[GMFunkytown]

Got a new video out that goes into an analysis of whether or not you should choose to Pre-Edge (Push the Limit) or Post-Edge (Second Chance) on a given test. Give it a look. Hope you enjoy and find it useful!

https://youtu.be/xARlZGjEQ_g

[falar]

But what about Post-Edge to Push the Limit?!?!?!?!

[Squirrel]

Bottom line:
The odds of using edge beforehand being the better choice do rise with the difference in size of you initial dice pool and your edge attribute declining.

[witchdoctor]

the video more or less confirmed what I had always thought, as your dice pool goes up the benefits of using edge before the roll goes way down and then there's the possibility that you're using edge on a roll you would have succeeded on without needing to spend any edge.

[grauetheorie]

Great video.

Could you please share the math?

I did the math on my own and got slightly different results.

[Reaver]

In the vid description, he has several links to his proofs if that helps you...

[Tecumseh]

UmaroVI hasn't been on the boards in two years but he was very active during Fourth Edition. His rule of thumb was to multiply your Edge attribute by 2.57 (or 18/7 more precisely) and the result was the cutoff between where you should pre-Edge or post-Edge. I grant that this was before Limits were introduced in Fifth Edition so that's an additional wrinkle to the consideration now.

I mention this because Umaro had a PhD in math and I figured he had done the math to his satisfaction. I've been using at a guideline since then with good results.

18/7 of your edge score is the break even point (where pre-declaring and rerolling misses are exactly equal in expected number of hits). Over that, you are better off rerolling; under that, you are better off predeclaring.

There are a few niche exceptions (if it's really close, but the amount of hits you need is above the expected value, you should predeclare because it's more volatile) but 18/7 is a very good rule of thumb. The actual numbers are 18/7*Edge rounded up (so for example, edge 2 would be 36/7, or 5 and a seventh. You can't have a seventh of a dice, so 5 or less you should predeclare, 6 or more you should reroll).

[Halinn]

Personally, I like just doing the math. Pre-roll edge gives 2/5 expected hits per die, post-roll gives 5/9. Doesn't take long to calculate if (pool+edge)*2/5 >= pool * 5/9

edit: had a number wrong

[FasterN8]

Great video.

Could you please share the math?

I did the math on my own and got slightly different results.

The 18/7 thing UmaroVI said is exactly right.

The expected value for second chance is easy(er) to figure and comes out to 5/9 x (Dicepool).  Just pretend you have 9 dice and use the 1/3 rule.

The expected hits from Pushing the Limit is trickier because it's the sum of an infinite geometric series.  Given a dicepool of (D+E) and a 1/3 chance for a hit on any given die and a 1/6 chance of any die exploding, you get an expected hits equal to (2/5)(D+E)  where D is dicepool and E is Edge score.

4-min tutorial on summing an infinite series here:
https://www.khanacademy.org/math/precalculus/seq_induction/infinite-geometric-series/v/infinite-geometric-series   
In our case the "a" value in that video would be 1/3 and the "r" value would be 1/6.  That summation gives the expected ratio of hits for any given number of dice.  (2/5)

The you just set  (5/9)D = (2/5)(D+E) and solve for D in terms of E.  That works out to D = (18/7)E which is where the expected hits from each method are identical.

[grauetheorie]

Thanks for explaining me how to maximize the number of expected hits on a given role.

I had a little more complicated infinite geometric series but it also approaches 2/5.

But all these calculations take only the first moment of the two different distributions into account. As he speaks about reaching a target number he has to take at least the second moments into account. I ran my own simulations taking the variance into account and got slightly different results. Therefore I just want to know if he adjusts for risk preferences or if the results change by looking at the skewness or kurtosis of the distributions.

Further, I am not sure how useful these charts are as most of the time you want to maximize your expected number of hits and not reaching a certain target number as you do not know this number. Hence I just stick to the following table:

Use second chance if you roll at least a dice pool of

1 Edge - Dice Pool: 3
2 Edge - Dice Pool: 6
3 Edge - Dice Pool: 8
4 Edge - Dice Pool: 11
5 Edge - Dice Pool: 13
6 Edge - Dice Pool: 16
7 Edge - Dice Pool: 18
8 Edge - Dice Pool: 21

[FasterN8]

But all these calculations take only the first moment of the two different distributions into account. As he speaks about reaching a target number he has to take at least the second moments into account. I ran my own simulations taking the variance into account and got slightly different results. Therefore I just want to know if he adjusts for risk preferences or if the results change by looking at the skewness or kurtosis of the distributions.

Further, I am not sure how useful these charts are as most of the time you want to maximize your expected number of hits and not reaching a certain target number as you do not know this number. Hence I just stick to the following table:

....

Well I'm not GMFunkytown, nor do I play him on TV, but I would assume he has adjusted for some risk tolerance since his google spreadsheet for hitting specific thresholds does not follow the 18/7 rule exactly, likely for the factors you mentioned and that the 18/7 rule is best applied to open-ended tests rather than binary pass-fail (threshold) tests.  Incidentally, your table is precisely the results of applying the 18/7 rule to the 8 possible edge scores.  That's a nice reference for those with more severe mathematical allergies.

If we really want to be rigorous with our decision making regarding edge there is not just 1 decision point, but 2 on whether (and how) to use edge on a test.. 

1) The first decision is whether to Push the Limit before the test or NOT.  This is a VERY important point, and one he pointed out in his video (7:21) - that if you wait to use edge, you might just do well enough without edge.  That's true but it's really only one of 4 cases at the second decision point which is after you decide not to use edge AND you see the results of the dice.  The 4 cases are as follows, starting with the one Bobby mentioned.

Case 1:  You do so well with your base dicepool that you don't need to spend any edge at all to get done what you needed to do.
Case 2:  You roll so abysmally poor that edge is not likely to help and you're better off just accepting failure than wasting precious edge.
Case 3:  Your limit is not a concern and rerolling failures is likely to give you a material benefit. (Most likely)
Case 4:  You roll so amazingly well that you're already over the limit and hits will be wasted unless you Push the Limit. (Least likely)

From a non-mathy perspective, the ancillary benefits of "wait and see" are significant enough (Case 1 and 2 saving you edge) that using edge before the roll is going to be the lesser advantage whenever it's a close call regarding which method to use, and his Edge Cheat Sheets bears that out as well.

[rednblack]

I agree with everyone else that re-rolling failures is the default for most Edge usage, but I disagree that the potential to blow past your Limit isn't that helpful, or that it doesn't show up that often.  For matrix chars, it can be especially important -- going up against a high rating Host, for example.  And your low LOG chars may want to use it for those really important Perception Tests -- you know, when the kindly GM gives a little wink-wink-nudge-nudge at an opportune time.

[FasterN8]

I agree with everyone else that re-rolling failures is the default for most Edge usage, but I disagree that the potential to blow past your Limit isn't that helpful, or that it doesn't show up that often.  For matrix chars, it can be especially important -- going up against a high rating Host, for example.  And your low LOG chars may want to use it for those really important Perception Tests -- you know, when the kindly GM gives a little wink-wink-nudge-nudge at an opportune time.

Well, the frequency of bumping up against limits may vary from character to character, and it's really important to note that none of the math performed above have taken limits into account.  They're all assuming a sufficiently high limit which may or may not be the case, so you've got to keep your current situation (limit) in mind when making your decisions. 

If you're rolling a sufficiently large dicepool that already approaches the limits of the given test, then yes, Second chance will be a poor choice.

[rednblack]

@FasterN8, my comment was directed more at the OP's video than the math discussion that followed.

[dicecrimes]

I don't have anything valuable to add to the math discussion, but I appreciated the video very much!