Know Your FoW: Counting Stones

Welcome to Know Your FoW! It’s the blog series here on Friend & FoW that deals with the decisions involved in the game and explores the probabilities and statistics behind each topic.

For this week’s topic, we look into what can be considered as the most straightforward aspect in the game: the Magic Stone. I call it straightforward because once you have determined that you need to call a Magic Stone, there is little to no decision making involved afterwards – the stone that gets put into play is at the mercy randomization. For the former magic players, there’s no mulligan for lands or sequencing of which to play first. For the former World of Warcraft players, there’s no need to identify which card is best to row as a resource. It’s as if we have no control whatsoever.

But we do have control. Not during the actual game, nor during shuffling (hopefully you do so as truly random as possible). We have control during deck-building. For mono colored decks this exercise is simple, just make sure you build our Magic Stone Deck with 10 of your chosen color and you’re all set. The challenge comes when one needs to find the right balance for multi colored decks.

The Ratio Method

A relatively simple method in balancing stones is to sum up the total number of attribute symbols found in the casting cost of cards in your deck and proportion your magic stones according to the identified ratio. Let’s take the following deck for example:

4x Brainless Scarecrow {G} 4x Cowardly Lion {G} 4x Heartless Tin Man {G} 4x Dorothy, the Lost Girl {G} (1) 4x Glinda, the Fairy {G} {G} 4x Deadman Prince {B} (2) 4x Pumpkin Witch {B} (2) 4x Jewels on Dragon Neck {B} (2) 4x Spire Shadow Drake {B} (2) 4x Mephistopheles, the Abysal Tyrant {B}{B} (2)

In our example, one could count up 24 {G} symbols and 24 {B} symbols, resulting in a nice 50:50 ratio. That would lead us to constructing our Magic Stone deck containing equal parts Wind and Dark, 5 of each stone without any duals, or 3 of each and 4 duals. While the ratio technique is good for a quick estimate of attribute balance, it overlooks a key factor in gameplay. Indeed half of the deck needs {G} and the other half needs {B}, but all turn one plays require {G} and only needs {B} by turn three at the soonest. This leads us to my preferred method:

The Curve Analysis Method

While taking just a bit more time and effort to implement, this method paints a clearer picture of how the deck would want to operate and the Magic Stone deck would hopefully reflect that. One would need to list out his deck by “ideal turn played” and identify what color requirements are necessary at each point in the game. Again for the example deck:

  • Turn 1: {G}
  • Turn 2: {G} (1) or {G} {G}
  • Turn 3: {B} (2)
  • Turn 4: {B} {B} (2)

On turn 1, this deck really wants to play a one-cost resonator. By this point, only one Magic Stone would have been called. The likelihood of calling a stone that can play a {G} casting cost resonator can be computed by the formula:

(# of stones that can produce wind) / (# of stones in magic stone deck)

The (# of stones in a magic stone deck) would be 10 at the start of the game, so there’s nothing we can do to improve our chances by manipulating the divisor. We can improve our probabilities by modifying the (# of stones that produce wind).

In a perfect world, we’d want to reach 100% certainty, which would require 10 out of 10 magic stones to produce wind. But as we shall see, we can’t always get 100% certainty, and maybe 90% or 80% or even 70% would be enough.

In order to cast Glinda, the Fairy on turn 2, two magic-stones should be called and both need to produce {G}. Thats actually two dependent events that need to have happened: turn 1 call a stone that produces {G} and turn 2 call a stone that produces {G}. In a formula: (# of stones that can produce wind) / (# of stones in magic stone deck) * (# of stones that can produce wind – 1) / (# of stones in magic stone deck – 1)

Note the “minus ones” in the second half of the equation, caused by the calling of the first stone decreasing the magic stone deck size. Again, to reach 100% certainty one would need 10 out of 10 magic stones to produce wind. While 9 wind producing stones amounts to 80%, 8 to 62.22%, 7 to only 46.67%. If we had followed the values specified by the ratio method we’d only be able to cast the turn 2 Glinda in our hand about half of the time.

While both Glinda and Dorothy are two-cost wind resonators, the difference in (1) {G} and {G} {G} is quite significant for your Magic Stone balance. For Dorothy, the Lost Girl, at least one of the two magic stone’s called would need to produce {G}. The event consideration is slightly more complex: if turn 1 call a stone that produces {G} then turn 2 call any stone if turn 1 call a stone that doesn’t produce {G} then turn 2 call a stone that produces {G} In formula:

(# of stones that can produce wind) / (# of stones in magic stone deck) * 100% + [(# of stones that can’t produce wind) / (# of stones in magic stone deck) * (# of stones that can produce wind ) / (# of stones in magic stone deck – 1)]

Fortunately for us, another way of considering this equation would be: what is the likelihood that we get both stones that do not produce wind and subtract that value from 100%, formula:

100% – [(# of stones that can’t produce wind) / (# of stones in magic stone deck) * (# of stones that can’t produce wind – 1) / (# of stones in magic stone deck – 1)]

This time, to reach 100% certainty we only need 9 wind producing stones. Even dropping to 7 wind producing stones, still provides us with a 93.33% certainty of being able to cast the Dorothy in our hand on turn 2.

As we move up the curve, we realize that representing the probabilities in a formula and calculating them accordingly gets more and more like a homework assignment instead of preparing for a fun card game. Worry not, because we have the internet to save us. Stattrek provides an effective hypergeometric calculator that we can use for these exact purposes (Available at: http://stattrek.com/online-calculator/hypergeometric.aspx). I’ll be referring back to this calculator many times in the future. To use it for our Magic Stone calculations, make use of the following inputs:

  • Population size = 10 (# of stones in magic stone deck)
  • Number of successes in population = up to you (# of stones that produce the attribute of your choice)
  • Sample size = Turn # (# of stones that have been called)
  • Number of successes in sample (x) = (# of attribute symbols are found in the casting cost of your card)

The “>=” result is relevant for our scenario because we want to meet the casting cost requirement but don’t mind going over.

So for Pumpkin Witch one would enter something like:

  • Population size = 10
  • Number of successes in population = 5
  • Sample size = 3 (assuming we want to cast her on turn 3)
  • Number of successes in sample (x) = 1

and be 91.67% confident that we’d be able to cast her on turn 3 even with just 5 dark magic stones.

For Mephistopheles, the Abyssal Tyrant on the other hand:

  • Population size = 10
  • Number of successes in population = 6
  • Sample size = 4 (assuming we want to cast her on turn 4)
  • Number of successes in sample (x) = 2

would give us 88.1% certainty that Mephistopheles can be played with 6 dark magic stones.

Now what do we do with all these numbers?!? With the Curve Analysis Method we’ve identified that it’s almost impossible to reach 100% certainty to cast all our cards in our multi colored deck, but we can come very close to it.

Assuming all the cards in our example deck are equally as important to be played as soon as possible, I’d probably balance the magic stones as 5 Wind stones 1 Dark stones and 4 dual stones. This leaves us with an 80% likelihood to cast Glinda on turn 2 and 73.81% chance to cast Mephistopheles on turn 4, all the other cards would be at least 90% certain that we would be able to cast them as soon as possible. Without any dual stones, I’d be very hesitant to try to make this deck work because the likelihood of successfully playing either Glinda or Mephistopheles would be too low for my preference.

It’s the cards with double casting cost requirement that are the most difficult to cast consistently in multi colored decks. However, not all cards are equally important to a strategy. Nor are all cards necessary to be cast as soon as possible. Knowing the strain Glinda and Mephistopheles are putting on our consistency, we now need to ask ourselves some questions:

1. Do we really envision casting Glinda on turn 2? Maybe we only need her later and lessen the wind stones needed.

2. Is Mephistopheles relevant enough in this deck to warrant the double dark requirement? Are we willing to play him at just 73.81% certainty, or can we remove him and lessen the dark stones needed.

3. Do we really envision casting Pumpkin Witch or Deadman Prince on turn 3? Maybe we only need them later and lessen the necessary dark stone count even further.

4. Why are there Jewels on the Dragon Neck in this deck aside from making the Ratio example all nice and balanced?

By analyzing the curve and attribute requirements, we can identify cards that don’t quite fit into the plan of our deck (even though they are powerful) due to restrictive casting costs. We also get to plan out when we envision each card to be played and increase the likelihood that we can play that card when we need it.

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