[5]dynamic pricing in the presence of strategic consumers...

Telecom Service/Policy5

Dynamic Pricing in the Presence of Strategic Consumers and Oligopolistic CompetitionConsumers and Oligopolistic Competition

Yuri Levin, Jeff McGill, Mikhail Nediak

20040391 Sungkyun Lee20081006 Arif l Islam20081006 Ariful Islam

1. IntroductionTelecom Service/Policy5

1. IntroductionDynamic pricing

consumers are able to track prices and available capacity

l d lExplores a modelstrategic behavior by both firms and consumers in a unified stochastic dynamic game oligopolistic firms selling perishable goodsfinite segment of consumers

1. IntroductionTelecom Service/Policy5

1. IntroductionNumerical experiment

measures the effect of strategic behavior

l iKey conclusionfirms may benefit more from limiting the information available to consumers

2. Basic Model ElementsTelecom Service/Policy5

2. Basic Model ElementsDecision period: t ∈{0, …, T-1}Firm j offers a product j at unit cost cj and price pj

Integer capacity Yj ≥ 0 → remaining capacity yj ≥ 0g p y j ≥ g p y yj ≥

N: total # of consumerss: # of consumer segmentsNr: initial segment r capacityNr: initial segment r capacity(N = ∑s

r=1Nr)A i i i i f fi dAs consumers acquire items, capacities of firms and

market segments are deployed

2. Basic Model ElementsTelecom Service/Policy5

2. Basic Model Elementsatrj: consumer perception of quality or value of the

dproductϵtrj: random variable with mean zeroj

Btrj = atrj + ϵtr : valuation of product j at time t by segment r consumerg

βr∈ [0, 1]: degree of strategic behaviorβ = 0: consumer completely disregards theβr = 0: consumer completely disregards the

possibility of future purchase (myopic)β 1 l h h hβr = 1: consumer values the current purchase the same as at any point in the future (strategic)

3. Modelling AssumptionsTelecom Service/Policy5

3. Modelling Assumptions(i) Information Availability

Perfect information: Firms and consumers have perfect knowledge of all market information (remaining capacities, market segments, and distributional and parametric characteristics)Zero information cost: Information relevant for decisions is available at no cost

(ii) RationalityP f t f i ht All k t ti i t hi ti t dPerfect foresight: All market participants are sophisticated (they anticipate the strategic behavior and compute resulting event probabilities and expected payoffs)event probabilities and expected payoffs)


3. Modelling Assumptions(iii) Demand Structure

Demand as a counting process:Aggregate demand from each segment is a counting gg g g gprocess with intensity dependent on market conditions such as time, price, and so on., p ,Each segment process is a sum of independent demand process originating from individualdemand process originating from individual consumers


3. Modelling AssumptionsShopping intensity control: Consumers respond to

k b lli h i i i imarket by controlling shopping intensities

Maximum shopping intensity: A maximumMaximum shopping intensity: A maximum probability of consumer’s eagerness to acquire a particular product It is same for all consumersparticular product. It is same for all consumers


3. Modelling Assumptions(iv) Choice Model

Intensity allocation as a choice model: Consumers allocate their shopping intensity among products according to a discrete-choice model with the outside alternative of delay in future

Valuation are known up to their distribution: Given the cons mer’s segment the state of his/her al ationthe consumer’s segment, the state of his/her valuation is known only as a distribution rather than a specific value (applies both to firms and all consumers)value (applies both to firms and all consumers)


3. Modelling AssumptionsConsumer perceptions are conditional on purchase:

f h h h di ib i f l iAfter the purchase, the distribution of valuations perceived by the consumers is the conditional

l i di ib i l i h h h dvaluation distributional given that the purchased product is preferred to any other option. Consumers

d fi f hi hif i di ib i iand firms account for this shift in distributions in their distributions in their decision making

4. The Competitive GameTelecom Service/Policy5

4. The Competitive Game(1) Firm’s Game and Consumer Response

At time t( ) i i iti f th fiy = (y1, …, ym) - remaining capacities of the firms

n = (n1, …, ns) - remaining # of consumers across segments

(y, n) - information state of firms’ decision(y n p) information state of consumer decision(y, n, p) - information state of consumer decision


4. The Competitive GameMove of the firms

p(t, y, n) = (p1(t,y,n), …, pm(t,y,n))

Sh i i t it ( f t )Shopping intensity (response of segment r consumers)λr(t,y,n,p) = (λr1(t,y,n,p), …, λrm(t,y,n,p), r = 1, …, s)

Probability that a unit of product j is sold to and individual consumer of segment rconsumer of segment r λrj(t,y,n,p)

Probability of no sale1 - ∑ n λ (t y n p)1 - ∑r,jnr λrj(t,y,n,p)


4. The Competitive GameMaximum shopping intensityλrj(t,y,n,p) ≤ λ for all t,r,j and (y,n,p)

Si f di ti tiSize of discretizationλmN (sufficiently small compared to 1: probability of more than one purchase in a given decision period is negligible)

Probability of segment r consumer choosing product j at given price p = (p p )given price p = (p1, …, pm)


4. The Competitive GameExpected surplus of a representative myopicconsumer in segment r

Certainty equivalent of a future purchase, at time t,Certainty equivalent of a future purchase, at time t,information state (y, n, p) as evaluated by segment r consumerconsumer

Qr(t, y, n, p) : certainty equivalent of a future purchase ( l f th ti f d l i h )(value of the option of delaying purchase)


4. The Competitive GameSegment r strategic consumer chooses product j with

probability P(Btr∈Arj(t, y, n, p))where

P i i 1 Th f f hProposition 1: The response of any of the nr segmentr = 1, …, s consumer in info state (y, n, p)are λrj(t,y,n,p) = λP(Btr∈Arj(t, y, n, p))


4. The Competitive Game(2) Expected Utility of a Future Purchase

Expected utility of a segment r consumer in info state (y n) and (y n p) at time t(y, n) and (y, n, p) at time tUr(t, y, n) and Ur(t, y, n, p)

At the end of planning horizonconsumers have the expected utility of 0consumers have the expected utility of 0Ur(T, y, n) = 0 for all r and (y, n)

If the firms run out of capacity, expected utilities are 0U (t 0 n) = 0 for all t r nUr(t, 0, n) 0 for all t, r, n


4. The Competitive Gamep*(t, y, n): equilibrium strategy profile Ur(t, y, n) = E[Ur(t, y, n, p*(t, y, n))]

To complete recursion, need to find Ur(t, y, n, p) as a function of expected utilities at time t+1function of expected utilities at time t+1

Gi k di i l d b l i hGiven a k dimensional vector z and by replacing thelth component z with z, we can obtain (z , z-l)


4. The Competitive GameProposition 2: The expected utility of a segment r consumer in information state (y, n, p) is

where


4. The Competitive GameValue of an Explicit Delay

For a segment r consumer at time t in state (y, n, p) the value of the option of explicitly delaying the purchase Qr(t, y, n, p) = βrUr(t+1, y, n)


4. The Competitive Game(3) Segment Demand, Firms’ Payoffs, and Equilibrium

Corollary 1: Under conditions of Proposition 2, demand intensity for product j from segment r indemand intensity for product j from segment r in information state (y, n, p) is


4. The Competitive GameEquilibrium expected payoff

Rj(t, y, p)

At th d f l i h iAt the end of planning horizonRj(T, y, n) = 0 for all (y, n)

If the firms run out of capacityR (t 0 ) 0 f ll ( ) h th t 0 0Rj(t, 0, n) = 0 for all (y, n) such that yj = 0 or n = 0


4. The Competitive GameExpected future profit of firm jRj(t, y, n, p)


4. The Competitive GameWe can rewrite Rj(t, y, n, p)

where


4. The Competitive GameAssumptions(A) Bounded Prices

The prices used are bounded by a sufficiently large p y y gconstant p

(B) Tie-breaker Mechanism(B) Tie breaker MechanismThere is a mechanism that selects an equilibrium to be implemented by the firms if G(t y n) has multiplebe implemented by the firms if G(t, y, n) has multiple equilibria. This mechanism ensures that equilibrium payoffs in G(t y n) are uniquely definedpayoffs in G(t, y, n) are uniquely defined


4. The Competitive GameTheorem 1

Under assumptions (A) and (B), there exists a Markov-perfect equilibrium in mixed strategies

5. Generalization of the Choice ModelTelecom Service/Policy5

5. Generalization of the Choice ModelGeneralized maximum shopping intensity

( ifi h i )q=1 (specific choice)

q=∞ (multiple choice)


5. Generalization of the Choice ModelAveraging Behavior

assume that a strategic consumers behaves as follows:first, a consumer determines his/her optimal shopping , p pp gintensitysecond consumer averages those intensitiessecond, consumer averages those intensities


5. Generalization of the Choice ModelProposition 3

decision period t, suppose that there exists a unique equilibrium in all subsequent periods.

Then the equilibrium response of any of the nrconsumers in segment r info state (y n p)consumers in segment r, info state (y, n, p)

(x*rj(t, y, n, p, Btr): optimal shopping intensity )


5. Generalization of the Choice Modelfor q=∞ (multiple)

x*rj(t, y, n, p, Btr) = I(Btrj - pj ≥ βrUr(t+1, y, n)),

j = 1, …, mj , ,

for q=1 (specific)for q=1 (specific)x*

rj(t, y, n, p, Btr) = I(Btr - pj∈Arj(t, y, n, p)),j = 1, …, m


5. Generalization of the Choice ModelCorollary 2

Let q = ∞ and consider decision time period t, and suppose that there exists a unique equilibrium in all subsequent time periods. Then, the demand intensity for product j from segment r in info state (y, n, p) isDrj(t, y, n, p)=nr P(Btrj – pj ≥ βrUr(t+1, y, n))

Corollary 3Under conditions of Corollary 2, expected future y , pprofit of firm j is separable in the components of p for all j = 1, …, mp j , ,


5. Generalization of the Choice ModelTheorem 2

For the case q = ∞ and under assumptions (A) and (B), there exists a Markov-perfect equilibrium in pure strategies

Theorem 3For the case q = ∞ s = 1 and for all t the distributionFor the case q ∞, s 1, and for all t, the distribution of Btr satisfies the assumptions of(C) Logconcavity: Marginal density of f (b) of each(C) Logconcavity: Marginal density of ftrj(b) of each component of Btr is logconcave, and(D) R l i Th i l di ib i f h(D) Regularity: The marginal distribution of each component of Btr satisfies the regularity condition

6. Managerial Insight from Numerical IllustrationsTelecom Service/Policy5

g g

When unit costs are negligible for each t and r, the Btrj, j i d d d id i ll di ib dj=1,…,m are independent and identically distributed according to the exponential distribution with mean arj

Some aspects of real markets can affect the results-information or waiting costinformation or waiting cost Consumer arrivals and departures without a purchaseRationing the product sales by firmsLimited information availabilityy


g g

Numerical experiments to study the effects of strategic b h i ilib i d h ff f d i ibehavior on equilibrium and the effects of deviation from equilibrium.

Percentage of revenue difference- effect of strategic g f ff gbehavior as the percentage difference in revenues between the strategic and purely myopic cases for each g p y y pfirm relative the myopic case.


g g

(1) Effects of strategic behavior

Considering 2 market segmentsβ=0 & β=1 (otherwise identical)β 0 & β 1 (otherwise identical)

Difference in percentage of revenues is much smaller in fully strategic consumers.g

Strategic effect is strongest when firms divide the market until N>50 (competition among consumers for a limited product supply reduces their expected utility

When, N<40, it has smaller effectPossible explanation- lower level of prices and revenues because of

more intense competition between the companies


g g

(2) Interplay of strategic behavior

Comparing symmetric equilibria under the multiple consumer choice assumptionconsumer choice assumption

Observation: strategic consumer behavior leads to lower revenues, and this effect is

d b haggravated by the presence of

titcompetitors.


g g

In competitive situations, the largest differences in revenue occur when the firms divide the marketwhen the firms divide the market

Reason- when consumers are myopic and the market size is increased proportionally with the number of firms there is aincreased proportionally with the number of firms there is a noticeable increase in revenues

Increases are made possible by the stochastic nature of the market because the firms have more opportunities to set higher prices when they face a market of proportionally larger size despite thewhen they face a market of proportionally larger size despite the presence of competitors

Dependency of resultOn the stochastic nature of the marketInformation availability


g g

(3) Losses from Deviating from equilibrium by wrongly i th t th iassuming that the consumers are myopic

Examining three situationsg


g g

ResultStrong firm is generally less affected by deviation than the weak firm or the firm in symmetric scenarioThe losses decrease when N increase and there is more intense competition between the consumersp

7. ConclusionTelecom Service/Policy5

7. ConclusionMost restrictive assumptions

“information availability”

i h i iTo improve the situationFirms to learn characteristics of the market as time progresses and continuously adjust the optimal pricing policyIncorporation of consumer learning will provide a more realistic representation of consumer behaviorp

ReferencesTelecom Service/Policy5

ReferencesYuri Levin, Jeff McGill, Mikhail Nediak - Dynamic Pricing in

the Presence of Strategic Consumers and Oligopolisticthe Presence of Strategic Consumers and Oligopolistic Competition

Yuri Levin Jeff McGill Mikhail Nediak Electronic CompanionYuri Levin, Jeff McGill, Mikhail Nediak - Electronic Companion – “Dynamic Pricing in the Presence of Strategic Consumers and Oligopolistic Competition”g p p

Anderson. S, A. de Palma, J.Thisse - Discrete Choice Theory of Product Differentation

[5]dynamic pricing in the presence of strategic consumers...

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