Abstract:

In order to gain a better understanding of the state of my cube and better learn how to improve various failings of the overall system, I analyzed my cube through comparative power and toughness across various CMCs. This method was selected over rigorous playtesting due to a lack of available play testers as well as reducing the variance in player skill level and preconceived perceptions of the cube.

Because of this analysis, the gameplay created from this designed system leads to an environment that rewarded incremental value and slower game plans, making fast and aggressive strategies far less viable than desired.

Terminology:

Magic: The Gathering is commonly abbreviated as Mtg or MTG and will be used as such throughout this paper. Additionally, this paper analyzes the structure of a larger system through various aspects of a MTG card. Specifically, the comparison between Converted Mana cost and the Power and Toughness of a single card.

            The Converted Mana Cost or CMC of a Magic: The Gathering card is found in the upper right hand corner of a card, and represents the total cost of resources that playing a card consumes. In the case of the Alpine Grizzly pictured above, it costs two resources – or mana – of any type, as well as specifically a green mana, for a total of three. As such, it’s CMC is three.

A creature’s Power and Toughness represent how strong the creature is, and is found in the bottom right hand side of a card. Specifically, the number on the left – power – represents how much damage the creature deals, and the number on the right – toughness – represents how much damage the creature can take before it dies.

Unless specified otherwise, creatures deal damage to each other simultaneously. In the event of our Alpine Grizzly, if it became engaged in combat with a creature with two power and two toughness, both would die. Similarly, if it became engaged in combat with a four power, four toughness creature, both would still die.

Finally, there is an important distinction between offense and defense within Magic: The Gathering. When on the offensive, each creature is an individual, but when defending, creatures can block as a group. For instance, a player on defense can pile fifteen creatures with one power and toughness each in front of a single creature with ten power and ten toughness.

Background:

Magic: The Gathering has many formats in which players can battle against each other using a common set of rules. One of these formats is called “Limited”, where the players must build decks from a limited set of MTG cards, hence the name. The way this functionally works is that players buy several packs of magic cards, and then build a single 40 card deck from only the cards they opened, and play against their peers who have completed the same task. There are many sub-genres within the limited Genre of Magic, but the focus of this paper is that of cube. A Magic: The Gathering Cube is a custom curation of MTG cards to create a unique system and environment for a specific style of gameplay. A cube changes the basic formula of limited, as players no longer need to physically purchase packs, but instead simulate packs through drawing from the Cube pool. While many cubes contain only the most powerful magic cards ever printed, many others vary drastically since a cube is custom to the creator, leading to many drastically different environments.

Specifically, my cube is only allowed to include common and uncommon rarity cards and is tailored to be a higher power level compared to most traditional sets, similar to that of a Masters Set. This allows me to showcase some of my favorite cards that are unfortunately not powerful enough for other formats of magic, or aren’t legal anymore. Additionally, this main restriction limits the complexity of the set, which is specifically tailored to my playgroup. Many of the other players that interact with this cube do not have prior experience playing Magic, so this allows for the creation of balanced play experiences, while still having large opportunities for memorable plays. Finally, on a practical standpoint, this, as well as several other restrictions, force me to be more creative with the cards that go into the cube.

My cube is designed to create a synergistic environment, with high choice and medium variance. As Mark Rosewater discusses regarding choice and variance in an article [1], experienced players prefer having high amounts of choice, and competitive prefer low variance, while inexperienced prefer high variance. As this cube is intended to be highly replay-able, the variance creates new situations every time it is played, while the choice creates more dynamic game states.

Furthermore, this cube attempts a heavily synergistic environment, where the combination of certain cards creates more powerful decks than what the card reads by itself. The danger of this type of construction is parasitism, where strategies have very little overlap with one another, greatly decreasing choice when constructing a deck. The most recent example of this effect in a traditional retail set was Ixalan. Ixalan was built as a Tribal set, where Pirates, Vampires, Merfolk, and Dinosaurs fought against one another. The problem was that each tribe had no mechanical overlap with each other, so any Merfolk in a Dinosaur deck weren’t particularly useful. As Nico Bohny says on Channel Fireball, “Ixalan is a draft format without real draft decisions” [2]. This discontent was not felt just from the professionals, but across the player base, with many citing it as one of the worst limited sets for a while. Additionally, the parasitic nature of the cards meant that very few competitive decks throughout Ixalan’s legality in a different format saw any play at all.

For the purpose of this analysis, I observed the power and toughness for cards across all converted mana costs to observe the relative strength and speed of my cube’s format. In magic, there are three large archetypes across any set, which are Aggro, Midrange, and Control, each of which prioritize different strategies to winning. Aggro decks are singularly focused on killing the opponent as fast as possible, and necessitate a large number of cards at one and two CMC. Midrange look to maximize value out of every card that they play, attempting to get more than a single card’s worth of value out of every card. Finally control decks look to abandon the early game, for the sake of overwhelming the other decks in the late game through sheer quantity of resources.

Each of these three archetypes are akin to that of rock-paper-scissors, where Midrange will usually beat Aggro, Aggro will usually beat control, and control will usually beat midrange. However, if there is an imbalance in this triangle, the most rock-like deck will win on average the highest percentage of the time. This became a concern of mine after observing matches taking longer and longer to play to completion.

Methodology:

For this analysis, I organized every card within the cube that created power and toughness, then plotted the difference between offense and defense in Excel, creating a bar-and-whisker graph. Each quadrant in a bar-and-whisker graph represents 25% of the sampling, and single dots qualify as outliers.

As mentioned previously in Terminology, some cards generate multiple creatures per card. These were assigned as multiple instances of their respective power and toughness while attacking, and a single instance of combined power and toughness while defending. For instance, a two CMC card that makes two creatures with one power and one toughness would count as two instances of one power and one toughness while attacking, and one instance of two power and two toughness while defending.

Although this distinction may not completely accurately reflect the ramifications of having multiple creatures from a single card, it presents a decent approximation. Furthermore, other limitations of this analysis include the exclusion of abilities in combat. Examples of these abilities include but are not limited to: first strike, flying, trample, menace, activated abilities, and deathtouch. These abilities change both offence and defense, but were not included within the scope of this analysis.

Data:

Results:

The first immediate conclusion from this statistical analysis is the impact of cards that generate multiple creatures, and how they change the implications of the closed system when. Across every CMC, the power and toughness of creatures are both reduced by about .25 when attacking as compared to blocking. Although this does not initially appear as a significant difference, this difference is amplified during gameplay.

For instance, when comparing cards of equal CMC, the attacking card usually dies, while the blocking creature usually survives. This is true across every CMC. This greatly disincentivizes attacking, as the aggressive deck will usually lose their creature,  preventing them from actually killing the player. This is then amplified if the aggressive deck is playing second. Most cards in Magic: The Gathering cannot attack the same turn that they are played, so if the aggressive player is going second, the defending player will have a card of one additional CMC to block with. In this situation, the attacker’s creature will virtually always die, and the defender’s creature will virtually always survive. In fact, the only way for an aggressive deck to gain an advantage in sheer power and toughness is if the defensive deck misses a resource, and cannot play a card with the next CMC.

The implications of this power/toughness imbalance only allows for aggro decks to exist if they have a large quantity of spells to continuously remove their opponent’s creatures. This then reduces the number of creatures that exist within the aggressive player’s deck, meaning that there were fewer creatures that the midrange and control decks had to remove before taking over the game state against the aggro decks. As such, any medium to high powered creature played later in the game often lacked an answer from the aggressive player, enabling those handful of cards to quickly take over the game state. This is the reason that previously cut cards from the cube such as Snare Thopter, Cinder Elemental, Tatyova Benthic Druid, and Rakshasa Gravecaller all felt far more powerful than they immediately appeared. Without an appropriate answer from the opposing player, these cards would quickly take over a game within a matter of two or three turns.

Furthermore, the lack of effective offensive individual cards placed a larger emphasis the cards that can make multiple creatures such as Jade Mage, Murmuring Mystic, and Belfry Spirit. When the individual cards cannot attack profitably, creating so many creatures that the opponent physically couldn’t block all of them was one of the few ways left to effectively deal damage to opponents. This importance of cards like these lead players of the cube to frequently complain about the strength of them, when the underlying issue was not just the strength of these cards.

The perceived issue of these cards was then compounded with the general type of spell used to remove your opponents creatures. Most of these removal spells were spot removal effects, which only removed a singular creature on the board. When a singular card creates two creatures, if the opponent spends an entire card to deal with one of the two, they essentially used an entire card to remove only half of yours. This meant that tokens were a powerful strategy for the midrange strategies, further weakening aggressive strategies. Furthermore, the best answers to token strategies are not very effective against the other cards in the cube. There are some cards that give all creatures minus two power and toughness, but these were unwanted by the aggressive decks, and didn’t have much efficacy against the other individual creatures. Specifically, 75% of 1 drops, 50% of 2 drops, 50% of 3 drops, and only 25% of four drops would die to this type of effect, meaning many of these cards would not see play at all, yet were still effective against the aggressive decks.

Conclusions:

In order to reduce the imbalance found within the cube, and to provide more to ensure aggressive decks are viable, an overall reduction of creature’s toughness across all CMCs is needed. Specifically, an increase in the number of three mana creatures with three power and two toughness would lead to an increase in aggressive strategies. In particular, it makes aggressively designed one and two CMC cards become more relevant, as a two power one toughness creature will still kill a three mana three power two toughness creature. This ensures that these one CMC creatures will not become irrelevant until much later in the game. Additionally, a three-power creature will end the game in only 7 turns of attacking, instead of 10 turns like that of a two-power creature.

The other impact of increasing the number of three-power two-toughness creatures is that it makes larger creatures more viable as well. A four-power four-toughness creature against two two-power three-toughness creatures cannot effectively attack, since only a single 2/3 will die in exchange for the 4/4. Against 3/2s however, both 3/2s will die in exchange for the single 4/4. This idea is supported by Ari Lax on his block, where he also advocates for the “average” creature to be worth three power and two toughness [3].

Specific investigation will have to proceed to determine which cards to replace with others. This investigation however, revealed how important small adjustments to a system can be, and the rippling ramifications of a slight imbalance in a system. Mathematical analysis of a closed system can provide insight to which components are over or under performing in relation to the others, as well as reveal underlying reasoning behind feedback that you receive as a designer.

 

Thank you for reading.

 

References:

[1] Rosewater, Mark. “Variance, Part 1.” Making Magic, Magic: The Gathering, 16 Dec. 2019, magic.wizards.com/en/articles/archive/making-magic/variance-part-1-2019-12-16.

[2] Bohny, Nico. “28 Days Later: Ixalan Limited.” ChannelFireball, ChannelFireball, 2 Nov. 2017, www.channelfireball.com/all-strategy/articles/28-days-later-ixalan-limited/.

[3] Lax, Ari. “Limited Design: 3/2 For Three Is the Place to Be.” Less Practical Magic, 17 Jan. 2019, armlx.blogspot.com/2019/01/limited-design-32-for-three-is-place-to.html.

An integral part of designing a game is following user-centric principles and iterating in order to provide the player with the best possible experience. However, some games have difficult components that are expensive in both money and time in order to iterate upon. This was the case with the game that I am the lead developer on that will be on Kickstarter later this year. Tiny Trees is a competitive Tree-building board game where unlike a large number of board games, it doesn’t lie flat on your table, but instead becomes a physical object in three dimensions. As you grow your tree, you have to try to earn the maximum number of points while also literally balancing your tree so it doesn’t collapse.

The game consists of 42 hexagonal cards that you slot together in order to grow a physical three dimensional tree. It was extremely time consuming to iterate on these components since the prototype needed high quality cardstock and had to be cut out by hand and individually drawn on. As such, the design process had to be predicated more on math and statistics rather than continuous playtesting in order to not waste valuable resources.

We had to determine what arrangement of cuts in the cards we wanted. The very first prototype had cuts on all six sides of the hexagonal cards, but I found myself growing roughly the same tree every time since there was no restrictions on what I could grow. Additionally, if each side of the hexagon had only one slit to reduce complexity, each side would have only two states: cut and not cut, represented below with a six digit binary equivalent.

When that six digit binary equivalent is converted to our standard ten digit unit of numbers, that gives a total of 63 possible arrangements of cuts on the hexagonal cards. However, this does not account for rotations or mirrored images. For instance, the leftmost hexagon shown above is still functionally identical when rotated 60 degrees. As such, when accounting for this repetition, there are in fact only 12 unique designs.

Although this number of unique designs seems innocuous, it was significant in my process since it established what could or could not be done with the cards. While we could create cuts that weren’t centered on each side of the hexagon or multiple cuts on one side, having knowledge of what options were available to us allowed for accurate design.

On a similar note, designer Mark Rosewater has repeatedly said that “restrictions breed creativity”, and I found that to be exceptionally true (Source). In the case of Tiny Trees, the very first paper prototype had cuts on all six sides. However, this did not lead to building any interesting trees since players would default to what they were familiar with rather than going out on a limb and trying a new structure. At the opposite end of the spectrum, if all of the cards had only two cuts, the players wouldn’t have enough choice in what they could grow. Having all of the cards with only two cuts didn’t provide enough options to the player, and six cuts provided too many, so the ideal must be somewhere in between. In the final version, there are five cards with two cuts, four cards with four cuts, and two cards with six cuts for each of three types of trees. We decided on this arrangement for three reasons: It gives an average of roughly 3.29 cuts per card, each number of cuts had a total roughly equal to 12 – the exception being the cards with two cuts – and the distribution of the values was appealing since each level has one fewer card. Additionally, by having more cards with only two cuts, it allowed the trees to become more interesting while the higher number of cuts allowed players to still have sufficient options in growing and balancing their tree.

While this math for achieving the average number of cuts per card (in which you add the total and then divide by the number of cards) is very simple, it still influenced our design decisions. Since we were aware of the average number of cards as well as the specific distribution, it gave us a much clearer understanding of the system that was in place and how that affected the player’s perception of the game. This in turn allowed us to quickly fix and understand any underlying issues that arose in playtesting. For instance, we were able to identify that even though playtesters didn’t directly address an issue with the distribution of number of cuts, we were able to more accurately identify it as the underlying issue due to the knowledge of the distribution.

At the end of a game of Tiny Trees, players earn points based primarily on two factors: the hexagonal cards that they grew onto their tree, and lifeforms that are also found on the cards. We added the lifeforms to increase the strategic depth of the game, as well as make the decisions more interesting.

The three types of lifeforms: Beetles, Mushrooms, and Birds

From the player’s perspective, without an additional incentive, there was little reason in selecting the cards with fewer cuts since it restricted their growth and made balancing their tree more difficult. By adding lifeforms as a mechanic, we had to balance three main factors: the number of lifeforms, the distribution of lifeforms, and the amount of points that the lifeforms were worth. In order to achieve this through math due to the limitations of our ability to iterate, we used a hypergeometric calculator liberally in determining these factors. Hypergeometric calculators are often used in card games where you draw some number of cards and then want to know the odds for drawing a certain card. In this context, we wanted to know and be able to control the odds more precisely rather than just intuition. An important design decision that we had made previous to adding lifeforms was that each option available to the player should be of equal value to a similar decision. This comes from the number of cards for each type of tree being completely equal, even to the distribution of number of cuts. Thus, we wanted to do the same for lifeforms, but be willing to alter that methodology given the numbers and math behind them. As such, we needed three types of lifeforms and more than one of each. We ended up settling on six of each type of life form, separated evenly between each type of tree. The advantage to this is that if a player wants to grow their tree with all of the birds available to them, then they aren’t shoehorned into one specific type of tree.

When it comes to the math, that means that roughly 43% of all of the cards have lifeforms, and that there is a 45% chance that exactly one of the top three cards that the players can choose from will have a lifeform, and a 25% chance that none will have a lifeform. Obviously that percentage changes as more cards are selected, but this knowledge helped us quickly refine the game – similar to the knowledge of the distribution of cuts. When it comes to what cards have the lifeforms, we focused on the cards with fewer cuts on them. Our reasoning was that if lifeforms are worth additional points, then the players should be incentivised for restricting their growth options, but not so much that it is obviously better than the ability for options and balancing your tree.

Then came the question of how many points should each lifeform be worth. Since the cards reward you for collecting more of the same type, so should the lifeforms. However, the scaling of points can be either linear or exponential. We decided on an exponential growth model, so that players are incentivised to collect the same type, but that collecting them incidentally doesn’t have that large of an impact on the overall point total. Specifically, the first two are worth one point each, the next two are worth two points each, and then the final two are worth three points each.

By designing our game while taking the numbers into consideration rather than just working off of intuition, we didn’t have to playtest or iterate on our designs as much as we would have to if we only gathered data from reactions from playtesting. While using this math obviously doesn’t eliminate the importance of playtesting and iteration, we were able to save a lot of time and resources from creating additional paper copies of the hexagonal cards and allowed each playtest to be more efficient since we were able to more accurately identify underlying issues and needed balance changes.

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For those who do not know, a M:tG Cube is a curated list of Magic: The Gathering cards to create a unique environment to play games of Magic. These curated lists are similar to be played like ordinary sets, where players open packs of cards and play games of magic using just the cards they opened. However, since cubes are curated lists, the powerlevel of cards can be much higher than what is found in normal sets, or create wild environments that also wouldn’t be seen in an ordinary set.

My cube is designed to feel similar to an ordinary set in terms of composition, but slightly higher power level than the average magic set.

The cube originated from a pile of excess cards that I had lying around, and I’ve been continuously updating it for one and a half years at this point, ensuring balance between each viable strategy.

The cube has 10 explicit strategies, with over 20 possible specific strategies that are viable and can win. With only 375 cards in the cube, nearly every card must fit into multiple strategiees in order to make the inclusion, while also being of a high enough power level to be considered in the first place.

To see the full list of cards, please click here.