Battlewise/Improved Init vs. Danger Sense

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So, my Runepriest just hit level 20. I'm sick of always going last, so I think I'm going to spend my feat improving my initiative. I have a negative DEX mod, and while I have a +4 Timeless Locket, that's the only bonus I've got. I'm sitting at +13 initiative at level 20 (before taking my feat). Since my WIS is 5 higher than my DEX, Battlewise would bring me up to +18 init . . . or I could take Danger Sense, ending up with +13 (roll twice, take better). In my situation, what's going to be better?

Second question. Obviously, if I go with Battlewise, it's in my best interest next level to retrain it to Superior Init (unless I just decide I REALLY want to go first and take Superior Init on top of Battlewise, but I think something like Divine Mastery is likely to be more important). If I go with Danger Sense, would the same be true?
The average initiative you get using:


  • Battlewise  = 28.5
    (you can get anywhere from 19 to 38 iniative, so (19+38)/2 = 28.5)
     

  • Danger Sense = 26.825
    (the average when taking the highest of 2 d20 rolls is 13.825)


So on average you are better of using Battlewise. Even when taking Superior Initiative on top of the aforementioned feats. (BW + SI = 36.5, BW + DS = 34.825). The only draw to danger sense is (in your case) is that it has less 'spread' (the standard deviation is lower). Read: The chances of you having an unlucky roll and having low iniatitive for that fight are greatly reduced.

Bottomline:
Battlewise is better, but danger sense can save you some "Ah crap, I rolled a 2" moments. 
Statistically speaking, Danger Sense gives you a 51% chance of rolling a 15 or higher, when normally you have a 50% chance of rolling an 11 or higher, so effectively it's a little bit better than a +4, so in terms of raising your Init, Battlewise is superior.

Since Init isn't a vacuume, it's a contest, you also have to consider what you're up against. At level 20, a Lurker will have a +21, a Skirmisher will have +19 or Soldier will likely have an Init mod of +17, Artillery and Controllers will likely have an Init mod of +16, and Brutes will likely have an Init mod of +15; roughly an average of +17. I'm really can't remember my statistics right now, but iirc, basically rolling twice only really helps you out if you're within 2 of their average as it gives you a slightly higher and more steady average.

But then, assuming at 21 you go up to a 22 Wis due to bump and/or ED, you'd be at +27 with both feats bonus feats, or only +22 (Since Dex will go up to 10, right?) with Superior+Danger Sense, versus the NPC average of now +18. In the case of the former, they need to beat your roll by 11 (since you'll win ties), that's rather significant, best of (2d20) +21 really just doesn't have a higher average.

edit: I just remembered some of it! At level 20 with Battlewise, you have a 57.25% chance of beating them, at level 21 with Battlewise and Superior Init, you'll have an 86.25% chance, and at level 21 with just Superior Init, you'll have a 70% chance. I don't have a calculator on me to figure out the Danger Sense percentages, but regardless, just Superior Init is clearly good enough.

If you have the patience, you can goto anydice.com and figure out the odds of each result with Danger Sense, and then take the sum of the odds of each result multiplied by their chance to beat you if you rolled that result, ie. if you roll a 1, you get a 23, which they'd need a 6 to get a 24 (you win ties), if you roll a 2, they need a 7, etc. so you'd do ((% Chance of a 1)*(Their % chance of a 6 which is 75%)) + ((%chance of 2)*(70%)) + ((%chance of 3)*(65%)) etc. The total is their chance of beating you, subtract that number from 1, and you'll have your chance to beat them.
"Invokers are probably better round after round but Wizard dailies are devastating. Actually, devastating is too light a word. Wizard daily powers are soul crushing, encounter ending, havoc causing pieces of awesome." -AirPower25 Sear the Flesh, Purify the Soul; Harden the Heart, and Improve the Mind; Born of Blood, but Forged by Fire; The MECH warrior reaches perfection.
Zathris was very much correct to mention the competion aspect, here is some extra data:

Level 20:


  • Initiative +13, against a monster with +17 and you get 1 roll (no feats)
    30.00% chance to win
     

  • Initiative +13, against a monster with +17 and you get 2 rolls (danger sense)
    44.5% chance to win
     

  • Initiative +18, against a monster with +17 and you get 1 roll (battlewise)
    57.25% chance to win


Level 21:


  • Initiative +22, against a monster with +17 and you get 1 rolls (SI)
    73.75% chance to win + 1 feat to spare
     

  • Initiative +19, against a monster with +17 and you get 2 rolls (BW+DS)
    77.69% chance to win
     

  • Initiative +27, against a monster with +17 and you get 1 roll (BW+SI)
    88.75% chance to win
     

  • Initiative +22, against a monster with +17 and you get 2 rolls (DS+SI)
    87.31% chance to win

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