Applications of Math in Guardians, Part 2: Deck Manipulation and Card Advantage

Well, the hiatus is officially over with this post.

If you need a refresher, you can find Part 1 here.


PREAMBLE

I've seen a game design philosophy for Guardians mentioned at least more than once, in the Guide to the Mid Realms and on the old message boards. And that philosophy is?

Guardians favors the aggressive player.

Seemingly based on the age-old mantra of "Fortune favors the bold", this philosophy is prevalent in the use of everything from flying decks and bribery to powerful channeling and movement strategy. Another component in which this is apparent is in Deck Manipulation and Card Advantage.


DECK MANIPULATION

First, let's define these terms. Deck Manipulation is being able to set up your deck so that you draw certain cards when you want to. The card text that enables this might read something like, "Search your deck for a Spell and put it in your Storage Hand." Or it might be, "Look at the top 6 cards of your deck and replace them in any order." Deck Manipulation can have 3 possible benefits:

1. You can immediately put into play a card that is useful in a specific situation; for example, maybe you need an Ancient Ogre to shut down your opponent's channel-heavy deck.

2. You can reduce your deck size. The benefit of reduced deck size is that as you pull specific cards out and the deck gets smaller, you increase the chance of drawing other types of cards in your deck later (think about hypergeometric distribution, which is a function of deck size).

3. Deck manipulation leads to card advantage.


CARD ADVANTAGE

Card Advantage is the ability to put more cards into play than your opponent by outdrawing your opponent, or forcing your opponent to lose cards in some way. A perfect example of this is the MDL & LDL modifiers, which are fitting for the "Guardians favors the aggressive player" mantra. If I have MDL and you have LDL, I am most likely drawing about 4 cards to your 2. Think of each card as an asset, or in other words, something I can put to use. Over the course of several turns, the more I draw and the less you draw, the more assets (creatures, shields, spells, terrain, etc.) I will be able to put into play. Thus I gain an advantage, as you could not hope to match the amount of assets I am able to bring to bear.

Probably the most famous examples of card advantage are 3 of the "Power Nine" cards from Magic the Gathering: Ancestral Recall, Time Walk, and Timetwister. These cards were considered some of the most powerful in the game, before they were restricted and then banned. Ancestral Recall allowed drawing three cards for a very cheap cost. Time Walk let a player take another turn, skipping the second player's turn and resulting in what could be considered a "double draw" for the first player. Timetwister allowed for all discarded cards to be shuffled back into the play deck, which gave an advantage to the player able to inflict instant damage. Why were they banned? The card advantage they provided was too great.

So how exactly does this relate to Guardians? Guardians has one restricted card, Champs, and yes it's restricted because of card advantage! Other than that one card, we must assume that there aren't any other cards that provide an unfair advantage. This means to gain true card advantage, a deck would have to use it as a theme (or a complimentary theme) rather than relying on 1 or 2 cards to provide it.


DRAW CLASSIFICATIONS

There are 4 classifications of cards that may be used to gain advantage through drawing or discarding:

1. Raw. Draw the top card(s) of your deck, or force opponent to discard top card(s) of deck. May require discard or Power Stone as cost to be paid.

2. Seek. Look through your deck (or your opponent's deck) for something and draw (or discard) it. May require discard or Power Stone as cost to be paid.

3. Conditional. Draw from your deck, or force opponent discards, *if* some specified event happens. May require discard or Power Stone as cost to be paid. You might not have control over the event.

4. Global. All players draw cards, or all players discard.

Below is a list of every card or game mechanic that fits the classifications above, with the classification listed next to it:

MDL: conditional
LDL: conditional
LUC: conditional

Ankh: raw
Argammond's Vision: raw
Assassin of Shadow: seek
Buster Scrimbo, conditional
Chant of the Osirans: conditional
Chickenhead McCracken: conditional
Eisnmir: conditional
Finn: conditional
Jonstollo the Seeker: conditional
K'Hutek: conditional
Planes of Entropy: raw
P'Tal: raw
Scarab of Bounty: conditional
Sebek, Queen of Magicians: raw
Shadow Strike: global
Shadow Warrior: conditional
Squibby: conditional
Tes Let: raw
The Hand of Chronos: global
Tookle: conditional
Vek-Nadra: conditional

What follows is an analysis of the cards above, whether they provide card advantage, and if so, how much.


CONDITIONAL DRAWS - UPCARDS


Buster Scrimbo, Chickenhead McCracken, Jonstollo the Seeker, and Squibby must be your upcard to gain an advantage. Let's say you have 4 Busters in a deck of 55. Using our hypergeometric distribution analysis, you are most likely to draw 1 Buster in your opening draw of 12. Of course, drawing him into your hand negates his ability (it can only be used when he is your upcard). So after you've drawn, you're ready to turn over your upcard, with 3 Busters left in a deck size of 43. Using our analysis again, you only have a 7% chance of drawing Buster as your upcard. Assuming you don't draw Buster as your upcard and you're drawing 4 cards on your next turn, since one of your cards drawn is your upcard, you're drawing 3 cards from your deck (which is now 42 because you pulled one card as your upcard last turn). Using our analysis once more, the most likely outcome (79%) is that you don't draw Buster. Now you turn over your upcard for this turn. To the analysis once more...you have a little over a 7% chance of drawing Buster, virtually unchanged from the previous turn.

There are two problems apparent here. One is that with such a low chance of drawing Buster (or Chickenhead or Jonstollo or Squibby) as your upcard, his special ability is largely useless. The second problem is that without his special ability, he becomes simply a 4 Vitality external with no combat ability. Compared with the abilities of other 4 Vitality creatures such as Amber Well, Cleric, or Haba Naba Kaba, Buster doesn't look so attractive. Even the repulsive Gorgal Skag, though it has no ability, is useful for the ability to fly.

CONDITIONAL DRAWS - GUARDIANS

Many of the Guardians in the game also have conditional draws. Eisnmir's draw is useless if the opponent plays very few magic items, and his LDL & MDL numbers are terrible. K'Hutek's draw only works well if you've been losing, a bad prospect. P'Tal's draw is raw instead of conditional, but it's also useless - you may be able to gain a card or two, but you're going to get kicked in the Power Stones for doing so.

Tes Let's ability, another raw, is useful but I'm not a big fan of throwing away cards to get cards - if you are throwing away cards, doesn't that mean that you could be using better ones?

This brings us to our most feasible Guardians for card advantage: Finn, Tookle, and Vek-Nadra.


At first Finn's ability seems extremely useful for a conditional draw - if you can keep at least two Swamps in play, you draw 2 extra cards, more than most other Guardians would get with just their MDL. But Finn only has a base draw of 2, so unless you can get 3 Swamps in play, you're only getting a one card (or no card) advantage over other Guardians, and it's even worse if you get the LDL, which is a -1 for Finn. Finn also locks you into one type of Terrain - Swamps - which limits your options on deck construction, and allows your opponent to stack up on Terrain bonus-busting cards.


Vek-Nedra's card advantage capability trumps Finn's on multiple levels. First, although Vek-Nadra also has a base draw of 2, he has no penalty from LDL, freeing you from worrying about disputed lands. Second, if you lose a battle but have at least one creature under your Shield, and you can retreat if required, Vek-Nadra keeps the card draw because the Shield is still alive. If you can keep 2-3 Shields in the disputed lands (which is entirely feasible), you'll gain 2-3 extra draws per turn. Remember that you can get that same draw with almost any other Guardian by winning MDL. Vek-Nadra simply removes the uncertainty of MDL & LDL from the equation.


However, when it comes to card advantage, Tookle has no equal. Small creatures & low upcard numbers go hand in hand. We just showed above how conditional draws for upcards are not an advantage. However, this changes with Tookle if all the creatures in the deck are small, because they represent a much higher percentage of the deck. Let's say that out of 55 cards, 30 are creatures. Using our hypergeometric distribution, we would expect to draw (on average) 7-8 creatures in our opening hand (let's use 8). Now we turn over the upcard. Our distribution shows that with 22 creatures left in a deck of 43 cards, we have a 51% chance of pulling a small creature, which gives Tookle an extra card draw. That's much better than the 7% chance of pulling Buster Scrimbo in our example above.

Fun fact: note that when you draw 1 card in the hypergeometric model, you can calculate the same 51% chance by simply dividing 22 creatures / 43 cards.

Anyway, back to Tookle's draw: Not only do you have a 51% chance of drawing a small creature, you have an excellent chance of also winning LUC, for another extra draw. With Tookle's base draw of 3, that means you have a 51% chance of drawing 5 cards per turn, and you can bump that to 6 cards if you win MDL. You can even increase those numbers up to 6-7 cards per turn by adding a Scarab of Bounty by Tookle's side, giving you +1 to LUC. Of course, Tookle's going to need that extra draw with all those wimpy creatures under his Shields. You can see now, though, how Tookle is the exception to any claim that there is no card advantage in Guardians.

PLANES OF ENTROPY


Finally , a discussion about probability! I love to argue probability with people, because they really don't understand how it works. There are several aspects of probability, but we are only going to cover chance, predictive probability, joint probability, unions, and averages. To illustrate this point, let's look at a simple coin-flip example. Let's say I flip a coin 10 times. Since I have a 50-50 chance of getting heads or tails, I should flip 5 heads and 5 tails, right?

Wrong.

Averages, when used for probability, do indicate that we will flip an equal number of heads and tails over time. This is called a normal distribution. If we assign heads a value of 1 and tails a value of 2, we could flip 10 times, add up the results, and divide by 10 to get an average. As you repeat this experiment several times, you will average a value of 1.5, which represents 50% heads and 50% tails. This is called approaching the mean. It involves a lot of complicated equations that I don't want to get into. Needless to say, it has no effect on a single coin flip. Why? Because a single coin flip is always heads or tails, and is not affected by the flip before it. For instance, say I flip heads. Now I'm going to flip again. My chance of flipping heads again is still 50%, since the only possible outcomes are heads or tails.

However, you can ask "what is the probability of flipping 5 heads in a row, before I start flipping?" This is called joint probability. If the chance of flipping heads is 1/2, or .5, then the chance of flipping heads 5 times is:

.5 x .5 x .5 x .5 x .5, or 3.125%.

This is predictive possibility - what *could* happen before you start. But once you start flipping, the probability changes.

Let's say I flipped tails. My probability of flipping 5 heads in a row is 0%, because I already got tails. But what if my first flip was heads? My first flip of heads will not affect my second flip - I can only get heads or tails. But looking at my predictive possibility of 5 heads in a row, I've already flipped my first heads. Now I need to flip 4 more heads in a row. The chance of doing this is:

.5 x .5 x .5 x .5, or 6.25%.

Say I was successful and flipped a second heads. Now to get 3 heads in a row the chance is .5 x .5 x .5, or 12.5%. Always remember that the past flips have no effect on the chances of your next flips.

The same concept applies to dice, but with 6 possible outcomes instead of 2. The chances of rolling a 1 are the same chances to roll a 6: 1/6, or 16.7%. The chance of NOT rolling a 1 are 5/6, or 83.3%. However, if we want to roll 2 dice, the chance of rolling snake eyes is 1/6 x 1/6, or 1/36 (2.8%). This is because there are 36 possible outcomes in this joint probability.

I could be wrong, but I can only think of one card that has you roll 2 dice in Guardians, which is Demorgan the Inciter. What is more useful to our discussion is a grouping within a single die roll, such as the chance to roll a 1 or 2.

This is called a mutually exclusive union - you cannot roll both a 1 and 2, but you can roll a 1 or 2, and combine the chance of doing so. Where our joint probability multiplied possible outcomes, our union adds possible outcomes. So rolling a 1 or 2 would be 1/6 + 1/6, equaling 2/6, or 33.3%. Now let's put that union probability to practical use.

Planes of Entropy allows you to ignore your draw for the turn and draw 1D6 cards instead. You would only want to do this when it was beneficial to you, unless you are a major risk taker. How do we determine when it would be beneficial?

Except in unusual cases, the worst a Guardian can draw is 1 (Finn with LDL and no Swamps in play would be a 1). In this case you would want to play Planes of Entropy, because you can roll no lower than 1.

But what if your card draw is 2? We established above that rolling a 1 is a 16.7% chance, and rolling a union of 1 or 2 is a 33.3% chance. This means you have a 33.3% chance of drawing the same or worse, but a 66.7% chance of drawing better. Not bad odds at all.

When you get to a 3 card draw, it would seem like your chances get a little risky, right? You have a 33.3% chance of drawing worse by rolling a 1 or 2. You have a 50% of drawing better by rolling a 4, 5 or 6. Looking at it another way, you have a 66.7% chance of drawing the same (3, 4, 5, 6) or better, still great odds. Sometimes it's all in how you look at your chances, because you really only lose if you roll a 1 or 2. Now that you know how to calculate the odds, you can decide if it's worth the risk.

Does Planes of Entropy give you card advantage? Remember that rolling a 5 or a 6 three times in a row, which you would need to gain a significant card advantage, is 2/6 x 2/6 x 2/6, or a predictive possibility of something like 3.6%. That's like flipping 5 heads in a row. What Planes of Entropy might do, however, is keep you from being at a disadvantage, which is still a good thing.

OPPONENT DISCARDS

There is one last concept I'd like to cover for card advantage, and that's opponent discards. If we defined card advantage as the ability to put more assets into play than your opponent, forcing an opponent to discard (and thus lose assets) meets this criteria. There are only a few cards that have the ability to force your opponent to burn cards. Are they worth using?

The short answer is no. When you force an opponent to discard a card, they do lose that card as an asset. However, that card is replaced by drawing the next card in their deck. A true card advantage through discard (and I'm going to drop another Magic the Gathering reference here) is "decking" your opponent. Decking means to run your opponent out of cards. The best example of this is the Millstone deck in Magic the Gathering. The Millstone card forced your opponent to discard two cards every turn; if you could get four of them in play, you could force your opponent to discard 8(!) cards per turn and rapidly run them out of cards, ending the game. While running out of cards doesn't end the game in Guardians, your chance of winning becomes slim if you're forced to rely on the cards you have remaining on the table. With that said, let's take a look at cards that discard.


Argammond's Vision, Assassin of Shadow, Shadow Strike, The Hand of Chronos: these cards have some usefulness...for instance, if your opponent has a Dragon Wing Lord as his upcard (or in his deck for the Assassin's ability), you can make sure it never gets under his Shield by forcing it into the discard pile. However, by playing one of these cards, you are giving up a card that could have been used as an asset, to force the opponent to also give up a card. Shadow Strike and The Hand of Chronos are even worse because you're giving up 2 potential assets (your upcard and Shadow Strike), or even 3 (for Chronos you lose 2 Power Stones as well!) to get rid of 1 opposing card. I'm not saying Chronos is useless; when you have an opportunity to win the game without letting your opponent get more cards or unturn Shields, it's extremely useful. What I am saying is that while all these cards are useful in specific ways, they do not lend themselves to card advantage.

Shadow Warrior, on the other hand, is exactly what we are looking for. It is an asset (a creature) that can also impact your opponent's deck. You *don't* have to attack your opponent's upcard, but think about this for a moment. In normal combat, you blindly put your attacker down, not knowing what your opponent is playing, and hoping for the best. In contrast, attacking an upcard is a thing of beauty: you can see the card you're attacking, and decide whether to attack or not. If you can beat it, you've just hurt your opponent twice...not only does he lose his upcard as a resource, but he will also draw one fewer card this turn. Remember that using our example above of 30 creatures & 55 cards, there's a 51% chance that the opponent's upcard will be a creature, great odds for the Shadow Warrior. The Warrior does have 2 areas of concern: if it can't beat the opposing upcard, it can't use it's ability; also, it takes up 30% of your Shield vitality (although that 9 Vitality is not bad for attacking with). So Shadow Warrior does provide a small amount of card advantage.


Chant of the Osirians is the closest card to a Millstone in the game. If you know you are going to be facing an Undead deck, pack at least 5 of these in your deck. Each time you play this, your opponent loses 5 cards, or 9% of a 55 card deck. Play it 5 times and they've discarded 45% of their deck, almost half! This card also works well with Sebek, Queen of Magicians - if you see your opponent has an Undead upcard and you've got an unused Power Lunch in your hand, put Sebek into play to discard the Power Lunch, go get the Chant of the Osirians, and force a 5 card discard. Nice! Though I don't like discarding cards to draw cards, that is worth it.

That wraps up Deck Manipulation and Card Advantage. I hope it was worth a read. Part 3 was going to cover cost/benefit of cards, but I think we'll expand our probability theory to include other cards that use dice rolls instead, and push cost/benefit ratios out to part 4.