Attack Dice Spreadsheet

[piousflea]

A lot of the math in the game is pretty opaque. For example, when is Aegis 1 better than Armor +2? How much weaker is a 3+3+3 triplestrike compared to a single 9-dice attack? How much better is the Emerald Tegu's 9 hp and 3 armor when compared to a Wolf's 10 hp and 2 armor? How many attack dice do you need to kill 14 HP and 1 Armor in a single Battle Fury?

This spreadsheet can help:
https://docs.google.com/file/d/0B4xG98ZoLE5sY3FwZUpnamItZFk/edit?usp=sharing

Note that this is an XLSX file so you cannot actually use the spreadsheet on Google; you have to download the file to your hard drive and open it with Microsoft Excel.

As far as I know the math is correct. If anyone finds an error in the spreadsheet please let me know.

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Answers:

1) When is Aegis 1 better than Armor +2?
If you are starting at 0 Armor, +2 Armor is better than 1 Aegis in every situation except for a 2-dice attack (even then it's marginal).
If you are starting at 1 Armor, 1 Aegis is better against 2-3 attack dice, equal against 4 dice, and worse against 5+ dice.
If you are starting at 2 Armor, 1 Aegis is better against 2-5 attack dice, equal against 6 dice, and worse against 7+ dice.
If you are starting at 3 Armor, just take the Aegis.

2) How much weaker is 3+3+3 versus 9 attack dice?
Against 0 armor, they both deal 9 damage.
Against 1 armor, 9 dice does 8.026 damage, 3+3+3 does 6.889. (-14.2%)
Against 2 armor, 9 dice does 7.111 damage, 3+3+3 does 5.444. (-23.4%)
Against 3 armor, 9 dice does 6.312 damage, 3+3+3 does 4.833. (-23.4%)
Against 4 armor, 9 dice does 5.665 damage, 3+3+3 does 4.569. (-19.3%)

3) How much better is 9HP/3AC versus 10HP/2AC?
Not much. There is a <5% difference in "Rounds to kill" for 2-10 dice attacks. However, healing 1HP on 9HP/3AC is ~10% better than healing 1HP on 10HP/2AC.

How many attack dice do you need to kill 14 HP and 1 Armor in a single Battle Fury?
With 5+5 dice you have a 1.76% kill chance.
With 6+6 dice you have a 11.59% kill chance.
With 7+7 dice you have a 32.32% kill chance.
With 8+8 dice you have a 56.93% kill chance.
With 9+9 dice you have a 76.94% kill chance.

[halvor]

David, I love the spreadsheet but I would like to see a reference section for each tab detailing your assumptions and a key to your formulas. I put together a spreadsheet my self and I found average damage to be useful but given the nature of armor, it is a bit more complex. More later...

[piousflea]

The spreadsheet does not use average damage for anything other than a single square ("HP/damage"). You can see the difference between "HP/damage" and "Rounds to Kill" by comparing the two values. Rounds to Kill is generally a larger number than HP/damage because of overkill; hitting for 9 damage against someone with 1 HP doesn't kill him any faster than hitting for 1 damage.

I've made the Rounds to Kill box a little more visible in the latest revision (new link above).

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Mechanically speaking:
The sheet tabulates health and damage as a set of 61 probability bins, one for each integer value from 0 to 60. For each attack (up to 5) it calculates the chance of every possible outcome from 0 to 20 damage, and sums this into a set of bins 0 to 60. (*) For example, a Hydra triplestrike has a nonzero probability of inflicting anywhere from 0 to 18 damage.

It then combines the damage dealt with the creature's health and regen to get "HealthLeft" and "Alive?" after Round 1. (in-game, this would be HP remaining after Upkeep phase of round 2, after Regen but before any additional attacks)

This process is repeated for 10 rounds of attacks. Therefore, the survival probability and rounds-to-kill is mathematically precise for the first 10 rounds.

At the end of the 10th round of attacks, the spreadsheet calculates two different longevity estimates and takes the smaller of the two numbers. These are estimates rather than precise numbers.

Linear Estimate: This assumes that every surviving creature has X health, takes Y damage per turn, and regenerates Z damage per turn. This is accurate for creatures with large numbers of health and no regeneration. It is inaccurate because a creature cannot survive with decimal amounts of health. (0.5 HP is either 1 HP or dead)

Logarithmic Estimate: This assumes that every additional round kills X% of remaining creatures, where X is equal to the relative risk of death of Round 10. This is accurate only when regeneration is a large percentage of expected health remaining - each round it will either die or go back to a similar amount of health that it had before.
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(*) The main assumptions that the spreadsheet makes are that no single attack can roll more than 10 dice and no creature can have more than 60 health. It is mathematically precise within these parameters (at least within the rounding error of Excel, which should be 32 bit float, I think).