# Critical hits for [Low-Dmg - High-Speed] vs. [High-Dmg - Low-Speed]

In the context of RPG (Role Playing Games), I was wondering about the difference between having low-damage attacks with high attack speed and having high-damage attacks with low attack speed, specially in the case in which critical attacks can be dealt.

To simplify things a little bit, let's consider that both players will successfully land a sequence of uninterrupted attacks for $$t$$ seconds, and that their attack speeds are such that: after $$t$$ seconds, both will be about to land their next attack. To be more precise, both land their first attack at time $$0$$ and they keep landing attacks in the time interval $$[0,t[$$. Their attack speeds are such that: at time $$t$$, they would simultaneously land an attack.

Let's call Fox the LDHS player (Low-Damage - High-Speed) and Brutus the HDLS player (High-Damage - Low-Speed). Fox attacks deal $$d$$ damage each and he lands $$N$$ attacks every $$t$$ seconds. Brutus attacks deal $$D$$ damage each and he lands $$n$$ attacks every $$t$$ seconds. As stated $$d < D$$ and $$n < N$$. In the time interval $$[0,t[$$, Fox would deal $$F$$ damage and Brutus would deal $$B$$ damage:

$$F = N\times d$$

$$B = n\times D$$

To focus on the advantages that may come from critical, let's also consider a case in which $$\boxed{F = B}$$. Thus, $$d$$, $$D$$, $$n$$ and $$N$$ are such that:

$$N\times d = n\times D$$

Under these settings, Brutus would be better considering low-HP enemies (potentially killing the enemy almost $$t$$ seconds faster than Fox). But, let's focus on enemies that will walk away alive after receiving all the damage, so none would have the advantage.

I also want to consider full cycles of attacks, for example: if Fox lands $$5$$ attacks per second and Brutus lands $$2$$ attacks per second, Fox attacks at $$t \in \{0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8,...\}$$ and Brutus attacks at $$t\in \{0.0,0.5,1.0,1.5,...\}$$. I would like to compare the total damage dealt by each one after the whole interval $$t\in[0,1[$$, or $$t\in[0,2[$$, or $$t\in[0,3[$$, and so on.

Now, let's include critical hits to the game. If, for both players, $$r \in]0,1[$$ is the critical rate and $$c$$ is the critical gain, Brutus would deal $$[1+c]\times D$$ damage per critical hit and Fox would deal $$[1+c]\times d$$ damage per critical hit.

For a very large number of hits ($$M$$ for Fox and $$m$$ for Brutus), the average damage dealt by each one would be: $$D\times[1+r\times c]$$ for Brutus and $$d\times[1+r\times c]$$ for Fox. Which would result in the same total damage when both perform averagely. However, even though the maximal achievable damage would also be the same, it should be much harder for Fox to perform maximal damage (landing all $$M > m$$ critical hits in a row) than it would be for Brutus (landing all $$m < M$$ critical hits in a row).

On the other hand, for a small number of attacks, it seems to me that Fox would have the advantage, since he would have more "attempts" of landing critical hits during the succession of attacks.

So, it should be clear that, whenever critical attacks are a possibility ($$r > 0$$ and $$c > 0$$), each one will have its own advantages, in different scenarios.

What I would like to know is how to perform a systematic comparison between these two players. Is there a way to visualize the scenarios in which one is better than the other? When will the ignorant muscle beast outperform Fox? When will the graceful machine gun shooting grapes beat Brutus? And how this analysis change when we increase or decrease the time (changing the total number of attacks performed by each player)?

• I don't really understand why the question was closed because "it needs to be more focused". The focus is on the effects of including critical hit mechanics on the comparison between these two players (that would be equivalent without crits). As I've tried to propose, only full cycles of attack should be considered (more focus); and we can assume that the enemy will survive (so we focus on total damage dealt, not on the killing or not-killing criterion). Commented Aug 29, 2023 at 20:48
• Moreover, no constructive comment has been made to improve the question, which makes me think that some people from the community are just trying to close questions that are not personally appealing to them... Commented Aug 29, 2023 at 20:50

1. Stronger-but-slower hits potentially waste more damage if you overkill weaker enemies; e.g. if you land a critical hit doing $$500$$ damage on a $$200$$ HP enemy you've wasted $$300$$ damage. So weaker-but-faster hits will generally be preferable if you're trying to take out multiple weaker enemies, depending on how the numbers work out (e.g. if the weak hits do $$199$$ damage against $$200$$ HP enemies you need two hits and the second one is almost completely wasted so that's still pretty bad). (On the other hand you're also correct that if the first hit always lands at time $$0$$ then stronger-but-slower hits could be better if they get the kill faster, again depending on the details of what kinds of enemies you're facing.)
• @Daniel: these will essentially be binomial distributions. As the time $t$ gets large they will approach normal distributions (by the central limit theorem) with the same mean but different variance. Commented Aug 29, 2023 at 0:49
• @Daniel: yes, that's similar, although strictly speaking those two have slightly different means ($6.5$ vs. $7$). For a small number of attacks it just depends on what your assumptions are; are you assuming they both get to perform full cycles or not? That's up to you. Commented Aug 29, 2023 at 1:27
• @Daniel: with the assumption that they both land their first hit at $t = 0$, these numbers never make the average damage the same. Brutus always does slightly more damage and it's entirely because of that $t = 0$ hit, which has nothing to do with crits. For example, at $t = 1.1$ Brutus has landed two hits (so $10$ damage) and Fox has landed six (so $6$ damage), ignoring crits. If you instead assume that Fox needs 1/5 of a second to wind up the first hit and Brutus needs a full second, then Fox does slightly more average damage except when their cycles line up at which point they tie. Commented Aug 29, 2023 at 1:44