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Monday, October 19, 2015

Star Wars Armada - Stats Analysis: Expected Damage

"Sir, our algorithms are failing!"
By Eric "R1H4" Riha

Now that I've completed my consciousness transfer from The Mathemagician into the R1H4 Battle Computer, it's time to delve into my favorite topic again: MATH.

Star Wars: Armada's unique dice arrangement - both in facing and in color - makes calculating "Expected Results" extremely difficult or super fun(!) depending on how you look at it. Defensive abilities that modify those dice directly (rather than simply modifying the final result) add a layer of complexity that makes Armada a highly variable (and super fun!) game to play.

Today, we are going to talk strictly about Expected Outcomes from each set of dice in the quantities we see from Capital Ships - specifically in terms of damage output. We will also look at these outcomes through the lens of a basic Pareto analysis; that is to say, which outcomes are greater than 80% reliable. We all know there's a chance we can roll 4 blanks on 4 black dice, but how common is it? Should the rare chance of that occurring impact how we make decisions? Using this "80% Rule" allows us to make decisions based on reasonable outcomes when chance/risk is involved.

Unlike the world of Flames of War or other games where the outcome of a die roll requires additional "calculations" (skill + modifiers = required value to hit), Armada's dice deliver results that are applicable across all (or nearly all) scenarios. For example, if I am firing out of the front arc of a Nebulon B using 3 Red Dice, the outcome of my attack roll is the same regardless of what my target is. Different targets may have different mitigation profiles (defense token sets), but the initial attack roll always has the same probability to damage, crit, accuracy, etc.

Let's start our analysis with a single Red Die.

Courtesy of BoardgameGeek
Based on the Die Faces, we can pick out our probabilities for damage pretty easily.

We have a:

3/8 (37.5%) chance for Zero damage
4/8 (50%) chance for 1 damage
1/8 (12.5%) chance for 2 damage

Based on these probabilities, we can say we have a 62.5% chance of dealing AT LEAST 1 point of damage with our attack roll. Using our "80% Rule", we would say 1 Red Die is not reliable enough to provide ANY damage, and we should not make any tactical decisions that rely on 1 Red Die.

This seems pretty straight forward - most of us wouldn't trust an attack with 1 Red Die to do any damage. That being said, the situation becomes significantly less clear as we add dice to the attack roll. If I need 5 damage to destroy a ship, should I plan to use one or two of my Assault Frigates to take it down? Should I use my second Assault Frigate or can my Nebulon B deal enough damage, allowing me to focus my Assault Frigate on another target?

The mathematical situation is further complicated by the existence of die with different facings, and multiple die of different colors used in the same attack. This makes a traditional Binomial approach difficult - especially when Red and Black die results can produce 2 damage per die.

Instead, we will be using the Monte Carlo Method to closely approximate our Expected Outcomes, specifically in terms of the amount of damage we should expect to roll given our "80% rule". For example, let's take a look at the Expected Outcome of 2 Red Dice based on 10,000 random rolls (courtesy of MS Excel).

Our chances of rolling 1 or 2 damage is just about the same, at about 35% each, while our chances of rolling zero damage is only about 14%. We can use the individual outcomes to produce a cumulative estimation, which is similar to saying "This is the probability of rolling AT LEAST X amount of damage."

Looking at our results, we see that we have about an 86% chance of rolling AT LEAST 1 damage when rolling 2 Red Dice. Since this is above our "80% Rule", we can safely lean on 2 Red Dice to deal 1 point of damage (before defense tokens).

Moving up the scale one more Red Die brings us to the basic Neb B shot out of it's front arc.

Our results show that, while it does have the potential to deal more damage, it's not very reliable in doing so.

Adding a Concentrate Fire Command to bring the die roll to 4 dice gives us.....

....a reliable method for rolling 2 damage.

Another option for that Nebulon B Frigate is to give it a new title: Salvation. Doing so changes the damage output of the Crit die faces to 2 points of damage. What effect does this have on our damage potential?

And with Concentrate Fire...

We always had a gut feeling Salvation was a good title for the Neb B - now we have the Math to back it up :)

Lastly, let's take a look at our good friend: the Gladiator I.

Front Arc:
Side Arc:
Side Arc w/Concentrate Fire Command (5 Black Die):

This analysis tells us that we can rely on delivering around 5 points of damage from a dual arc shot; 6 points if we've timed that Concentrate Fire command.

But this only leaves us with half of the Gladiator picture. 9 times out of 10, we'll see the Advanced Concussion Missile upgrade card on that Gladiator, which means we'll need to know if we roll a Crit to activate the upgrade.

In fact, several upgrade cards rely on Critical Effects. To make good use of those upgrade cards and the points we spend on them, we should make sure the ship rolls enough dice to reliably trigger them.

We'll cover the math behind Criticals in my next article.

Until then, 01100110 01101100 01111001 00100000 01100011 01100001 01110011 01110101 01100001 01101100 00100000 01111001 01101111 01110101 00100000 01110011 01100011 01110010 01110101 01100110 01100110 01111001 00100000 01101100 01101111 01101111 01101011 01101001 01101110 01100111 00100000 01101110 01100101 01110010 01100110 00101101 01101000 01100101 01110010 01100100 01100101 01110010 01110011.

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