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

Known Poisson Arrival Rate λ

HJB Equation

• Consider continuous-time dynamic system

• Objective to minimize a function

• : Optimal cost to go at time t and state x.

• The necessary condition is

With boundary condition

Applying to our problem

Here,

• 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

Heuristic

• Why Heuristic?

• Certainty equivalent

• Greedy Pricing

• Decay Balancing

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.

Computational Study

• Performance relative to Clairvoyant Algorithm

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

Heuristic

• Certainty equivalent

• Greedy Pricing

• Decay Balancing

Questions?