Chunyang TongSriram Dasu

Information & Operations Management

Marshall School of BusinessUniversity of Southern California

Los Angeles CA 90089

Dynamic Pricing under Strategic Dynamic Pricing under Strategic ConsumptionConsumption

A FrameworkA Framework

Consumers Strategic Consumers Strategic

The Seller Capacitated

No Yes

No

Yes X ( case IV)

Single priceis optimal (case II)

Price discriminatingis optimal (case I)

Literature (case III)

Strategic Consumers: Anticipate prices and buying strategies of other buyers

Are Buyer’s Strategic?Are Buyer’s Strategic? Empirical Evidence: “The price reduction  Empirical Evidence: “The price reduction 

occurrence can sometimes mean a more occurrence can sometimes mean a more reliable source to come back to, time and reliable source to come back to, time and time again.” (time again.” (http://www.buyersale.com/sale_info.htmlhttp://www.buyersale.com/sale_info.html ))

Experimental Evidence: Posted price Experimental Evidence: Posted price market buyers withhold demand (Ruffle, market buyers withhold demand (Ruffle, 2000)2000)

Problem SettingProblem Setting A Seller (risk-neutral monopolist) has A Seller (risk-neutral monopolist) has KK units of product units of product

to sell in finite horizonto sell in finite horizon A pool of A pool of NN risk-neutral consumers with heterogeneous risk-neutral consumers with heterogeneous

valuation. Commonly known is the cdf valuation. Commonly known is the cdf G(v)G(v), , Single-unit demand per consumerSingle-unit demand per consumer Consumers have anticipation of future prices and Consumers have anticipation of future prices and

maximize their expected surplusmaximize their expected surplus Excess demand is resolved via proportional rationing Excess demand is resolved via proportional rationing

(inefficient rationing) mechanism(inefficient rationing) mechanism Since initial capacity Since initial capacity KK is exogenously given, cost is is exogenously given, cost is

treated as sunktreated as sunk

Period 2 Period 1

Pricing Schemes & Information Pricing Schemes & Information StructuresStructures

Two pricing schemes: Two pricing schemes: Upfront pricing and contingent pricing Upfront pricing and contingent pricing

Two information structures: Two information structures: Common posteriors and common Common posteriors and common priorspriors

Common priors: buyer has a Common priors: buyer has a conditional probability distribution conditional probability distribution based on his own valuation and a based on his own valuation and a common priorcommon prior

Uncapacitated Seller, Uncapacitated Seller, Strategic ConsumersStrategic Consumers

Upfront Pricing Scheme ( P2, P1) Since consumers face no risk of stock-out, they simply choose

min(P2, P1)

Single-pricing Optimal

Randomized Pricing ( P2, F(P1)) Due to common knowledge of market information ( K,N,G(v)), Single Pricing Optimal

Upfront Pricing (Upfront Pricing (PP22,P,P11), ), deterministic demanddeterministic demand

With limited supply single pricing may not be optimal:

Example:

K=2, N=10, with valuation ( 100,40,35,30,28,26,25,23,21,20)

Optimal Single Price Scheme: P*=100, with revenue = 100

Two Price Scheme ( P2, P1)=(82, 20), with revenue = 82 + 20 = 102

Period 2 Period 1

Upfront Pricing Scheme, Upfront Pricing Scheme, Deterministic demandDeterministic demand

Two-price scheme is optimalTwo-price scheme is optimal

Price

time time time

Price Price

P3

P2

P1

Pc

Pc

Pc

P3

P2

P1

Lemma: The optimal pricing scheme consists of two prices ( P2, P1).The clearing price Pc is located in between.

Upfront Pricing Scheme, Upfront Pricing Scheme, Stochastic demand (common Stochastic demand (common

posteriors)posteriors)Consumers’ Symmetric Bayesian Nash Equilibrium Strategy:

Threshold Policy: Only buyers with valuation v y* will buy in the second last period. Others will defer to the last period.

y* solves the following equation: 2(y)(y – p2) = 1(y)(y- p1)

where:i(y) : probability of the buyer getting the object in period i.

Period 2 Period 1

A unique structure of equilibriumA unique structure of equilibrium

Conjecture: Threshold is uniqueConjecture: Threshold is unique Provide sufficient conditions for the Provide sufficient conditions for the

threshold to be uniquethreshold to be unique Numerically verified that it is unique Numerically verified that it is unique

for U(0,1)for U(0,1)

Upfront Pricing Scheme, Upfront Pricing Scheme, Stochastic demand (common Stochastic demand (common

posteriors)posteriors)

Computational result for Computational result for uniform distributionuniform distribution

For uniform distribution (0,1), N=5-50, K=1-(N-1), P2 is increasing Function of y.

Computational result for Computational result for uniform distributionuniform distribution

The more scarce the product is, the larger gap between prices

Impact on ProfitabilityImpact on Profitability

K/NK/N 1/41/4 2/42/4 3/43/4

Buyers’ Buyers’ strategy strategy consideredconsidered

Threshold value/PThreshold value/P22 0.7076/0.7076/

0.61290.61290.6235/0.6235/

0.46090.46090.5590/0.5590/

0.34600.3460

Total expected Total expected revenuerevenue

0.53200.5320 0.78690.7869 0.84740.8474

Ignoring Ignoring buyer’s buyer’s

strategystrategy

Actual y*/PActual y*/P22 0.9632/0.9632/

0.73790.73791/0.6821/0.68211

1/0.6551/0.65555

Total expected Total expected revenuerevenue

0.35860.3586 0.57250.5725 0.76800.7680

% of revenue % of revenue lossloss

32.6%32.6% 27.3%27.3% 9.4%9.4%( P1 is fixed at 0.3 )

Asymptotic ResultsAsymptotic Results Prices are monotone non-increasing Prices are monotone non-increasing Upfront pricing scheme leads to a Upfront pricing scheme leads to a

valuation-skimming process. It is valuation-skimming process. It is strategically equivalent to declining price strategically equivalent to declining price auction when the number of price changes auction when the number of price changes approaches infinity.approaches infinity.

  When number of buyers approaches When number of buyers approaches infinity, a single price ( close enough to the infinity, a single price ( close enough to the upper bound of valuations) almost upper bound of valuations) almost guarantees a near-optimal profit.guarantees a near-optimal profit.

Contingent Pricing Scheme Contingent Pricing Scheme (common posteriors)(common posteriors)

Buyers and sellers have common Buyers and sellers have common knowledge knowledge

Seller determines price based on Seller determines price based on sales in previous periodsales in previous period

Contingent Pricing Scheme Contingent Pricing Scheme (common posteriors)(common posteriors)

Consumers in period 2 will buy immediately if and only Consumers in period 2 will buy immediately if and only if if

K

ii

i PxiPx1

*112

2 )()Pr()(

The curves of LHS and RHS have only one crossing point y*

The consumers’ equilibrium strategy is again a threshold policy

P2P1

RHS

LHS

y*

RHSofslopeijj

Ki

K

i

iN

kj

K

i

1

111

2 )Pr()Pr()Pr(

Impact on Profitability – Value of Impact on Profitability – Value of Dynamic PricingDynamic Pricing

Valuation uniform between 0 & 1K (number of units for sale) 5 4 3 2 1N (number of participants) 10 10 10 10 10

Single Period Optimal Revenue 2.26625 2.043808 1.719728 1.281258 0.715201Two Period Optimal Revenue 2.3330371 2.184288 1.860892 1.395448 0.779945% Change 2.95% 6.87% 8.21% 8.91% 9.05%

Single Period optimal Price 0.58 0.62 0.67 0.79 0.79

Two Period Pricing

Second Last Period Price 0.57 0.62 0.693 0.768 0.838Cut off (Buying Threshold) 0.8 0.74 0.79 0.83 0.88

Last Period Price When Inventory is1 0.576 0.548 0.6004 0.6391 0.69522 0.536 0.51 0.561 0.59763 0.504 0.481 0.52934 0.48 0.465 0.464

Contingent Pricing, Common Contingent Pricing, Common PosteriorPosterior

Prices may increasePrices may increase

Relative value of Dynamic Pricing Relative value of Dynamic Pricing depends on level of scarcitydepends on level of scarcity Best for “moderate” levels of scarcityBest for “moderate” levels of scarcity When N is very large in the limit a single When N is very large in the limit a single

price is adequate price is adequate

Extensions of Common Extensions of Common Posterior CasePosterior Case

Threshold policy and approach for Threshold policy and approach for computing optimal prices extend to:computing optimal prices extend to:

Multiple periodsMultiple periods New buyers entering each periodNew buyers entering each period Valuations changing over time Valuations changing over time

(provided expected valuations are (provided expected valuations are convex functions of time)convex functions of time)

Limitation of Common Limitation of Common Posterior ModelPosterior Model

Static pricing policy is near-optimal if:Static pricing policy is near-optimal if:1) the support of distribution is 1) the support of distribution is

bounded;bounded;2) Common posteriors;2) Common posteriors;3) Large number of buyers 3) Large number of buyers

To relax the assumption of common To relax the assumption of common posteriors, we can assume just common posteriors, we can assume just common priors on distribution of distributionspriors on distribution of distributions

Common PriorsCommon Priors

Distribution on distributionsDistribution on distributions (i) is the pdf for G(i) is the pdf for Gii(v) (distribution of (v) (distribution of

observed valuations)observed valuations)

Posterior distribution of each buyers Posterior distribution of each buyers depends on his/ her observed valuedepends on his/ her observed value

Assumption: GAssumption: Gii(v) G(v) Gjj(v), if i > j, where (v), if i > j, where i and j are the observed valuations of two i and j are the observed valuations of two buyers.buyers.

Example of Common PriorsExample of Common Priors

Prior distribution: N(Prior distribution: N(, , pp)) Buyer with observed valuation Buyer with observed valuation vv, believes , believes

that true mean that true mean ’ = ’ = vv + + , , where where is N(0 , is N(0 ,ee).). The posterior distributions are: N(The posterior distributions are: N(vv

where where vv = v*( = v*(22

pp/(/(22ee++22

pp)) + )) + *(*(22ee/(/(22

ee++22pp))))

= = 22ee22

pp/(/(22ee++22

pp)) ( The more you value the product, the ( The more you value the product, the

more you believe others value)more you believe others value)

Common PriorsCommon Priors

If , then a threshold If , then a threshold policy is a symmetric Nash Equilibrium.policy is a symmetric Nash Equilibrium.

),()( vi NvG

Work in ProgressWork in ProgressTerminal consumption: uncertain valuation until the final period

Capacity Control along with pricing Seller can strategically reduce supply

Multiple unit purchase Multi-firm competition

Experimental Studies