Prescriptive Data Science 101: Types of Problems

Photo by Franki Chamaki on Unsplash

Welcome back to a series of articles on prescriptive data science! In my last post, I talked about descriptive, predictive, and prescriptive data science and required skillsets for prescriptive data science. Today, I will help your journey by defining different types of prescriptive data science problems.

Please note that the methods described in this article is called in many different names across multiple disciplines — mathematical optimization, control theory, single agent dynamics, multi agent dynamic game, dynamic programming, Markovian decision process, or reinforcement learning. In the subsequent article on key elements of problem definition for a single agent case, I will point out the key constructs which are core to the problem definitions and methods. Please note that the underlying ideas are common across multiple disciplines in Operations Research, Economics, Marketing Science, Computer Science, Machine Learning, and Artificial Intelligence.

By the end, you will have a basic knowledge of:

• What prescriptive data science is.
• Two key dimensions to define prescriptive data science problems: number of agents and dependency across time periods

How Do We Define Different Problem Types in Prescriptive Data Science?

Prescriptive data science is a subfield of data science that helps us to obtain answers to questions such as “What should we do (among the choices of potential X’s)?” It is not well defined due to multi-disciplinary nature. Thus, I intended to define it in a more structured way and introduce the key dimensions which will help you to identify different prescriptive data science problem types.

There are two key elements that you need to consider to classify different types of prescriptive data science problems: (1) number of agent and (2) dependency across time periods.

1. Number of agent. The number of agent who is relevant for the given problem can be either (1) single (i.e. only 1 agent), or (2) multiple. The key distinction here is whether you have to think about and incorporate (strategic) interactions between agents to solve your problem. Single agent problem is simpler since you don’t have to think about actions (i.e. decisions) of other agents. In contrast, when you have multiple agents, (strategic) interactions among agents need to be a part of the problem solving. This will introduce the notion of “equilibrium” in game theory. Moreover, defining the information sets (what’s known vs. unknown for each agent) will be important to properly specify multi-agent problems. To say the least, you can immediately see that multi-agent problems are much harder to solve compared to single agent problems.
   
   As an example, if you are considering category profit maximizing pricing for soft drink for the only supermarket in a small town, you don’t have to worry about responses from other supermarkets (i.e. monopoly situation.) In this case, as long as there is a demand model (i.e. output from your predictive data science, which maps unit volume (as a target) to pricing and promotion features with other control features ), you can try non-linear optimization to find out a set of prices, which maximizes the category profit of the focal supermarket for each product (i.e. SKU) in the soft drink category for the given timing of the year (e.g. 4th of July week.) In contrast, if you have 2 supermarkets in small town, things get much complicated and you now need to think about how other supermarket would respond with its pricing decisions.
2. Dependency across time periods. Dependency across time periods can be either (1) static (i.e. no dependency) or (2) dynamic. The key here is whether you can solve each period problem in isolation (i.e. separability) or whether you need to think across multiple time periods to capture dependency over time.
    
   As an example, for the case of category profit maximizing pricing decisions of a single supermarket in the town, one can argue that pricing decisions for each period can be solved one by one, since the prices as of now at the point-of-sale only matters, not the prices from other time periods. Please note that I am assuming away potential complication of forward looking consumer behaviors where consumers can expect future promotion and stock up when prices are favorable (so called purchase acceleration or forward buying) for simplicity. In a simplified world without forward buying from consumers, pricing of frequently purchased goods can be viewed as a “static” problem, since you can solve a sub “pricing” problem for each week independently (i.e. no dependency.)
    
   In contrast, if you consider the only mobile phone (i.e. consumer durables) store in a small town (e.g. the only Verizon store), you can easily realize that the pricing decision of iPhone 12 Pro Max in the current period is not independent of pricing decision in 24 weeks from now. Consumers are very familiar with the fact that prices for durables decreases over time. Moreover, there are heterogeneous consumer segments: Some consumers are willing to pay premium to adopt the new product quickly (i.e. early adopters). Other consumers will wait until prices become favorable (i.e. laggards). Because of these, the prices which will maximize the current period profit is not equal to optimal prices which will maximize the total profit over the entire time horizon. Here, due to dependency over time, you need to solve a more “dynamic” version of the problem. We will revisit “dynamic” problems in the subsequent version of the article series.

In case you have training in econometrics or time series, dependency over time can be introduced when you have lag terms (i.e. variables from previous periods. In the case of structural econometrics (in empirical industrial organization), structural consumer or firm behavior models with expectation also naturally creates dependency over time. In operation research, computer science, machine learning, and statistics, hidden (i.e. latent )state models such as hidden Markov models, Kalman filtering, or Sequence models such as Recurrent Neural Network, attention models, or Transformer architecture also naturally lead to long and short term dependence over time. The way time dependency is modelled also have significant implication on the solution algorithm for subsequent optimization or reinforcement learning problems.

Summarized below are different prescriptive data science problems along the dimension of (1) number of agents and (2) dependence across time periods.

In a subsequent article, I will focus on “single agent” case and (1) describe key constructs to properly define the problem, which can be solved by optimization or artificial intelligence algorithms. In addition, I will provide (2) high-level overviews of key methods to solve the “single agent” problem.

In case the readers are interested in the links across different academic discipline, shown below is an article, which maps structural econometrics models in IO to Artificial Intelligence methods such as AlphaGo.

Artificial Intelligence as structural estimation: DeepBlue, Bonanza, and AlphaGo: Prof. Mitsuru Igami at Yale University