Variance, Part 1

“We can't act without knowing what's happening.”


—Bumblebee to Prowl, More than Meets the Eye #23

Introduction

As white cards add to our attack/defense only in a indirect way, there is the risk of overlooking them, and being led to think that they dilute the effectiveness of our flips in a similar way as blank cards. This is incorrect, at least as long as we stay within a very reasonable limit on the number of white cards we play. The aim of these notes is to clarify the difference between white and blank cards, and to highlight the unexpected similarity between white and single-color, double-icon cards. 

In order to quantify the effect of flipping white cards, we first need to understand the variance of our flips, and that having non-zero variance isn't necessarily bad. Variance can have both a positive and a negative effect. And in building our deck, we need to make sure that we are taking advantage of it more often than we suffer from its negative consequences.

Let's proceed step by step, and first consider the simplest possible deck composition: 40 cards all having the same single-color battle icon. In the following we will assume that color to be orange. But focusing on blue would lead to the same conclusions.

40 🄾 cards

P ( n < 2 ) = P ( n > 2 ) = 0 % 

This is very simple. We always flip 2 orange icons. Our flips have zero variance. This will be our benchmark scenario. 

Acquiring some chances of flipping more than 2 icons will be considered positive, while exposing ourselves to the possibility of flipping less than 2 icons will be negative.

The previous histogram doesn't really add much to our understanding of this trivial case. But this way of visualizing the likelihood of our flips will be useful as we move on to consider less heterogeneous deck compositions.

Individual effect of double-icon, white, and blank cards 

Let's now consider how the distribution of our flips is affected by the inclusion of double-icon, white, or blank cards. We will first consider these different kind of cards in isolation. Later on we will consider their interplay. In order for their effect to be visible in our histograms, we will include them in a reasonable amount. We are going to consider three different decks. In each one of them we will replace 6 🄾 cards with 6 cards of one of those three kinds.

34 🄾 , 6 🄾🄾 cards

P ( n > 2 ) = 28.1 %

Because of the occasional presence of 2 icons on the same card, our flips acquire some variance: i.e. our result is no longer always 2. Two orange icons is still our most likely result. But when it is not, our result is greater than 2. The number listed as P ( n > 2 ) is the probability of flipping more than our 2 original icons, and it is reasonably high. About once every four times we flip cards, we should expect to see either one or two double icon cards. This P ( n > m ), and the analogous P ( n < m ), notation will be adopted in all the cases we'll consider from now on.

34 🄾 , 6 blank cards

P ( n < 2 ) = 28.1 %

Blank cards also add variance. But this time it only affects our flips in a negative way. There is a non-zero probability   P ( n < 2 )  of flipping one or two blank cards. We shouldn't be surprised to see that this is exactly the same probability of flipping more than 2 orange icons in a deck with 6 🄾🄾 cards. 

More interesting is the fact that the individual probabilities in the two histograms are the same: We have now the same chances of not flipping any orange icon than we had before of flipping 4 of them. We have the same chances of flipping just one icon than we had of flipping 3 before. And our chances of flipping 2 icons are unaffected. We can summarize this by saying that blank cards have exactly the opposite effect as double-icon cards in modifying the distribution of our flips.

34 🄾 , 6 🅆 cards

P ( n < 2 ) = 0.7 % vs. P ( n > 2 ) = 20.0 % 

  Let's now replace 6 🄾 cards with the same number of 🅆 cards. How do they affect our flips? If we flipped 2 🅆 icons, and then a third one and an 🄾 (or even two more 🅆's and no 🄾), that would be worse than our original 2. But how often does that actually happen? This  P ( n < 2 ) is less than 1%. On the contrary, we will flip more than 2 orange icons about 20% of the times .

This scenario is remarkably close to the one in which we played 6 🄾🄾 cards instead of 6 🅆 cards. Which makes perfect sense, as we are almost always replacing 1 🅆 with 2 🄾, which is equivalent to flipping 1 🄾🄾 instead of 1 🅆 during our first two flips.

Technical note: All these plots are generated by flipping from the top of 100'000 simulations of their corresponding deck. All the numbers listed in what follows  as "probabilities" are actually normalized occurrence numbers.

Interaction among double-icon, white, and blank cards

Now we should have a clearer picture of the effect these different kind of cards have.  We also saw how to quantify their individual effect in terms of how much, and in which direction, they shift our flips. Let's now consider their interaction, starting from putting together 🄾🄾 and blank cards.

28 🄾 , 6 🄾🄾 , 6 blank

P ( n < 2 ) = 23.5 %    vs.    P ( n > 2 ) = 23.5 %

Same effect, opposite direction. Symmetric distribution.

28 🄾 , 6 🄾🄾 , 6 🅆

P ( n > 2 ) = 45.6 %    vs.    P ( n < 2 ) = 0.6 %

White cards increase our chances of flipping 🄾🄾, and boost our result to values grater than 2 more than 45% of the times.

By the same line of reasoning, should we expect 🅆 cards to negatively affect our flips when played in conjunction with blank cards? Let's see. 

28 🄾 , 6 🅆 , 6 blank

P ( n < 2 ) = 28.4 %    vs.    P ( n > 2 ) = 10.7 % 

This is our first non completely trivial result. Let's compare these numbers to the scenario in which we were just playing 6 blank cards, and no 🅆. We see that our chances of flipping less than 2 orange icons are unchanged (28.4% vs. 28.1% is not a real difference). But we now gain a small chance (10.7%) of flipping 3 orange icons.

Why aren't our flips worsened by the presence of white cards? Aren't they increasing our chances of flipping blank cards? Of course they are. But the main point is that 🅆 cards are replaced by two additional cards. Even if one of the two new cards is blank, the remaining one is very likely to be an 🄾 card which just replaces the original 🅆, setting back our result to the most likely value of 2. 

Playing white cards in addition to blank ones doesn't make our flips any worse. If anything, it mitigates the negative effect of the blank card, by sometimes increasing the total orange in our flips.

Let's now put the three kind of cards together in the same relative amounts. The following histogram refers to what we might call an "archetypal mono-color deck", in which blank card are often very difficult to avoid, double-icon cards are used to counterbalance the negative effect of blank cards, and white cards tilt the distribution in a way that favors us.

Remember that "blank" doesn't necessary mean no battle icon in this plot. It just mean no icon of the color we are counting. For example, in orange cars deck, they might be 3 Start your Engines (blue icon) and 3 I Still Function! (no icon).

22 🄾 , 6 🄾🄾 , 6 🅆 , 6 blank

P ( n < 2 ) = 22.4 %    vs.    P ( n > 2 ) = 34.6 %

"Too Much White"

What happens if we keep increasing the number of white cards? The following plots show that we slowly increase P ( n < 2 ) and decrease P ( n > 2 ). But this happens so slowly that the difference between playing 6 and 9 🅆 is a negligible 2%. We might consider the 6% increase in P (n < 2 ) from playing 12 white cards not negligible anymore, and that might set our upper limit. But consider that we need to go as far as playing 18 🅆 cards to double our chances of flipping less than two cards. Even if we do that, our chances of flipping more than 2 is still as high as 30%.

Everything said so far holds whenever we don't have Bold. And our intuition probably tells us that we should stay away from white icons if plan on flipping many cards while battling. Our intuition is wrong.

19 🄾 , 9 🄾🄾 , 9 🅆 , 6 blank

P ( n < 2 ) = 24.6 %    vs.    P ( n > 2 ) = 36.1 %

16 🄾 , 6 🄾🄾 , 12 🅆 , 6 blank

P ( n < 2 ) = 28.3 %    vs.    P ( n > 2 ) = 35.8 %

13 🄾 , 6 🄾🄾 , 15 🅆 , 6 blank

P ( n < 2 ) = 33.5 %    vs.    P ( n > 2 ) = 33.4 %

10 🄾 , 6 🄾🄾 , 18 🅆 , 6 blank

P ( n < 2 ) = 39.5 %    vs.    P ( n > 2 ) = 30.0 %

High values of Bold/Though

We should check our distributions by increasing the value of Bold one unit at a time. But if our result doesn't get worse at Bold 6, that's enough for it not to be affected at lower values either. The following two histograms shows the probability distributions at Bold 6 before and after adding 6 🅆 cards to a deck in which we have 6 🄾🄾 and 6 blank cards.

28 🄾 , 6 🄾🄾 , 6 blank cards (Bold 6)

P ( n < 8 ) = 36.1 %    vs.    P ( n > 8 ) = 36.1 %

22 🄾 , 6 🄾🄾 , 6 🅆 , 6 blank cards (Bold 6)

P ( n < 8 ) = 35.7 %    vs.    P ( n > 8 ) = 40.8 %

Playing those white cards is actually a good idea! Our new most likely value is 8.  P ( n < 8 ) is slightly decreased (less than 1%) by the presence of our 6 🅆 cards, while P ( n > 8 ) slightly increases (about +5%). The bad news is that for higher numbers of 🅆 cards this would be no longer the case, at least for values of Bold as high as 6. We would eventually start seeing the negative effect of diluting our orange component. But 6 is a safe (sometimes a recommended) number of white cards, even if we plan on boosting our attacks as far as to Bold 6. For lower values of Bold the upper constraint becomes less and less relevant, with 12 🅆 cards as a reasonable upper value with no Bold.

This is probably even more relevant to blue decks than to orange ones. As typical values of Tough are about half the values of Bold we can reach, we don't really need to be too conservative in playing white cards in our blue decks.

We can summarize what we've seen so far this way: