Solving Wordle using information theory
Audio Brief
Show transcript
This episode explores information theory and entropy, using the popular word game Wordle as a practical example to illustrate core concepts.
There are three key takeaways from this discussion.
First, to solve problems with incomplete information, focus on choices that maximize expected information, efficiently reducing possibilities.
Second, an event's information content is inversely related to its probability; highly unlikely outcomes are very informative.
Third, the best initial guesses in games like Wordle are not just common letters, but words that mathematically partition the set of possible answers into many small, manageable groups.
Information theory quantifies uncertainty, where the amount of information gained from an event relates to its surprising nature. The standard unit, a "bit," represents a halving of possibilities.
Entropy, defined as expected information, is a weighted average of information from all possible outcomes. A guess with higher entropy more effectively reduces uncertainty on average.
Highly likely outcomes provide little new information. Conversely, rare or surprising outcomes are very informative, fundamentally explaining what it means to be informative. This principle guides optimal strategy: prioritizing choices that reveal the most, not necessarily the most common.
The goal in games like Wordle is to select guesses that provide the most information to narrow the list of answers. Optimal initial words like CRANE or TARES excel because they divide the remaining possibilities into the smallest, most evenly sized groups, regardless of the feedback received.
Effective algorithms balance maximizing information gain through high entropy with choosing words likely to be the actual answer. Information gain is initially prioritized, shifting to probability as options narrow and possibilities become limited.
Ultimately, understanding information theory provides powerful tools for navigating uncertainty and making optimal decisions in complex scenarios.
Episode Overview
- The episode uses the popular word game Wordle as a practical example to explain the core concepts of information theory, particularly entropy.
- The host walks through the process of building a Wordle-solving algorithm, starting with simple strategies and progressively refining it using mathematical principles.
- Key concepts like "bits" of information, expected value, and entropy are defined and illustrated through the lens of choosing the best possible guess in the game.
- The algorithm's performance is tested and improved across several versions, demonstrating how incorporating probability and maximizing information gain leads to a more optimal strategy.
Key Concepts
- Wordle Strategy: The goal is to choose guesses that provide the most information to narrow down the list of possible answers. The best initial guess is not necessarily a word with the most common letters, but one that best partitions the remaining possibilities into the smallest, most evenly-sized groups, regardless of the color pattern returned.
- Information Theory: The amount of information gained from an event is related to how surprising or unlikely it is. The standard unit of information is the "bit," which represents a halving of the space of possibilities. The formula for information (I) of an event with probability (p) is
I = log₂(1/p). - Entropy: Defined as the "expected information" from a probability distribution. It is calculated by taking a weighted average of the information of every possible outcome. A guess with higher entropy is better because it reduces uncertainty more effectively, on average.
- Optimal Algorithm: The most effective algorithm balances two goals: maximizing the information gained from a guess (high entropy) and choosing a word that is likely to be the actual answer (high probability). Initially, information gain is prioritized, but as possibilities narrow, guessing likely answers becomes more important.
Quotes
- At 00:06 - "It occurs to me that this game makes for a very good central example in a lesson about information theory, and in particular a topic known as entropy." - The speaker introduces the core thesis of the video, connecting the popular game to a complex mathematical concept.
- At 02:15 - "If you're wondering if that's any good, the way I heard one person phrase it is that with Wordle, four is par and three is birdie." - The speaker provides a simple, relatable analogy to benchmark the performance of a Wordle player or algorithm.
- At 06:07 - "In fact, what it means to be informative is that it's unlikely." - The speaker explains a fundamental and somewhat counter-intuitive principle of information theory: rare outcomes provide more information than common ones.
- At 21:12 - "You should call it entropy!" - The speaker recounts John von Neumann's advice to Claude Shannon on naming his measure of uncertainty.
- At 21:23 - "Nobody knows what entropy really is. So in a debate, you'll always have the advantage." - Continuing the anecdote, this quote explains von Neumann's humorous and practical reasoning for using the term "entropy."
Takeaways
- To solve problems with incomplete information, focus on making choices that maximize the expected information you'll receive, thereby reducing the space of possibilities most efficiently.
- An event's information content is inversely related to its probability. Highly likely outcomes provide little new information, while highly unlikely outcomes are very informative.
- The best starting words in Wordle (like 'CRANE' or 'TARES') are not just collections of common letters; they are words that are mathematically proven to divide the set of possible answers into many small, manageable groups on average.
- Entropy provides a quantitative measure for uncertainty. A guess that leads to a probability distribution with higher entropy is one that is more likely to significantly narrow down the remaining options.
