So last time we introduced what we termed equivalence, reducing all the information of a card or a deck to a single statistic (a number, usually a modification of the base stats) which represented not only the base stats BUT ALSO the relative merits of the abilities (including its USAGE and EFFECT). We noted that effect varied depending on size and configuration of deck, but we did not take the time to note then that this game may have been designed for a 3-card deck to begin with and we probably won’t get into that at this time. For DECK EQUIVALENCE we discussed combos in a limited fashion using simple cases. (We also talked about standardization, which just means setting a base for equivalence to something like 100, which is what they do with IQ scores and other standardized tests, in case you’re wondering).
In this installment, we’re going to get to the promised bit where we introduce how usage (probability) affects our modification equations. This will help us answer questions such as “Is it better to have a 7% chance of a 25% boost or an 8% chance of a 20% boost? AND HOW MUCH BETTER!?
So, probability intimidates a lot of people and can be confusing, but the basics are as simple as flipping a coin! (see what I did there?) Remember simple rules like independent trials/gamblers fallacy, the commutative and associative properties, that dividing is just multiply the inverse, etc… or don’t – I’ll remember them and you can just come along for the ride. Let’s talk game application:
In-game, abilities (cards) and combos (decks) have usage (probability % of activating). The usage of the later is given, and the former has been researched and is given in-game as very, relatively, or adjectivelessly, low, average, or high. In general, game balance dictates that the better the usage, the worse the effect, or the lower the base stats. It’s like the old wag, “You can have cheap, fast, or easy—pick two.” In this game generally you can have stats, usage, or ability (pick two). Hence, PD Strange is a relatively low usage and SP Spider-man has lesser stats. There are exceptions and the formula were going to look at now will help us discern between cards that are close too (e.g. when deciding between high stats or good ability, just how high should the stats be?)
Let’s take a single card first. Typically a card has three important values [stats], [effect], and [usage]. Stats will be easy, we’ll just use the stats value as is (using standardization was helpful for abstracting away when we didn’t know the value of a particular card, for example, comparing relative combos in absence of the particular cards used for them… it’s less helpful here), so we have stats; let’s use algebra and set stats = to a variable we’ll call S (for “Superman” – just kidding, S is for “stats” :P). Next, let’s look at effect. Now typically we see effect written like “Boosts ATK of Bruisers” or “Degrades DEF of opposing Team” etc. with different effect sizes based on rarity and skill (e.g. 16% for a skill 1 UR+ alignment boost, 30% for a skill 10 L+ alignment boost, 32% equivalence on a skill 10 L+ degrade; see charts). Because multiplying two percentages is scary for some (and unnecessary), we’re not going to do that here. Effects are not probabilities, they are proportions. They are the kind of percents that are merely fractions of a whole.* So this % is not a “chance of” anything, it’s simply another integer number. We’re going to call it “E” for “Enough already!”** E is going to be the value from the chart (C) times the size of the deck (D) used (NOTE: drop this term for self-boosters***), TIMES the stats (S) of the card in question {we could also use our standardization here, but won’t}, | So, E=mc... wait, that’s not right, E= C*D*S which, for say, a survival deck of with FL Cyclops would give you C=.3 D=3 S=15564 E=.3*3*15564=14008**** and in a 5 card deck (again, assuming all bruisers of averaged out equal stats) E=23346.*****
Now for usage (or variable “U”). We’re going to assign this variable the percentage equal to its chance of activating (which is exactly what usage means). So for our FL Cyclops (Average, at skill 10) this means at .5 (50%). Now a quick tutorial on probability: a 50% chance of $100 is equal to a 25% chance of $200 and a 100% chance of $50 – that is, over time, repeatedly taking any of those 3 chances over and over again the same amount of times (say, a million times), will get you an expected $50 million – they all average out or “regress to the mean” and become, for our purposes, equivalent. Now, of course, an individual draw may be different ($50 isn’t $200 after all), but over repeated draws, these are the same chance. It’s the same in battle math – Cyclops won’t activate every time (sometimes he will with his full ability, sometimes he’ll give you 0 boost), for this card, at skill 10, you have a 50% shot at it when on the wing of a survivor deck, so over time, the expected value of his boost is going to be U*E > in this specific case, in survivor mode, 14008*.5=7004. {Maybe I’ll make a note about what you’d want to do for the 5-card version, but frankly, if you’re just comparing two cards, it doesn’t matter, as long as you do 3 card to 3 card – the only issue is with self-boosters which are way more powerful comparatively in a smaller deck, or contrawise, team boosters are more powerful with a larger team to boost).
SO – back to card equivalence. We have three numbers, S (stats), E (effect), and U (usage). To compare two cards, the formula you should use is S+E*U à this will give you an adjusted stat that takes into account effect AND usage J
In our Cyclops example the Card Comparison Adjusted Stat (CCAS) would be 15564+14008*.5=22568
Savvy readers may note that you can get this by calculating S*(1+DCU) or S(1+3*.3*.5). If so, they can just wink at me knowingly.
So let’s take our card comparison on the road! Let’s use PF Phoenix and PS Cable, calculate their adjusted stats and see how they compare with Cyclops!
- PF Phoenix
- S = 15052
- E = .3*(3*15052)
- U = .6
- CASS=23180
- PS Cable
- S = 14578
- E = .345*(3*14578)
- U = .7
- CASS=25140
So, as a strict wing card in a survival deck (composed of similarly stated hero bruisers) we see that Cable > Phoenix > Cyclops. Although, that last one, just barely. You may, in the market see Cyclops trading higher than Phoenix (but neither higher than Cable), that would mainly be because of COMBOS.
Which brings us to decks.
The same thing done above can be done on decks. Before we talk combos, let’s talk deck positioning. Let’s say I happen to have picked up a Cable, Phoenix, and Cyclops (as described above) with respective high, relatively high, and average usages. They make a nice set and have a combo to boot! Where should I put them in my survivor deck? To calculate this we’ll make the same sort of calculations as above and adding them, but double the proc for the center ~and stuff, Trust me, I’m a doctor~
- Option 1 Cyclops Center: 86070
- Option 2 Phoenix Center: 83882
- Option 3 Cable Center: 82487
^Add the stats, multiple the Usages by the Effects (center gets double), add the original stats to stats times Usages time Effects and you’ll have your pre-combo deck compares!
To add a combo, simply add the combo usage*effect. In this case add .8*.05*(S sum or Adjusted Deck Stats) where ever you’d like (as long as you’re consistent).
OK, did I do everything I promised to do??? This is feeling long and I’m not getting a lot of audience feedback until I press submit… So I’ll go ahead and do that and you guys can let me know if I need to clarify anything.
At this point you should have a method for comparing any two cards of any rarity or ability anywhere in the game and, indeed, any two decks (almost as good as a simulator eh?!). There are some nuances we skimmed over, but this should be “quite enough to be getting on with.”
*all percents are, but whatever.
**Seriously, I shouldn’t have to annotate this.
***As an alternative quick standardization, you can divide self-boosters by the deck size and compare that directly to the boost terms as is. You might find, in that case, that a L+ skill 10 self-booster gives 90% boost which is the EQUIVALENT of a 30% boost to 3 cards (of similar stats), or in other words, a Skill 10 L+ Alignment boost. Similarly, a Skill 10 L+ ATK/DEF self-booster, gives 72%, or 24% equivalent, comparable to the skill 10 L+ Boost/Degrades (right between them in a sense, but degrades are better when you work out the equivalence J)
**** I shall be rounding appropriately throughout for convenience.
***** I’m actually lying here. 5 card decks suppress procs and are a little more complicated, but my PsySense is telling me I’m already pushing the envelope for keeping my audience so let’s just pretend it’s this simple :D