Category Archives: Good Math

Zero Sum Games

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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.

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Introduction to Linear Regression

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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.

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Basic Statistics: Mean and Standard Deviation

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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.

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Introducing Game Theory

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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?

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Meta out the wazoo: Monads and Monoids

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Since I mentioned the idea of monoids as a formal models of computations, John Armstrong made the natural leap ahead, to the connection between monoids and monads – which are a common feature in programming language semantics, and a prominent language feature in Haskell, one of my favorite programming languages.

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Monoids and Computation: Syntactic Monoids

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While doing some reading on rings, I came across some interesting stuff about
Monoids and syntax. That’s right up my alley, so I decided to write a post about that.

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Ideals – Abstract Integers

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When I first talked about rings, I said that a ring is an algebraic
abstraction that, in a very loose way, describes the basic nature of integers. A ring is a full abelian group with respect to addition – because the integers
are an abelian group with respect to addition. Rings add multiplication with an
identity – because integers have multiplication with identity. Ring multiplication doesn’t include an inverse – because there is no multiplicative inverse in
the integers.

But a ring isn’t just the set of integers with addition and multiplication. It’s an abstraction, and there are lots of thing that fit that abstraction beyond the basic realization of the ring of integers. So what are the elements of those
things? They can be pretty much anything – there are rings of topological spaces,
rings of letters, rings of polynomials. But can we use the abstraction of
the ring to create an abstraction of an object that resembles an integer, rather than an abstraction that resembles the set of all integers?

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Full Circle: the Categorical Monoid

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By now, we’ve seen the simple algebraic monoid, which is essentially an
abstract construction of a category. We’ve also seen the more complicated, but interesting monoidal category – which is, sort of, a meta-category – a category built using categories. The monoidal category is a fairly complicated object – but it’s useful.

What does a algebraic monoid look like in category theory? The categorical monoid is a complex object – a monoid built from monoids. If we render the algebraic monoid in terms of a basic category, what do we get? A monoid is, basically, a category with one object. That’s it: every algebraic monoid is a single object category.

But we can do something more interesting than that. We know what a monoidal category looks like. What if we take a monoidal category, and express the fundamental concept of a monoid in it?

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This is getting fun! On to Monoidal Categories.

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In the last post on groups and related stuff, I talked about the algebraic construction of monoids. A monoid is, basically, the algebraic construction of a category – it’s based on the same ideas, and has the same properties; just the presentation of it is different.

But you can also see a monoid in categorical terms. It’s what we computer scientists would call a bootstrapped definition: we’re relying on the fact that we have all of the constructs of category theory, and then using category theory to rebuild its own basic concepts.

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