Post on 13-Jul-2015
Dynamic Pricing with a Prior on Market ResponseVivek F. Farias, Benjamin Van Roy
• Why Dynamic Pricing?
• Price as a tactical lever to influence demand.
• Traditionally to maximize revenue with changing inventory at hand
• More useful with demand learning
• What is the Paper about?
Model
• Limited Inventory
• Uncertainty about demand with learning
• Infinite time horizon
• Customers: Poisson Arrival with i.i.d.reservation price
• Plugging in HJB
• First order optimality condition for prices gives
• Assumption 1 and existence of solution
• Also for computing,
• Lemma 1. is decreasing in x (on N) and non-decreasing in . .
• Lemma 2. For all x in N, is an increasing, concave function of .
Unknown arrival rate, Prior• Prior on Arrival rate is a finite mixture of Gamma
distributions.
• Kth order mixture is parameterized by vectors and a vector of K weights that sum to unity.
• The density and expectation for such a prior is given by
• The posterior at time t is
,
Unknown Arrival Rate
• Let denote the set of states reachable from
• HJB equation for this gives
where ,
, and
Certainty Equivalent
• Each point in time computes the expected arrival rate conditioned on observed sales data.
• Known arrival rate model is then used to compute price. This solves
• Arrival uncertainty plays no role
Greedy Pricing
• A policy is said to be greedy if
• The first order condition gives the greedy price by
• Approximations to could be or
Decay Balancing
• HJB equation gives
• First order optimality condition implies
• Optimal Policy characterization
• Holding , and fixed increases as decreases.
• For a fixed inventory level , the optimal price in presence of uncertainty is higher than case when arrival rate is known.
• Approximating by the delay balancing approach chooses a policy that satisfies
• Holding , and fixed increases as decreases.
• For a fixed inventory level , the optimal price in presence of uncertainty is higher than case when arrival rate is known.
Multiple Stores and Consumer Segments
• Model with N stores and M consumer segments.
• Consumer of class j arrive according to Poisson process
• distributed according to Gamma distribution a0,j and b0,j.
• Updating process