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Dynamic Pricing of Information Goods under Demand Uncertainty Eric Cope The Sauder School of Business University of British Columbia June 10, 20044 th INFORMS Conference on Pricing and Revenue Management

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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

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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

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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

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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

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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 γ3n3p3γ3n3p3 A simple model of an e-commerce site Time 1 2 3 4 5 6 7 8 9 Price p1p1 p2p2 p3p3 Revenue γn1p1γn1p1 γ2n2p2γ2n2p2 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 γ5n5p5γ5n5p5 γ6n6p6γ6n6p6 γ7n7p7γ7n7p7 γ8n8p8γ8n8p8 γ9n9p9γ9n9p9

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Demand uncertainty can be captured using Dirichlet priors Vendor chooses prices from a fixed set P = {z 1,…,z k } Multinomially distributed reservation prices Place a Dirichlet prior over probability vector (α 0,…, α k ) Conjugate prior for multinomial distribution Vendor cannot observe reservation prices directly Sales data are assumed to be binomial Censored reservation prices Price z 0 =0z1z1 z2z2 z3z3 zkzk α 0 α 1 α 2 α 3 α k

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The Dirichlet prior is easy to specify and interpret Dirichlet distribution defined by c, (β 0,…, β k ) α i is distributed as Beta (cβ i, c(1 - β i )) Mean β i, variance β i (1- β i ) / (c+1) β i is prior expectation of α i c is a certainty parameter Higher values of c lower variance of α i s Demand values λ(z i ) = j=i k α i are also Beta Sample demands are a.s. decreasing Greater generality possible Mixtures of Dirichlets Dirichlet processes

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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 βs Can be hard to set c properly Idea: Scale Dirichlet process so λ(p) < 10% a.s. Price z 0 =0z1z1 z2z2 z3z3 zkzk α 0 α 1 α 2 α 3 α k Fixed: e.g., α 0 = 0.9 Dirichlet Distribution scaled by 1-α 0 : (α 1,…,α k ) / (1- α 0 ) ~ D(c,β 1,…,β k )

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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

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The approximate update methods produce highly accurate results Gibbs Sampler

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The approximate update methods produce highly accurate results Exact Observation Approximation (Modified)

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Optimal pricing strategies are hard to compute Maximize total expected discounted revenues Solve a dynamic program? State space = set of distributions on (α 0,…,α k ) Far too large to solve exactly Find near-optimal, computationally tractable strategies

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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)

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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?

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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)pλ(p) Yes No

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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

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Simulation results using the Gibbs sampler approximation

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Simulation results for the exact observation approximation

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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

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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), 771784 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), 97106 Marn, M.V., and R.L. Rosiello. Managing Price, Gaining Profit. Harvard Business Review, September/October 1992, 8493. Mockus, J., Bayesian Approach to Global Optimization: Theory and Applications, Kluwer, 1989 References

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