The curse of dimensionality in hotel management, and the surprising connections with Occam’s razor

Simplicity of control should be the goal of every hotel manager, this is particularly true for Hotel Revenue Management.

Consider these two numbers.
1,000,000,000,000,000,000,000,000,000,000,000,000
432,000,000,000,000,000

The second number is the age of the universe, the number of seconds from the Big Bang (according to this theory). An impressive number indeed, but the first number is much bigger. What does it represent? Think about it for a minute and contact us for a special prize if you guess it right.

Let’s assume that an hotel manager needs to decide which discounts to apply to children of age less than 18. This problem requires filling a table with 18 values (one for each age, from 0 to 17) of the discount to apply (a percent value, from 0% —no discount at all— to 100% — children stay for free).

When asking manager about why they selected specific discount values, answers range from “Why not?” to “We imitate what our neighbour is doing”, to “We do not know, the software required some values and we had to fill them in in some way”. Determining an optimal solution for the above problem, one leading to the maximum possible profit, is not difficult because of laziness or ignorance but because there are just too many possibilities. There are 100 possible discount values for each age, to be multiplied to get the above amazing number of different ways of filling the complete table with percent values, that is, 1 followed by 36 zeros. Even if you analyse a possible way of filling the discount table in one second, the entire life of the universe would permit to consider only a minuscule fraction of possibilities, and your professional life is less than the life of the universe.

face

Imagine that the hotel owner or revenue manager starts considering possible discounts immediately after the Big Bang, well, after recovering from his personal big shock and after some years of evolution to create the human species. He observes the creation of stars and galaxies, and keeps trying discounts, observes the formation of the sun from dust and its ignition to create light and heat up the cold matter, and keeps trying discounts, observes the creation of planet Earth and Moon and keeps trying discounts, observe his personal life pass without excessive emotions and keeps analysing discounts.

This explosion of possibilities is known also as the “curse of dimensionality”. Smart optimization schemes can deliver optimal solutions in reasonable times despite enormous numbers of possibilities only for a selected list of special problems, which unfortunately excludes almost all complex problems of real-world interest. If hotel managers want to apply a scientific, principled manner to choose business settings, and not black magic, a much smaller number of possibilities must be considered.

For a simple example, the manager can decide among two possibilities, and set up an experiment to decide whether a 50% discount for children of all ages will improve profits w.r.t. no discount at all. Of course, a children discount may be desirable if the hotel targets families and wants to discourage couples or singles, but the decision should be based on measurable results. For example, an experiment can be done with split testing in the hotel website, or by trying the two possibilities for comparable analogous periods. Similar conclusions hold for different decisions, like for setting proper prices for each type of room and for each day depending on demand, market evolution, competition, type of customer, groups, agencies, etc., problems which are much more difficult and more crucial than deciding about children discounts.

William of Occam
(William of Occam and friend, painting by Andrea di Bonaiuto da Firenze)

The above fact related to the impossibility of dealing in an effective manner with decisions with too many parameters is deeply rooted in the scientific approach. A good scientific theory is characterized by brevity, and concision is the source of power enabling a solid scientific theory to conquer the world. Occam’s razor (a.k.a. “law of parsimony”) is the problem-solving principle stating that in science the simplest theories must be preferred. The idea is attributed to William of Ockham (c. 1287–1347), an English Franciscan friar and philosopher. The same “law of parsimony” is at the basis of machine learning (ML) or learning from data. Because the objective of ML is to generalize for new and unseen data, if the model is too complex, the examples presented during learning will be reproduced, but learning degrades to a sort of superficial memorization of the examples, not extracting the deep underlying rules. For revenue management, a crucial aspect is to forecast demand by customers and hotel occupation levels. Data about previous months and years, together with data from similar hotels, can be precious to “machine learn” a forecast. Again, simple models with a small number of free parameters need to be used to predict the future (before it happens).