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knives1990

Shaddoll/Invoked Fusion Spell Calculations

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knives1990    334

If someone could even just point me in the right direction to find this out, that would be helpful or explaining the process would be much appreciated but here is the question:

 

id like to know how many cards I should main deck to get the highest probability of opening 1 or 2 fusion spells and no more than that amount. This is critical for my deck-building purposes (and this can include any of the searchers and Aleister himself, because I'm building around the idea that my normal summon needs to be reserved for Aleister every turn) here are the cards I'm using to see a Fusion spell:

 

3 Terraforming

3 Shaddoll Fusion

3 Aleister

2 Magical Meltdown

1 El Shaddoll Fusion

 

thank you to anyone who can help

Edited by knives1990

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the4got10-1    17

I used this: http://yugioh.party/

 

Set deck size to 40, hand size to 4, 12 cards, open 1 to 2. Occurs 70.31% of the time.

 

Set deck size to 41, keeping everything else the same. Occurs 69.76% of the time.

 

So optimal deck size is still 40 if you consider yourself going first. Just for a side note, the actual optimal deck size is 32 going first.

 

Going second though (ie hand size 5), a 40 card deck gives 70.20%, a 41 card deck gives 70.21%, a 42 card deck gives 70.16%.

Edited by the4got10-1

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knives1990    334

Hmm it seems like I open more than 2 a lot more than I'd like, is there a way to reverse that and determine how often I get more than 2? That's the main thing I'm trying to avoid. I play enough draw spells and deck thinners to see at least one in my opening hand a good amount of time

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Oh_The_Irony    772

It's unclear what you mean when you say you want the highest probability of opening two fusion spells and no more than that amount, since increasing the probability of one means increasing the probability of the other. Heuristically, you could just let the probability assigned to each number of fusion spells be P(Open 1 or 2 fusion spells) - P(Open 3 or more fusion spells), and then maximize that probability. Or you could set a maximum limit on the probability of opening three or more fusion spells, and find the maximum number of fusion spells such that you don't go over that limit. Or you could set a minimum probability for opening one or two fusion spells, and find the minimum number of fusion spells such that you reach that limit. All of these strategies are simple to execute using yugioh.party (once you set the appropriate limit for the latter two strategies).

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knives1990    334

It's very hard for me to understand this kind of mathematics and I'm doing my best on my own by individually increasing deck size, then min/max of 1-2, then changing that to 3-3. It's hard to comprehend and I'm sorry that I am not understanding it. I want to open 1-2 max, but it's not nearly as important as not seeing 3 or more.

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+ACP+    34066

Basically we want to choose a integer N that maximizes the hypergeometric equation: (K!/(1!*(K-1)!)*(N-K)!/((n-1)!*((N-K)-(n-1))!))/(N!/(n!*(N-n)!))+(K!/(2!*(K-2)!)*(N-K)!/((n-2)!*((N-K)-(n-2))!))/(N!/(n!*(N-n)!)), for K=12, n=5 (if you're going first), subject to the constraints 40 <= N <= 60. Using the standard optimization tools that come with any math software package, we get the solution N=41. Note that 40 and 41 cards almost the exact same probabilities. For 40 cards, the probability is .70199. For 41 it's .70213. They're effectively the same.

 

However, if you're going 2nd (so n=6), this optimization scheme yields the solution N=49. This is a very different number, but this makes sense. Because we're starting with an extra card, we've increased our chances of opening more than 2 copies of our fusion card, so we need to add extra cards to our deck to counteract that.

 

The correct number of cards to run overall then would depend on how often you expect to go first. Assuming that you expect to go 1st 50% of the time, it would be correct to play 45 cards. If you expect to go first 60% of the time (maybe some people in the meta are blind-picking 2nd), then it would be correct to play 44 cards.

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+mark    3096
5 hours ago, knives1990 said:

Hmm it seems like I open more than 2 a lot more than I'd like, is there a way to reverse that and determine how often I get more than 2? That's the main thing I'm trying to avoid. I play enough draw spells and deck thinners to see at least one in my opening hand a good amount of time


With yugioh.party you fill in decksize, starting hand and how much copies of the card you play

 

then you fill in 'min 1 max 1' = odds of drawing exactly 1

'min 2 max 2' = odds of drawing exactly 2

'min 3 max 3' = odds of drawing exactly 3

 

do this until you feel satisfied

 

your question is vague though, do you want the highest odds of opening 1 or 2, while opening 0 or opening 3 are equally bad? or is drawing 0 better than drawing 3, since you say 'no more than that'. anyway you can set min to 1 and max to 2, this gives you the odds of opening exactly 1 or 2.

If you play 40 cards no drawspells/deckthinning etc: the optimal number is:


11 or 12 going 1st

10 or 11 going 2nd

 

so 11 is the optimal number, it gives around 70% odds of opening 1 or 2 fusions spells regardless of going 1st or 2nd. adding more will result into drawing 3 spells to often, and adding less will result into drawing 0 too often

 

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+mmf    23263

you missed invocation in the list in the op, right? if you're including that yugioh.party won't work and you'll need to use the overlap calc instead. neither are necessary though because Allen did all the work above

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knives1990    334

Drawing 0 is not equally as bad as drawing 3 or more. My worst case scenario is opening 3 or more, bad but not as damaging is opening 0. Best case is opening 1 or 2. That's my criteria and it's probably still difficult. I do play 3 allure and 3 wonder wand (granted if I wonder wand the same turn that I'm not opening one of the 12 cards, that decreases my ability to fuse that turn drastically even if drawing into one of the twelve)

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+Satchmo    3226

whygodwhygodwhygodwhygod

 

What you're actually looking for is the point of diminishing returns, where the increase in the chance to open your ideal number (1 or 2 in this case) is less than the increase in your chance to draw extra copies you don't want (3+) up until a certain point. First you need to note what that point is- for example playing a 40 card deck with 12 fusion spells you'll draw 3 14% of the time, minus the % of the time you draw 3 and can still make a strong play. In the case of 40 cards, the point of diminishing returns is 12, as the increase of drawing 1 or more (3.1x%) is the same as the increase in the chance to draw 3 or more.*

 

You're essentially working backwards from the standard fix of this. Instead of decreasing the amount of copies in your main deck, you're increasing the size of your main deck. Which leads to this:

cards in deck-Chance to draw 1+-chance to draw 3+-percentage decrease
40 - 70.20%  -14.86%-0/0 [In other words, we're comparing how increasing the deck sizes decreases both numbers]
41 - 70.21% - 13.94% - negative .01(???)/.92
42 - 70.16% - 13.09% - .05/.85
43 - 70.04% - 12.30% - .12/.79
44 - 69.88% - 11.58% - .16/.72
45- 69.66% - 10.91% - .22/.67

Note all those numbers are going first. But the changes between 43 and 44; 44 and 45 are also very small for opening 3+. So the point of diminishing returns by increasing deck sizes isn't actually 45 (Allen was doing a different kind of math, so he never claimed it was). But you can see the issue that increasing deck size gets more harmful than helpful very quickly. However you've already discounted the alternative to this, which is "play less fusion spells"

So the real questions are, as opposed to "how can I play more mediocre cards to draw 3 fusion spells less often" is "What percentage of games can I reasonably allow myself to draw 3 fusion spells in exchange for the most optimal deck" and "what percentage of games can I draw 3 fusion spells and still win (no the number is not 0)".

The second question can be answered by accounting for all the combinations of cards you can draw in addition to excess fusion spells. The first question is one that can't actually be calculated by anyone, it's an arbitrary number you will have to determine that is you "Comfort zone". Once you can determine what your comfort zone is and what % of games get shaven off the total number of games you draw 3+, then you can determine what number of cards in deck/fusion spells is optimal.

Also- Invocation is a fusion spell, even if you don't consider it one when you only draw it or Invocation + (fusion Spell), you absolutely need to calculate it in draw 3+. Drawing 2 Aleister and Invocation is, by your account, the same as drawing 3 Aleister.

 

*There's a point of contention, where you could argue that because the increase in both is the same, that 11 is actually correct, under the premise a net 0 increase is the same as a negative increase. I'm not a hyper math theorist and I don't have the ability to debate something so specific one way or the other.
--------------------------------------------------
Someone please spell/grammar/logic check this

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knives1990    334

You're completely right, I forgot to add the 2 invocation to that number above making the total amount 14, thanks for catching that. I noticed that in most of my games tonight I was opening with 3+ a large portion of the time at 40.  Maybe anecdotal 

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knives1990    334

It's a hard problem to tackle in a 40 card deck because I have to play enough shaddoll monsters and different attributes to be able to fusion summon in the first place. Then I noticed that even when I'm making big plays, I need some kind of traps to reinforce the field or to help fight back if my play gets stopped. I'm going to try and digest what's been said here and go back to yugiohparty to see if I can figure it out for myself. Didn't expect so many replies, I'm appreciative of you all trying your best to give me understanding

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+mark    3096

Not really an answer to the question of OP, but rather an extension:

Honestly I think it's better to work in percentages, rather than black and white thinking (0 or 3 is bad, 1 or 2 is good). I mean yes, drawing 1-2 fusion spell is probably more optimal then 0 or 3, but that doesn't mean drawing 1-2 fusion spells + 3 traps will do you any good, for example. You could work in categories, the first one obviously being going first or second, which requires an estimation of how opponent will blind 1st/2nd. For example, if you blind first and assume 90% of people to blind first, you can begin at:

55% - Going first
45% - Going second

Now that we got that out of the way, let's start with going first:

There's probably the 'ideal' kind of hand. Opening a fusion play + a trap would probably be it. This hand should have a certain win ratio, say, 90%. Then there's the 'good but not broken' hands, where you can fusion play but don't have a trap, and maybe you win 70% when you open these. Then there are the brick kind of hands - but some are better than others. Some hands may win you literally 0-5% of games, but others may still give you 20-30%. Knowing how often you win when you open a certain kind of hand is useful because then you can actually calculate what ratio's you need. You may find hands with 1 fusion to be better than hands with 2 fusion spell, because when you draw 1 fusion spell you're more likely to also see a trap, etc. 

So now you can multiply all these scenario's and add up the percentages: this gives your total win rate. It's probably your win rate for game 1, but you can account for siding and do the same for game 2 and 3 as well. So to give a quick example to show what I mean:
If going first you're 40% to draw hand A, and 60% to draw hand B, and hand A gives you a 90% win ratio while hand B gives you a 40% win ratio, and going second you have a 40% chance of drawing hand C and a 60% chance of drawing hand D (I assume going 2nd you're looking for different cards etc such as board breakers, but these are random numbers anyway to show the process), and hand C gives you a 40% winrate but hand D gives you a 30% winrate, then your current win rate is:

55% * 40% * 90% (=19,8%)
+
55% * 60% + 40% (=13,2%)
+
45% * 40% * 40% (= 7,2%)
+
45% * 60% * 30% (=8,1%)
=
Total win rate = 48,3%

Looking at the when diminishing returns will arise is helpful, knowing what ratio will give you the highest chance of opening 1 or 2 fusion spells is helpful, but these aren't telling you how good the rest of your hand is, and, whether you're winning the game or not. Obviously how often you win when you draw certain hands depends on your match-ups against certain decks, and how much people play these decks, etc. But you need these numbers first, before you can look at maximizing your win rate: calculating the odds of opening certain hands is basically the last step you take when determining your deck, not the first.
 

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»victor    6400

Aside from the consistency of Fusion Summoning, isn't it better to diversify rather than double down on 2 Fusion Summoning themes?

 

I mean Dimension Barrier is a blowout, so you don't want to just stop. That's why WWIA, have Wind Witches for Synchros, and Artifacts so they can play despite DBarrier. And Zoodiacs play a lot more backrow than a Shaddoll Invoked deck. Paleozoics have Traps to fall back on, and you can XYZ and Fuse in that deck.

 

You don't really distinguish yourself from Fluffals in that case, with folks considering Purple Herald for DBarrier, just saying.

 

-----------------------

 

I ask myself, what can the deck do if its disrupted, what is the best field you can end with?

 

That's the line of thinking that leads me to consider Jigabyte (WATER), Inari Fire (FIRE), and Nefarious Archfiend Eater of Nefariousness (EARTH).

 

So Shaddolls and Aleister are both Spellcasters, so you can SS these whenever to Rank 4, and they double as Atrributes you can Fuse with, or SS and Wonder Wand :)

 

I wonder if something like Dark Renewal is worth considering now, that Pendulums are a lot less prevalent.

 

DARK Monster Reborn or Burst Rebirth are interesting as well, but the key factor to consider is Shaddoll Dragon is 1900 ATK, stronger than Daigusto Emeral. 

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+ACP+    34066

@Satchmo, if you're going 1st 50% of the time, the point of diminishing returns is 45 (with the numbers that he gave in the OP). Optimization and finding the point of diminishing returns are essentially the same thing.

 

When we add 2 Invocation for 14 fusion spells, then it becomes correct to play 48 cards when going 1st, and 58 cards when going 2nd. If you expect to go 1st 50% of the time, then you should play 53 cards.

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knives1990    334

I cut terraforming to 1 so I'm still playing 12, and was basically just guessing and using 45, now 46 because of victor's suggestion about the lvl 4 special summon guys they're coming in pretty handy. Trying a mix of those and kaijus for a couple extra fire and earth elements to make shek and grysta live a little more often, as well as giving myself some other options each turn outside of fusion summoning. Extra deck is insane atm tho 

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knives1990    334
On 3/9/2017 at 10:13 AM, victor said:

Aside from the consistency of Fusion Summoning, isn't it better to diversify rather than double down on 2 Fusion Summoning themes?

 

I mean Dimension Barrier is a blowout, so you don't want to just stop. That's why WWIA, have Wind Witches for Synchros, and Artifacts so they can play despite DBarrier. And Zoodiacs play a lot more backrow than a Shaddoll Invoked deck. Paleozoics have Traps to fall back on, and you can XYZ and Fuse in that deck.

 

You don't really distinguish yourself from Fluffals in that case, with folks considering Purple Herald for DBarrier, just saying.

 

-----------------------

 

I ask myself, what can the deck do if its disrupted, what is the best field you can end with?

 

That's the line of thinking that leads me to consider Jigabyte (WATER), Inari Fire (FIRE), and Nefarious Archfiend Eater of Nefariousness (EARTH).

 

So Shaddolls and Aleister are both Spellcasters, so you can SS these whenever to Rank 4, and they double as Atrributes you can Fuse with, or SS and Wonder Wand :)

To branch off of the Cryston thread, one of the main things I think has caused such a decline in ygo posting activity is the fact that no one seems to discuss anything anymore in depth. Victor is one of those few posters that may mention a lot of mediocre ideas on the surface, but he continues to post and provide examples of different things that can work in whatever deck he is talking about at the time. And with these kinds of posts in general from anyone, there is value to be gained in some way. Maybe you need to really need to dig through and apply some of the thought processes to your own ideas or theory, but it will resonate with some people and this particular part of his post resonated really well with me and I think gave me another breakthrough in the deck.

 

I'm still fine tuning this and will be taking this to a regional in a few weeks, but cards that provide me with useful Attributes that can also extend my plays or like Victor mentioned give me something to do if I get Barrier'd are exactly what i need. Some combination of these types of cards and Traps, while being able to keep the original engine and plays intact is what will give myself the best chance to build the best version of this deck. I decided to play 2 Archfiend Eater out of those mentioned as well as 2 Kumongous for now since EARTH is the most useful Attribute as a secondary suite to the DARKs. What gave me another idea is to play Photon Thrasher over Ghost Ogre too since he is more useful as an all around card at the moment in going first or second.

 

At the moment, Id say I'm on the right track though. Tackling the extra deck issue is hard, but to begin I think side decking a few extra deck cards to suit different matchups would be ideal.

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»victor    6400

For the record, I often tack on card interactions for the sake of it, there's a main point I want to make, but I like mentioning other cards, sometimes for fun, because nobody else will, or dares to.

 

E.g. I was going to mention Fortune Ladies (how Metalfoes can pop LIGHT->LIGHT->FF-> Pend WATER, draw 2, the modern LV 4 WATER interactions with Norden and being Pendulum Summoned, and Fortune's Future with Invoked, especially with Allure of Darkness on DARK) but decided against it earlier.

 

I mean that deck is good now, because in the past, you just accumulated a ton of cards with WATER and Fortune's Future, but you played terrible monsters, and couldn't do anything as a deck (no win condition), but now you have Invoked to make use of the advantage. And you have Pot of Desires now, so it's more consistent for you to string the 2 card combos together.

 

I'm sure I could be more prudent in posting, but I honestly don't care or have anything to prove.

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»victor    6400

I mean, think about it, Fortune's Future, aside from Pot of Desires, is the only other +1 draw card in the game, and FF Water is Destiny Hero Disc Commander.  What other cards draw 2 right now, and give you a body or a recycled resource for an Invoked Summon, on top of it?

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knives1990    334

No I really appreciate your posts in general. It gets the thoughts flowing and there are a lot of times where I've read your posts over the years and wondered why no one else was thinking about some of the things you delved into. Hoban said it himself that having edges in deckbuilding is always superior to simply playing a standard deck and relying on technical skills. They go hand in hand, and I'm going to be catching a lot of people off guard with my deck.

 

The main problem with fortune lady's is they only serve a draw engine and a fusion engine for the invokeds. I don't even main the invoked WATER fusion either because there's simply not enough extra deck space for a fusion with a bad effect and low attack too. There's also the problem of taking up a lot of deckspace. Your other idea serves as fuel for both fusion engines and also gives me another play outside of fusion summoning. I'm already playing 3 allure 3 wonder wand and 2 desires on top of card searchers

Edited by knives1990

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