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Lilt
I looked back over the previous topics and found a few mechanics-probability related ones, but none really addressed this question directly so I thought I might give it a shot:

Say I have a dice pool of 13 and an edge of 5. I know that I really want to make this next skill roll, and want to do it really well, so I think I'm probably going to use edge on the test in some way.

There are a number of ways to do this, however, so which one do I use?
  1. Spend Edge before the test, getting 5 extra dice and exploding 6es on all 18 dice.
  2. Spend edge after the roll for 5 extra dice with exploding 6es.
  3. Spend Edge to reroll failures.
I can also spend edge to cancel a glitch, but as glitches don't really have mechanical impacts I'll leave that out for now.

Now the average number of successes per die using exploding 6es has been worked-out to 2/5 by Ellery here. The probability of success on a normal die is 1/3 (perhaps obvious that one), and the probability of a success on a die which is rerolled if it fails is the probability that it succeeds plus the probability that it succeeds given that the first roll was a failure, or (1/3)+(1/3)*(2/3)=5/9.

So, with an intitial pool of D and an edge of E, the expected number of successes for each of these edge expenditures are:
  1. 2(D+E)/5
  2. D/3+2E/5
  3. 5D/9
Given these formulae, we can now evaluate wether or not we're best to spend edge before or after the roll given our example:
  1. 2(13+5)/5=7.2
  2. 13/3+10/5=6.3333333333333333333333333333333
  3. (5*13)/9=7.2222222222222222222222222222222
It seems that in this case, the largest expected number of successes overall comes from waiting until after the initial roll and rerolling failures, but only just. As one might expect, the system by which 18 dice are rolled, all with exploding 6es, came-out above the system by which only 5 of the 18 dice explode.

This assumes that we have made-up our mind about which system to roll already, and it's an important note that if we wait until after the roll then we have a choice of wether to add 5 exploding dice or to reroll failures. If you only failed on 4 dice, wouldn't adding 5 exploding dice ad a higher number of successes to the expected result? More on this tomorrow! Please, feel free to add your own evaluations and observations.
buddha
So for how small a dicepool will rolling an extra 5 dice awith exploding 6's be the best alternative and how will a lower Edge alter the numbers??
DireRadiant
For the re rolling of failures, " You may re-roll all of the dice on a single test that did not
score a hit.". Is thsi taken into account in the probability calculation? I am nto sure it's as simple as 5D/9.
nezumi
It seems to me that you ignore the pleasant combination of spend one edge for an extra dice, plus reroll all 6's, if successes are few, use a point of edge to reroll failures, then use the remaining 3-4 dice to roll normally.
FrankTrollman
Or we could use the search function.

QUOTE (FrankTrollman)
Spending a point of edge gives you extra dice and gives you the rule of six - which makes every die contribute an average of .4 hits (including the base dice pool, if you use your edge ahead of time). But spending a point of edge to reroll failures brings each die up to a 5/9 hit contributor, and has the additional advantage that it works at full value even if used after the fact, and it has the added benefit of eliminating glitches most of the time.

That being said, from an averages standpoint, and assuming the failure rerolling option's benefits for being used after the fact are unquantifiable, the two options are equal if 7/45 of your original dice pool is equal to 2/5 of your edge. Which means that even if you have an Edge of 7, you're better off just rerolling failures if your original dice pool is 19 or more dice (the breakeven point is 18 dice).

With an edge of 6, you are better off rerolling failures with a 16 die pool.
With an edge of 5, you are better off rerolling failures with a 13 die pool.
With an edge of 4, you are better off rerolling failures with an 11 die pool.
With an edge of 3, you are better off rerolling failures with an 8 die pool.
With an edge of 2, you are better off rerolling failures with a 6 die pool.
With an edge of 1, you are better off rerolling failures with a 3 die pool.

So if you have any sort of min/maxxer's dice pool to start with, each edge point is actually worth less than the one before it. After all, at any given time you may or may not want to use edge between now and the next time your edge refreshes. And the less time there is before it refreshes, the less likely you are to need an edge expenditure before rollover. And the more edge you have, the more times you've used it (and thus the less time until rollover) before you get to your last one.

When you buy a new point of edge, you are purchasing your last point, not your first one, and since most of the time you're going to be using it to reroll failures to save your hoop - there's a substantial chunk of diminishing returns.

---

Now, if they let you buy an Edge of 9, that would be something. Suddenly you'd be better off rerolling failures only with a dice pool of 24 dice - which means that for almost all tasks you'd be pulling ahead on buying extra dice, which in turn means that your edge would start to really matter for things other than uses per adventure. But unless you're a spirit, you can't have an edge of nine, so that's that.

-Frank


-Frank
nezumi
Because I'm lazy (and a bad person because of that, but since I'm not actually playing SR4 I won't feel too guilty about it) could someone please summarize their findings in one or two sentences? Just glancing over, it sounds like the final verdict is it's best in most cases to just roll your dice and use karma pool to reroll or buy successes as need be (I presume that 1 guaranteed success is about equal to rerolling 3 dice in regards to value).
blakkie
Something to keep in mind though Lilt is that average might not be what is important to you, and as Frank points out those breakpoints are evaluated by the average number of hits.

Comparing adding Edge dice to the pool vs. rerolling, rerolling produces a more consistant result (lower variation coefficient in statistical terms). So while you tend to have less downside rolls of only a few or simply no hits, your total hits are capped by the initial pool side and the probability of rolling at least given number hits drops as that number approaches the initial pool size. Plus the extra benefit of delaying the Edge use past the initial roll to avoid Glitches.

So generally those are the breakpoints, but they can shift somewhat when you are attempting very difficult tasks (where the sum of the Threshhold and required net hits is high), or a spectacular success is worth the risk premium. (EDIT: Shift the other way if you just need a few hits relative to pool size to guarantee success, but then you aren't likely to be using Edge.)

P.S. Possibility of Glitches also create tricky decisions for rerolls at the lower end on that scale depending on what you rolled the first time. But at least you get a choice which you don't if you added Edge directly to the initial pool.
Teulisch
has anyone looked at the impact of the 'bad luck' negative quality? Im looking at edge 1 with bad luck, and while i can see it makes risk higher in some cases (edge 1 you save for when its do or die), the 20 BP is very tempting.
Feshy
QUOTE (Teulisch)
has anyone looked at the impact of the 'bad luck' negative quality? Im looking at edge 1 with bad luck, and while i can see it makes risk higher in some cases (edge 1 you save for when its do or die), the 20 BP is very tempting.

The problem with bad luck is, as you say, you spend that edge when it's "do or die." 1/6 odds, you die. That's pretty harsh.

When I was testing the functionality of the die roller I wrote, Bad Luck would often startle me. Sometimes, it'd hit two or three times in a row, and the results on the die roll where very painful. Then there would be long streaks where it wouldn't show up at all. Basically, it's what you'd expect: sooner or later, your bad luck will catch up with you, and chances are, you'll die. I'd spend half the BP it gives you on a second Edge, just in case you have to hand of god...

That said, all this talk of statistics makes me want to add some statistical calculators to my die roller so that *it* can tell me how best to spend my edge. Let the computer do all the math, I say...
Azralon
Unfortunately, taking 4 Incompetences in active skills that you're unlikely to use is a much better allocation of Negative Quality points than Bad Luck.

At least you can avoid situations where you'd have to use those Incomp'd skills (hopefully having teammates who can do it instead). Bad Luck all but destroys SR's only global panic button.

And yeah, yeah, Notoriety. Different discussion there.
Lilt
@buddha: I had largely meant giving the formulae to let people ork-out what was best for their character. FrankTrollman's post covers this, however, so shoudl give you most of the info that you need for each character.

@DireRadiant: The average number of hits from a die with the option to reroll is 1*(1/3) (1 hit, times the probability of a success when it's first rolled) plus 1*(2/3)*(1/3) (One hit given that the first roll was a failure and the second roll scored a hit. Sum this together and you get 5/9 successes per die.

@nezumi: I can't quite get what you're saying. Are you talking about the fact that you can leave the decision on wether to roll failures or roll extra dice afterward until after the roll, and base your choice on the numbers that come up? If so that's what I was planning on going into tomorrow, or rather later today after I sleep.

Simple summary: Roll edge extra dice (invoking the exploding dice rule) if your pool is small or your edge is large. If you have a large pool or low edge, reroll failures. The exact break-points can be found in FrankTrollman's post.

@FrankTrollman: I did perform a search, that's how I found Ellery's post which I linked, but evidently my search-fu is weak.

@blakkie: True. I'll have to look at that later, allong with an evaluation of wether or not the average number of successes increases if you differ choice on reroll failures or edge extra duce until after the roll.

@Teulisch, Feshy, and Azralon: I'd consider bad luck on a 1-off character, as I coudl see it killing the character but being interesting. I'd rather not run with a bad-luck character as one bad roll can (as I'm sure we've all seen before) hose the whole run.
blakkie
QUOTE (Lilt @ Nov 16 2005, 09:19 PM)
@blakkie: True. I'll have to look at that later, allong with an evaluation of wether or not the average number of successes increases if you differ choice on reroll failures or edge extra duce until after the roll.

If you roll the Edge dice after the initial pool the 6's in your pool don't explode, therefore they are still at the average of .33 hits, only the Edge dice are at .4.

Choosing between rolling Edge separate and rerolling depends on how many dice you have at that point that are NOT hits. The base calculation is Dice_Not_Hits * 0.33 compared to Edge Dice *0.4. So an Edge (6) is comparible to 6*0.4 / 0.33 = 8 rerolling dice. (in average number of hits)

You then temper that with two things. Whether or not, and how close it is to occuring, the initial roll had enough 1's (or 2's if this is a Rush Job on an Extended Test) to create a possible Glitch (obviously not a Critical Glitch though) when you add the Edge dice in.

And of course the desire for the possibility of a larger number of hits which the exploding 6's can give.

EDIT: Note that at this point the Edge dice have the higher average hits per dice. So when you have equal numbers of dice they both, barring a current Glitch in the already rolled dice, the Edge are going to be the way to go because they have the most upside AND better average.
Lilt
Yeah, the next step however is to see if that affects the initial strategy much. You see, the fact that you might be better-off doing something other than re-rolling, and that you get to make the choice after the initial roll, may well increase the average number of successes of the 'wait and see' tactic which sets the numbers in FrankTrollman's post off. Well, actually FrankTrollman's post is correct as far as adding edge initially versus rerolling failures goes, but I don't think he took the second choice into account.

The 'wait and see' tactic really has 3 sub-choices (re-roll failures, add your edge attribute in exploding dice, or cancel a glitch) of which you're obviously going to choose the most favourable one.

Right now, trying to work-out how to calculate all of the expected numbers ETC in a spreadsheet for all cases is giving me a headache, as I've not done stats for a while. frown.gif
FrankTrollman
Well, there's actually a fourth option for characters who wait-and-see: do nothing at all. If you roll your five dice and get five hits, you'll keep your edge, because you are already doing fine.

However, the option "cancel a glitch" is usually a bad option and only barely ever worth considering. In most cases, rerolling failures will cancel a glitch anyway. If you roll 5 dice and get 2 hits and 3 ones, that's a glitch (and a result that'll happen just over 3% of the time). But if you reroll failures, the glitch comes up only 1 time in 216. Where the glitch was originally if half the dice came up 1s, in the reroll it's only a glitch if ll of the dice you are rerolling come up 1s.

Sure, there are times when you've already succeeded and your dice pool is small when you just want to take the safe way out and cancel a glitch, but in general you are better off just rerolling failures and hoping that you don't get a grip of ones on the reroll.

That being said, I would say that the advantages of wait-and-see are so great that even if it would be slightly advantageous on average to add extra dice before rolling vs. waiting and rerolling if necessary, that you should still probably wait.

-Frank
blakkie
@Lilt

You mean comparing the odds from the optimal choice between rerolling, Edge dice in pool, and Edge dice outside pool?

Ok, that would mean calculating the results of making the optimal rerolling/"Edge dice outside pool"...which can vary somewhat based on those tempering choices. And you have go beyond just the simple averages to get it, and for the results to have more value than what you have now. As the precision rises the outside influences rise in importants. You have entered into the region of codifying multiple variables, such as assigning a value to each of the possible number of hits you could roll.

Now my instincts tell me that the results of the optimum choice between rerolling and Edge dice outside pool will shift that breakpoint in the opposite direction that Edge dice in the pool did when that initial shift favoured using Edge dice in the pool. But never with a greater magnitude, and sometimes much less magnitude. But instincts can be a bit off sometimes. They are based in unconcious thought and thus subject to errors, sometimes huge errors, when they overlap with a mental distortion. Which brings us to....

Or do you plan to commit to memory these data. Or is your GM going to be cool with you bringing a laptop, or a nice thick booklet of tables/graphs. Because this is multiple variables to cover different senarios. Me? I'd be cool with someone showing up with a laptop or booklet to calculate probabilities in the same way i'm cool with a blind person bringing special blind-person dice, or someone rolling up in a wheelchair when their other option is dragging themselves their on their belly.** I'll willing to reasonably accomidate the handicapped. nyahnyah.gif More so because for it to significantly help you in the game you'll need to learning, and i wouldn't want to stand in the way of that.

Learn? Why yes, at the very least you'll upgrade that existing model of SR probabilities in your head. From there you might even be inclined to try understand why this model is the way it is. That sort of learning tends to leak through to elsewhere too (because there are a lot of other situational variables to influence the dicision that are outside of what we have talked about). You are also likely to learn, if you don't already know, WTF Sky's point was with all that babbling about cider.

** Hell i'm cool with someone showing up in a wheelchair "just because" as long as they patch any divets they put in my walls. smile.gif
blakkie
QUOTE (FrankTrollman @ Nov 17 2005, 11:33 AM)
That being said, I would say that the advantages of wait-and-see are so great that even if it would be slightly advantageous on average to add extra dice before rolling vs. waiting and rerolling if necessary, that you should still probably wait.

Only at times the Edge in the pool advantage isn't particularly slight at all, expressed in relative terms.

I would add that Glitches can be, depending on your GM's take on them, not even close to the worth of a point of Edge. If i had the number of hits i think, given what i know at that point, provides an acceptable level of success then i'd be inclined to just eat the Glitch (unless i was a metagaming shmuck that was confident a refresh was coming up wink.gif ). It's the Critical Glitches that can really hurt, which are something that you are likely to eat only when really low on Edge. Critical Glitches are also very unlikely with high single digit total sized pools and larger.

P.S. I personally would have liked the "reroll" option to have been reroll all the 4's, 3's, and except for Rush Jobs 2's. The current reroll is a little on the strong side at low levels, and becomes innane at higher levels.
Lilt
Actually, in an extended test a glitch means -1d6 hits. That averages at 3.5 which is worth around a 10-11 dice reroll or 8-9 exploding dice.

I didn't have as much time as I was hoping for today to do the rest of the spreadsheet. Stay tuned for more wacky statistical fun!
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