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First, I'm pretty sure that you can't really calculate how good an attack bonus is without knowing an opponent's AC. If that logic holds true, then we can't specifically calculate how good a +2 initiative bonus is without knowing your opponent's initiative modifier.

Typically, perhaps these values are best represented on a table. Is there a way to mathematically evaluate how much more often you will go first in a battle, only considering the effectiveness of a +2 initiative bonus from the Reactionary trait?

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closed as unclear what you're asking by YogoZuno, Trish, KorvinStarmast, T.J.L., Pyrotechnical Oct 23 '17 at 13:33

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This is probably one of the most variable things in the game. Even ignoring the fact that going first is not always better, and that the tactical situation might well change whether or not it is better to go before the bad guys, there are innumerable different combinations of creatures you can encounter, which have many different sets of special abilities and stats. In addition, with 2-5 other PCs in the party, you may want to go before or after some of them.

Assuming you always want to go first, the next issue is with determining what your chance of going first is. In a d20 contested roll, if both sides have the same modifier, and there is a roll-off for a tie, then both sides have an exactly 50% chance of going first. For the first +1 modifier of one side over the other (which, incidentally, also ensures that side breaks ties), the chance of winning initiative goes from 200 in 400 to 229 in 400. From here, every +1 modifier changes the odds by 20, less the difference + 1. So, for a 2 greater modifier, the chance is 247 in 400, for 3 greater, it's 264 in 400, and for 4 greater, it's 280 in 400. You can convert this to a percentage by dividing. This continues until a modifier of 19, at which point, the lower side cannot possibly win.

So, to work out how much benefit +2 is going to give you, you need to know what range of modifiers you will compete against. For a given average party level (APL), in theory, you should face single creatures with a Challenge Rating of 1 lower than your APL, to 3 higher. Of course, if the GM puts you up against more creatures, they will have even lower CRs.

If you look at the d20pfsrd Monster DB, you can see some stats for creatures, and you could use that list to filter to a specific CR, and get some idea of the rough initiative ratings. The initiative modifier is not explicitly on the sheet, but you can use the Dexterity score to get an idea, and look for Improved Initiative in the feat list. Of course, doing this totally ignores custom creatures, templates, or NPCs with class levels. This also gives no indication of the distribution of these creatures during the encounters you will face.

For even the simplest case, of a level 2 party, the single creatures they can face range from CR1 to CR5. The creatures in that range have initiative modifiers ranging from -4 to +11. These entries will change at every level, with no guarantee that the distribution of creatures will ever match what is in the table, or any guarantee of what creatures the GM will use in encounters. NPCs with class levels built by the GM could vary this even more.

For the sake of some illustration, take creatures with a +0, +2 or +4 modifer. The chance of going first against those is shown below, with Initiative Modifiers of +0, +2 and +4 (10 Dex, 10 Dex with Reactionary, and 10 Dex with Improved Initiative).

$$ \large{\text{Chance for PC to go first}} \\ \begin{array}{r|lll} & \rlap{\rightarrow \text{PC initiative bonus}} \\ \text{Creature initiative bonus} \; \downarrow & +0 & +2 & +4 \\ \hline +0 & 50\% & 61.75\% & 70\% \\ +2 & 38.25\% & 50\% & 61.75\% \\ +4 & 30\% & 38.25\% & 50\% \\ \end{array} $$

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The difference between attack rolls and initiative checks, in terms of statistics, is that an attack roll is 1d20+modifier v.s. static number while initiative is 1d20+modifier v.s. 1d20+modifier. This means that, unlike AC and attack modifiers, you can just compare the values directly to see who will go first on average. The 2 d20s can be thought of as 'cancelling out', in terms of the average result, since they are added to both sides.

If you are interested in more than just who goes first more often on average, then the math is more complicated. You said you are interested in only the +2 from reactionary, so I've made an AnyDice program for you, so you can see how affects things based on what the preexisting difference was. Note that you can't critical an initiative check, so the programming is actually significantly simpler. However, accounting for the triple-tie check (viz. when you tie initiative and your modifiers are equal so you roll off with a chance of (n^2-1)/(2n^2) for either you or your opponent to go first and 1/n^2 for undefined behavior where n is the number of sides on the die you each roll) was too much so I just assume you lose those cases.

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