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Chunyang Tong Sriram Dasu Information & Operations Management Marshall School of Business University of Southern California Los Angeles CA 90089 Dynamic Pricing under Strategic Consumption

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A Framework Consumers Strategic The Seller Capacitated NoYes No Yes X ( case IV) Single price is optimal (case II) Price discriminating is optimal (case I) Literature (case III) Strategic Consumers: Anticipate prices and buying strategies of other buyers

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Are Buyer’s Strategic? Empirical Evidence: “The price reduction occurrence can sometimes mean a more reliable source to come back to, time and time again.” (http://www.buyersale.com/sale_info.html) Empirical Evidence: “The price reduction occurrence can sometimes mean a more reliable source to come back to, time and time again.” (http://www.buyersale.com/sale_info.html)http://www.buyersale.com/sale_info.html Experimental Evidence: Posted price market buyers withhold demand (Ruffle, 2000) Experimental Evidence: Posted price market buyers withhold demand (Ruffle, 2000)

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Problem Setting A Seller (risk-neutral monopolist) has K units of product to sell in finite horizon A Seller (risk-neutral monopolist) has K units of product to sell in finite horizon A pool of N risk-neutral consumers with heterogeneous valuation. Commonly known is the cdf G(v), A pool of N risk-neutral consumers with heterogeneous valuation. Commonly known is the cdf G(v), Single-unit demand per consumer Single-unit demand per consumer Consumers have anticipation of future prices and maximize their expected surplus Consumers have anticipation of future prices and maximize their expected surplus Excess demand is resolved via proportional rationing (inefficient rationing) mechanism Excess demand is resolved via proportional rationing (inefficient rationing) mechanism Since initial capacity K is exogenously given, cost is treated as sunk Since initial capacity K is exogenously given, cost is treated as sunk Period 2Period 1

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Pricing Schemes & Information Structures Two pricing schemes: Two pricing schemes: Upfront pricing and contingent pricing Two information structures: Two information structures: Common posteriors and common priors Common priors: buyer has a conditional probability distribution based on his own valuation and a common prior

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Uncapacitated Seller, Strategic Consumers Upfront Pricing Scheme ( P 2, P 1 ) Since consumers face no risk of stock-out, they simply choose min(P 2, P 1 ) Single-pricing Optimal Randomized Pricing ( P 2, F(P 1 )) Due to common knowledge of market information ( K,N,G(v)), Single Pricing Optimal

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Upfront Pricing (P 2,P 1 ), deterministic 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 ( P 2, P 1 )=(82, 20), with revenue = 82 + 20 = 102 Period 2Period 1

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Upfront Pricing Scheme, Deterministic demand Two-price scheme is optimal Price time Price P3P3 P2P2 P1P1 PcPc PcPc PcPc P3P3 P2P2 P1P1 Lemma: The optimal pricing scheme consists of two prices ( P 2, P 1 ). The clearing price P c is located in between.

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Upfront Pricing Scheme, Stochastic demand (common posteriors) Upfront Pricing Scheme, Stochastic demand (common 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 – p 2 ) = 1 (y)(y- p 1 ) where: i (y) : probability of the buyer getting the object in period i. Period 2Period 1

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A unique structure of equilibrium A unique structure of equilibrium Conjecture: Threshold is unique Conjecture: Threshold is unique Provide sufficient conditions for the threshold to be unique Provide sufficient conditions for the threshold to be unique Numerically verified that it is unique for U(0,1) Numerically verified that it is unique for U(0,1) Upfront Pricing Scheme, Stochastic demand (common posteriors) Upfront Pricing Scheme, Stochastic demand (common posteriors)

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Computational result for uniform distribution For uniform distribution (0,1), N=5-50, K=1-(N-1), P 2 is increasing Function of y.

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Computational result for uniform distribution The more scarce the product is, the larger gap between prices

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Impact on Profitability K/N1/42/43/4 Buyers’ strategy considered Threshold value/P 2 0.7076/0.61290.6235/0.46090.5590/0.3460 Total expected revenue 0.53200.78690.8474 Ignoring buyer’s strategy Actual y*/P 2 0.9632/0.73791/0.68211/0.6555 Total expected revenue 0.35860.57250.7680 % of revenue loss 32.6%27.3%9.4% ( P 1 is fixed at 0.3 )

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Asymptotic Results Prices are monotone non-increasing Prices are monotone non-increasing Upfront pricing scheme leads to a valuation- skimming process. It is strategically equivalent to declining price auction when the number of price changes approaches infinity. Upfront pricing scheme leads to a valuation- skimming process. It is strategically equivalent to declining price auction when the number of price changes approaches infinity. When number of buyers approaches infinity, a single price ( close enough to the upper bound of valuations) almost guarantees a near-optimal profit. When number of buyers approaches infinity, a single price ( close enough to the upper bound of valuations) almost guarantees a near-optimal profit.

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Contingent Pricing Scheme (common posteriors) Contingent Pricing Scheme (common posteriors) Buyers and sellers have common knowledge Buyers and sellers have common knowledge Seller determines price based on sales in previous period Seller determines price based on sales in previous period

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Contingent Pricing Scheme (common posteriors) Consumers in period 2 will buy immediately if and only if The curves of LHS and RHS have only one crossing point y* The consumers’ equilibrium strategy is again a threshold policy P2P2 P1P1 RHS LHS y*

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Impact on Profitability – Value of Dynamic Pricing

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Contingent Pricing, Common Posterior Prices may increase Prices may increase Relative value of Dynamic Pricing depends on level of scarcity Relative value of Dynamic Pricing depends on level of scarcity Best for “moderate” levels of scarcity Best for “moderate” levels of scarcity When N is very large in the limit a single price is adequate When N is very large in the limit a single price is adequate

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Extensions of Common Posterior Case Threshold policy and approach for computing optimal prices extend to: Multiple periods Multiple periods New buyers entering each period New buyers entering each period Valuations changing over time (provided expected valuations are convex functions of time) Valuations changing over time (provided expected valuations are convex functions of time)

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Limitation of Common Posterior Model Static pricing policy is near-optimal if: Static pricing policy is near-optimal if: 1) the support of distribution is bounded; 2) Common posteriors; 3) Large number of buyers To relax the assumption of common posteriors, we can assume just common priors on distribution of distributions To relax the assumption of common posteriors, we can assume just common priors on distribution of distributions

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Common Priors Distribution on distributions Distribution on distributions (i) is the pdf for G i (v) (distribution of observed valuations) (i) is the pdf for G i (v) (distribution of observed valuations) Posterior distribution of each buyers depends on his/ her observed value Posterior distribution of each buyers depends on his/ her observed value Assumption: G i (v) G j (v), if i > j, where i and j are the observed valuations of two buyers. Assumption: G i (v) G j (v), if i > j, where i and j are the observed valuations of two buyers.

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Example of Common Priors Prior distribution: N( , p ) Prior distribution: N( , p ) Buyer with observed valuation v, believes that true mean ’ = v + , Buyer with observed valuation v, believes that true mean ’ = v + , where is N(0, e ). where is N(0, e ). The posterior distributions are: N( v The posterior distributions are: N( v where where v = v*( 2 p /( 2 e + 2 p )) + *( 2 e /( 2 e + 2 p )) = 2 e 2 p /( 2 e + 2 p ) ( The more you value the product, the more you believe others value) ( The more you value the product, the more you believe others value)

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

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Work in Progress Terminal 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

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