Variance, Part 2

“Glad we were naive enough to bring that extra fuel.”

—Perceptor to Topspin, Last Stand of the Wreckers #3 

In this second part we're continuing to explore the properties of the variance of our flips. In the first part we considered simple deck compositions in order to isolate, and understand the effect of single-icon, double-icon, "blank", and white cards. In doing so, we focused on either attacking or defending. Motivated by a recent article by Scott Van Essen on the design process of the white icon, we'll look at our deck as a whole, aimed at functioning both on attack and defense. At the time of this writing, this is a completely unexplored territory, at least by us, the players. And −with the sole exception of Metroplex− mixed-icon decks haven't really made an appearance so far. The complexity of the problem we face in building a mixed deck comes from the need of reaching conflicting objectives, as every icon which adds to our attack is effectively blank on the defense, and vice versa. But, even when focusing solely on an aggressive or defensive strategy, there are cards we'll always play because of their primary effect, on-color or not. No matter how aggressive we want to build our car or Dinobot deck, we'll always space for Start your Engines and Dino Chomp! Very few blue players would not at least consider the inclusion of Sparring Gears, etc.  It's a problem intentionally rooted at the core of the game, to prevent it from becoming some sort of chess with spells. The main focus of these notes will be on trying to figure out if and how well we can turn off-color cards from an hindrance to our flips, to a more useful resource.

Attack-Defense Tradeoff

In his article, Van Essen points our attention towards an unexplored aspect of the white cards, i.e. their potential to increase the rate at which we flip our off-color cards when we actually need them for their icons. He does that with a realistic example: an aggressive deck with 3 🄱, 3 🄾🄱 cards, and 3 🄱🄱 cards splashed to improve our defense. (As a side note, Handheld Blaster and Improvised Shield aren't dead draws in opposite color decks, and their intended value will probably become more and more evident with the rise of balanced-color decks.)

In the first part of these notes, we're reexamining Van Essen's example, in order to start understanding what the tradeoff is between offense and defense when replacing orange cards with white cards. Clearly, the same argument applies to blue decks after the replacements Bold ↔ Tough and orange ↔ blue. We'll plot the actual values of the attack and defense bonuses, as opposed to normalized values, to resist the temptation to compare quantities varying on very different scales. There's a problem with normalizing the attack and defense bonuses with respect to their zero white case in a heavy orange deck, as any improvement with respect to the −approximately zero− initial defense bonus will look like significant, even when very small. While losses in the attack bonus might look like modest, when they're actually significant. Therefore, in figure 1 we're plotting the attack (horizontal axis) and the defense (vertical axis) bonuses in terms of their common unit of measurement, i.e. one battle icon. Every deck composition corresponds to a point in this attack-defense plane, obtained by taking the average of the outcomes of 100'000 simulated attack flips and 100'000 simulated defense flips. Starting with 25 🄾 and zero 🅆 cards, we progressively replace 🄾 with 🅆. This way we draw a curve in the plane. A drift towards the right/left side corresponds to an increased/decreased average attack bonus. Similarly,  a drift towards the upper/lower part of the plane corresponds to an increased/decreased average defense bonus. Figure 1 shows four such curves, for different values of Bold. Each value of 🅆 corresponds to a different dot, with multiples of 3 🅆 enhanced and labeled to help readability.

Figure 1 (100'000 simulations per point)

Before discussing figure 1, let's consider the general behaviors these curves can exhibit. The four scenarios are summarized in figure 2. We might be improving both our attack and defense, case 🅐. This is the best scenario, and we should almost always make changes to our deck which put us in this situation. Alternatively, we might be trading attack for defense, case 🅑, or defense for attack, case 🅒. Whether to make these changes depends on our goal, and on how convenient the tradeoff is. The remaining case, 🅓, in which we decrease both our attack and defense, is something that we should consider only if we need the effect printed on the cards worsening our flips. This is what happens when we add actual blank cards (no battle icons) to our deck. Blank cards often have the strongest effects. They are of the like of I Still Function! and One Shall Stand, One Shall Fall. In case 🅓 we are trading both attack and defense for these effects. 

🅐 increasing attack and defense

🅑 trading attack for defense

🅒 trading defense for attack

🅓 decreasing attack and defense

Figure 2

If we now look again at figure 1, we can see that the kind of deck changes we're making, 🄾🅆 replacements, initially correspond to case 🅐 (increased attack and defense), and then become a trading of attack for defense, 🅑. The position of the turning point can be easily understood, as it approximately corresponds to the number of white cards maximizing our probability of flipping the first, but not the second white icon. It's the same quantity we've considered when dealing with Lionizer, that we're plotting again in figure 3 for more values of Bold. This probability is fairly simple to derive, and very useful, as it depends only on the number of white cards in our deck. Adding white cards up to the turning point corresponding to our target value of Bold (or Tough) is a no-brainer. It's after that point that things became less intuitive. That's when we need to figure out whether we're gaining more than what we start losing. 

Figure 3 (200'000 simulations per point)

Let's consider a rather extreme example, in which our target Bold count is 2, and we don't rely on Tough. We replace 18 🄾 with 18 🅆 , and our new attack bonus is 80% of what it was. We've lost a 20%. On the other hand, our defense bonus is 170% of its initial value. It looks like a good trading, but it's not, as we've lost 20% of 4 on attack (=0.8), to gain 70% of 0.6 (=0.4) on defense (figure 1). Referring to these actual values of +0.8 attack and −0.4 is only better at a very first glance. Once we compare these averages to the intrinsic variance of our flips, they lose most of their statistical relevance. The horizontal arrows in figure 1 show the standard deviation (SD) of our attack flips for different values of Bold and white. It ranges from about 1 to about 2, which is already greater than these little changes. (The SD of our defense flips is not even shown, as ±1SD would be already greater than the range shown in the figure. Besides, the SD is not even the best way of quantifying our variance when the expected value is often zero.) 

We might conclude that this kind of analysis is either misleading (20% loss vs. 70% gain still being disadvantageous), or totally pointless (a +0.4 gain on an average affected by an uncertainty at least twice as big). The truth is that we're just referring to quantities that are only loosely connected to our perception of the way we flip. We're much better at understanding integrated quantities. We never flip +0.4 blue icons. What we perceive is how often our flips are better or worse after we've changed our deck. We might still need to normalize our flips (the comparison to our previous version of the deck), but we also need to integrate them over many flips. And we better keep the single icon as our unit of measurement. But not fractions of icons, as they don't exist.

The integrated quantity we're going to adopt is as simple as the probability of flipping better than our old average. We know we would flip 4 orange icons on average, with Bold 2 and no white. How often do we flip more than 4 after we change our deck? Is it greater or smaller than it was before? This is the same measure we adopted in the first part of these notes, and we're using it again here to determine how much white we should actually play before starting to harm our flips.

Diminishing Returns

Let's consider the case in which we expect to attack with an average Bold count of 2, and defend with no Tough. At zero white, our attack bonus is most often 4, and our most likely defense bonus is zero. Therefore, we'll evaluate the probabilities of flipping more than 4 🄾 on attack and at least 1 🄱 on defense after each replacement of an orange card with a white one. These two probabilities are shown in figure 4.

Figure 4 (100'000 simulations per point)

We can increase our odds of attacking with a better bonus with about 5÷7 white cards, just like we know from figure 3. We're increasing our odds by about 10%. This is the kind of information we can use in deckbuilding. 

At the same time, we're also increasing our chances of flipping at least 1 🄱 on defense by the same amount. Our defense bonus is still zero half the times, but 1 (and sometimes more) the rest of the times. As long as we don't count on attacking very often with Bold much greater than 2, this represents a strict improvement to our flips. We're in case 🅐 of figure 2. 

Pushing the amount of white past this point means trading some of the additional attack we've gained for our defense, case 🅑. But we can now exploit a useful feature of maxima. As our P (attack bonus > 4) has a maximum at 6 white, it also shows a pseudo-plateau around 6. On the contrary  P (defense bonus > 0) keeps increasing. What we lose on the attack is less than what we gain on the defense, and 9 white cards still approximately correspond to +10% chances of flipping more than 4 🄾  , but our P (defense bonus > 0) has now increased by +15%. 

Three more white cards, and we've traded everything we gained on our attack, for an irrelevant defense bonus, and we shouldn't go that far. This is what's meant with pushing a little bit past the point of diminishing returns.

This conclusion could've been already reached by just looking at figure 1. But recasting it in terms of integrated probabilities allows us to make deckbuilding decisions based on quantities that we can immediately relate to our games.

These specific boosts of +10%, +15%, etc., depend on how many orange and blue cards we play. But the amount of optimal white does not. Most of the times, figure 3 is everything we need. In an aggressive/defensive deck, it shows the recommended amount of white optimizing our attack/defense bonus at the expected count of Bold/Tough. A good rule of thumb at low values of Bold/Tough is that we can safely play an additional playset of white cards on top of that amount. More white than that might quickly drag us into a scenario in which we trade relevant attack for negligible defense or vice versa. 

External Links

Scott Van Essen - Secret Origins: The White Battle Icon

Scott Van Essen - Coming Into Focus (Spoilers)