  Three Houses

# Relic Equipment Stacks With Pavise/Aegis

All Hero's Relic equipment has the property that damage taken may be halved if the user also bears the linked Crest. For example, Thyrsus may halve damage taken if the wielder bears the Crest of Gloucester. There are also two abilities, Pavise and Aegis, which give a chance to halve damage taken against melee and ranged attacks respectively. Through manual testing, it was determined that units with Relic Equipment (and the necessary Crest) as well as either Pavise/Aegis will have two rolls to activate the damage halving effect, one for the equipment and one for the ability.

The results of the manual testing are captured in this spreadsheet.

# Chance of Hero's Relic Halving Damage

Again from manual testing, it was determined that the percentage chance for a Hero's Relic to halve damage is equal to the unit's Dex stat. This chance is halved if the unit obtained the Crest through an NG+ crest item. It was also determined that using a Crest item on a unit that already has the same Crest will further increase the chances (for example, giving Lorenz the Craft Dragon Sign). It is not entirely clear how this is increased, although there are two main possibilities (the test results are more similar to the first one).

• The percentage chance of activating the Relic effect is increased to $$Dex + \frac{Dex}{2}$$
• The game will roll a second time to see if the Relic effect activates, with a chance of $$\frac{Dex}{2}\%$$

# Calculations

The probability of NOT activating the regular Pavise/Aegis ability is equal to $$1 - \frac{Dex}{100}$$. Similarly, the probability of not activating the Hero's Relic half damage effect is $$1 - \frac{Dex}{100}$$ for Major/Minor bearers, or $$1 - \frac{Dex}{200}$$ for Crest Item users. Hence, the probability of both the regular Pavise/Aegis ability not activating AND the Relic half damage effect not activating is $$(1 - \frac{Dex}{100})^2$$ for natural Crest bearers and $$(1 - \frac{Dex}{100}) * (1 - \frac{Dex}{200})$$ for Crest Item users. The probability for the half damage effect activiting can therefore be calculated by the below formulas (top is natural Crest bearer, bottom is item user).

$$p = 1 - (1 - \frac{Dex}{100})^2$$
$$p = 1 - (1 - \frac{Dex}{100}) * (1 - \frac{Dex}{200})$$

If the unit has both the natural Crest, as well as the relevant Crest item applied, the two possible formulas are the following (top is increased Relic chance, bottom is second roll for Relic). Neither has been confirmed to be correct, and they both produce somewhat similar percentages.

$$p = 1 - (1 - \frac{Dex}{100}) * (1 - \frac{Dex + \frac{Dex}{2}}{100})$$
$$p = 1 - (1 - \frac{Dex}{100})^2 * (1 - \frac{Dex}{200})$$

For example, let's say that NG+ Dorothea with 30 Dex used the Craft Dragon Sign to obtain the Crest of Gloucester. She also mastered the Paladin class and gained the Aegis ability. With Thyrsus equipped, she gets attacked by an enemy Archer. The probability and percentage chance that the damage is halved is calculated as follows.

$$p = 1 - (1 - \frac{Dex}{100}) * (1 - \frac{Dex}{200})$$
$$p = 1 - (1 - \frac{30}{100}) * (1 - \frac{30}{200})$$
$$p = 1 - 0.7 * 0.85$$
$$p = 0.405$$
$$\textrm{HalfDamage}\% = 40.5\%$$