Math and the Game Designer

In addition to Zzzzzz's excellent suggestions (which I need to follow up on myself), I'll mention that I asked Knizia himself that question recently. He said, to my surprise, that he does not use mathematical techniques to answer those kinds of questions. He just does cycle after cycle of playtest-and-tweak, until he thinks it's done.

I then asked him how he knew when it was "done"; he said that when a game had gone a week's worth of near-nightly playtests without any rule changes, he would set it aside for a month. Then he'd get it out again, and if he still saw no need for changes, it was ready for publishing.

I found this most depressing, as unlike Dr. Knizia, I do not have teams of experienced playtesters playing my games five nights a week!

But then there's this article by Bruno Faidutti on how to make a game "self-balancing" that might help.

I think that the most instantly useful math skill in game design is a good knowledge of Probability. especially as it pertains to dice.

Undestanding the occurence frequency of a specific number being rolled allows you to tailor in-game events to those frequencies and thereby "balance" the game. (assuming that you are using dice at all.)

In my fledgling days as a game designer, I rolled hundreds of pairs of dice and recorded the results, just to discover what the most common roll would be, and therefore determine the basic "to-hit" number in the combat game I was working on at the time. Ahh, what a chump! It's kind of embarrassing, now that I think about it.....

Games designed purely from a balanced mathematical foundation tend to end up as Abstract Strategy games... "AbStrats", I'm calling 'em, and they can function well, but sometimes seem more like 2-player puzzles than true games.

~Josh

Most of the math in my designs comes down to what's called Combinatorics, a branch of Discrete Math. It helps you answer questions like "How many ways are there to stack n poker chips, each of which can be red, white, blue or green, such that each red chip is adjacent to at least 1 green chip?" Once you have a handle on how many times certain events can occur, or how many types of card XYZ are important, you can start looking at ratios, get some probabilities and start balancing from there.

I searched Amazon for a good intro book on Combinatorics (one where the first line isn't "Theorem 1.1" or "Let G be a graph with E edges and V vertices such that..." and found Counting: The Art of Enumerative Combinatorics which looks interesting with lots of examples for the non-math major people. (The poker chip example is from the first page of this book.) Of course like stark said, playtesting can negate all the math if it's just not fun.

Despite having an honours degree in statistics, I only used maths to a limited extent . I find the main use is to get an idea of averages and variability.

For example, on average how many draws of the deck will I need to draw a particular type of card or get a set of three. This is used to assign an initial card distribution which is tweaked through playtesting.

Similarly, an understanding of maths and spreadsheeting can be good to compare average number f actions required to the expected victory point pay-off.

To me, the maths is useful for helping set the initial starting points (e.g. card distributions, vicotory points, etc.), while play-testing is essential in making sure the final distributions are right.

Cheers,
Bill

For one game I've worked on, my buddy and I spent a good deal of time trying to use graph theory to prove that various board configurations were not possible. It was very difficult so we gave it up, but we have tried such things at other points :)

Probability somewhat obviously plays a big part. Similar to what someone above mentioned, we use it a lot to try and figure out how often a card might come up, how much effect some bonus actually has, etc.

I use a lot of geometry and trigonometry in creating game boards, firing patterns, etc, both for the art & the more basic game elements.

I think however that of greater influence than actually applying math to our designs, is that we tend to come at the task with math-oriented minds. For example, although the graph theory didn't work out, we have applied very basic induction and contradiction to show that various board configurations and other game properties could or could not happen. We also employ a loose sense of utility & decision trees in evaluating the quality/richness of a player's decisions. Also, we use a lot of symmetry to try and balance things out. For example, more powerful actions being more rare to counter lesser, more common events. That reasoning may or may not have actual numbers associated with it.

It's worth noting perhaps that I'm a computer science grad student, and the buddy with whom I do the most game design work has a master's in math, so we're always looking for ways to apply our education to gaming :)

I guess none of that's really helpful, but it does say that there's lots of interesting places to apply math in game design, and that there's people here who'd be interested in discussing such things!