Which Is Stronger Deathfire Touch or Thunderlord's Decree on APs?

Hi everyone, today you're getting an extra post regarding a hotly contested keystone mastery choice. As the title of the post suggests, it revolves mainly around the selection of Deathfire Touch or Thunderlord's Decree for your keystone mastery.

Since this will be mainly math and graph based post, I won't beat around the bush. Let's dive right in!

Introduction

Hi everyone, today you're getting an extra post regarding a hotly contested keystone mastery choice. As the title of the post suggests, it revolves mainly around the selection of Deathfire Touch or Thunderlord's Decree for your keystone mastery.

Since this will be mainly math and graph based post, I won't beat around the bush. Let's dive right in!

The Keystone Masteries

Deathfire Touch
Your damaging abilities cause enemy champions to take 6 + 50% of your Bonus Attack Damage and 20% of your Ability Power in Magic damage over 3 seconds (Area of effect and damage over time abilities apply half of this damage over 1.5 seconds instead)

Thunderlord's Decree
Your 3rd attack or spell on an enemy champion shocks the area around them, dealing 10 damage per level plus 20% of your Bonus Attack Damage and 10% of your Ability Power as Magic damage to enemies in the area (30 second cooldown).

Comparative Amounts

In simple terms, both deal magic damage, and will receive no attack damage scaling on the majority of AP casters. At maximum over the course of 30 seconds (the cooldown of Thunderlord's Decree), the damage is as follows:

Deathfire Touch: (6 + 20% of ability power) / 3 * 30 sec

Thunderlord's Decree: (10 * level) + 10% of ability power

It will be very rare for a champion to be able to damage an enemy champion once every 3 seconds for 30 seconds, so we'll assume that a champion like Cassiopeia has used her dot on an extremely tanky enemy champion.

Since DOT and AOE damage are halved but apply twice as fast, this is essentially the same total damage. Therefore, we'll leave that as is for now since we're assuming 100% uptime for the same duration as Thunderlord's cooldown.

Variables

In this case, we're looking primarily at the following variables:

1. Bonus Ability Power
2. Seconds of effect
3. Champion Level

Graphing the Data

Since there are so many variables to graph out, let's pick a relatively neutral point for a few of the variables to make things fair and keep it simple. Obviously if you have low AP but a high level, Thunderlord's will do more and vice versa.

Instead, let's just chart out Ability Power using some reasonable assumptions with varying champion levels.

Mid Game Chart (Champion level 12)

The breakeven time is calculated as Thunderlord Damage / (Deathfire Touch / 30) - 3. The minus 3 is included since the damage will continue ticking for 3 seconds after you finish your fight (at most).

As you can see from the chart, it seems that during mid-game, around 200~300 AP is enough for a 5 second fight to deal comparable damage with both masteries. Higher AP would give the advantage over to Deathfire Touch, and lower durations would depend on how long the fight lasts.

Early Game (Level 4)

Unsurprisingly, at early levels the advantage clearly goes over to Thunderlord unless you happen to have an insanely good early game or somehow manage to drag out a fight with your mediocre HP levels.

Late Game (Level 16)

This chart is actually quite interesting. Late game, if you happen to be playing a Rylai + Liandries type build (180 AP), it seems that you still need a relatively long time before Deathfire Touch beats out Thunderlord's Decree. Meanwhile, once you get the +120 AP and 35% Deathcap to total around 400 AP, Deathfire beats out Thunderlord handedly.

Other Variables to Consider

Other Mastery Points

Selecting a keystone also gives you one of the following depending on your tree:

Piercing Thoughts: 7% Magic Penetration

Precision: 3 + 0.3 per level Magic/Armor Penetration

For simplicity's sake, I've left it out.

Champion Synergy

Champions like Cassiopeia and Brand can keep up Deathfire Touch for longer, and will synergize with that better in most cases. 

Champions like Ahri with her instant 3 spell hit via her W may find Thunderlord's Decree very good for early game harass.

Difficulty Landing 3 Spells

There's also the rare point to consider that landing three spells every 30 seconds on an enemy champion may prove difficult for certain players, but I've also left this variable out as it's difficult to plot out.

Conclusions 

Given enough uptime on the Deathfire Touch mastery, it will always beat out the Thunderlord's Decree mastery. However, as an average "skirmish" may only last 5 seconds on the high end, it will depend on how much bonus AP you can get.

Deathfire will be the better mastery if you can get about this much AP during these points in the game:

• Early Game: about 60 AP
• Mid Game: about 200~300 AP
• Late Game: about 400 AP

Again, this analysis leaves out any natural champion synergies.

TLDR; Thunderlord's is better early game and/or before you pick up some AP, Deathfire is better late game unless you're still behind in items. Specific champion synergy matters.

Think I missed some parts of the analysis or have some ideas to improve on the calculations? Comment below!