1. Expanding the Scope of Dynamic Pricing Yuri Levin, Tanya Levin, Jeff McGill, Mikhail Nediak Queen’s University, School of Business Kingston, Ontario, Canada Sixth Annual INFORMS Revenue Management and Pricing Section Conference Columbia University June 6, 2006

2. 2 Introduction l traditional RM and Dynamic Pricing –risk neutral –service sector focus –business-to-consumer l expanded scope –retail sector –risk-sensitive managers –strategic customers –business-to-business –volatile customer behavior

3. 3 Outline retail sector risk-sensitive managers strategic customers volatile customer behavior price guarantees incorporation of loss- probability in dynamic pricing strategic customers in monopoly and oligopoly online learning of customer attributes

4. 4 Price Guarantees l very common in retail sector –internal or external –forced by ‘free returns’ policies –typically free of charge l potential benefits –customer: reduced risk of opportunity loss –company: encourage immediate purchase, improve customer satisfaction l surprisingly little prior analytical work

5. 5 Price Guarantees Model Elements finite time horizon: [0, T] finite time horizon: [0, T] l policy variables –dynamic prices: p(t) –guarantee strike price: k(t) –fee for guarantee: f(t) l demand processes –Poisson inquiry process: N(t), rate –Poisson inquiry process: N(t), rate –probability of item purchase: u[p(t), k(t), f(t), t] –item purchase process: N p (t), rate u[·] –probability buy guarantee: v[p(t), k(t), f(t), t] –guarantee purchase process: N f (t)

6. 6 Price Guarantees Objective Function Maximize expected revenues due to item sales plus revenue due to sales of price guarantees minus losses due to compensation payments

7. 7 Price Guarantees Approach l motivated by continuous time formulation l discrete-time analogue l nonlinear programming approach to solution l structural properties of the model l lower bound heuristic l numerical experiments with discrete model –exact for small problems – NLP –lower bound heuristic for larger problems via dynamic programming

8. 8 Price Guarantees Main Results l existence of optimal policy l necessary optimality conditions via NLP formulation l intuitive monotonicity results for value function l sufficient conditions for fixed policy with price guarantees to dominate dynamic policy without price guarantees l in case of free price guarantee the demand is more sensitive to changes in price than in strike price l useful lower bound heuristic

9. 9 Price Guarantees Example - Heuristic Policies vs. Time Before First Sale

10. 10 Price Guarantees Example - Heuristic (2) States with Nontrivial Expected Guarantee Payments

11. 11 Risk in Dynamic Pricing l traditional RM models are risk-neutral –objective to maximize expected revenues at end of disposal period –appropriate for transportation and accommodation services –pricing strategies are implemented over hundreds or thousands of problem instances l not the case for other applications –major event management –‘big-ticket’ item clearance seasons

12. 12 Risk Model Elements finite time horizon: [T, 0] finite time horizon: [T, 0] initial inventory: Y T initial inventory: Y T dynamic prices: p(t) dynamic prices: p(t) l demand processes –nonhomogeneous Poisson demand process: N′(t), rate (t, p) N′(t), rate (t, p) –sales process: N(t) = min{ N′(t), Y T } N(t) = min{ N′(t), Y T }

13. 13 Risk Model Elements (2) l risk-neutral objective l loss-probability risk constraint z is desired minimum level of revenues and  0 is the minimum acceptable probability with which we want this level z is desired minimum level of revenues and  0 is the minimum acceptable probability with which we want this level l revenue process

14. 14 Risk Model Elements (3) l If  0 is varied, problem has different optimal solutions -- efficient frontier in the plane of optimal l Alternative way: solve for range of values of l penalty parameter C -- cost of not meeting the revenue target z

15. 15 Risk Approach state variable (vector): [ Y(t), R(t) ] state variable (vector): [ Y(t), R(t) ] l discrete state space [ Y(t), R(t) ] intensity-controlled, nonhomogeneous, finite-state, continuous-time Markov Chain [ Y(t), R(t) ] intensity-controlled, nonhomogeneous, finite-state, continuous-time Markov Chain l introduce randomized policies to convert to form of deterministic optimal control problem l particularly convenient form: bilinear control problem l feasible for problems of realistic size –e.g. 250 items, 25,000 time periods

16. 16 Risk Main Results l existence of optimal policy l necessary optimality conditions via optimal control l intuitive monotonicity results for value function l interesting phenomena: the price can drop following a sale l highly efficient computation produces solution for all values of initial inventory and desired level of revenue simultaneously l generalizations include: –salvage and disposal costs –extended risk-neutral horizon –moving revenue target

17. 17 Risk Example: Risk/Return Frontiers z = target revenue

18. 18 Strategic Consumers l traditional dynamic pricing models –consumer behavior myopic l strategic consumers can increase utility by timing their purchase decisions to periods with lower price l consumers aided by third party brokers l company may increase its revenues by modelling the strategic nature of consumers explicitly

19. 19 Strategic Consumers Consumer Population l most dynamic pricing models assume infinite customers or customers sampled with replacement l modeling strategic customers requires considering customers individually l in reality: –populations are finite –durable items or ticket sales, customers sampled without replacement l possible competition between customers when product supplies limited

20. 20 Strategic Consumers Model Elements l company(s) offer a perishable product in a finite number of consecutive time periods l customer population is stochastically homogeneous –random valuations exchangeable l valuation distribution known to company(s) and all customers but realizations are not l price may change with time, inventory level, and customers’ presence in the market

21. 21 Strategic Consumers Model Elements (2) l customers fully rational - can anticipate pricing policies of company(s ) l customer utility increasing in (valuation – price) surplus l utility of acquiring product in future discounted by a factor β per time step 0  β  1 is strategicity parameter β = 0 – myopic customers β = 1 – fully strategic customers

22. 22 Strategic Consumers Model Elements (3) l in each time period –company(s) announce price –customers observe individual budgets –customers express eagerness to purchase l successful purchases resolved probabilistically –one purchase per time period –probability proportional to eagerness l each customer maximizes expected present value of utility of acquiring product l company(s) choose pricing policy to maximize expected future revenues in each planning period

23. 23 Strategic Consumers Oligopoly Elements l competition among companies l consumer choice behavior: choosing between different brands –two possible choice rules –specific choice customers have to allocate eagerness towards a specific product –multiple choice customers can be equally eager to purchase several of the available products i.e. any with positive surplus i.e. any with positive surplus

24. 24 Strategic Consumers Approach l stochastic dynamic games l seek Markov-perfect equilibria l dynamic programming formulation for both customers and company(s)

25. 25 Strategic Consumers Oligopoly Case l stochastic dynamic game with asymmetric information and hierarchical equilibrium structure

26. 26 Strategic Consumers Results: Monopoly l unique equilibrium solution l optimality condition for consumers -- probability of purchase: l monotonicity results: full supply case –initial number of customers initial inventory –initial number of customers  initial inventory –β general expected future revenues linear in remaining inventory expected future revenues linear in remaining inventory price constant in remaining inventory price constant in remaining inventory –β=1 – price is decreasing in time

27. 27 Strategic Consumers Results: Monopoly (2) l general supply case, with β = 0 –revenues concave in remaining inventory –price decreasing in time –price decreasing in remaining inventory l β = 0 (myopic) case interesting since finite population dynamic pricing model not previously reported l computational procedure is tractable and efficient for realistic-size problems

28. 28 Strategic Consumers Results: Oligopoly l existence of unique Markov-perfect equilibrium –‘multiple choice’ consumer case –logconcave valuation distribution l similar optimal decision rule for customers l fewer provable structural results relative to monopoly case l also computationally feasible for realistic problem sizes

29. 29 Strategic Consumers Example (Monopoly) Price vs. Time Before First Sale

30. 30 Strategic Consumers Example (Oligopoly) each company has supply of 20 units

31. 31 Online Learning for Dynamic Pricing l dynamic pricing requires model of consumer behavior (usually parametric) –cycles of parameter estimation and optimization l desirable to have a learning method –integrates periodic updates of parameter values into pricing policy selection procedure –should be independent of particular functional form of demand model l two lines of research –demand learning with strategic customers –myopic consumers with general valuation distribution

32. 32 Strategic Online Learning Model Elements l consumer game: similar to set up of monopoly strategic consumer model except… l customers cannot compute company’s price policy –‘anticipated price’ Markov process l customer attributes to be estimated –general valuation distribution –expected (valuation – price) ‘surplus’ proxy for β and future purchase behavior proxy for β and future purchase behavior

33. 33 Strategic Online Learning Approach l based on ‘aggregating algorithm’ of Vovk(1999) –very general game-theoretic methodology for ‘machine learning’ –players: nature, advisor pool, decision-maker l in present setting, resembles Bayesian estimation –start with a prior distribution for all parameters –observe sales and price histories –use posterior parameter distribution to estimate the expected customer response, probabilities of sample paths, and expected revenues l learning unfolds over several selling horizons

34. 34 Strategic Online Learning Approach (2) l periodic updates during each horizon l approximate valuation distribution by discrete sample update –accept-reject method with bootstrap resampling –similar to selection of the fittest in genetic algorithms with likelihood as a fitness function l avoid degeneration to a few discrete values with perturbation by random walk (random mutation of solutions)

35. 35 Strategic Online Learning Approach (3) l restrict to reasonable policy class, e.g. –piecewise-constant in time –or, threshold-based in inventory, linear in time l derivative-free simulation-based numerical method for optimizing a policy in its class up to the end of the horizon l simulation by sampling parameter vectors from the posterior followed by simulation of sales paths

36. 36 Strategic Online Learning Main Results l existence and uniqueness of customer game equilibrium l monotonicity results for equilibrium customer surplus l customer response model l strategicity cannot be ignored in online learning l online learning feasible for –strategic customers –myopic customers with general valuation distribution

37. 37 Strategic Online Learning Example

38. 38 Conclusions l controlled price guarantees can protect consumer goodwill and increase revenues –special cases particularly accessible –numerical approximations required l risk in dynamic pricing can be incorporated in loss-probability form l possible to account for strategicity in both monopoly and oligopoly settings l online learning is feasible l more work required!