Author Topic: Dice vs. flat symbols - which is better?  (Read 7266 times)

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DittmanRat

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Dice vs. flat symbols - which is better?
« on: February 22, 2015, 01:12:15 PM »
"Shining Steel Sword" offers +1 sword, and "Battle Axe" offers +1 die for the same priceā€¦. but which is better? At first I thought it would be +1 die because of the chance to explode, but I did some calculations and now I'm not sure.

 I'm about 90% sure my math is correct- if someone else could analyze this it would be great!

For this analysis I'm going to assume that gaining a sword is equal to gaining a shield, since gaining a shield will cancel out one of the opponent's swords.  I will call both swords and shields "S", meaning "Success".

When we roll a single die we can get the following outcomes:

 Sword: +1 success (S)
 Shield: +1 success (S)
 Sun: +1 success (S)
 Moon: failure (F)
 Wyld: +1 success (S) and roll again
Rot: failure (F)
(Sun and moon can be switched, as can Wyld and Rot, but you effectively get one success vs. one failure)

Result:
3/6 S
2/6 F
1/6 S + ?

Without going any further, there's a 3/6 chance for success and 2/6 for failure.  This is absolute.  But the final 1/6 (the Wyld die) is going to be split into six outcomes as well, which means:

3/36 S
2/36 F
1/36 S + ?

And then, with another Wyld roll:

3/216 S
2/216 F
1/216 S + ?

Let's stop here for now.  Obviously we can keep rolling Wylds but the likelihood of having a certain number of Wylds decreases the higher you go.  So, let's put this all into a single list:

3/6 S                                  1S         108/216
2/6 F                                  0S         72/216
3/36 S+S                         2S         18/216
2/36 S+F                        1S          12/216
3/216 S+S+S                3S          3/216
2/216 S+S+F                2S          2/216
1/216 S+S+S+?           3S+?     1/216

Thus, we've reduced the number of outcomes to 7, and the right column adds up at 216/216, which equals 1.  Now, let's multiply the right column with their respective number of successes:

108/216    *1    108/216
72/216       *0    0/216
18/216       *2    36/216
12/216       *1    12/216
3/216          *3    9/216
2/216          *2    4/216
1/216          *3    3/216
Total:                    172/216  =  0.7963  or about  0.8

A single die will roll an average of 0.8 successes,  compared to the fixed 1 success that a flat symbol will give you.  Thus, my conclusion is that a flat symbol, on average, is better than a die.  However, the trade-off with a flat symbol is that there is zero chance to get anything more, whereas it's still possible to get more than one success with a die.

Kletian999

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Re: Dice vs. flat symbols - which is better?
« Reply #1 on: February 22, 2015, 02:07:58 PM »
Yes, it's only the prospect of card burning that makes the Dice vs. Static symbol comparable.

DittmanRat

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Re: Dice vs. flat symbols - which is better?
« Reply #2 on: February 22, 2015, 02:26:21 PM »
Yes, it's only the prospect of card burning that makes the Dice vs. Static symbol comparable.
AH.

Yes, that will make dice more desirable than before.  Sheesh- but I have no idea how to add that to the math. -.-

This game is complicated.

Kletian999

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Re: Dice vs. flat symbols - which is better?
« Reply #3 on: February 22, 2015, 02:32:13 PM »
Well, each card can be 1 of 6 symbols, the average player will have 2-4 cards in their hand at the start of a battle, maybe presume someone always has either 1 and just 1 Wyld or Sword to burn (50/50).  Note that burn cards cannot influence exploded dice.

Hobbes

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Re: Dice vs. flat symbols - which is better?
« Reply #4 on: February 22, 2015, 03:01:09 PM »
The quick and dirty goes like this :

Symbols (fixed) have the advantage that they cannot be interfered with by other cards (such as Mirror Image), and have their outcome predetermined, this is useful if you wish to plan out your initial layout for shields and swords and have an edge before card burns. However, what you see is what you get. One sword is just that. It's important to note static symbols are declared first though, which can be useful for metaplay (if you have two shields out right away, that can force the opponent to burn cards on swords before anything else happens).

Dice (dynamic) can explode, can be positively improved by other equipment (Moon Scythe, Royal Shield), have a higher potential than fixed symbols, however this is offset by the risk of getting "misses". Dice can also be card burnt in order to fix them into specific symbols for combat, usually wyld or swords. However, this tends to be the case only for the -first- combat, once a player has shot their load on cards, then dice tend to be weaker without a very optimised equipment loadout.

Lienn

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Re: Dice vs. flat symbols - which is better?
« Reply #5 on: February 22, 2015, 06:59:14 PM »
Nice topic, Rat! ^^

Yes, it's only the prospect of card burning that makes the Dice vs. Static symbol comparable.

Meh i got a perfect example of where the differece between them is...but it's locked at private screenshot until the patch goes live later on. A very nice screenie of a guard (that happened to only have 1 die since he actually had 2 but the other one busted) rolling 6 wyld and then a shield.

That's the main difference here. Flat symbols offer linear performance while dice offer variable performance. The card burn just allows you to manipulate this variable performance, which is the main strength of it.

Yet, for the sake of pointing out it's a matter of opinion lemme add another 2 screenies here from a pal that cannot card burn! ^^ 13 dice resulted in 13 symbols w/o card burn.
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Kletian999

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Re: Dice vs. flat symbols - which is better?
« Reply #6 on: February 23, 2015, 12:17:34 AM »
He had 3 rot symbols, so ultimately 16 dice became 13 symbols.  It's true that as dice # gets above 6, the probability of getting at least 1, and then subsequent explosions, increases exponentially.  This really only happens with Rot advantage.  Getting a roll like your screenshot requires extreme luck (4-5 dice would have missed in an average roll), whereas 13 static symbols can never be worse than 13 static symbols.

Hobbes

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Re: Dice vs. flat symbols - which is better?
« Reply #7 on: February 23, 2015, 04:12:22 AM »
Lienn, edge cases are pretty but functionally useless for the debate. Everyone will eventually see edge cases. This is dealing in general probability.

Lienn

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Re: Dice vs. flat symbols - which is better?
« Reply #8 on: February 23, 2015, 12:41:30 PM »
He had 3 rot symbols, so ultimately 16 dice became 13 symbols.  It's true that as dice # gets above 6, the probability of getting at least 1, and then subsequent explosions, increases exponentially.  This really only happens with Rot advantage.  Getting a roll like your screenshot requires extreme luck (4-5 dice would have missed in an average roll), whereas 13 static symbols can never be worse than 13 static symbols.

Now that test is over i flagged the screenshot as public too. Check out the Guard rolling 6 Wyld and a shield from 1 die! ^^ The 2 faces of the same coin. Usually the King would've rolled 20+ symbols with this many dice but, as you pointed out, only 3 Rot exploded.

But what i meant with these screenshots (and the one at Steam) is just to reinforce my point...they're even w/o the card manipulation, which is why i think flat symbol items should be cheaper than die ones...because in the end roll manipulation is an one-sided option.
« Last Edit: February 23, 2015, 12:46:59 PM by Lienn »
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Hobbes

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Re: Dice vs. flat symbols - which is better?
« Reply #9 on: February 23, 2015, 03:12:23 PM »
The math doesn't bear you out Lienn, I could have said this, and Dittman's numbers pass muster, so I'll just state the obvious, I've double checked them myself against my simsheets to be sure of them. Fixed symbols might lack the flexibility, but overall they are mathematically more stable than "die" because of their ability to provide what you want, when you want.

At best having that extra dice will give you a bit of extra flexibility, and the possibility of a wyld when it comes to rolling, at worst, it will miss and give you absolutely nothing. A sword will always give you a flat gain of one sword, one hundred percent of the time. That's something you have to bear in mind.

It may not matter to how you elect to play and that's your good right, but attempting to assert your playstyle as fact flies in the face of the math, so don't.

Edit : We keep discussing card burning as if that's relevant to the additional dice, it's not except in very specific edge cases, more often than not, burnt cards and the fixing of die will only have an effect on die that would ordinarily have been rolled as part of the fight statistic, you'll have a couple of extra dice left over to roll, but functionally all you're doing is extending the pool. It only becomes relevant when you're facing die constrained circumstances (such as mirror image on 4 fight) or similar.

Edit 2 : Also, Lienn, you're not considering the fact of other players using things that may reduce your dice pool when you're arguing the value of having those extra die. Skipping facts to support your argument is bad mmkay? Did you not consider that those fixed symbols are immune to things like Mirror Image in the first place?
« Last Edit: February 23, 2015, 03:29:17 PM by Hobbes »

Alvorn

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Re: Dice vs. flat symbols - which is better?
« Reply #10 on: February 23, 2015, 04:15:21 PM »
Right off the bat, Maths is not my strong suit in fact its not even my second or third suit, Im just really bad at it :) (like that suit pun).
BUT I have to ask (and will probably be told that yes I don't know maths, at which point I will go back to hiding in the game guide and wiki threads  ;) )
When we roll a single die we can get the following outcomes:

 Sword: +1 success (S)
 Shield: +1 success (S)
 Sun: +1 success (S)
 Moon: failure (F)
 Wyld: +1 success (S) and roll again
Rot: failure (F)
(Sun and moon can be switched, as can Wyld and Rot, but you effectively get one success vs. one failure)

Result:
3/6 S
2/6 F
1/6 S + ?

Without going any further, there's a 3/6 chance for success and 2/6 for failure.  This is absolute.  But the final 1/6 (the Wyld die) is going to be split into six outcomes as well, which means:

3/36 S
2/36 F
1/36 S + ?

Shouldn't that be
4/36 S
2/36 F
1/36 S + ?

The reason being that the Wyld (or Rot) while causing another dice roll still counts as Success itself ? i.e. 3S, 2F, 1 Variable, but if rolled it actually makes it 4/S 2F. The exploding dice gives you extra chances for ANOTHER success but does't change the probability of the single first die from 4S 2F.

Its been years since I looked at this stuff so am probably wrong, just curious is all :)

Alvorn

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Re: Dice vs. flat symbols - which is better?
« Reply #11 on: February 23, 2015, 04:20:08 PM »
The Battle Axe vs Shining Steel Sword debate just like this one happens in the studio as well! We have people on either camp.. I prefer.. both?  ;D

Hobbes

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Re: Dice vs. flat symbols - which is better?
« Reply #12 on: February 23, 2015, 04:41:46 PM »
Shouldn't that be
4/36 S
2/36 F
1/36 S + ?

The reason being that the Wyld (or Rot) while causing another dice roll still counts as Success itself ? i.e. 3S, 2F, 1 Variable, but if rolled it actually makes it 4/S 2F. The exploding dice gives you extra chances for ANOTHER success but does't change the probability of the single first die from 4S 2F.

Its been years since I looked at this stuff so am probably wrong, just curious is all :)

Alvorn

(Pulled the relevant bit for brevity)

You're -sortof- right, but Dittman's analysis is correct for probability distribution because he's treating it as a success in it's own right but it's a success with an additional condition, you'd be accidentally double dipping the success if you added it in to the success on the 3S side as well. In reality it's 4 possible successes (one of which is crit) and two fails, but you split out the crit success because that can branch out into it's own separate set of rolls.

Alvorn

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Re: Dice vs. flat symbols - which is better?
« Reply #13 on: February 23, 2015, 05:10:21 PM »
(Pulled the relevant bit for brevity)

You're -sortof- right, but Dittman's analysis is correct for probability distribution because he's treating it as a success in it's own right but it's a success with an additional condition, you'd be accidentally double dipping the success if you added it in to the success on the 3S side as well. In reality it's 4 possible successes (one of which is crit) and two fails, but you split out the crit success because that can branch out into it's own separate set of rolls.

Thanks Hobbes for indulging my little curiosity :) I have to admit I don't completely understand the answer, but I suspected I might not when I asked the question :D But thats me and maths, I left it all behind when I finsihed high-school and haven't looked back since  ;)

Lienn

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Re: Dice vs. flat symbols - which is better?
« Reply #14 on: February 23, 2015, 08:39:53 PM »
OK, i can't believe i actually did this. This time i'll use some visual support to explain it and not going past this because it would require one of these things:

a) advanced numerical methods, which is boring as hell and takes alot of time to do.
b) coding it...but i don't code anything for at least 5 years so it also may end taking way more time than it should.

Anyway, neither of above options are worth doing just to brag rights so, instead, i made easy-to-understand graphs (that are very crappy so pardon me if not everything is aligned or keeping the right proportions...)

Here we have the graph. At X is the input. The number of symbols or dice. Y is the result, what you get from that input.



When we do symbol input we have this:



It reflects the linear growth of it...1 sword will always result in 1 sword, 2 swords in 2 swords and so on. This, though, cannot be used on die. When we make a graph for dice we may see something like this:



Just imagine that it's a spreadsheet of results. The size of dots reflect the number of results you got from a sample at that input. The bigger the dot the more often that result happened. So when we calculate the average result and make a graph of dice it would be like this:



The pointy corners reflect the variable aspect of it. So, now we placing both together we have:



Notice that in the end the results are pretty much the same. The sample is what defines how "pointy" the dice graph is...the bigger the sample the closer it gets to symbol graph.

But now the question...and what wyld cardburn do to it? It does this:



In this image B is the biggest distance it can get from symbol graph, which represents the overburn (all dice were burned with Wyld so instead of "0" the symbol graph now is the minimum result dice roll can offer).

Notice that, though, as you add more non-burned dice the graph slowly gets closer to symbol again because the impact the cardburn causes when compared to the number of dice starts dimishing. When you burn 1 Wyld and have 1 die you actually caused overburn...the minimum result is the symbol graph. If toy have 2 dice, the same carburn would affect only 50% of the dice...if you have 10 dice it would affect only 10% of them. That's why it moves back closer to symbol graph.

If you feel like testing it yourself, just get a paper and, couple dice and write down the results as follow:

a) 1, 2 = adds +0 to total
b) 3, 4, 5 = adds +1 to total
c) 6 = adds +1 to total and roll that die again.

You'll notice that the more you roll for that input the closer it gets to symbol result (number of input = number of result, like rolling 2 dice ending with 2 symbol) because every time you roll you're increasing the sample that will be used to calculate the average point.

Now believe me if you wish or try it yourself and you'll see it's correct. As said at start i'm not going any deeper in this subject than this and also not really try to convince anyone that it's correct anymore.
« Last Edit: February 23, 2015, 08:53:59 PM by Lienn »
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