rational numbers are ratios:

\begin{equation} \mathbb{Q} = \left\{\frac{a}{b} \middle| a,b\in \mathbb{Z}, b\neq 0\right\} \end{equation}

Motivation

Suppose we would like to say that “we prefer all to well \(A\) more than bad blood \(B\)”

\begin{equation} A \succ B \end{equation}

No right or wrong answers in this statement by itself, but we can check whether or not your preferences are inconsistent with itself.

von Neumann and Morgenstern Axioms

Axioms for checking if a set of preferences are rational. The axioms allow you to check if a set of decisions are Rational Preferences.

\(\mathbb{R}\) real numbers are numbers generatable by a possibly infinite sum of powers of 10.

plan to depth \(d\), take action, replan

Recommender System is a system that provide recommendations due to search; it combines Information Retrival with another goal:

• Editorial/Hand-Curated lists: “list of favorites”, “essential items”, etc.
• Aggregates: top 10 lists, most popular, recent uploads
• (hardest) Individual tailors: user-based recommendation

Formal Model

• \(X\) the set of users
• \(S\) the set of things to recommend
• \(R\) the set of distinct and totally ordered ratings (stars 1-5, real number 0-1, etc.)
• Utility function: \(U:X \times S \to R\) (“how much

Three key problems: