Improved Initiative vs Danger Sense

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I created a spreadsheet to compare rolling twice and taking the largest (Danger Sense), and just adding 4 to your initiative (Improved Initiative).

Improved Initiative had an average of 14.5, while Danger Sense had an average of 13.8.

If initiative matters to you and you need to choose between these, it looks like Improved Initiative, which is available in the Heroic Tier, is the preferable one.
Incorrect, because the distribution for Danger Sense isn't uniform -- it's heavily slanted towards the higher numbers. For example, 20 is the most common result.

Sometimes mean ≠ mode.

That said: take both, or take Quick Draw and Danger Sense.

Cheers, -- N
However, Danger Sense gives you zero results above 20, while Imp. Init. has 21-24 each occurring equally as often as 5-20.

Consider it this way: at what N added to a single roll, will the result be statistically higher than the maximum of two rolls? 1? 4? 10? 20?

I don't have the math to calculate that formally. But experimentally, with 10,000 trials in Excel, +1 and +2 are lower than the max of two, +3 is almost equal (within about 3%), and +4 gives greater results around 25% of the time, or a ratio of 0.8:1 in favor of I.I.

Improved Initiative is the better deal of the two.

You're right that the bare average isn't necessarily informative: I found that at +3, the average for the high value is slightly higher than that of the single roll, but the single +3 still "beat" the other method a slight majority of the time. Without knowing how to calculate the odds, the experimental method consistently reveals which is actually more benefit.
Ziana, the point is that while Improved Initiative is clearly better for getting a higher initiative roll most of the time, Danger Sense is (for obvious reasons) better at preventing a low roll. Some people think that's important, and there's been an ongoing discussion as to how much it matters.

In any event, taking both is clearly the right call.
Danger Sense is (for obvious reasons) better at preventing a low roll.

DS will statistically skew to 20, but it's still possible to roll a 1. With II you can't roll lower than 5.

Some people think that's important

Isn't statistically having the higher roll more of the time of the most value? That's what II offers.

taking both is clearly the right call.

Only if you consider initiative that essential, and you're willing to give up two feats to maximize. If someone only has room for one, then if they want to be higher on the initiative list, more often, then II appears to be the way to go.
The average for 2dK drop lowest is:
SUM (n*(2*n-1), n, 1, K) / K^2
which works out to: (K+1)*(4*K-1)/(6*K), for any die size K.

d20+4, avg = 14.5
2d20, drop1, avg = 13.825

Here's how you can work it out yourself:
Use 2d4, make a grid:

....1 2 3 4
1| 1 2 3 4
2| 2 2 3 4
3| 3 3 3 4
4| 4 4 4 4

The top row is the result of die 1, the column is die 2. The number in each grid represents the value of the roll. If you rolled a 2 and a 3, you drop the 2 and have 3. The odds of getting any grid result is 1/16. So, just add up all 16 numbers in the grid and divide by 16.

You can figure out the formulas for 3dK drop 1 etc this way, it just takes some time.

Cheers,

The Hanker Chief
DS is better if you already have a high initiative modifier (relative to enemies).
II will be better if you have a low modifier.

..but people spending a Feat here will often already have a high modifier or want both.
It doesn't help that you apparently usually fight enemies that have a couple of levels on you.

Initaitive is a lot more important to some classes than others. The main one to care that won't have a massive Dex is Wizards.
DS will statistically skew to 20, but it's still possible to roll a 1. With II you can't roll lower than 5.


Isn't statistically having the higher roll more of the time of the most value? That's what II offers.


Only if you consider initiative that essential, and you're willing to give up two feats to maximize. If someone only has room for one, then if they want to be higher on the initiative list, more often, then II appears to be the way to go.

The only thing initiative boosters are good for is getting a higher result than your enemies. Pretty obvious, but necessary to point out.

The only stats you're looking at are min roll, mean roll, and max roll. That's a good starting point, but it doesn't tell the whole story. Improved Initiative is a better feat when the number you need to roll to win initiative is between 2 and 6 or 16 and up. In between that, DS is better.

Does anyone have numbers on what sort of init bonuses you're looking at from monsters at various ranges? If we have that we can compare average rolls from monsters and how likely a character with each is to win initiative.
While you're at it you might look at how important winning initiative is to those monsters.
The only thing initiative boosters are good for is getting a higher result than your enemies. Pretty obvious, but necessary to point out.

The only stats you're looking at are min roll, mean roll, and max roll. That's a good starting point, but it doesn't tell the whole story. Improved Initiative is a better feat when the number you need to roll to win initiative is between 2 and 6 or 16 and up. In between that, DS is better.

Does anyone have numbers on what sort of init bonuses you're looking at from monsters at various ranges? If we have that we can compare average rolls from monsters and how likely a character with each is to win initiative.

I worked out the math for a few cases and then did a numeric simulation using matlab at work to check (a million opposed rollls for each initiative value from 20 above to 20 below the character's) and I found that II is always better than DS. By better I mean a higher probability of beating a monster's initiative.
By better I mean a higher probability of beating a monster's initiative.

That's a pretty narrow definition of "better". How about:

Beating the most important monster('s)(s') initiative(s).

And how about taking into account the character's dexterity mod and whether or not that character has taken the Quick Draw feat or class features that increase initiative? The larger an initiative bonus the character already has, the larger the benefit of Danger Sense.
Here's the deal:

If you need a 7 or better to win initative (say, you have a +13 base initiative modifier somehow and thus will always win initiative versus a straight d20 roller) you're better off with Danger Sense than Improved Initiative, as you will win initiative 91% of the time rather than 90%.
Initaitive is a lot more important to some classes than others. The main one to care that won't have a massive Dex is Wizards.

http://forums.gleemax.com/showpost.php?p=16033553&postcount=8
I worked out the math for a few cases and then did a numeric simulation using matlab at work to check (a million opposed rollls for each initiative value from 20 above to 20 below the character's) and I found that II is always better than DS. By better I mean a higher probability of beating a monster's initiative.

Titanium Dragon does not agree.. which is good because I was pretty sure you were wrong and now I don't need to work it out ;)

EDIT2: my maths was wrong. I'll have another look tomorrow if no one has produced some conclusive numbers.
Titanium Dragon does not agree.. which is good because I was pretty sure you were wrong and now I don't need to work it out ;)

EDIT2: my maths was wrong. I'll have another look tomorrow if no one has produced some conclusive numbers.

The problem is that he took a specific result and generalized it.

Improved Initiative is at its best when you don't have a very good initiative bonus relative to your foes, or when you have an overwhelmingly large initiative bonus relative to your foes.

Danger Sense is at its best when you have a good but not overwhelming initiative bonus relative to your foes.

Here's the deal:

If a result of 15+ on your initiative roll will always win you initiative, you're better off with Danger Sense. So, for instance, if you have a +5 bonus on your initiative check, and your foe has no bonus to their initiative check, you will ALWAYS win initiative 51% of the time, versus 50% of the time for Improved Initiative.

As it turns out, there's a range of values over which one is superior to the other. If your intiaitive bonus is less than 5 greater than that of your enemies, Improved Initiative is superior. If your initiative bonus is between 5 and 13 higher than your enemies' bonuses, then Danger Sense is superior. If your initiative bonus is at least 14 points higher than that of your enemies, then Improved Initiative is better agian. This simply falls out of the table of probabilities for Danger Sense:

DS Total X or higher
1 0.0025 1
2 0.0075 0.9975
3 0.0125 0.99
4 0.0175 0.9775
5 0.0225 0.96
6 0.0275 0.9375
7 0.0325 0.91
8 0.0375 0.8775
9 0.0425 0.84
10 0.0475 0.7975
11 0.0525 0.75
12 0.0575 0.6975
13 0.0625 0.64
14 0.0675 0.5775
15 0.0725 0.51
16 0.0775 0.4375
17 0.0825 0.36
18 0.0875 0.2775
19 0.0925 0.19
20 0.0975 0.0975

Compare to that of Improved Initiative:

II total X or higher
1 1
2 1
3 1
4 1
5 1
6 0.95
7 0.9
8 0.85
9 0.8
10 0.75
11 0.7
12 0.65
13 0.6
14 0.55
15 0.5
16 0.45
17 0.4
18 0.35
19 0.3
20 0.25
21 0.2
22 0.15
23 0.1
24 0.05

And now the difference between the two:

Difference
0
0.0025
0.01
0.0225
0.04
0.0125
-0.01
-0.0275
-0.04
-0.0475
-0.05
-0.0475
-0.04
-0.0275
-0.01
0.0125
0.04
0.0725
0.11
0.1525
0.2
0.15
0.1
0.05
I think you've done what I did and failed to take the variation in enemy roll into account ..and I think it matters: it's not just about the chance of getting a roll that always wins but also about the chance of winning even when you don't get a 'safe' roll.
I think you've done what I did and failed to take the variation in enemy roll into account ..and I think it matters: it's not just about the chance of getting a roll that always wins but also about the chance of winning even when you don't get a 'safe' roll.

You don't understand Titanium Dragon's math. He took everything relevant into account. Those numbers are true for value X, where X is a target number. This number can be anything, from a somewhat good initiative roll to a very good initiative roll.

The math says that Improved Initiative is better unless you have a decent initiative bonus already. Then, get Danger Sense, unless your initiative bonus is truly massive. In that case, you probably want to invest your feat elsewhere rather than in Improved Initiative in the interest of optimization in other areas.
Ranix: looks like you think there's a fixed target. Enemy initiative isn't a fixed target, it's a variable roll. The probability curve looks very different. I'll get round to working it out soon ..I still may be wrong about it mattering but I think it does. (his conclusion is in line with what I was expecting though)
Ranix: looks like you think there's a fixed target.

Variables are fixed? OK guys, math is over. Permanently. Someone divided by zero. I'll be in the bomb shelter.
I'll try to explain more clearly.

The fight: PC 'P' against single monster 'M'
Suppose M has Inititaive modifier +0 (we only care about the relative modifier)
P has modifier +X (X could be negative)

Both roll for initiative.
Call P's intiative 'Q'
M now has an initiative 'N' but that could be anything from 1 to 20 and for the purpose of analysing the benefits of these feats you don't know what it is. All you know is that it has an even distribution across the 20 possibilities. You cannot just look at the fixed initiative roll for M, you must look at all the possible rolls for M for each value of X -and for each of those values of M you must look at each possible value of Q.

What I think TD did was to say: "ok M=20-X".
I hope all that follows is correct.. it's definitely close but someone should check.
I used a table because Excel was handy and it was easier on the brain than working our formulae; ther are also some edge conditions that can make formulae tricky.

Considering Improved Initiative

You can easily draw up a table of PC's advantage (X-0) against P's target initiative to go first (N or N+1 depending on whether X>0) with each entry showing the probability that the PC goes first for that roll by the Monster and that difference in initaitve modifiers. This table will have a lot of 1s in one corner and a lot of 0s in another corner. In between these each entry will be 1/20 different from it's neighbours.

In the band where Q can land either side of N with or without Improved Initiative, you will get a 20% improvement from having it. That improvement doesn't extend to the edges though: If the monster rolls particularly low against a PC with a higher initiative modifier then the PC is gaining no advantage from the feat that time.

From the table you can easily calculate the advantage granted by II by taking the difference between the sums of two rows 4 points apart. Since that was summed across the 20 possible rolls by the monster, we then multiply by 5* for the percentage gain from Imp Init.

(eg the rows for X=7 and X=3)
Sum for X=7: 16 2/20
Sum for X=3: 13 4/20
Benefit: 2 18/20
In this case we see that the PC with a base +3 advantage over his enemy has an extra 14.5% chance of winning initiative after he takes Improved Initiative.
NB: That means that in 14.5% of case he will now win where he would have lost.
His chance of winning initiative with +3 was: 13.2/20 = 66% so Imp Init represents nearly a 22% improvement in his chances.

In fact the advantage to the PC with -1 is 18.5% and it drops by 1% for each point of relative initiative bonus right up to the +16 bonus that means he automatically wins.


* Divide by 20 then multiply by 100 to give a percentage (100/20=5)
whether or not that character has taken the Quick Draw feat

Since Imp. Init. doesn't stack with Quick Draw and you'd only gain an additional +2 initiative, picking Danger Sense in that case is a given.
Summary: Take Improved Initiative

ok I've done DS too now and the comparison. (e-mail me at yahoo dot co dot uk if you want me to send you the excel workbook)

The upshot was:
- DS is indeed better than II if you have a +2 to +10 initiatve advantage (X above) on the opponent*.
- BUT in that range DS yields only an extra 1% of wins over II; +13.56% v +12.5% at the +5 advantage
- II is much better over all and yields up to +7.95% absolute over DS at -3.

The point where II puts you ahead of the opponent instead of behind makes a big difference

So you don't ever lose much by using II over DS but can gain a lot. Since you will face a variety of oppeonents, and have an 80%+ chance of winning Init with II even when it's outperformed, Imp Init is much the better choice over all.


I think that's probably the final word on these two ;)


* I'm not about to work it out for multiple opponents -not having generated proper formulae- but I expect the results to be very similar
M now has an initiative 'N' but that could be anything from 1 to 20 and for the purpose of analysing the benefits of these feats you don't know what it is. All you know is that it has an even distribution across the 20 possibilities.

You don't need to know what a variable is to make an equation. You only need to know that the variable exists, and that it is something.

In this case, we know that for no matter how many opponents there are on the field there is an optimal initiative to beat (the highest). There is also a spread of high-efficiency results that end in you beating the initiative of the majority of combatants or the most significant combatant.

N would not be simply 'an initiative', but a function representing a weighted spread of possible high-efficiency scenarios.
Improved Initiative had an average of 14.5, while Danger Sense had an average of 13.8.

Ok so I got bored at work and Ran a statistical analysis on a random sample of 10,000 Initiative rolls. (So yes there were a total of 20,000 rolls for Danger sense) I got these numbers. DS=13.9774 ImpInit=14.6948 Both=17.9774 however those numbers don't mean much of anything.
Sal was correct in how to find the actual averages but again that doesn't mean much and it doesn't cover the Markov Chain but the "real" numbers are/would be really close.
Titanium Dragon's math is correct for what you percent chance of getting a result on the d20 roll.
Timlagor did a bit more work and involved a variable monster outcome and he... Came to the same conclusion! The main difference is that he stated how much better DS is better at those ranges.
Some people were whining about DS being skewed toward 20, and this is true. In order to get a 1 with DS you have to roll 2 1's any other number you have a wider range to roll. II has similar frequency at all numbers it gets DS does not. I exported my data to excel and made some pretty frequency distributions and Charts.

Danger Sense
Die Roll Frequency
1 ---------22
2 ---------69
3 ---------125
4 ---------167
5 ---------212
6 ---------250
7 ---------306
8 ---------358
9 ---------398
10 --------473
11 --------547
12 --------548
13 --------619
14 --------663
15 --------739
16 --------787
17 --------826
18 --------890
19 --------988
20 --------1013

Improved Initiative
Die Roll Frequency
1 ---------0
2 ---------0
3 ---------0
4 ---------0
5 ---------472
6 ---------482
7 ---------469
8 ---------500
9 ---------476
10 --------482
11 --------506
12 --------516
13 --------464
14 --------476
15 --------543
16 --------482
17 --------491
18 --------520
19 --------524
20 --------460
21 --------519
22 --------525
23 --------546
24 --------547

If you want to know the precents move the decimal 2 places to the left and put a % after it, there was 10,000 trials.

To save space I am not posting the Both chart, it is the same as the Danger Sense one, just make 1=5 and 20=24. For the Charts well I can't paste them so if you want to see them copy the numbers above go to excel and make a scatter chart. They really won't tell you anything this didn't.

Overall if you care enough to do the math or read these dry posts then take II. Personally I believe die rolls are not random but governed by your subconscious (ever noticed how timid players seem to roll low? and Seasoned Veterans who are not timid seem to roll high?) I have more thoughts on this but my boss should be getting in soon.
You don't need to know what a variable is to make an equation. You only need to know that the variable exists, and that it is something.

In this case, we know that for no matter how many opponents there are on the field there is an optimal initiative to beat (the highest). There is also a spread of high-efficiency results that end in you beating the initiative of the majority of combatants or the most significant combatant.

N would not be simply 'an initiative', but a function representing a weighted spread of possible high-efficiency scenarios.

You know there is a variable. You don't know what the result is. To tell anything you need the whole distribution of the variable.

It's worthless to know how good you are at hitting a point if you have no idea whether how likely your target is to be at that point.

TD was completely ignoring one of the variables and considering only one row of my 38-column table (and not taking into account the possibility of a tie on equal modifiers).
You do know how likely your target is to be at that point based on the frequency of high-initiative monsters your DM throws at you and how high your own initiative is relative to that frequency.
From a choosing standpoint, you get Improved Initiative, if only because Paragon feats are better than Heroic ones, so why waste a paragon feat slot for something that gives about the same effect on average as Improved Initiative: unless they stack, which they do. You're not really going to be choosing between them.

And really, I don't think II doesn't make any classes top 6 heroic feat list.

Flipping through the Monster Manual, it looks most monsters have a 1/2 level +3-+8, which means that if anything, they'll probably beat the average player without II and/or a high dex.
You do know how likely your target is to be at that point based on the frequency of high-initiative monsters your DM throws at you and how high your own initiative is relative to that frequency.

No that gives you a guess at the difference in initiative modifiers ..though I doubt you can actually get a good guess from that information.

The 'target' is based on the enemy's modified d20 roll and so has a nice big variation.
And really, I don't think II doesn't make any classes top 6 heroic feat list.

I disagree, II makes top six on my feat list for Wizards. I would even go so far as to ALSO get the initiative reroll, either through a feat or via the Divine Oracle.
From a choosing standpoint, you get Improved Initiative, if only because Paragon feats are better than Heroic ones, so why waste a paragon feat slot for something that gives about the same effect on average as Improved Initiative: unless they stack, which they do. You're not really going to be choosing between them.

ImpInit gives much better benefit than DS across the whole range and only marginally less than DS when DS is at its best. I believe you can swap Heroic Feats for Paragon feats once you reach paragon levels so which tier they come in only determines when you can get them.
Flipping through the Monster Manual, it looks most monsters have a 1/2 level +3-+8, which means that if anything, they'll probably beat the average player without II and/or a high dex.

That pushes ImpInit more ahead of DS.
You missed the double negative (apologies). Every character I can think of wants II.

Also, looking at this thread, Initiative averages 3/4 level +1.27
You missed the double negative

Blast!
IMAGE(http://risse.com/james/Init.JPG)
http://www.domenicdenicola.com/Initiative.pdf

That is all.

Pretty ...but irrelevant :P
(for reasons I've already tried to explain)
I think it's important to remember what your goal is. Assuming there are at least 3-4 enemy initiative rolls you can expect a pretty decent range of initiatives. So is your goal to

1) Go before as many enemies as possible (Favors Danger sense slightly if you have a good init bonus)

2) Go first as often as possible (Favors Improved Init)

If you're not sure which one to take always take Improved Init as it will do more good for the most people in the the most situations.
You know there is a variable. You don't know what the result is. To tell anything you need the whole distribution of the variable.

It's worthless to know how good you are at hitting a point if you have no idea whether how likely your target is to be at that point.

TD was completely ignoring one of the variables and considering only one row of my 38-column table (and not taking into account the possibility of a tie on equal modifiers).

Here's the problem: you're making incorrect assumptions.

First off, if you have a higher initiative modifier than your opponent, it doesn't matter WHICH you're using - you win ties regardless. As that is where DS is better than II, ties are therefore irrelevant in our calculations.

Now, you're acting like we cannot know what monster initiatives are, but why can't we? Its not that hard to go through the MM and look at monster initiatives and see where they are clustered.

You don't need a static target. I was too lazy to really do the deeper calculations, but I knew you were wrong :P

Basically, you're acting like there's this huge range of enemy initiatives, but I haven't really seen evidence for that at a particular level in comparison to you. If you've got a +8 bonus to your initiative on top of your +1/2 level bonus, how often do you fall into that range where DS is better as compared to II versus monsters? From my cursory examination of the MM, it looks like it happens pretty often.

We don't care about the entire range; the entire range is irrelevant. What matters is what range the character who is faced with this choice is likely to fall into.
EDIT: Oops

Apologies to TD for poorly understanding his post.
S'okay.
Here's the problem: you're making incorrect assumptions.

Send me your e-mail and I'll send you my spreadsheet. You clearly haven't understood what I've done.


Now, you're acting like we cannot know what monster initiatives are, but why can't we? Its not that hard to go through the MM and look at monster initiatives and see where they are clustered.

No that tells you their likely initiative MODIFIER. Their initiative depends what they roll. When I'm talking about modifiers I say so (and there it's the relative modifiers).

While it's true that you can try to get yourself into the initiative sone of modifier compared to monsters of your level where DS is better, you will need ImpInit to get there* and there will still be enemies outside that range (one way or that other). Bear in mind that some enemies you face will be higher level that you (or lower).

Comparing yourself to a single monster with a fixed initiative modifier requires you to consider all the possible d20 rolls both you and he can make.

Comparing your initiative modifier to monsters in general requires you to compare all the rolls you can make against all the rolls the monster can make with each modifier he might have (which can be simplified a little by just taking the difference in initiative modifiers).

I hope my other posts make more sense now..

* max +10 from Dex at 30th level and everyone gets the level/2.
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