Dynamic Pricing of Information Goods under Demand Uncertainty

Eric CopeThe Sauder School of BusinessUniversity of British Columbia

June 10, 2004 4th INFORMS Conference on Pricing and Revenue Management

Understanding demand response to price can be critical to effective pricing

• Retailer profits can be highly sensitive to price

“For a company with average economics, … a 1% improvement in price, assuming no loss of volume, increases operating profit by 11%.Improvements in price typically have three to four times the effect on profitability as proportionate increases in volume.”

(Marn and Rosiello, 1992)

• Customers’ price sensitivity is key to effective pricing

The Internet makes it easier for retailers to assess customers’ price sensitivity

• Price sensitivity research has been difficult to conduct effectively • Expensive• Time-consuming• Fear of alienating customers• Difficult to assess customer base in advance

• Price tests can be performed in e-commerce channels • Prices can be adjusted quickly and cheaply• Price information can be carefully controlled• Customer behavior can be tracked

Information goods are especially conducive to dynamic pricing

• Examples• Digital media documents

• Software products

• Subscriptions to real-time data streams

• Access to database content

• Information goods are characterized by• Low-to-zero marginal costs

• Differentiated market

• Nonperishable products

• No major issues with inventory & production capacity

Two key methodological issues in dynamic pricing

1. How to represent demand uncertainty• Nonparametric Bayesian representation of demand

• Dirichlet priors

• Priors are flexible, easy to specify and interpret

• Exact and approximate solutions for posteriors

2. How to set prices• Achieve high revenues over the sales period

• “Index function” strategies

• Exploration-exploitation trade-off

• Balance revenue vs computational efficiency

• Single-item good is for sale in each of N time periods• Price fixed during each period

• Customers have privately held reservation prices r• Will buy one unit if r ≥ p, otherwise will leave the site

• Demand λ(p) = Probability that customer buys good at p• “Conversion rate”

γ3n3p3

A simple model of an e-commerce site

Time 1 2 3 4 5 6 7 8 9Price p1 p2 p3

Revenue γn1p1 γ2n2p2

p4 p5 p6 p7 p8 p9

γ5n5p5 γ6n6p6 γ7n7p7 γ8n8p8 γ9n9p9

Demand uncertainty can be captured using Dirichlet priors

• Vendor chooses prices from a fixed set P = {z1,…,zk}

• Multinomially distributed reservation prices

• Place a Dirichlet prior over probability vector (α0,…, αk)• Conjugate prior for multinomial distribution

• Vendor cannot observe reservation prices directly

• Sales data are assumed to be binomial• Censored reservation prices

Pricez0=0 z1 z2 z3 zk

α0 α1 α2 α3 αk

The Dirichlet prior is easy to specify and interpret

• Dirichlet distribution defined by c, (β0,…, βk)

• αi is distributed as Beta (cβi, c(1 - βi))

• Mean βi, variance βi(1- βi) / (c+1)

• βi is prior expectation of αi

• c is a “certainty parameter”

• Higher values of c → lower variance of αi’s

• Demand values λ(zi) = ∑j=ik αi are also Beta

• Sample demands are a.s. decreasing

• Greater generality possible• Mixtures of Dirichlets• Dirichlet processes

Specifying the Dirichlet parameters

• Values of β should be set low• Conversion rates are typically very small (< 10%)

• High coefficient of variation of λ(p)’s for small β’s• Can be hard to set c properly

• Idea: Scale Dirichlet process so λ(p) < 10% a.s.

Pricez0=0 z1 z2 z3 zk

α0 α1 α2 α3 αk

Fixed:e.g., α0 = 0.9

Dirichlet Distribution scaled by 1-α0 : (α1,…,αk) / (1- α0) ~ D(c,β1,…,βk)

Three methods of updating the prior based on binomial sales data

1. Exact analytical formulas• Posteriors are mixtures of Dirichlets

• Mixture weights can be hard to compute

2. Gibbs sampler (Kuo & Smith, 1992)

• Possible to sample from mixture

3. “Exact observation” approximation• Observed demand per period ≈ true demand

• Posterior demands remain Beta

Few customers

Many customers

The approximate update methods produce highly accurate results

Gibbs Sampler

The approximate update methods produce highly accurate results

Exact Observation Approximation (Modified)

Optimal pricing strategies are hard to compute

• Maximize total expected discounted revenues

• Solve a dynamic program?• State space = set of distributions on (α0,…,αk)

• Far too large to solve exactly

• Find near-optimal, computationally tractable strategies

N

iii

iD pnE

1

‘Index function’ strategies are more tractable

• Figures of merit computed from the marginal distributions• Avoid working with joint distribution

• Choose the price where the index value is largest

• Index functions have been in use for some time• Dynamic Allocation Indices (Gittins, 1989)• Interval Estimation (Kaelbling, 1993)• Quantile comparison (Kushner, 1963)• Response Surface Bandits (Ginebra and Clayton, 1995)• Improvement functions (Mockus, 1989)

Characteristics of index function strategies

• Index functions usually increase with both the mean and the spread of a distribution

• Examples• Upper bound of a 100(1-a)% confidence interval for the mean

• Mean + k standard deviations• k=0: certain equivalent policy

• Can we avoid tuning parameters?

A ‘one-step lookahead’ method

• Appropriate when many customers arrive per period

• Assume that revenue r can always be obtained per period as an alternative to testing at any given price

Test p?

Stay at p?pλ(p)

rγN

ni

i

rγN

ni

i 1

)(1

ppλγN

ni

i

Yes Yes

No

No

An index function for ‘one-step lookahead’

• Largest value of r that makes you indifferent between choosing price p and the alternative that brings r.• Can be solved numerically

• Value is sensitive to number of periods remaining

• Similar to a formulation of the Gittins index

Total revenues, % of Max Achievable

60%

65%

70%

75%

80%

85%

90%

95%

100%

0 6 12 18 24 30Period

Static

Cert. Equiv.

RSB(k=0.5)

RSB(k=1)

RSB(k=2)

Simulation results using the Gibbs sampler approximation

Simulation results for the exact observation approximation

Efficiency Gap in Total Revenues

0%

5%

10%

15%

20%

25%

30%

35%

γ = 0.9, c=25 γ = 0.9, c=100 γ=0.99, c=25 γ=0.99, c=100

OSLR

IERSBCert. Eq.

Static

Index function strategies work well with nonparametric priors• Index functions favor prices with high revenue potential

• Uses marginals rather than joint distributions of demand • Marginals are readily available from Dirichlet priors

• Significant computational savings

• Performance does not appear to suffer much as a result• Demand dependencies are localized in our model

• Local demand information contained in marginals

• Parametric models often assume strong dependencies• May result in underexploration of prices

• Unjustified in most contexts

Cope, E. “Nonparametric Strategies for Dynamic Pricing in E-Commerce.” Working paper, University of British Columbia, 2004.

Ginebra, J., and M. K. Clayton, “Response Surface Bandits,” Journal of the Royal Statistical Society, Series B 57 (1995), 771—784

Gittins, J. C., Multi-Armed Bandit Allocation Indices, Wiley, 1989

Kaelbling, L. P., Learning in Embedded Systems, MIT Press, 1993

Kushner, H., “A New Method of Locating the Maximum of an Arbitrary Multipeak Curve in the Presence of Noise,” Journal of Basic Engineering 86 (1963), 97—106

Marn, M.V., and R.L. Rosiello. “Managing Price, Gaining Profit.” Harvard Business Review, September/October 1992, 84—93.

Mockus, J., Bayesian Approach to Global Optimization: Theory and Applications, Kluwer, 1989

References