Dynamic Pricing with Demand Learning and Reference Effects

We consider a seller’s dynamic pricing problem with reference effects: the phenomenon that sales is not only influenced by the current price, but also by an expected, so-called reference price constructed in the minds of potential customers based on the seller’s price history. There is substantial empirical evidence that customers are loss averse: that means that the demand reduction when the selling price exceeds the reference price is larger than the demand increase when the selling price falls behind the reference price by the same amount. Consequently, the expected demand as a function of price has a time-varying “kink” and is not differentiable everywhere. The seller neither knows the underlying demand function nor observes the time-varying reference prices. In this setting, we design and analyze a policy that (i) changes the selling price very slowly to control the evolution of the reference prices, and (ii) gradually accumulates sales data to balance the tradeoff between learning and earning. We prove that, under a variety of reference-price updating mechanisms, our policy is asymptotically optimal; i.e., its T-period revenue loss relative to a clairvoyant who knows the demand function and the reference-price updating mechanism grows at the smallest possible rate in T. We also extend our analysis to the case of gain-seeking customers, and show that `difficulty of the learning problem’, measured by the asymptotically optimal growth rate of the regret, is parameter-dependent.

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