by James Ernest and Glen Barnett
Players love the art and characters of Button Men. Players also love the strategy. In fact, some of them even engage in long conversations about the ideal ways to play. I’ve constructed this article from a recent e-conversation between myself and Glen Barnett, a PHD in statistics working in Australia. Glen’s comments are in blue...
I was disappointed that no one took up your challenge of discussing Button Men strategy on rec.games.abstract. But I’m always up for it.
Glad to hear it! Anyway, I was interested in anyone’s thoughts on good heuristic strategies for the game. Some things are pretty obvious, but the obvious simple strategies are clearly suboptimal often enough to worry me.
I think the basic play strategy evolves something like this:
1: I must take more points
2: I must therefore lose fewer points
3: In a given situation, I should choose an attack which simultaneously lets me roll up my most vulnerable dice, and scores the most points for me.
Yes. I actually had this much by the time I started my first game.
My current comfort level is with what I’m calling 2nd-level strategy. This introduces two new factors: Score Differential, and Change Killing.
Score Differential says this: With any set of characters, there is probably a “bigger” character. That player must keep some number of sides to win. For example, in Avis(10) vs. Hammer(10), Hammer must keep more than 18.7 sides (i.e., his 20) to win the game.
The equation for sides-you-keep is 2(you-them)/3. This equation is instrumental in choosing appropriate Swing Dice versus a particular opponent. For example, if Avis is playing vs. Hammer(10), she’s better off choosing a 6. Now Hammer must keep 21.3 sides to win.
Yes, we figured this one out a few weeks ago. It makes a big difference to how quickly you can figure out the best rolls, *and* it helps with choosing swing dice, as you note. I mentioned this approach to some of your Demo Monkeys who I am friends with.
Change Killing is how Hammer will protect his 20’s. Suppose the situation is this:
Hammer: 3/6, 3/12, 7/20, and 10 points.
Avis: 3/4, 4/4, 2/10, 6/12, and 30 points.
“Basic” strategy might tell Hammer to take the 6/12, either with his 7/20 or by adding the other two. However, that’s going to hurt him. Hammer’s best move is to take Avis’ 3/4, since that prevents her from making 7, and therefore protects the 7/20.
Taking the 6/12 would leave his 20-sider vulnerable, even if he rolled it. If he rolls the 20-sider to take the 6/12, he runs about a 40% chance of losing the 20-sider on the next turn, and with it the game, since it’s already understood that Hammer must keep 20 points to win.
Yes, we make similar calculations ourselves. We quickly figured out the importance of leaving your rival with multiples of the same number (like 3,3,6), and of killing combinations that can take your good dice. Often there are two combinations that will get you, and then you need to figure out what dice they have in common and try to kill those.
e.g. you have 17 on a d20 (which I write as 17(20) myself), and your rival can get that with a 7,6,4 or with a 10,7 (and he only has one 7!). Then if you kill that 7, you have zapped both combinations.
Right. I call that die the “keystone,” if it’s there. Sometimes the opponent has two independent ways to take your big die, so there’s no keystone. Then you have to roll the target die to have a chance at protecting it.
I’m curious if there are situations in which that’s so risky that there’s a smarter thing to do. In other words, leave the die that’s probably lost in favor of trying to protect something else, or make a more devastating attack.
Of course there are – there are situations where you have to give up a neat attack in order to reroll your forked dice (two independent attacks that add up to your die is like a fork in chess, right?). If that reroll is going to be into nasty territory, you will be better going for the good attack and giving up your die. for example, consider (these may not be real characters, but I could make one up that was):
2/4 3/6 12/12 11/20 vs 3/4 5/6 6/6 8/8 12/20
The first player listed is to move. The d20 is gone unless he rerolls it. Say he takes the d8. The values in the shade are 13 16 19. Erk. Much better to go for an exchange of d20s. Even if that last die of his opponent’s was a d12 he still might be slightly better off taking it in this case.
This work led me to an early realisation that if you had highish odd values, your opponent couldn’t get them if he only had lower even numbers. Of course that then led straight to what you call change killing – which you *can* do in a game, since it isn’t so calculation-heavy.
On the subject of Swing Dice, I’ve made a first draft of a very clunky chart of Avis’ ideal swing dice against all opponents.A point on the chart dictates what Avis should pick against a certain character with a certain Swing Die. “X!Y” means that Avis can either go big or small, and these are the ideal ways of doing either.
“Ideal” is pretty unscientific right now, though. Here’s how I did it:
For smaller characters: If Avis is against a large character, she knows he must keep a certain number of sides to win. Avis will choose the largest die she can without reducing that number of sides. For example, in Avis4 vs. Bauer20, Bauer must keep more than 24 sides to win. This means keeping a 20 and one other die. If Avis upgrades to an 8, Bauer needs only 21.3 sides , but this is equivalent to 24 in terms of his actual dice. Avis gains attack power with no compromise in points. However, if she chooses a 10, the break point is 20 exactly, so Bauer can now tie by keeping a 20. So, vs. Bauer20, Avis chooses 8.
For larger characters, the goal is to choose the largest die possible without being forced to keep more dice. This is essentially the opposite of small characters, though you’re still going as large as possible within a given zone.
For roughly equivalent characters, the goal is to choose the smallest die possible. This is the argument: Let’s say Avis is fighting Karl4. Between the range of Avis4 to Avis12, the game is the same: either player must keep one die to win. Within this range, I propose that it’s preferable to go small, with the intent of going first more often. Players who go first are more likely to retain a die, and in this case that’s all it takes. You’ll notice that this is the only zone where I go low instead of high.
That top-left-corner item strikes me as a bit odd; if 4 is “ideal” against Avis4, that means the best you’re expecting to do with a free choice of swing die against Avis4 is break even!
In Blackjack, hitting 16 vs dealer 10 is the best move, too. Even though it loses a majority of the time, it still beats the alternative (barely). I won’t try to prove that 4!10 is the right solution to that set, because my math is admittedly mushy. But it -is- true that sometimes the ideal response to a given situation still loses more than half the time.
I’m just starting to get into Shade Theory, which deals with true probability of rolling a number the opponent can’t catch, sort of the refined equivalent of “rolling high.” If a 20-sided die is rolling into (2,3,4), as described above, then it is in the shade on an 8, or a 10 through 20. That’s 12 spots out of 20, or 60 percent safe.
If a 20-sider is rolling into (7,7), then it’s 60% safe. Rolling into (5, 4, 3) is 55%. Rolling into (9,3) it’s 50%. Here’s a neat one: (7,7,7,7) is still 60%.
I looked at explicit probabilities long before we worked out the 2/3 rule – for me that probabilities was second stage – but the problem is that it’s hard to do the calculations during a game.
(One thing I did get from playing around with the probabilities is just how important Skill attacks can be in the early game.)
With the probabilities, it isn’t just rolling what your opponent can’t catch but also setting up rolls that can catch more of your opponent next time around – and rolling a nice low 1 or 2 can often be very important in leaving you with options even after your opponent has done his worst.