Fun With Maths

This post started off just as notes that I was taking for myself, and so I’ve presented those thoughts, slightly edited for clarity. I thought it might be interesting to actually lay out the way that I think about things in more or less “real” time as I was thinking about them and taking notes. Note the rhetorical questions throughout.

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So let’s do the math.

How many cards per color should we see in the normal 5 tier playthrough?

First though, how many cards should there be in a deck towards the end of the game? Let’s say for the sake of argument we would be looking at a deck of 30 cards. Since we start with 5 cards, that would mean that we would collect 25 more cards. That would imply that there would be a total of 5 cards added per tier – which is dumb.

Let’s also take into account that there are card sinks – places where cards can be trashed to. If, over the course of an average game, the player has leveled up to level 5, that means that they have trashed 11 cards. Which brings our total up to 36 added cards, or 6 cards per tier. That’s still not right.

We must also consider the fact that the colors do not come out all at once. Here’s the rub I think. In that case, if the about example was a Blue deck, then in order to have 36 cards, they would have need to have seen 108 total cards, or 21 per tier. Now we’re getting somewhere.

But that’s also making certain assumptions, but I think that those assumptions hold true for multicolor too. The only real difference would be that a multicolor deck would have spent more cards in order to level up.

So if they got to level 4 in both colors then they would have spent 22 cards to get there, meaning that they would have drafted 52 cards (the 30 for the optimal deck size and the 22 for leveling up). They would have had to see 78 (26 red, 26 green and 26 blue – due to how the cards come out). Now, this is also under the assumption that the other player has decided to play a mono-blue deck and only has 32 cards in it.

If one of those colors was “cut” by another player who drafted half of the cards seen, then they would have needed to see twice that number of cards in order to get the same level value. So if we needed to see 52 green cards (to draft half of them), then our total number of seen cards would be 156.

Now then, from a game design perspective I think that I would rather these cards appear earlier than later, so that coming into mid enemy tier 4 and 5 your deck is more or less complete. That way players can enjoy having a powerful deck. that’s something I can sort out in the encounter design though. But that number – 156 I think is the magic number to look at.

If we go by that assumption – that we will see around 52 cards per color per game. So that’s where I think I should be looking. (coincidentally, with 52 cards a monocolor 30ish card deck would be Level 7/8 depending on mainstat – beefy).

Right then. Since we do not want the players to see every card in every game, that means that each color needs to have more than 52 cards. Breaking that down (and rounding a little bit) we end up with 27 commons, 18 uncommons and 9 rares in that collection. That’s fine. Although that’s not a lot of rares (and honestly, that would just be 9 commons too, with 3 copies each). If we add 50% (ish) more we get to 14 rares, 28 uncommons and 42 commons for a total deck size of 84. Although at that point let’s just round up and enjoy the symmetry of 90 cards.

With 90 cards we have 15 rares, 30 uncommons and 45 commons, but only 45 actual cards per color, or 135 distinct cards. That’s not that bad actually. Let’s dive into more math though. Looking at the hyper-geometric calculator, working with our givens of 52 cards per color per game and 90 card draft deck, there would be odds of drawing a specific rare card in 57% of games, and an uncommon at 89%, and common popping up 92% of the time. That’s interesting, since it implies that we don’t need to worry all too much about not creating linchpin rares that are required for a deck to work. It does tell me though that linchpin uncommons really should be onboard effects.

That also gives me a rough design skeleton as follows :

15 Red Commons
15 Red Uncommons
15 Red Rares
15 Blue Commons
15 Blue Uncommons
15 Blue Rares
15 Green Commons
15 Green Uncommons
15 Green Rares

But that’s only half the story. You see, I don’t want to build out the card file all willy and/or nilly. We need to think about the curve and how commonly we would like things to appear. For example, I don’t think that there needs to be a 5 drop at common for any color. Mainly because it is basically impossible to throw out a 5 drop until late in the game anyway. That means I don’t want there to be a too many of them. I also think that commons should be weighted ever so much towards the lower end of the spectrum since more decks are more likely to splash those colors and play them. So a preliminary breakdown would looks like this:

Common:
1 Drop – 5
2 Drop – 4
3 Drop – 4
4 Drop – 2
5 Drop – 0

Uncommon:
1 Drop – 3
2 Drop – 4
3 Drop – 4
4 Drop – 3
5 Drop – 1

Rare:
1 Drop – 2
2 Drop – 3
3 Drop – 4
4 Drop – 4
5 Drop – 2

This leaves us with an average cost of 2.2 for Commons, 2.67 for uncommons and 3.1 for rares. Which seems like as good a place as any to start. The average costs (for those of you keeping track at home is 2.65 – which means that that level 3 is an important threshold).

Right then. Next up I just need to look at the cards that I’ve designed already for playtesting and see how they can slot into that framework. It feels now, in a weird way, like I’ve finished building the engine, and now I actually need to develop all of the content. This is oddly familiar place to be.