A Basic Metric of Complexity Creep

I get it, you got memes to make and takes to serve piping hot, here are the click-bait worthy “deets” you can tweet.

1. Complexity Creep is “the biggest danger to the game,” yet part of WotC’s method to abate it - rotating formats - might be ineffective as complexity creep is alive and well.
2. Mono-white is.... the least complex color. Despite having the second most “hidden complexity” (excluded reminder text), even when added in mono-white is still the least complex.
3. Mono-white’s missing complexity is not made up with an increase in keywords.
4. Mono-white appears as the least complex mono color per set more than random chance, especially in recent sets.
5. The missing complexity missing from mono-white is greater than the total complexity mDFCs as a whole add to a set.

1.1 Nosh-Sized Nuance

Ah, I get ya, someone tweet first or you want to chime in to the conversation with some of value without reading this. No worries, just read below:

1. This definition of complexity is painting in very broad strokes. Thus while its use is limited, if anything stands out as fishy at this resolution it might warrant further investigation. Indeed mono-white’s abysmal complexity is curious.
2. Complexity is not equal to power (by this definition and likely any other). A five word, one-mana instant with oracle text “Target player wins the game.” is objectively powerful and not complex.
3. Mono-white’s missing complexity is more extreme recently (≈M19 forward) but also a historic issue (starting ≈ ONS to ALA).
4. Complexity of a group of cards is based on an average. The number of mDFCs in a set is small comparatively (hence while the comparison is provocative, it is not necessarily informative).

2. Introduction

Today’s topic takes the form of power creep’s twin: complexity creep. For those unfamiliar, complexity creep is the game getting more complex over time. While Magic: the Gathering (MtG) players often like to herald any new MtG set as the decline of the game, Mark Rosewater (MaRo) has stated he thinks complexity creep is “the biggest danger to the game” [7, 6]. By MaRo’s own admission complexity will continue to go up, so part of the strategy to combat the creeping complexity once again lies in rotating formats like standard [7]. Yet with Ikoria’s recent mutate mechanic, one might start to wonder if entire blocks of standard are rising in complexity despite their attempts to abate it.

One might dare say mutate is more complex than the mechanic devotion. Yet how does one go about quantifying this? By the number of rules introduced or modified? By the number of judges who cry during pre-release? While such metrics might be useful, when we returned to Theros in early 2020 Mark Gottlieb wrote an article addressing complexity creep. Foremost Gottlieb starts be introducing a basic definition of complexity, which we can formalize a bit here [5].

3. Core Definitions, Terminology and Notation

Let the pool of cards under consideration be represented as 𝒞. Then we can define a card, C ∈𝒞, as the ordered set of faces:

Additionally, let 𝒲_facei^C be the multiset of words fromo the oracle text on a face face_i^C as defined on Gatherer, and 𝒲_C the corresponding multiset for all faces of the card [2].

Then we can further define the reminder text of a face r_facei^C as the multiset of words in the reminder text of the oracle text on a face face_i^C as defined on Gatherer, where correspondingly

and naturally ∅ ⊆ r_C ⊂𝒲_C. Let r_C^max be the maximal reminder text a card could have given its oracle text e.g. a card with the oracle text “Flying, first strike” with maximal reminder text would become “Flying, first strike (This creature can’t be blocked except by creatures with flying or reach. This creature deals combat damage before creatures without first strike.).” Then we will let r_C^est estimate the cardinality of r_C^max i.e. r_C^est ≈|r_C^max|. I’ll break down how this estimate is produced in section 5.

Now with the pieces in play we can define complexity as explained by Gottlieb [5]:

For those indisposed to math, this is a fancy way of saying that the complexity of a given collection of cards is the average text on said cards. Now as I’ve eluded to above with the definitions of reminder text, Gottlieb also gave the definition of complexity sans-reminder text as:

Additionally, I’m expanding Gottlieb’s definition with the estimate maximum complexity as:

i.e. for each card, removal all reminder text then add as much reminder text as appropriate using the estimated text measure defined above.

4. Caveats

Now before we go any further, there are a series of caveats we must address:

1. As stated by Gottlieb this is a very basic measure of complexity. It is a broad stroke to get a sense of the complexity landscape. If something odd arises when using this metric it suggest it ought to be investigated further, not as a death knell to signal a rally against Wizards of the Coast (WotC).
2. By this metric complexity does not equal powerful. Please see table 1 for two stark counter examples (this is why complexity creep is a twin of power creep, not an alias).
3. In the article Gottlieb refers to values with and without reminder text. By his own admission “if including a mechanic’s reminder text would drop the card’s text below the minimum font size, we pop that trap door and poof—it’s gone.” This means cards with reminder text are just the cards as printer rather than “maximal” reminder text (when is the last time you’ve seen first strike’s reminder text outside of a core set?).
4. In the article Gottlieb acknowledges that WotC is aware of wordiness and has taken a variety of steps to try and reduce it. In other words, this metric is biased as WotC attempts to design less wordy cards. Consequentially more wordy cards could be seen as more damaging than they appear.
5. In this article I say rules text a lot. This is a bit of a misnomer. I tend to use rules text (what some associate with the actual words printed on the card) and oracle text (the text you will see in Gatherer or Scryfall with any errata) interchangeably. I am always, unless otherwise specified referring to the oracle text.
6. Gottlieb mentions that they use complexity in general for commons and uncommons but recently applied to rares and mythics. All cards are part of the game, so we shall apply it to all.

As stated in caveat 3 they do not, at least in that article, consider the maximal reminder text e.g. the “Flying, first strike” example I previously mentioned. It is my personal opinion that this might be the biggest omission in their “crunched” numbers. Why? Well the omission of reminder text when it fits is WotC having its cake and eating it too. It leverages the “entrenched playerbase,” the players of which have internalized these snippets, whilst allowing such basic metrics of complexity to shave off a few words all (it twas indeed a heist of words). This circle backs to MaRo’s sentiment about new players always starting the game at the same place and the game continuing to get more complex. The more keyword actions and abilities that forgo reminder text the even more extreme the complexity becomes (although by this metric it “reduces” complexity).

5. Estimating Maximal Reminder Text

I would not be surprised if, but nonetheless doubt, WotC has a database of the card’s oracle text with the maximum reminder text in it. However, we as players do not. Therefore before we proceed we must generate it.

After Ikoria released, I prototyped a Scryfall syntax extension Scry-Soup which added keywords to Scryfall as outlined in table 2. While not all of these were later implemented - independently - by Scryfall, they did add keywords.

Thus the first step towards r_C^est was to, for each keyword, grab every card with said keyword. Why? Well whether or not reminder text is included is subjective and time dependent. With every card that has a given keyword we can then filter out cards that lack parens, as that is where reminder text is housed. Then we can search for reminder text that is keyword specific. This can be done by splitting the oracle text into lines, searching for the line the contains the reminder text and then checking if the parens from before are also in this line. If we have a hit, we will associate the reminder text with the keyword (regardless if it is or this specific keyword or not).

Now that we have the multiset of speculative reminder text per keyword, we can whittle it down. We will keep only the elements with the most or tied for the most multiplicity in the reminder text (the text that occurs the most) and any text that contains the keyword directly regardless of how often it occurs. This produces a nice manageable 315 pairs of (keyword,remindertext) which I could then validate as correctly belonging to that keyword or not.

The reason there is more speculative pairs is partially because of a few false hits (only around 5), but also many alternate versions of reminder text:

• “to mill a card, put the top card of your library into your graveyard” versus “to mill a card, a player puts the top card of their library into their graveyard”,
• “look at the top two cards of your library, then put any number of them into your graveyard and the rest on top of your library in any order” versus “to surveil 2, look at the top two cards of your library, then put any number of them into your graveyard and the rest on top of your library in any order”, or
• “when this creature enters the battlefield, target player may put zndrsplt into their hand from their library, then shuffle,” “when this creature enters the battlefield, target player may put sylvia into their hand from their library, then shuffle,” etc.

We can then average these representative reminder text snippets per keyword (not weighted). Now for each keyword (as indicated by Scryfall) in a given card’s oracle text we can simply add the average as another coarse “good enough” estimate after removing any preexisting reminder text e.g.

6. Experiments

With our three definitions of complexity (with, without, and maximum reminder text respectively), we can investigate complexity creep. Additionally we can look at the “per face” level (average number of words on a single face of a card) or on the “per card” level (average number of words on all faces on a single card). This is becoming more relevant with Modal Dual Face Cards (mDFCs) becoming more prevalent. For these experiments we will be using “all” sets, as indicated by Scryfall with at least 100 cards, and “new” sets (M19 forward, non-silver-bordered). This arbitrary 100 card threshold is because Scryfall indexes many sets that most players would not consider a set (e.g. the oversized commander face cards from pre-cons).

6.1 An expansionist view

6.1.1 All together now

With our basic metric of complexity and out experiment we can take a look at complexity across all cards in a set (not just commons / uncommons or rares / mythics) without reminder text, with it, and with an estimate of the maximum reminder text. Whether looking since alpha (fig 1) or since m19 (fig 2), complexity creep is on the rise. Looking at Kaldheim, we see a massive spike whether look per face or as one. Additionally, treating the mDFCs (a fraction of the cards in the set) as one bumps the average complexity by around two words.

6.1.2 A prismatic viewpoint

Now that we know that complexity creep is on the rise (along with an ever rising player resentment of player’s feeling white is getting the short end of the stick), perhaps we should look at complexity through a mono-colored lens. Same experiments as above, but now we are looking at exclusively mono colored cards.

Well, well, well look at those fitted lines (figures 3, 4). Red is clearly the winner of recent designs. Impulsive draw effects are quite wordy. What is surprising isn’t that white is currently the least complex color, but how vast the gap is. In tables 3 and 4, you will find the average number of words missing in mono-white from the closest next color across recent and all sets respectively. Further in tables 5 and 6, we see that the percent of time mono-white is the least complex color is starkly above random chance.

While many may take this data to leap on the “mono-white bad” train, what is of interest here is how much of a difference the estimated maximum reminder text makes (table 7). White plays second fiddle to blue for the most hidden complexity. Further, and I can not emphasize this enough, even with the hidden complexity revealed (estimate maximum reminder text), mono-white is still the least complex color.

So now that we know mono-white has some hidden complexity, where might all of these words come from? Well I suspect it might be in the deluge of keywords that we often see in white (flying, first strike, vigilance). So lets take a look at that next.

6.1.3 The key is in the keywords

I thought, perhaps, mono-white’s hidden complexity could be explained in keywords. Perhaps the “keyword soup” strategy WotC seems to be sprinkling in white (“Odric, Lunarch Marshal,” “Odric, Master Tactician,” “Akroma, Angel of Wrath,” and “Akroma, Vision of Ixidor”) may be paying dividends. So again we will look at “all” sets and recent sets using Scryfall’s keyword annotation. Additionally we will look at the average and total keyword “mentions.” A keyword is mentioned if the keyword is on the card, not necessarily that the card has the keyword i.e. “Creatures you control with flying get +1/+1.” As shown in figure 5 white is neither the most keyword heavy color of even keyword-matters (as shown in the mentioned subplots) color.

6.1.4 Kaldheim

While the regression lines may not be too hopeful, lets zoom in even further into Kaldheim. Once again doing the complexity and keyword experiments. Here we see a shift (figures 6 7). Yes mono-white is still the least complex... however it is second in keywords.

But can we trust these values? Many MtG expansions do not solely consist of new cards; rather, they often include many reprints. Could mono-white’s past be boosting its stats?

6.2 Novelty only

Here we will repeat every experiment again, but only using the card in a set if that is their first printing. As expected whether across all of MtG or just the recent sets, complexity is going up (figures 8, 9). Switching to a prismatic view we see red and green shooting up, with white still at the bottom (although catching up) (figures 10, 11). Looking at the average missing words from the closest color (tables 9, 8) we see generally strictly increases. As for the percent of times mono-white is the least complex color (tables 10, 11) we see general more of a mixed bag. When considering reprints mono-white was always the second most hidden complex color, however now mono black is attempting to claim that spot (table 12). Turning now towards keywords we see that white’s stagnation is recent (figures 12). Kaldheim’s stats don’t change too much since there are not too many reprints in the set (18 cards out of 285 in the main set 10 of which are lands).