1. Dynamic Pricing with a Prior on
Market Response
Vivek 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?

2. Model
• Limited Inventory
• Uncertainty about demand with learning
• Infinite time horizon
• Customers: Poisson Arrival with i.i.d.
reservation price

3. Known Poisson Arrival Rate λ

4. HJB Equation
• Consider continuous-time dynamic system
• Objective to minimize a function

5. • : Optimal cost to go at time t and
state x.
• The necessary condition is
With boundary condition

6. Applying to our problem
Here,

7. • Plugging in HJB
• First order optimality condition for prices gives
• Assumption 1 and existence of solution
• Also for computing,

8. • 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 .

9. 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
,

10. Unknown Arrival Rate
• Let
denote the set of states reachable from
• HJB equation for this gives
where ,
, and

11. Heuristic
• Why Heuristic?
• Certainty equivalent
• Greedy Pricing
• Decay Balancing

12. 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

13. Greedy Pricing
• A policy is said to be greedy if
• The first order condition gives the greedy
price by
• Approximations to could be or

14. 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.

15. • 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.

16. Computational Study
• Performance relative to Clairvoyant Algorithm

17. 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

18. Heuristic
• Certainty equivalent
• Greedy Pricing
• Decay Balancing

19. Questions?