1 Introduction

This course centers on a semester-long project to build a working algorithmic decision tool. As used here, the term algorithmic decision tool means: a computational system that empowers a user to make data-driven, near-optimal (or at least, better) decisions in a specific domain area.

In the broadest terms, you can think of a decision tool as having two components, or modules:

  • A prediction component that generates information (possibly imperfect) about the state of the world;

and

  • An optimization component that identifies the best action to take, given the information provided by your prediction tool.

We introduce the basic concepts of statistical decision theory — a rigorous framework for taking optimal decisions under conditions of uncertainty. In the context of a bare-bones simple model, we address how to formalize a decision problem in terms of uncertain random variables, the menu of possible actions, the potential payoffs from choosing alternative actions, and the decision-maker’s ultimate objectives.

We then analyze this model to derive optimal decision rules: rules for choosing actions optimally, given the decision-maker’s beliefs, possible actions, payoffs, and ultimate objectives. We consider how improvements in information can lead to better decisions: with better information, the decision-maker is more likely (though not guaranteed) to opt for actions that improve her results.

We also derive formal expressions for the expected value of information, defined as the improvement in the decision-maker’s expected payoffs when the she can make choices on the basis of better information. The expected value of information offers a quantitative measure of the value-add of the decision tool.

1.1 Motivation: Data-driven decision-making

Humans make billions of decisions every day. Many of these decisions are not optimal. Many are based on unexamined replication of whatever procedures were used historically. Sub-optimal decisions create waste.

Explosion of access to data, analytic tools, & computing power opens new opportunities for decisions to be more data driven.

1.2 The general form of the problems we’ll take up

  • There is a decision-maker who must choose one option from a menu of options.

  • The decision-maker has objectives.

    • These objectives can be quantified in the form of an objective function (equivalently, a loss function).

  • The payoffs to different choices depend on the values of certain state variables or unobserved parameters. The payoff function expresses the costs and benefits the result from the decision-maker’s chosen action.

    • In some applications, it makes sense to model the payoffs as depending on the realized value of a random state variable.
    • In other applications, it makes sense to model the payoff’s as depending on the value of the unobserved parameter.

  • The decision-maker is in general uncertain about the values of the relevant state variables or parameters. In general, we will represent the information the decision-maker has in the form of probability distributions over the relevant state variables or parameters.

  • In most applications we will consider, tools are available to reduce these uncertainties. Depending on the application, the tool might be derived from a statistical model, a machine learning model, or a forecasting system based on a physical model. We will refer to these informational tools collectively as predictive models.

1.3 Building decision models

When building a decision model, there are essential questions you must answer early in the process:

Who is the decision-maker?

What decision does this agent confront?

What is the set of options (or potential actions) from which this decision-maker chooses?

What are the stakes of the decision? What real costs and benefits are realized from making better vs. poorer decisions?

The predictive tool:

What information does your chosen predictive tool provide?

How will the decision maker use the information generated by this tool to make better decisions?