Estimating true ratings

If you take a rating website, say IMDB, or Goodreads, and you sorted items purely by review scores, the stuff that would float to the top would be pretty obscure. That’s because the easiest way to maintain a perfect score is to have a very small sample size.

So, a math question: what is the statistically “correct” way to handle this?

In this analysis, I will assume there exists a “true” average review score, and we are trying to estimate it. The “true” average is the average that would be attained if there were a sufficiently large sample of reviewers. We’re not imagining that everyone in the world is reviewing the same book (for example, we don’t expect book reviews to reflect the opinions of people who don’t like reading books period). But we could imagine, what if there were a billion identical yet statistically independent Earths, and we averaged all their review scores.  Obviously it’s very hard to come across a billion identical yet statistically independent Earths, and that’s why we use math instead.

This premise may be fairly questioned. I once discussed the philosophical problems with review scores, including questioning the very idea of taking averages. But here, I’m just focusing on the math for math’s sake.  And, I really mean it, it’s hardcore math.  If you don’t want math, just skip to the last section I guess.

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My economic simulation of spacefaring kittens

Kittens Game is a clicker game that you can play in your browser. It makes a strong first impression, as it tempts you into choices that will kill off your kittens within twelve minutes. But I’m not here to review the game, I’m here to talk about spreadsheets!

Clicker games often support passive gameplay (e.g. leave it running overnight), active gameplay, or any combination of the above. On the very active end, you could try to optimize it, setting up spreadsheets to run calculations. So, I spent a thousand years tinkering with spreadsheets, and I liked it. There’s a story there, a mathematical story.

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The information theory of Mysterium

A question that keeps me up at night is “What is the theoretical best you can do in Mysterium?” I’m exaggerating a bit, but it is a pointless question about a silly board game that I nonetheless spent too long thinking about. I went so far as to watch a series of lectures about information theory–listening to it in the background while in dance class, as one does. I never solved the problem, but let me at least explain what the problem is.

Mysterium

Mysterium is a cooperative board game where the players are trying to solve a murder mystery via psychic communication with the victim. One player takes the role of the ghost, and the rest take the role of psychic mediums. The ghost is not allowed to speak, and may only communicate through cryptic visions. The visions are represented by cards with surreal artwork. For example, one card has two people climbing into a giant fish mouth, another has a tarantula-like thing over a chandelier. After the mediums receive their visions, they discuss what they mean and make their guesses; and the whole time the ghost giggles about how wrong they are.

Example visions: two people climbing into a fish's mouth; a polar bear and spirit owl read a book; a chandelier hanging from strings from a tarantula's mouth

Examples of vision cards.  Source: Mysterium rulebook.

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Infinite Fractal Mazes

My previous post, “Solving fractal mazes” is a prerequisite to this one. Fair warning, this will be long and dense.

Fractal mazes contain infinite paths, but the only solutions permitted are finite. Some people find that disappointing. What’s the point of all that extra maze if we don’t get to traverse it? So my goal is to come up with a variant ruleset for fractal mazes that permits and formalizes infinite solutions. In fact, I will propose two distinct rulesets, provocatively titled Countably Infinite Fractal Mazes and Cantor Fractal Mazes.

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Solving Fractal Mazes

What are fractal mazes?

Fractal Mazes are a type of maze popularized (or invented?) by Mark J. P. Wolf, published in the Mathpuzzle blog in 2003. A fractal maze is a maze that contains nested copies of itself.

small fractal maze

“Small Fractal Maze”. Credit: Mark J. P. Wolf. Source: Mathpuzzle

Fractal Mazes are typically visually represented as a sort of circuit diagram. In the above image, the goal is to find a path between the “+” and “-” by following the colored wires. The wires are color coded in order to clearly indicate where paths cross over/under each other. The three modules, labeled A, B, and C, are each copies of the entire maze. However, the start and finish only exist in the largest copy of the maze. So however deep you go into the fractal, you must eventually climb all the way back up again.

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