Phantom Roll: Knowing the Odds (long, math)

[Karinya]

This is a probabilistic analysis of Phantom Roll, some possible strategies and the outcome probability distributions for each. It is long; while a full understanding of the methods involved will require some knowledge of math, reading the conclusions does not.

Phantom Rolls can be divided into four groups based on their lucky and unlucky numbers (thanks to Spider-Dan for the complete list). Strategies and their probabilities are a little different for each group. In each group I identified several different strategies which I characterize as "safe", "moderate", and "risky" based on their bust risk. Strategies that reroll more often have a higher bust risk, but also have higher average numerical outcome and a higher chance of hitting 10 and 11. (The chance of hitting the lucky number does not change for any strategy examined in this post since there is no reason not to keep rolling as long as you are below the lucky number, and no good reason to roll if you are *on* it. The lucky number chances I calculated coincide exactly with Icemage's calculations in another thread.)

Methodology (technical): The game was modelled as a digraph in which vertex (n,k) represents the probability of having a total of n after precisely k rolls. Based on the strategy being used (formally defined as a vector of 1's indicating numbers that will be rerolled, and 0's indicating numbers that will be kept), some vertices are terminal while others have outgoing connections to (n+1,k+1) through (n+6,k+1). In order to shorten the calculations, it was assumed that a maximum of 6 rolls could be made. Calculations were performed by an OpenOffice 1.1 spreadsheet using formulas of the form (innerproduct( [p(n-6,k-1),p(n-1,k-1)] , [reroll(n-6),reroll(n-1)] ) / 6 (using dummy entries with 0 probability when necessary), and then summing across rows for terminal results. All strategies have 99+% probability of proceeding to completion (either a number you want to keep in that strategy, or a bust) within 6 rolls.


Group 1 - 2/6 Rolls (Magus, Choral, Samurai)
Safe strategy: Reroll any number 6 or less, except 2 (which is lucky).
Lucky 2: 19.44%, 7: 19.36%, 8: 16.58%, 9: 16.58%, 10: 13.34%, 11: 9.56%, Bust: 5.15%

Moderate strategy: Reroll any number 7 or less, except 2.
Lucky 2: 19.44%, 8: 19.80%, 9: 19.80%, 10: 16.56%, 11: 12.78%, Bust: 11.60%

Risky strategy: Reroll any number 8 or less, except 2.
Lucky 2: 19.44%, 9: 23.10%, 10: 19.86%, 11: 16.08%, Bust: 21.49%

Group 2 - 3/7 Rolls (Healer's, Monk's, Drachen, Gallant's)
Safe strategy: Reroll any number 5 or less, except 3 (which is lucky); also reroll 7, which is unlucky. Keep 6.
Lucky 3: 22.69%, 6: 30.88%, 8: 13.80%, 9: 10.56%, 10: 10.56%, 11: 6.78%, Bust: 4.74%

Moderate strategy: Reroll any number 7 or less, except 3.
Lucky 3: 22.69%, 8: 19.80%, 9: 16.56%, 10: 16.56%, 11: 12.78%, Bust: 11.60%

Risky strategy: Reroll any number 8 or less, except 3.
Lucky 3: 22.69%, 9: 19.86%, 10: 19.86%, 11: 16.08%, Bust: 21.49%

Group 3 - 4/8 Rolls (Ninja, Hunter's, Chaos, Puppet, Beast, Warlock's)
Safe strategy: Reroll any number 5 or less, except 4 (which is lucky); also reroll 8, which is unlucky.
Lucky 4: 26.47%, 6: 30.88%, 7: 14.21%, 9: 10.10%, 10: 6.32%, 11: 6.32%, Bust: 5.72%

Moderate strategy: Reroll any number 6 or less, except 4; also reroll Unlucky 8.
Lucky 4: 26.47%, 7: 19.36%, 9: 16.10%, 10: 12.32%, 11: 12.32%, Bust: 13.43%

Risky strategy: Reroll any number 8 or less, except 4.
Lucky 4: 26.47%, 9: 19.86%, 10: 16.08%, 11: 16.08%, Bust: 21.49%

Group 4 - 5/9 Rolls (Corsair's, Evoker's, Rogue's, Fighter's, Wizard's)
Safe strategy: Reroll any number 4 or less, and Unlucky 9.
Why is this "safe" when you're rerolling a 9? Because a bust isn't much worse than an unlucky number (compared to turning it into a 10 or 11, which is a major improvement), and your chances of getting *either* of the bad outcomes is quite low due to the overall small amount of rerolling.
Lucky 5: 30.88%, 6: 30.88%, 7: 14.21%, 8: 11.43%, 10: 5.78%, 11: 1.37%, Bust: 5.46%

Alternative safe strategy: Reroll any number 6 or less, except for 5; if you get stuck on Unlucky 9, keep it as preferable to a 2/3 chance of busting.
Lucky 5: 30.88%, 7: 19.36%, 8: 16.58%, Unlucky 9: 13.34%, 10: 9.56%, 11: 5.15%, Bust: 5.15%
As you can see the bust chance doesn't change much, but there is now a significant unlucky chance too. However the 10-11 chances are slightly improved.

Ultrasafe strategy: Only reroll 1 through 4.
Lucky 5: 30.88%, 6: 30.88%, 7: 14.21%, 8: 11.43%, Unlucky 9: 8.19%, 10: 4.41%.
Bust is impossible under this strategy, the worst that can happen is getting stuck with unlucky 9; however, 11 is also impossible and the chance of a 10 is under 5%. It pretty much amounts to giving up, and probably getting a weak result, if you don't get your 5.

Moderate strategy: Reroll any number 6 or less, except for 5; also reroll Unlucky 9.
Lucky 5: 30.88%, 7: 19.36%, 8: 16.58%, 10: 11.78%, 11: 7.37%, Bust: 14.04%

Risky strategy: Reroll any number 7 or less, except for 5; also reroll Unlucky 9.
Lucky 5: 30.88%, 8: 19.80%, 10: 15.54%, 11: 11.13%, Bust: 22.63%


It's up to you which strategy to choose for any given roll and situation; there isn't necessarily any one right answer to how much risk to take. But it's better to do so with knowledge of the potential risks and benefits of each course of action.

[Taskmage]

You win. Sticky for you.

[fastcart]

/em 's brain explode!

Thanks for the analysis. I can't stop myself from re-rolling on 8s. "I have a 50/50 chance of not busting!"

[Icemage]

There's something wrong with your methodology. I'm not sure quite what it is yet, as I haven't finished duplicating your calculation model, but off the cuff your entries for should show variance in the percentage of results of Lucky Numbers based on the rolling hueristics.

I'll post more info as I discover it. I'm checking your calculated results against a program I just wrote to model 1 million attempts under these same conditions, and I'm not quite seeing the same values you are.


Icemage

[fastcart]

Dayum. Your geek is strong. Both of you. And, I mean that as compliments.

[Icemage]

Hmm.. I just finished writing the code for the modeling. Looks like the lucky number percentage doesn't vary based on rolling hueristics. However,

I see where my numbers differ. I wrote my code to limit the calculation to 6 rolls, since you can only double up 5 times. I had thought the difference was in the lucky number terminators, but it's not - it's the cases where you "skip" the lucky number by rolling a 2 or 3 (thus not terminating the roll series) and end up either below your rerolling cap or on an unlucky number after 6 rolls. The difference is minimal in those cases.

I do see a larger difference in the bust chances with you choose to not to reroll them, however. While having an unlucky result isn't particularly useful, it also doesn't lock down one of your roll slots; this choice significantly skews the results.


Icemage

[Karinya]

I omitted the probability that you would end up (after 6 rolls) on a number you would have wanted to roll again to get off of, because it was always less than 0.1%.

Lucky number percentages don't change because the different strategies are only different in what they do for numbers that are already *higher* than the lucky number. By the time the strategies differ, you have already either gotten the lucky number, or are in a state where it is impossible to get it no matter what you do.


It seems to me that wasting a slot on an unlucky number "buff" that isn't doing anything is essentially as bad as wasting it on a bust, unless it's your second bust; but that may depend on the exact mechanics of busting, replacing your rolls, etc. As a non-COR party member, I didn't get a bust icon when the COR busted, so it seems that they could have put two subsequent rolls on *me* even if they didn't get the benefit of both *themselves*... too little data to be sure of that at this point, though. The factor-of-5 difference between PR's cooldown and duration certainly seems to argue for the intention (on SE's part) for CORs to keep up more than 2 rolls total, even though only 2 can remain in effect per member. If you are doing so, it won't exactly be cheap to get rid of that unlucky number with a replacement roll.

It's entirely possible that you should change strategies when you already have a bust in effect, to avoid double-busting, though. Possibly even to the extent of keeping unlucky 9s or even 8s (you may even want to change which rolls you are using so you won't risk being put in that kind of situation in the first place). Double busting won't kill your party, but it will turn you into a thief without SATA (plus the bust debuffs) for the remainder of the duration of your first bust.


One interesting result is that for any of the four games, it's possible to get an approximately 50% chance of one of the top three results (10, lucky, 11) with only 10-15% chance of bust and the remainder are all pretty decent rolls. So from the party's perspective it isn't really "we don't know if we're going to get anything" - they actually do get buffs, probably of a substantial amount, at least semi-reliably, if the COR is playing smart.

[Spider-Dan]

Essentially, all your first Bust does is hurt you (the COR); you can still give double rolls to the PT; but you'll only keep one. It's double Bust that cripples you, but under current game mechanics, you can clear all roll effects (including Bust) by logging out. I don't expect this to stay in the game for long.

The other problem with Busting is desynching songs. For example:

0:00 - COR Busts on +EXP roll
1:00 - COR rerolls +EXP to PT
2:00 - COR rolls +ATK to melee
3:00 - COR rolls +MP (or whatever) to mages
...
5:00 - Bust wears. (1:00 remaining on PT +EXP roll)

If COR immediately rolls a song that PT currently has active (say, +EXP) then you'll get it, PT won't, and when their +EXP wears in 1:00, you won't be able to reroll it.

The solution here would be to change roll mechanics so that COR can't reroll an effect they already have (preserving the spirit of the roll "gamble" by not allowing the tactic of "reroll existing effects every 60s until you get a # you like"), but allowing any new roll to renew existing effects on PT members.

[Feba]

I'm not good at math, but i'll take your word on this o_o



Very well done. Shows the advantages and disadvantages of playing certain ways

[ClydeArrowny]

Hmm...I would check the math on this, but someone stole my stats book as i was turning in my test. If I come across $80 to buy another one, I'll try to do the math as well, and show the equations and work done for the nay-sayers.

[Matera]

Hmm...I would check the math on this, but someone stole my stats book as i was turning in my test. If I come across $80 to buy another one, I'll try to do the math as well, and show the equations and work done for the nay-sayers.

Poor guy.

That's so lame, sorry to hear about the Stats book.

Anyways, yeah... Good to know, thank you.

[IfritnoItazura]

Kinda off topic (please don't hurt me, moderators. >_< ), but where do I go (on web) to educate myself so I can understand the math behind this dicussion?

[Jei]

I think when you bust you get negative effects. So it may not be worth rerolling any unlucky above 8. A little plus is always better than minus.

[Caspian]

Kinda off topic (please don't hurt me, moderators. >_< ), but where do I go (on web) to educate myself so I can understand the math behind this dicussion?

right here, but I think its expensive to get in.
There's a reason I only went through Cal I.

It is interesting that it almost seems like a dice version of blackjack, with the odds of busting and such. Atleast thats the way it seems at first glance.

[Matera]

Now, what would be really amazing if somebody got the stat boosts for each roll and figured out which of these strategies is the best.

(IE: Penalty of bust and chance weighed with gains and chance.)

Whoever does that gets a cookie and half of mine. Chocolate Chip, I may throw in milk if the table is organized.