of feedback prices)

(i) The optimal prices are decreasing in the

inventory level.

(ii) If demand is strongly increasing in time then

the optimal prices are increasing in the time.

(iii) The optimal prices are increasing in the

number of waiting customers, especially if

time-to-go is small and the inventory is small.

Figure 3 illustrates evaluated average prices E(a

∗

)

(orange curve), which are, on average, increasing over

time. Compared to E(a

∗

) the average reservation

prices

(blue curve) are smaller in the beginning and

higher at the end of the sales period. Hence, sales are

likely to occur late. The green curve depicts the pre-

determined reference prices a

(re f )

, which are overall

consistent with the evaluated offer prices E(a

∗

0 10 20 30 40 50

100

200

300

400

(ref )

Figure 3: Average evaluated price paths E(a

∗

) (orange

curve), average reservation prices

(blue curve), and ref-

erence prices a

(re f )

(green curve); Example 3.1.

Remark 3.3. In order to make the model fully consis-

tent, the resulting average prices obtained (cf. E(a

∗

))

can be used to deﬁne the expected reference price

paths a

(re f )

, cf. (3). This way, adaptive reconﬁgu-

rations and iterated model solutions can be used to

obtain converging equilibrium price paths where the

evaluated optimized solution (E(a

∗

)) coincides with

the underlying expectations of customers (a

(re f )

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

184

3.4 Strategic vs. Myopic Customers

In this section, we study the impact of different setups

of strategic customer behaviors, cf. (4) and (6).

Example 3.2. We assume the setting of Example 3.1.

We consider different combinations of (time consis-

tent) parameters η

= η (return probability), and γ

= γ

(share of forward-looking customers) characterizing

the customers’ strategic behavior.

Table 4: Expected total proﬁts E(G

) for different strategic

customer behaviors; Example 3.2.

Set ting η γ E(G

)

I 0 0 3 455

II 0 1 3 390

III 1 0 6 241

IV 1 1 4 940

V 0.5 0.5 3 924

Table 4 summarizes the expected proﬁts for ﬁve

different customer behaviors, cf. Example 3.2. As

a reference, setting I represents the classical myopic

customer behavior without any kind of strategic ef-

fects. In setting II, all consumers are forward-looking,

cf. (4), but do not recur. In setting III, all consumers

recur, cf. (6), but are not forward-looking. In setting

IV, all consumers are forward-looking and steadily re-

cur. Setting V corresponds to the mixed setup of Ex-

ample 3.1. Comparing the expected proﬁts in Table 4,

we observe that the strategic behavior has a signiﬁcant

impact on a seller’s sales results.

0 10 20 30 40 50

100

200

300

400

500

Figure 4: Expected evolution of optimal prices for different

strategic behaviors (settings: I blue, II orange, III green, IV

red); Example 3.2.

Figure 4 illustrates the expected price paths as-

sociated to the ﬁrst four settings I - IV, cf. Table 4.

While the increasing shape of the four paths is overall

similar, the level of prices differs signiﬁcantly. The

highest (lowest) prices correspond to setting III (set-

ting II). The moderate setting V, cf. Example 3.1 and

Figure 2, corresponds to a mixture of setting I and IV,

i.e. the blue and the red curve.

For setting I - IV, Figure 5 depicts the associated

inventory levels over time. All four curves are of con-

cave shape. Most of the sales are realized at the end

of the time horizon, which is due to the increasing de-

mand (i.e., reservation prices). The low return proba-

bility of setting I and II (blue and orange curve) leads

to sales that occur comparatively early. Instead, in

setting III and IV (green and red curve) with recurring

customers sales can be realized later and the number

of unsold items can be reduced.

0 10 20 30 40 50

Figure 5: Expected evolution of the inventory level for dif-

ferent strategic behaviors (settings: I blue, II orange, III

green, IV red); Example 3.2.

Further numerical experiments led to similar re-

sults. Finally, we summarize our observations in the

following remark.

Remark 3.4. (Impact of strategic behavior)

(i) Compared to myopic settings a higher share (η)

of patient recurring customers leads to higher prof-

its. The (predictable) higher number of interested cus-

tomers results in higher prices and delayed sales.

(ii) A higher share (γ) of forward-looking cus-

tomers leads to lower proﬁts. Customers strategical

time their purchase in order to increase their con-

sumer surplus. Further, due to suspended decision-

making, sales – that would have been realized in my-

opic settings – may even get lost. Hence, a higher

number of anticipating customers forces the seller to

lower prices which, in turn, leads to earlier sales.

(iii) Finally, the two effects, i.e., the quantities η

and γ, are counteractive. When comparing both ef-

fects, we observe that the impact of anticipating cus-

tomers is overcompensated by those of recurring cus-

tomers.

Stochastic Dynamic Pricing with Strategic Customers and Reference Price Effects

185

4 PRICING STRATEGIES WHEN

WAITING CUSTOMERS ARE

NOT OBSERVABLE

In real-life applications, the number of returning cus-

tomers is typically not exactly known. Further, it

might not always be possible to distinguish between

new and recurring customers. In this section, we show

how to adjust the model presented in Section 3 by

using probability distributions of waiting customers.

This makes it possible to still compute effective strate-

gies although less information is available.

4.1 Return Probabilities

While the number of recurring customers is often not

observable, the average return probability can be eas-

ily estimated. Based on average return probabilities it

is possible to derive a probability distribution for the

number of recurring customers.

The key idea is to exploit the model with full in-

formation, cf. Section 3, and to use probabilities for

waiting customers (cf. Hidden Markov Model).

The probability that k customers return in period

(t,t + 1) is denoted by π

(k)

:= P

= k). We can

estimate π

(k)

based on the observable number of in-

terested customers v, i.e., the sum of old and new cos-

tomers, cf. v := k + j, see Section 3.1. Assume in a

period (t − 1,t), we observed v interested customers

and i buyers. Hence, we have, cf. (11), k = 0,1,...,M,

t = 0, 1,...,T , v = 0,1,...,J + M, i = 0,1,...,v,

(k)

:= P(K

= k|v,i)



v − i



· η

· (1 − η

)

v−i−k

(15)

For all k > v − i, we obtain π

(k)

:= 0. Note, as (15)

is based on the observable number of interested (v)

and buying customers (i), we do not need to be able

to distinguish between new and old customers.

4.2 Computation of Prices

Next, we compute optimized strategies. We use given

(state) probabilities π

(k)

, cf. (15), and the value func-

tion V

(n,k), cf. (13), of the scenario with full in-

formation, see Schlosser, Richly (2018) for a similar

HMM approach. We deﬁne the following heuristic

pricing strategy denoted by ˜a

(n) for the scenario with

unobservable recurring customers, n = 0,1,...,N, k =

0,1,...,M, t = 0, 1, ...,T ,

˜a

(n) = argmax

a∈A











∑

k=0,...,M

(k)

∑

j=0,1,...,J

i=0,1,..., j+k

m=0,1,..., j+k−i

(new)

( j)

·p

(demand)

a, j , k ) · p

(wait)

j, k,i )

·((a − c) · min(i,n) + δ ·V

t+1

(max(n − i, 0), m))

}

(16)

Algorithm 4.1. Use (13), (15), and (16) in the fol-

lowing order to compute ˜a

), t = 0, 1, ...,T − 1:

(i) Compute the values V

(n,k) for all n =

0,1,...,N, k = 0,1,...,M, and t = 0,...,T via (13).

(ii) In t = 0 let π

(k)

:= 1

{k=0}

. Compute the price

˜a

(N) using (16) for the initial inventory X

:= N.

(iii) For all t = 1, ..., T − 1 observe the number v of

interested customers in period (t − 1,t) and the num-

ber i of buying customers. Given v and i compute π

(k)

k = 0, 1, ..., M, via (15). Let X

= X

t−1

− i. Use π

(k)

to compute the price ˜a

) for the current inventory

level X

, cf. (16).

4.3 Numerical Examples

In the following example, we demonstrate the appli-

cability and the quality of our Hidden Markov ap-

proach, cf. Algorithm 4.1.

Example 4.1. We assume the setting of Example 3.1.

We assume that recurring customers cannot be ob-

served. We consider η

= 0.5 and γ

= 0.5.

0 10 20 30 40 50

100

200

300

400

E(a

)

(ref )

E(a

)

Figure 6: Average evaluated price paths E( ˜a

) (blue curve),

cf. Algorithm 4.1, compared to optimal price paths E(a

∗

)

(orange curve) of the model with full information, cf. Fig.

2, and reference prices a

(re f )

(green curve); Example 4.1.

ICORES 2019 - 8th International Conference on Operations Research and Enterprise Systems

186

Figure 6 illustrates average evaluated price curves

of our heuristic strategy ˜a

compared to the opti-

mal strategy a

∗

, which, in contrast to the heuris-

tic, takes advantage of being able to observe waiting

customers. We observe that both curves are almost

identical which indicates that realized prices of both

strategies are similar.

0 10 20 30 40 50

1000

2000

3000

4000

)

E(G

)

E(G

)

Figure 7: Average accumulated proﬁts E(

| ˜a

) (blue

curve) of strategy ˜a

, cf. Algorithm 4.1, compared to av-

erage accumulated proﬁts E(

∗

) (orange curve) of the

optimal policy a

∗

of the model with full information, cf.

(14); Example 4.1.

Figure 7 shows the corresponding evolutions of

accumulated proﬁts up to time t (denoted by

). The

curves verify that the performance of the heuristic

strategy is close to optimal. For other settings of the

customer behavior we obtain similar results.

5 CONCLUSION

In this paper, we proposed a stochastic dynamic ﬁ-

nite horizon framework for sellers to determine price

adjustments in the presence of strategic customers.

Compared to classical myopic setups, we consider

customers that (i) recur and (ii) compare the current

consumer surplus against an expected future surplus,

which is based on anticipated future prices.

Given an initial inventory level our pricing strat-

egy maximizes expected proﬁts by accounting for

time-dependent demand and the number of returning

consumer, which constitute a (predictable) additional

demand potential.

The strategic behavior of the customers has a sig-

niﬁcant impact on prices and expected proﬁts. We

ﬁnd that recurring customers lead to higher average

prices, delayed sales, and most importantly higher

proﬁts. The presence of forward-looking customers

has opposing effects. However, it turns out that the

latter impact is overcompensated by the effect of re-

curring customers.

Using a Hidden Markov model (HMM), we also

consider the case in which returning customers can-

not be observed by the seller. By comparing solutions

of the extended model and the previous model which

exploits full information, we veriﬁed that the perfor-

mance of the HMM model is close to optimal.

Our framework is characterized by the average

evolution of customers’ reservation prices and the

predetermined reference prices. In future work, we

will study the case in which the reference prices are

updated by the evaluated average offer prices of the

model. To this end, the current model can be adap-

tively resolved until reference prices and evaluated

average prices are fully consistent. Alternatively, it

might also be possible to endogenize the impact of a

seller’s price adjustments on current reference prices.

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APPENDIX

Table 5: List of variables and parameters.

T time horizon / number of periods

t time / period

N initial inventory level

random number of items to sell in t

random number of waiting customers

random future proﬁts from t on

c shipping costs

δ discount factor

individual reservation price in t

average reservation price in t

(re f )

reference price in t

A set of admissible prices

n current inventory level

j number of new customers

i number of buying customers

k number of waiting customers

(n,k) value function

a offer price

individual consumer surplus in t

(buy)

(a) purchase probability for price a

(new)

( j ) probability for j new customers

(wait)

(k) probability for k waiting customers

(demand)

(i) probability for i sales in period t

∗

(n,k) optimal prices (full information)

(k)

beliefs for k waiting customers

˜a

(n) heuristic prices (HMM model)

share of waiting customers

share of anticipating customers

random accumulated proﬁts up to t

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