"Applications of Math in Guardians" is a 3 part series of articles I'm doing to take a look at how you can use math to look at Guardians in a new way.
Today I'm covering Part 1, Probability and Distribution
CCG designers can sometimes be described as math wizards, from Richard Garfield, the graduate student/math professor who created Magic, to the designers of today. Every aspect of a game can be expressed as a formula, a statistic, a ratio, or a cost-benefit relationship. This is necessary to achieve balance in the game and make sure no faction, class, or game mechanic is more powerful than any other.
There are 2 major topics I’m going to explore:
1. Deck makeup, resources, and probability
2. Distribution analysis of cards during a game
This article is about probability, not certainty. CCGs have often been greatly affected by chance: drawing the right card at the right time, good or bad draws by your opponent, and even a good or bad matchup during a tournament. This is what makes the game different from chess, where only the players’ skill comes into play. The goal of the CCG is to maximize skill while introducing enough chance to make each game unique. The information presented here can be of value, but should not be used as a crutch. Chance is what makes each game different; the possibility of beating the odds is what keeps things interesting. My hope is that this article will help you to minimize the impact of chance so that you can focus more on honing your skills. At the very least it should be quite educational, if you’re into that sort of thing.
DECK MAKEUP AND PROBABILITY
Let’s take a look at a sample deck taken from Phil’s site, the “No Secondary Attacks Deck” by Andre:
Stronghold: Khnumian Stronghold
Shields: 4x Demonic
Standard Bearer: 2x Brom's Demonic
8x Shield Terrain
3x Dispel Magic
5x Babes Bribery
2x Gold Bribery
5x Beer Bribery
5x Shroud of Grahzue
2x Rocks of skull cracking
4x Hammer of doom
2x Sacrificial altar
5x Roving Force Inferno
5x Mist Veiler
5x Roaming Steam Geyser
5x Tangle Web
5x Vesuvious Rex
3x Great Ba'te
2 Stronghold Upgrades
8 Shield terrains
2 Standard Bearers
Total = 70 Cards
Something we should get out of the way right off the bat is Guardian’s (intended) lack of resources. If you’re unfamiliar with the concept, a resource is basically a card (or cards) you must have to play a second card. That second card has a cost to play it, and if you don’t have enough resources to draw from, you can’t play the card. Games like Magic the Gathering, which uses Land as a resource, and World of Warcraft TCG, which uses Quests, are examples of games with resources. In Guardians, the designers have stated a few times that Guardians is a game that is not resource-dependant...you can technically play any card you want without using resources. However, I feel that is a case of semantics and is slightly inaccurate. By my count, Guardians has 3 mechanics that can loosely be defined as resources: Power Stones, Terrain, and Shields.
Power Stones are primarily used for channeling, but there are cards that require Stones to be spent to use them. Altar of Takuli is an example of this. You can put the card into play without paying any cost; however, the effect of the card, heal 1 creature for 1 Stone, is worthless if you don’t pay the Stone. Therefore, because the effective use of the card requires using the Stone, it has a cost, which makes Power Stones a resource.
Similarly, a Shield can technically be considered a resource because without one, you can’t move creatures anywhere. Since winning the game can only occur by venturing out into the disputed lands (or into an opposing Stronghold), a Shield becomes a resource despite the fact that there is no “cost”.
Finally, Terrain is what I call a “reverse resource”. This means that if you do not place a Terrain after winning a space, you must place a card face down. In other words, you incur a cost by using a card that loses its intended value when playing it as a Terrain card.
Now that we’ve defined what constitutes a resource in Guardians, we must accept that the Shield is the most valuable resource in the game. Why? Because you start the game with your full amount of Power Stones, so you’re not limited there; also, since you can place any card face down as a substitute for Terrain, as long as you have at least 1 card, you have Terrain. However, you’re not going anywhere without Shields. So let’s look at how you get Shields from your deck into your Storage Hand.
Probability tells you that any card can be pulled as the top card of your deck, expressed as a percentage: how many of a given card is in a deck divided by total number of cards. Think of your standard poker deck. You have 4 aces and 52 cards, so your chance of pulling an ace, assuming the deck is shuffled, is 4/52, or 7.7%. Applying the same principle, in our deck above we have a total of 14 Shields. Since our Guardian and Strongholds start the game already in play, our effective deck size is 66 cards, so 14/66 = 21.2%...this is the probability that if you draw a single card of the top of a deck, it will be a Shield.
With a base draw of 3 and a LUC of 1, you will draw 3-4 cards per turn until LDL or MDL is established. Let’s say LDL and MDL bounce back and forth between -1, 0, and 1 from turn 3 to turn 10, and you get the LUC 50% of the time. By your 10th turn, you should have drawn about 40 cards (28 cards + 12 card opening draw), so you should draw about 8 Shields in those 10 turns. How did we figure that? Remember that we have a 21.2% chance of drawing a Shield. If we draw a Shield 21.2% of the time, when we draw 40 cards, take 21.2% of 40 to give you about 8 Shields drawn. However, there is a problem: the randomness of shuffling means the Shields are not guaranteed to be evenly distributed throughout the deck. We could draw 4 in a row, or not draw any for several turns. That number is an average, not a guarantee.
Is 8 Shields enough? It depends on how many you need. We'll cover an analysis in the next section you can use to decide. Obviously you don’t want to add too many Shields to your deck, simply because drawing Shield after Shield means you’re not getting the Creatures, Spells, Terrain, and Magic Items you need. However, if you don’t draw enough, you have a bunch of unusable creatures sitting around on Strongholds or the Creature Pen.
The numbers we have established above apply to any cards in your deck. Using our above example of 10 turns and 40 cards drawn, if you have 5 Tangle Webs, 5/66 is 7.6%, and 7.6% of 40 is 3.03, so we would draw on average 3.03 Tangle Webs over 10 turns. Since you can’t draw .03 of a card, you’ll probably draw 3 Tangle Webs.
And how is this useful for deck building, you might ask? Well, let’s say you only put 1 copy of Tangle Web in the deck. 1/66 is 1.5%, so we would draw .6 Tangle Webs (1.5% of 40) over 10 turns– that’s not even 1! Dropping a card from a quantity of 5 in this deck to 1 makes it likely it won’t even show up during the first 10 turns. It doesn’t mean it won’t show up – it just means the chance of it showing up is low. And if the game is over before turn 10, the chance is even lower.
Looking at our deck above, let’s use our Shields again. Can we predict when you are most likely to draw them? We can try, but remember - we’re talking about probability here, not certainty. You could draw one first turn, or not draw one at all. And this is where the math becomes a little more complicated. Luckily we can use Excel for this.
Excel can perform what is called a hypergeometric distribution analysis. To do this, you’re asking Excel “if I draw an opening hand of 12 cards, and I have 14 Shields in a deck of 66, what is the chance a Shield will be (or will not be) one of the 12?”
Here is how to use the distribution. In Excel, select any cell and type the following:
The first number in parenthesis is the number of cards we’re looking for. Here it’s 0, which you would use to find out your chance of drawing 0 Shields.
The second number in parenthesis is the # of cards drawn – in this case our opening hand of 12.
The third number is the total quantity of the card in our deck – in this case 14 Shields.
The last number is the deck size before the draw (66 cards).
Excel returns a result of .041922, or 4.2%. So for every 100 games you play, 4 games you won’t draw a Shield in your opening hand. When you consider that you can re-shuffle and re-draw for 1 stone, you can feel fairly confident you will always have a Shield on your first turn.
Our next question is how many are we likely to draw?
Our chance to draw exactly 1 Shield: =hypgeomdist(1,12,14,66), = 17.2%
Our chance to draw exactly 2 Shields: =hypgeomdist(2,12,14,66), = 29.2%
Our chance to draw exactly 3 Shields: =hypgeomdist(3,12,14,66), = 27.2%
Our chance to draw exactly 4 Shields: =hypgeomdist(4,12,14,66), = 15.3%
As you can see, our most likely outcome is drawing 2 Shields, and we are starting to trend down at 3-4 Shields. If you want to find out what the chance to draw more than 4 Shields is, just take 100% (which represents all possibilities) and subtract the totals of all the other results up to the 4 Shields result. So 4.2% + 17.2% + 29.2% + 27.2% + 15.3% = 93.1% (our sum of results), and 100% - 93.1% = 6.9%. This means there is a 6.9% chance to draw more than 4 Shields.
Let’s make a logical leap by looking at our second turn. Say we followed the percentages and drew 2 Shields in our opening hand. What is the chance we draw a Shield on turn 2? Well, let’s ask Excel like we did above. “What’s the chance we draw 0 of a card, while drawing 3 cards (Gaar’s base draw), with 12 Shields still in the deck, out of a deck size that is now 54 (66 minus our opening draw of 12)”. Using our function:
Excel will answers with .462869, or 46.3%. That’s our chance to draw 0 Shields on turn 2. Subtract that from 100% to get all other possibilities, and you have a 43.7% chance of drawing 1 or more Shields. That’s pretty high! While it ensures you will probably have a steady flow of Shields coming in, remember our disclaimer above: “drawing Shield after Shield means you’re not getting the Creatures, Spells, Terrain, and Magic Items you need.” If you have a creature-heavy (and Spell/Magic Item-light) deck, this might be good so you can keep those creatures moving out into the disputed lands. But if you’re looking for Spells, Magic Items, or bribery cards, maybe you have too many Shields. Just remember that if you reduce the number of Shields in the deck (to be able to draw other cards), you will reduce the distribution on first turn draw as well. Use the hypergeometic function to run different quantities and percentages to find numbers you are comfortable with.
You can perform this same hypergeometric calculation for any group, as long as you know the group total…for instance, you could calculate the chance of drawing your Tangle Webs, or the Fairies in your Fairy deck, or even the total number of creatures in your deck (like we did with total number of Shields). Now that you know how to calculate the distribution, you can decide if you have enough of a particular card or a group of cards in your deck so that you will be satisfied with the chance to draw 1 or more in your opening turn or subsequent turns.
During the next post, I’ll talk about Deck Manipulation and present a sample deck that uses Card Advantage, and after that I’ll wrap up the series with a look at cost/benefit ratios of cards.
I hope you enjoyed this article. Leave me some comments or questions if you liked it, or even if you didn’t.
Some source material for this article was drawn from:
The Mathematics of Magic: The Gathering - A study in probability, statistics, strategy, and game theory by Jon Prywes