Exploding Dice!


1. INTRODUCTION:
Written By GAMESLAYER989

Helloeveryonethisisme: Gameslayer989 aka NooBrainer and welcome to my first ever guest article!

One thing that’s near and dear to my heart in this game is the original Yoda, the Legacies version, the guy who cost me at least £80 and that's totally not the reason why I love him so much. I don’t love people for the money, let’s make that clear! Even though he is absolutely fantastic at bringing home the big bucks with his Special chaining and all, but you know what? This article isn’t even about him! It’s instead about another great thing that Star Wars Destiny uses: DICE. Not just any dice though, oh no, I want to discuss the dice of infinite potential: EXPLODING DICE!

2. EXPLODING DICE:
Exploding dice are dice where IF you roll their maximum value, you get to roll the die again and add up the results. If that next roll also rolls its maximum, you get another roll, and this keeps going on until you stop rolling that value. What does this have to do with Star Wars Destiny? Well, we kinda have a similar thing with a couple of dice in the game, and two prominent characters with conditional exploding dice have started to regularly feature at top tables across the globe.
Reylo1jpg

I am of course, talking about Spark of Hopes Rey & Kylo - Bound By The Force. These two characters both have Specials which allow you to resolve the die and reroll it instead of removing from the pool. Yes they’re conditional, but these can technically go Infinite - especially if pitted against each other! Rey's Special gives 2 Shields, Kylo's Special breaks 2 shields. Rey's Special gives 2 Shields... LONGEST. FIGHT. EVER!

rey kylo fightjpg

So this begs the question, what’s the average value of their dice? Obviously we can’t say the value of each symbol is equivalent, but for the sake of simplicity in our maths we shall assume that every Symbol is equal in value. This means that Rey and Kylo's dice each have a 1/6 chance to roll a Blank, 3/6 a 1 value side, 1/6 for a 2 value side, and 1/6 for a Special, which is 2 and a reroll. Without the reroll this would give us an average value of 7/6, about 1.2 per die.
kylo rey mathjpg

However, that Special is obviously worth more than a 2. It’s worth 2 + the average value of the die again, which has a 1/6 chance of hitting another Special, which is 2 + the average value of the die again, and again, and again; you see the problem here. The Special is more tricky to evaluate since it includes itself in its own definition, and we know that this can be up to infinite in value.

With that in mind, how can we get an average value when one of the possibilities is infinite? Well, the chance you roll multiple Specials in a row is highly unlikely. At some point the probabilities are so insignificant as to be inconsequential, they won’t affect the final result in any meaningful way. To solve this we can use a geometric series, but that involves maths that I once understood but now can no longer remember. Never fear though, the internet is here! Apparently, the average value of an exploding die simplifies to:
exploding dice1jpg

To put that into perspective, it then averages more than a die with three 2 sides, such as AtG Jyn Erso, who has an average value of 8/6: 1.3333, but less than a character with a 3 side and two 2s, like SoH Maul, 9/6: 1.5.
exploding dice2jpg
Evidently, Exploding dice really adds a good amount of average value to a die, in this case adding effectively 1.4 to a single side of the die which… hey hang on, that added value is on the Special! The logic checks out, the Special has a value of 2 + 1.4, 3.4. We can also understand it in this was: Every time you resolve the Special and it explodes you are functionally getting another die roll, which has an average of 1.4.

3. ANYDICE:
Of course, you can’t trust everything you read on the internet. Especially when it involves maths that gets complicated enough that someone, like me, can easily make an error. The thing about dice rolls and probability though, is that it’s very easy to simulate. I used a handy program on the internet called ANYDICE.com that allows you to simulate dice rolls, including exploding dice. You can check the code here, make sure to select the Summary tab in data when you output, as those are the values we care about. Yes, I know I could have written my die a lot simpler, but this is to illustrate the point! The result is that on average, the die has a value of 1.4. Perfect! Our formula and our simulation line-up!

probability review THUMBNAILjpg

4. THE CASE OF BLACK ONE:
Now, you might be worried that this is already getting a little too divorced from reality, because after all: These dice explode only if specific conditions are met. Alright, I’ll bring it down a little and talk a little bit about a card I really liked and used back before it got rotated out, a card I swore was effective, and yet no one else seemed to play. The original, Awakenings BLACK ONE.
gameslayer quotejpg

Black One is a vehicle that costs 4 resources and ONLY has a 2 Ranged and a 3 Ranged damage side. Naturally, you want your dice cards to be cost effective, and once Legacies dropped Black One was competing in a world with T-47 Air Speeders, Fang Fighters and Y-wings! The Fang Fighters in particular was a stern competition having identical damage sides, 2 and 3 Ranged, but costing just 3 resources. All the competing vehicles each have maximum potential damage ratios equal to or higher than the resources spent playing them; in an ideal world the FANG FIGHTER is 3 damage for 3 resources, T-47 AIRSPEEDER is 2 damage for 2 resources and so on. Also, the more your resources are spent on a single die, the weaker you are to removal; that card that removed your Black One stopped you using 4 resources worth of die for this round! There was however one thing Black One did that all other vehicles didn’t: It exploded! The die that is, not the vehicle.

Black One has a Special that does 1 damage and explodes the die. Without the exploding die the average value of Black One would be:
(3+2+2+1+1+0)/6 = 1.5

Meanwhile the Fang Fighter is:
(3+2+1+1+1+0)/6 = 1.333
exploding dice3jpg

The difference in value between the 2 dice is 1/6, for a full resource. Just not worth it! Counting in the exploding die however, Black one becomes:

1.5 / (1 - 1/6) = 1.8

That’s almost a difference of 3/6! The Black One Special has a value of 2.8, meaning it has a 3 damage side, a 2 damage side and a 2.8 side. Does that make it worth the resource? Well I don’t think we quite have enough information to decide that.
exploding dice5jpg

The thing is, this game also includes discard-to-rerolls and Focus sides. The Fang Fighter is actually a pretty weak die value-wise, but it’s ability to have a 1 for 1 damage ratio (1 damage for each resource spent) can really help maximise the value of Focus sides. The best support decks typically have a lot of Focus sides, and aim to ignore this whole average value nonsense, instead forcing the maximum value side, and there Fang Fighter has Black One beat, as they both have a maximum side of 3. Or does it?

5. MR. POSTER BOY:
Yoda poster boyjpg
Hah, and you thought my earlier rambling had no point to it!

YODA is perhaps heroes' biggest poster-boy for focus shenanigans, with his Special sides allowing him to focus dice and so Special chain. The deck I have been alluding to many of you would know as Drive-by-Shooting, taken to last year's Worlds 2018 by Joe Colon aka HonestlySarcastc from TheHyperloops. This was my version of that deck:
Drive-byjpg

This deck, similar to Joe Colon's original deck, was designed to have a good amount of focus and rerolling in order to maximise the value of our large number of dice. However, I would often have situations where I had a lot of excess focus, and Black One helped resolve that issue. Since Black One has a non-conditional exploding die (you always get to resolve it and reroll it), you can think of resolving any Focus side to turn its die to a Special as if it is in fact converting focus into damage at a 1:1 ratio. It’s not great, especially in a situation where you’re turning Black One from a 2 or a 3 Ranged damage side to its Special just to "benefit" from the Focus side. However, if we assume there will be no rerolls or focusing afterwards, then as the Black One Special has an average value of 2.8, that isn’t particularly good seeing that the alternative would be a 3 Ranged damage side!

6. THE MISSING LINK: REROLLS
There’s been a missing link to the equation I have set out earlier however, NAMELY, the more you decide to reroll, the higher your average value becomes. This is obvious, when you look at it this way: If we reroll every time we hit a Blank then obviously we are increasing a die's expected average value. This is where the maths gets really complicated, so instead I’m going to go the simulation route for this.

Let’s assume that on a die there are only 2 sides we are willing to immediately resolve. For the Fang Fighter that would be its 2 Ranged and 3 Ranged damage sides, for Black One it's the Special and 3 Ranged damage side. Any other result we will reroll once, and then we take what we can get.
exploding dice7jpg

exploding dice8jpg

Do you see that? Calculating with one reroll the Fang Fighters expected value increased by 0.39 (from 1.33 to 1.72), whereas the Black One's increased by 0.44 (from 1.8 to 2.24). The more rerolls, the greater the average becomes with exploding dice over non-exploding dice. If we decide to take the 2 Ranged instead of rerolling, we actually go up to 2.28. Fancy that, I should not have been rerolling those 2 sides when I only had a single reroll and no focus.
7. ADDING FOCUS SIDES:
Speaking of Focus sides, let’s add them into our model. This program assumes we have 1 Focus and 1 reroll available to us: If we don’t hit anything with our reroll we instead Focus into the maximum side. If we do hit our maximum side though, then we’ll have a Focus left over that can be used to gain a value of ... let’s say 1.

exploding dice10jpg

For a Fang Fighter the average value becomes: 3.28. A consistent value of 3, but sometimes we get to keep our Focus. Without rerolls, our average value becomes 3.17, the reroll really doesn’t do much, as we are turning to the 3 Ranged damage anyway, there’s just an extra 1 in 6 chance we get higher than a 3 Ranged damage because we get to keep our Focus.

exploding dice9jpg
For Black One the average value becomes: 3.72. Why is this increase so significant? Well, the more rerolls the greater the chance of rerolling into a Special, meaning more explosions, and the Focus inherently rerolls. At this point we actually go down to 3.64 if we take the 2 Ranged damage when it’s immediately offered, and with no rerolls at all we go down to 3.40. A stark contrast to Fang Fighter, which benefits with a reroll by only 0.11 more compared to 0.24.
OPRAHjpg


With a second Focus and a Reroll? 4.28 vs 4.98. For the Fang Fighter this makes logical sense, we cannot do anything more with 2 Focuses than we could with 1, so the total value only increases by 1, the value of the Fcous. On Black One however every extra Focus has a total value of 1 + (1/6) chance of another explode by way of a Special. At this point it is correct to only ever take 3s, otherwise Focus to the Special and resolve. I had a play around and 4.98 was the best I could get, which means an extremely likely chance to be able to convert 2 Focuses and a Reroll into a 5 off of the back of a Black One. It bears repeating however the assumption that Focuses can always go somewhere else for a value of 1, and only 1, not focusing a Blank into a 2 or anything else.

8. ONLY SLIGHTLY WRONG:

fang fighter vs black onejpg

Now the real question
, is a 0.7 value difference worth a full resource to play the vehicle? Well remember how earlier I mentioned the 1:1 ratio? Well now, we can see that in our scenario here of 1 Reroll and 2 Focuses, the Fang Fighter has an average value of 4.28, costing 3 resources, as opposed to Black One's 4.98, costing 4 resources. By this measure, Fang Fighter is still 0.3 resources more efficient. I guess I might have been wrong after all. Still though, it’s better than people give it credit for! Under the right circumstances it is only slightly inefficient compared to the Fang Fighter!

9. REY?
So, back to Rey. I mentioned Yoda for a reason dammit, and that’s because his Special is a perfect fit to provide Rey with the shield necessary to make her Special explode! When you turn one of Rey's dice to a Special after grabbing a Shield with a Yoda Special, you are getting an average value of 3.4 from Rey's die exploding. This is because the explodey bit is worth 2 instead of 1, massively increasing the value of focuses (in this instance Yoda's Special) used on it.
yoda reyjpg

What about with the rerolls and a second Focus?

Well, from Rey's die that gets aggressively rerolled and focused, 6.60! Of course, we are having to spend part of a Yoda Special that could have been something else, like a Resource; if we include that in the calculation then it drops  to a ‘humble’ 5.59. It’s unlikely we get 2 Yoda Specials though, more likely we only get 1 and instead have an extra reroll because of Rey's Power Action, bringing us to 4.49 or 3.59 if we include the opportunity cost (the fact that it could have been something else instead of a Shield) of Yoda focus. Still though, when most character dice have a maximum of 2 or 3, then Rey is quite a cut above most of the other characters in the set, provided she can always meet her explosion condition, such as by having Yoda as her partner. Pity it’s all in Shields...
Since you got this far, it would be remiss to give you all this theoretical knowledge without a deck to apply it to, so here is a Reyda deck that I have been tinkering with lately, attempting to abuse those exploding dice as much as possible. Remember, be greedy, and HAVE FUN EXPLODING.

10. DECK LIST:

REYDA deck listjpg
8. FINAL THOUGHTS:
Thishasbeenme
, Gameslayer989akaNooBrainerthankyouforreading, I hope you all learnt something from this, because I certainly did. AnyDice is an awesome program and I highly recommend you try it out for yourself. And thank you to YOUR Destiny for hosting and editing this article, because oh boy you guys did not want to be stuck reading the first draft :P