Thursday, September 23, 2010

When 2+2 doesn't equal 4

A friend of mine recently told me that I post too much math to this blog. Well, today I get to disappoint him. Don't worry, the math isn't hard at all.

We're going to be discussing the cases where 2+2 doesn't necessarily equal four.

Now, before you ask, I'm firmly seated on my rocker, in possession of a full set of marbles and deck of cards, and my lid remains unflipped. Let me get an example for you, so you may be as persuaded of these facts as I am. Well, I shouldn't expect that sort of miracle.

In Magic: the Gathering you win by reducing your opponent's lifetotal from 20 to 0. (A quick side note for the quibblers. Yes, there are alternate win conditions. Yes, there will be exceptions to other things I say about the game. Ignoring the complications for now to keep it simple.) You reduce their life total by attacking them with creatures. So it'd take twenty turns to kill someone with a one power creature, and ten with a two power creature, assuming they don't do anything that stops you. In fact, let me make a table of that for you:

1 power: 20 turns
2 power: 10 turns
3 power: 7 turns
4 power: 5 turns
5 power: 4 turns
6 power: 4 turns
7 power: 3 turns
8 power: 3 turns
9 power: 3 turns

Contrary to my previous assertion, we see that it takes exactly two bears to win the game in the same time as one Serra Angel. But look at three power, if you work out the ratio it takes 40% more hill giants to kill in the same time span a Serra Angel could, when 4 is only 25% larger than three. How does that work? Even worse, take a look at the six power guys. Despite being slightly better than a five power guy, they still kill your opponent in the same amount of time. Same thing with seven, eight and nine power, it's all three turns.

This works, not because in some situations three is actually smaller than others, but because three doesn't divide evenly into 20. You only need to deal 20 damage, there's no bonus for going over. Since doing three damage a shot doesn't stop until you hit 21, that last point of damage is wasted. Attacking for five gets to twenty exactly in four turns, but attacking for six gets to twenty four, with surplus pain. Relatively speaking, five is just as good as six despite the fact that our starting postulates make six objectively better.

(The example is pretty obviously not that relevant in a game, but I have noticed that sort of thing come up playing an aggro deck against control. Looking at a 6/6, counting the turns I had to draw a lightning bolt to finish him off and realizing that it was still four.)

So where else does this apply? Well, I can't think of another game off hand, but it is the mechanism that allows computer worms to work. Y'know, like the one in Office Space? Or that old Superman movie they referenced in Office Space? Anyway, the programs treat dollars as if they're continuous, which means that when they divide them they sometimes get fractional pennies that can't be expressed in real world money. So they round them off, because who cares about a fraction of a penny anyway? All it took from there was someone to notice that money was disappearing and figure out a way to make that money his.

Another way this sort of thing happens is when you're talking about relative power levels. Again with the magic example. Let's say that on my side of the board I've got three glory seekers (a 2/2 with no abilities.) On your side of the board you've got one Stone Golem (a 4/4). If I try to attack with my glory seekers, one is going to get beaten by the angel and then I'll have traded a creature for a lousy couple damage-- not a very good exchange. So I've got to sit back and wait until you decide to do something, which means that your one 4 power guy is superior to my three two power guys, despite the math saying something different. (Of course, all this can be changed by either of us having any one of a number of combat tricks, but we're doing the default right now.)

More generally, we can look at a set and try to figure out what the relevant toughness is. Take Magic 2011, the most recent core set. My brother did the analysis, by the time he was drafting his first packs he already knew to stay away from two toughness cards like glory seeker. Why? There's an abundance of creatures that can profitably hold off an army of glory seekers in the set. Like that Stone Golem. Or a Giant Spider. Or an Azure Drake. Or any number of commons. That and Pyroclasm, as a kill spell that hits multiple creatures of two toughness, could really wreck his day. A single 3/3, by virtue of that one more point of power, can fight it's way through quite a few more combat situations than a 2/2. Leading me to make such inane but accurate statements like "Three is SO much more than two!"

Ok, another example. Let's talk hearts. (If you don't know hearts, your windows computer has it as one of it's built in games. Learn how, it's a good game.) Ok, say that I'm playing in a four player game, and it's my lead. Hearts has just been broken, and I want to get rid of the lead, so I want to lead a low heart. I have the 5 and the 9. How likely am I to take the trick if I lead one versus the other?

Let's say that I lead with the five. There are exactly three cards that get under it (the 2, 3 and 4). If I lead the Nine, there are three more cards (6, 7 and 8; we know where the five is already.) The problem is is that taking it with the nine isn't twice as likely, it's several times as likely. I'd give you the actual numbers but I royally confused myself trying to remember how to do a simple calculation and so you're going to have to suffer through inexact figures.

But think of it this way. I lead the five, West has to have one of the three, North has to have one of the two remaining and East has to have the last one. It's not that likely. If I lead with the nine, even if the other two have cards East has any of four possibilities to duck under. Or each of them might have two. The only way the nine doesn't take it is if some sad sack has missed six chances to get one of those cards in his quarter of the deck. Those aren't very good odds. In this way we see that the Nine is more than twice as dangerous as the five, even against the logic of our first intuition.