Zero Sum Games

In game theory, perhaps the most important category of simple games is
something called zero sum games. It’s also one of those mathematical
things that are widely abused by the clueless – you constantly hear
references to the term “zero-sum game” in all sorts of contexts, and they’re
almost always wrong.

A zero-sum game is a game in which the players are competing for resources, and the set of resources is fixed. The fixed resources means that any gain by one player is necessarily offset by a loss by another player. The reason that this is called
zero-sum is because you can take any result of the game, and “add it up” – the losses will always equal the wins, and so the sum of the wins and losses in the result of the game will always be 0.

Continue reading Zero Sum Games

Introduction to Linear Regression

Suppose you’ve got a bunch of data. You believe that there’s a linear
relationship between two of the values in that data, and you want to
find out whether that relationship really exists, and if so, what the properties
of that relationship are.

Continue reading Introduction to Linear Regression

Framing and Expelled: Why the Framers are Mis-Framing

I’m going to jump into the framing wars again. As I mentioned last time,
I think that most folks who are “opposed” to framing really don’t understand what they’re talking about – and I’ll once again explain why. But on the other hand,
I think that our most prominent framing advocates here at SB are absolutely
terrible at it – and by their ineptitude, are largely responsible for
the opposition to the whole thing.

Continue reading Framing and Expelled: Why the Framers are Mis-Framing

Basic Statistics: Mean and Standard Deviation

Several people have asked me to write a few basic posts on statistics. I’ve
written a few basic posts on the subject – like, for example, this post on mean, median and mode. But I’ve never really started from the beginnings, for people
who really don’t understand statistics at all.

To begin with: statistics is the mathematical analysis of aggregates. That is, it’s a set of tool for looking at a large quantity of data about a population, and finding ways to measure, analyze, describe, and understand the information about the population.

There are two main kinds of statistics: sampled statistics, and
full-population statistics. Full-population statistics are
generated from information about all members of a population; sampled statistics
are generated by drawing a representative sample – a subset of the population that should have the same pattern of properties as the full population.

My first exposure to statistics was full-population statistics, and that’s
what I’m going to talk about in the first couple of posts. After that, we’ll move on to sampled statistics.

Continue reading Basic Statistics: Mean and Standard Deviation

Nim

As an introduction to a mathematical game, and how you
can use a little bit of math to form a description of the game that
allows you to determine the optimal strategy, I’m going to talk a bit about Nim.

Continue reading Nim

Introducing Game Theory

Lots of people wanted game theory, so game theory it is. The logical first question: what is game theory?

Game theory is typical of math. What mathematicians like to do is reduce
things to fundamental abstract structures or systems, and understand them in
terms of the abstraction. So game theory studies an abstraction of games – and
because of the level of abstraction, it turns out be be applicable to a wide
variety of things besides what you might typically think of as games.

Game theory starts with the fundamental idea of a game. What is
a game?

Continue reading Introducing Game Theory

Requests?

As you’ve probably noticed, things have been rather slow around here lately. I’ve got more posts in the works on group theory and abstract algebra – but they take a lot of time to research and write, so they’ll be coming out slowly – one a week or so.

In the meantime, I’m looking for other topics to write about, and I’d like to know what you, my faithful readers, are interested in hearing about.

Some things I’ve considered:

  • Cellular automata: CA are very cool. I’ve been wanting an excuse to read my copy of Wolfram’s text.
  • Data structures: my programming-related posts have always been very popular; and there’s a collection of unusual data structures that have interesting mathematical properties.
  • Game theory: a pretty cool area of math.
  • Conway’s games: basically the second half of Conway’s ONAG.

Or any other mathematical subject that you’re interested in learning about. Suggest away in the comments.

And keep those bad-math links coming!