Introducing Information into RM to Model Market Behavior INFORMS 6th RM and Pricing Conference, Columbia University, NY Darius Walczak June 5, 2006

Outline Introduction Market Priceable demand Introducing Market Information into DP Bellman’s Equation Numerical Examples Conclusions

Introduction A simple optimization model for a single-resource control problem that includes important elements of competition A significant portion of demand buys from the seller with lowest price in the market We are optimizing from a point of view of one of the competitors Not a game-theoretic formulation, but… Inputs to the model can be obtained from such considerations.

Introduction, continued We use the concept of market priceable demand to model behavior of consumers who seek the lowest price in the market for an otherwise identical product A seller (controller) captures a portion of that of demand that depends on the state of the market The state of the market is described by an information variable We show how to include the information variable in a relatively simple DP formulation of a pricing optimization problem We present single-resource numerical examples.

Elements of the Model Introduce a new state variable, market indicator X market indicator X is a random variable with known stochastic dynamics x, a realization of X, becomes known to the controller before setting price tier for a product X can be a null Introduce a new demand type: Market Priceable Demand Dynamic Programming formulation with one resource (single-leg) and a product (seat) that can be offered at a finite number of price tiers (price points).

Elements of the Model: Product One product offered in the market (easily extendable to a finite number): The product can be offered at a number of tiers The demand for the product has three components Each competitor has a similar tier structure.

Demand Model Three components: Yieldable demand Priceable demand Market Priceable demand It is assumed that forecasts for each type of demand are available.

Yieldable & Priceable Demand Yieldable Demand: Demand at different price points is independent The product can be offered at different price points simultaneously Dedicated to tier and seller (requests a specific tier and will not buy from competition) Priceable Demand The demand at the different price points is dependent Will buy at the lowest price if more than one price point offered at any given time Dedicated to seller (will not buy from competition).

Market Priceable Demand Demand at different tiers is dependent Will buy at the lowest tier available in the market Coin toss if all competitors price the same (same tier offered).

Demand Example Assume the usual nesting structure so the decisions are of the form: open class 1, open class 1 and 2, open class 1, 2, and 3, or open all classes If the decision is to open classes 1-3, then the expected demand is captured as follows: Yieldable: 8.65+22.55+19.80 = 51 Priceable: 25.50 Market Priceable: 25.50 but only if class 3 is the lowest class in the market.

Distributing Market Demand Rules for distribution of the market priceable demand given each state of the market are assumed known Consider very simple case with three possible market states: Noncompetitive (all market priceable demand captured) Competitive (fair coin-toss, e.g. roughly half captured on average) Very competitive (completely undercut, nothing captured).

Market Dynamics We consider very simple market dynamics (but it is fairly easy to model a more complex process, time-dependence etc.): When a booking request arrives the state of the market (value of the market indicator X) is chosen independently from everything else with given probabilities (e.g. 1/3 for each state, as used later on in the numerical examples).

Bellman’s Equation with Information Variable Continuous-time formulation with general X Plus initial and boundary conditions.

Bellman’s Equation, Discrete Time

Bellman’s Equation, Discrete Time The is the probability of the market being in state x The is the probability of selling a unit of inventory at tier j and in the market state x. This probability is obtained from the three demand components: yieldable, priceable and market priceable The is the revenue obtained at tier j from all three demand components (an average, since yieldable demand will buy at higher-priced tiers even if tier j has lower price).

Numerical Experiments: Objectives   1. Determine feasibility of running a single-leg DP with the additional state variable 2. Estimate drop in expected revenue due to not correctly modeling the market priceable demand in the optimization, i.e. by using an inadequate demand and control model.

Numerical Experiments: Methodology Several scenarios of redistribution of the Market Priceable demand (for the incorrect model) considered: The market priceable demand not explicitly modeled but portion of it treated as the yieldable or the priceable demand For example, one simple way is to assume that yieldable demand will get 50% of the market priceable demand and the priceable demand will get the same (i.e. both increase due to market priceable demand).

Methodology For each redistribution scenario generate controls (bid prices) from a DP (incorrect model) Feed them into the DP-based recursion to calculate expected revenues under those controls (with the true market priceable demand present) Compare the revenues to the revenue under optimal policy with market priceable demand explicitly modeled Note: This is fully equivalent to running a simulation (Monte Carlo) and comparing revenues under different controls The advantage is that we do not have any variability issues, disadvantage is that it is only possible in simple settings such as single-leg problems and that we do not model forecasting process.

Numerical Example: Data Capacity is 136; Fares and Demands: Market Information Distribution

Numerical Example: Results

Numerical Example: Summary Based on this and other similar examples it appears that: Revenue decreases as more market priceable demand is modeled as yieldable Shifting more of the market priceable to the priceable demand improves performance (smaller decrease) Best results when 100% of market priceable modeled as priceable Magnitude of differences is very data dependent

Conclusions The run time increase is proportional to the number of states of the information variable (worst case) Modeling market priceable as priceable demand appears to be better than splitting market priceable Caveat: we assume forecasts are given Need to try more complex random dynamics for X

Future Research Apply models in competitive simulations Multiple carriers with their own RM Systems (RMS) competing for the same demand Initial results show significant positive revenue lifts Solution more robust to different data conditions and other carriers’ RMS

References Talluri, K. T., G. J. van Ryzin (2004), The Theory and Practice of Revenue Management, Kluwer Academic Publishers. Walczak, D., “Semi-Markov Information Model for Revenue Management and Dynamic Pricing,” OR Spectrum (Springer), special issue on RM and Pricing, 2006.

Introducing Information into RM to Model Market Behavior INFORMS 6th RM and Pricing Conference, Columbia University, NY Darius Walczak June 5, 2006