Button Men Theory

by Simon McGregor

Button Men Theory is tough. Leaving aside Swing Die theory and character selection (and I have some ideas of my own on that one), ordinary play strategy isn’t simple.

I’m currently working on “endgame analysis” (i.e. play strategy when there are only a few dice left), and it’s evident that rules of thumb aren’t infallible by any means. Nor is it always obvious when they don’t apply. And it gets worse! Not knowing what “best play” is, it’s really hard to tell either from theory or experience how likely the various endgame scenarios are in the “perfect play” case.

Yesterday, I whipped up a small Button Men computer game for personal use. I know you’ve got your own one in development, but if you want a look at mine, you’re welcome to it.

Anyway, here are some (unsolved by me) questions about Button Men strategy:-

• How much of an edge, in general, does optimal play have (vs. reasonable but sub-optimal play) in a single round? And in a three-round match?
• Are there situations where sub-optimal play has a greater edge over sub-optimal play than optimal play does? (i.e. are there ever times when, knowing the “best” strategy versus “best play”, you can choose another strategy and better your chances by assuming your opponent won’t spot “best play”?
• Assuming “best” swing die selection, is there a “best” character? Or are we playing scissors-paper-stone?

I am looking at some brute-force computational approaches right now. Button Men Theory is amenable to brute-force solution in two areas:-

1. First-go odds. A complete (60 * 60) table of first-go odds would be useful, and is straightforward to calculate.
2. Play strategy. With five dice, Button Men is simple enough to exhaustively calculate the best play strategy from a given starting position, and the chances of winning.

I’ll let you know when I’ve calculated the first-go table.

If Button Men takes off like it should, I guess we’ll see a whole lot of mathematical analysis. And I don’t think it will be solved quickly. Or maybe ever.

I’ve come up with some simple (and potentially conflicting) heuristics for the game in general. I’m sure I’ve missed some, so feel free to point them out.

1. Keep the highest number you can for your next turn. Sometimes your highest number is going to be taken in your opponent’s turn, so this means maximising your next highest number.
2. Force your opponent to re-roll their best numbers to lower ones.This usually makes them easier to take later.
3. Lock your opponent (i.e. leave them without a legal move) whenever possible.The bigger characters usually depend on a lock to win. Locking requires you to have the highest, though not necessarily the heaviest, die on the board.
4. Take your opponent’s heaviest dice (i.e. the ones with most faces). This increases your chances of locking later on, and decreases your opponent’s chances of locking.
5. Maximise your spread (i.e. the range of different numbers you can take) and minimise your opponent’s. A large spread protects against locking.

In general, smaller numbers give better skill plays, because you can add lots of them together without going over 20. But if your numbers don’t add to more than the weight (i.e. number of sides) of your opponent’s heaviest die, you are better off with larger numbers. The chance of getting a skill play is the same, and your power plays are better.