Maths and Strategy of using LUCK in battles

[thealmightymudworm]

This is one of those topics I feel a bit daft raising because surely somone must have analysed it to death at some point, and I never did Maths A-Level (to use John Brawn's criterion). Nevertheless I've seen very little analysis of this question compared to e.g. how combatants of different SKILL and STAMINA are likely to fare against each other, so I'll get the ball rolling and then someone can show me up at leisure.

Let's consider for example using LUCK to save you when your STAMINA is running out. If you are hit on STAMINA 2, then obviously it's always worth Testing your LUCK as even if you are on LUCK 2 that gives you a 1/36 chance of surviving to the next round versus certain death.

But on STAMINA 3, you'd be unwise to use LUCK even if you have LUCK as high as 7. If you're unlucky, you die immediately when otherwise you'd have had another round, teetering on STAMINA 1, whilst even if you're lucky, that only leaves you on STAMINA 2, which only saves you a round if you successfully Test your LUCK again with your LUCK on 6 to teeter on STAMINA 1 for another round. The risk of LUCK killing you whilst looking for a chance for LUCK to save you is too high.

On STAMINA 4, having a LUCK of 7 means it's arguably still too low to be worth rolling (I think). If you don't use LUCK, you are two rounds from death. If you use LUCK and fail, then you don't lose anything combat-wise as you still get the second round even if on STAMINA 1. But if you are lucky (and end up on STAMINA 3) you gain just one round unless you test your LUCK again on STAMINA 3 LUCK 6 which isn't worth it. So you might as well wait until you're on STAMINA 2 before testing your LUCK. But having a LUCK of 8 might make it worth it.

Starting on higher LUCK scores it's clearly more likely that you can be lucky two times (three, four...) in a row, so worth trying earlier, but someone else can probably do something more statty about how high LUCK versus how many times worth testing. (Probably assuming this is a fight for which there's little or no point trying to save LUCK points – either final boss or a fight that's near certain to kill you otherwise).

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It's worth bearing in mind the degree to which your chances of passing a LUCK roll drop with each point loss. Perhaps most people are aware that the biggest drop is from LUCK 7 to 6, as 7 is the most likely number to roll on two dice. Your chances of passing a LUCK roll drop from 21/36 (7/12) to 15/36 (5/12) with that single point drop.

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Likewise, how do you play off extra damage against limiting damage? This is partly a matter of how your SKILL and STAMINA compares to your opponent of course. If you are a SKILL 9, STAMINA 14, LUCK 9 character battling, say, Hawkana after the Scroll of Agonising Doom, (SKILL 12, STAMINA 6, I think), then clearly it's worth rolling (once) for extra damage, as the odds are against you in every round and this will reduce the number of rounds you need to win from 3 to 2.*
Against a SKILL 7, STAMINA 30 fighter (let's say a sluggish animated statue) it's probably worth just slogging away and limiting the damage it inflicts on you. But again, has anyone done any proper analysis on where the crossover point might be?

*(Even with Hawkana, I think it may be worth waiting until you hit her a second time before using LUCK. If that second hit never comes, you wouldn't have won anyway – there's no benefit in going early. Whereas if she beats you all the way down to STAMINA 2, it might be worth having a favourable-odds LUCK roll still available.)


Anyway, thoughts?

[Peter]

When I started reading and playing the books properly, I reduced luck-in-battle decision-making to one simple statistical point: use luck on yourself and you gain 1 or lose 1 stamina, use luck on your enemy and they lose 2 or gain 1 stamina. So, assuming a 50-50 chance of success vs non-success, the average impact when using luck when you lose an attack round is zero stamina change to you, but the average impact when using luck when you win an attack round is -0.5 stamina to your enemy.

So in the long run, using luck when you win an attack round will give you, overall, an increased chance of winning battles. Using luck when losing an attack round will have a neutral impact when averaged out in the long term.

Therefore, I developed the habit, when fighting tough foes, of using luck only when I won an attack round. Because of the 2-point effect unique to being lucky on a winning round, I decided this was the most effective use of your limited luck. Only when I was down to 3 stamina would I start using it on myself, in a last-ditch attempt to stay alive.

In terms of deciding when to use it and when not to, you would need to combine the odds of success with the odds of winning a round and your likely stamina loss from the battle overall. The first two of these are easy to determine, and I believe the third has been calculated by someone here. Add them together (possibly in a 4-dimensional table) and you will find your "tipping-points".

[CharlesX]

You gain luck after the Hawkana battle, which makes using luck an easy decision (if you've read TOD and know the true path). I'm predictably going to say it often isn't worth spending a luck point against an enemy because of the regular luck tests, which are often more important, unless that is your enemy has a special attack or something bad\catastrophic will occur if you don't kill them in enough rounds.
   Is Crypt's odds algorithm calculated without the use of luck in battle?

[daredevil123]

Feb 25, 2022 8:43:41 GMT CharlesX said:

   Is Crypt's odds algorithm calculated without the use of luck in battle?

The algorithm accounts for the use of Luck. Basically you should only use Luck against Razaak, and only while your Luck is >6. I dread to think how low the odds would be otherwise!

[scouserob]

Great question and probably one which we have all wondered.  

I've had a go at creating an Excel sheet to calculate this.
I've assumed that you will choose to test your luck if, and only if, doing so increases the probability of winning the battle.

This first chart is for fighting an opponent of equal skill.
The numbers represent the minimum luck points for which to use test your luck on that won or lost round.

I like the unexpected patterns that have cropped up in the round won chart.
They go down in groups of 4, which must match the 4 point deduction from your enemies health if you are lucky.
If seems if the enemy's health has a remainder of 3 on division by 4 (3 mod 4) then using testing for luck is good even with low luck points. (?)
If the opponent has 2 mod 4 stamina then it seems better not to test luck unless you have have many luck points. (?)

Either that or there is a bug in my Excel sheet, which is more than likely.



There are some weird entries that I've glossed over in the chart.
Such as when winning a round with your stamina 5 and his 5 you should only test your luck in the range 8-10 luck points rather than 8+.

[scouserob]

Feb 25, 2022 5:52:32 GMT thealmightymudworm said:

If you are hit on STAMINA 2, then obviously it's always worth Testing your LUCK as even if you are on LUCK 2 that gives you a 1/36 chance of surviving to the next round versus certain death.
At least my chart agrees on this.  That gives me some confidence.

But on STAMINA 3, you'd be unwise to use LUCK even if you have LUCK as high as 7. If you're unlucky, you die immediately when otherwise you'd have had another round, teetering on STAMINA 1, whilst even if you're lucky, that only leaves you on STAMINA 2, which only saves you a round if you successfully Test your LUCK again with your LUCK on 6 to teeter on STAMINA 1 for another round. The risk of LUCK killing you whilst looking for a chance for LUCK to save you is too high.
It seems you need high 9+ luck to make testing your luck better here.  (unless the opponent has very high stamina).
I guess then the benefit of a possible two extra rounds outweighs the possibility of instant death.

[scouserob]

Here is what I get for a skill advantage of 3.
It seems to be more geared towards testing your luck in defence.



Here is what I get for a skill disadvantage of 3.
It seems to be more geared towards testing your luck in offence.

[nathanh]

I've checked a bunch of those cells with my own calculation and I agree with you, so I think you've done it right. I find that enemy 18, player 2 cell in the 0-difference version intriguing!

[scouserob]

Feb 26, 2022 16:52:56 GMT nathanh said:

I've checked a bunch of those cells with my own calculation and I agree with you, so I think you've done it right. I find that enemy 18, player 2 cell in the 0-difference version intriguing!

I am finding rationalising these results a bit difficult.
Why test your luck here at 2/18 but not at 2/14??
Don't know.

The advantage that I calculate for testing your luck here with 12 luck points is only an extra 0.007%.  
So you'd win an extra one in about 14,000 fights.
Every little helps I guess. 

[thealmightymudworm]

I don't have much to add here, only to say I'm really happy that this is a topic that isn't known for being done to death, that I was largely right in my original thoughts and especially that it's been extended so much further than I could have hoped to do. Big thanks to scouserob

I love how counterintuitive a lot of it is. If I'd been thinking aloud about a fight in FF, saying "I'm going to use a potion of Fortune after this anyway – I wonder whether it's worth using LUCK now that I've won this round with them on 7 STAMINA..." and someone had said "Well clearly you should, provided that your LUCK is at least 3, allowing that your own STAMINA is an odd number, otherwise it needs to be a point or two higher", I'd probably have tried to have them committed. Who knew that FF's famously bare bones combat system had such complexity tucked away?

[schlendrian]

Very interesting

What's your method, scouserob - do you have a formula (if so, would you care to share?) or do you use some kind of battle simulator to fight a lot of battles?

[scouserob]

For each decision whether or not to test luck, I think you can work out the probabilities of winning the battle for each choice using the probabilities of winning the battle following the optimal strategy with the resultant lower stamina/luck values after you are lucky/unlucky/don't test your luck.

So you can just calculate your way up from low stamina values.

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In a bit more detail and just concentrating on the decision if you won the round (as the calculation when losing the round is similar).

Say, with a certain, fixed, difference in skill, your stamina is YS, his stamina is HS, your Luck is L.
Define:

The probability of winning/losing a round as P(WR)/P(LR) respectively. [Calculated from the skill difference.]
The probability of being lucky/unlucky as P(Lucky,L)/P(Unlucky,L) respectively. [Easy to calculate for each luck value L.]
The probability of winning the battle with these stats following the best strategy as P(YS,HS,L).
(Assigning a probability of 1 for this last one if HS is less than or equal to zero and a probability of 0 if YS is less than or equal to zero.)

Now assuming your have won this round, define:
The probability of winning the battle if you test your luck as P(TL,YS,HS,L).
The probability of winning the battle if you don't test your luck as P(Not,YS,HS,L).

Then:
P(TL,YS,HS,L) = P(Lucky,L)*P(YS,HS-4,L-1) + P(Unlucky,L)*P(YS,HS-1,L-1)
P(Not,YS,HS,L) = P(YS,HS-2,L)

Then the best strategy here is to test your luck if P(TL,YS,HS,L) > P(Not,YS,HS,L) and not to test your luck otherwise.

Of course you need those pesky three parameter probabilities P(YS,??,?) to calculate these two for the comparison so....

Let the probability of winning the battle with the optimal strategy if you won/lost the round be P(WR,YS,HS,L)/P(LR,YS,HS,L) respectively.
[So P(WR,YS,HS,L) = max(P(TL,YS,HS,L),P(Not,YS,HS,L))]

Then:
P(YS,HS,L) = P(WR)*P(WR,YS,HS,L) + P(LR)*P(LR,YS,HS,L)

You just have to calculate from low stamina up.
For example:
P(-2,HS,L) = P(-1,HS,L) = P(0,HS,L) = 0
P(YS,-3,L) = P(YS,-2,L) = P(YS,-1,L) = P(YS,0,L) = 1
P(1,1,L) = P(WR)
etc.

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Not sure I've written that clearly, but please feel free to point out any errors I've made or ask questions on it.

[schlendrian]

No worries, written very clearly and great approach

[schlendrian]

Feb 25, 2022 23:10:50 GMT scouserob said:

If seems if the enemy's health has a remainder of 3 on division by 4 (3 mod 4) then using testing for luck is good even with low luck points. (?)

I take the question mark as an invitation to try to explain this behaviour, so here's my thought: At a stamina f.e. of 7 you can be more generous than at 6 because the negative effect of a failed luck test is mitigated.

At stamina 7, if you hit him and don't test your luck, he goes down to 5st so you still have 3 rounds to win.

If you do test your luck and fail, he goes down to 6 stamina, so you also have 3 rounds still to win - the only effect of the failed luck test is a lost luck point, the combat itself stays unaffected.

At stamina 6, if you hit him and don't test your luck, he goes down to 4st so you still have 2 rounds to win.

If you do test your luck and fail, he goes down to 5 stamina, so you still have 3 rounds to win - in addition to losing a luck point, the failed roll basically negates your successful combat round.

[scouserob]

Yes please, any insights are welcome.     

I agree with everything you are arguing, and without the maths I'd be pretty confident you are right.
Except that if we add two onto his stamina for the values in the example (9 and 8 rather than 7 and 6) then the logic is the same but the results go against that logic.

The odd stamina number seems to be more generous with luck tests in these pairs of stamina values:
(7,6), (11,10), (15,14), (19,18)

Yet it seems to flip to often being slightly more generous with the even value for these pairs:
(4,5), (8,9), (12,13), (16,17)

So a pattern of four rather than just an even/odd pattern of two.

I can't grasp why this is.
I'm guessing it must be something to do with the fact that you knock off four from his stamina if lucky.
But that unconvincing connection between fours is about all I have.

[schlendrian]

Sorry, you're right, a bit of selective perception on my part...

I agree with you, that a hypothetical connection between the 4st damage and the 4-pattern is unconvincing. I don't really see how the extra damage in case of a successful luck test would give any pattern in the table, as it's effect is so uniform. Regardless of stats (exception of enemies with 1 o 2 st) it's basically an additional combat round you don't need to win.

[someone]

Sorry for performing necromancy with this post, but I did not see it before.
Well, for the standard damage part of the question, the problem has been addressed by Professor Iain G. Johnston of Bergen University (Norway).
You can find the answer at: arxiv.org/abs/2002.10172

However, since a gamebook can only be solved as a whole and the described calculations were made for a single fight, the described strategy only strictly applies for fights after which there is no more possible stamina losses or luck tests, that is tipically the last fight of the book.
Hope it helps.