Alice in Numberland: Why 3n+1 is not a good play

Since the release of this design mistake back in Nov 2022 , I have noticed that most of the Alice-related contents seemed to only focus on deck builds and card choices, both on JP and EN side. When it comes to the foundation of this deck i.e. bringing out multiple 2/1 Alices on turn 2/3, these are some of the more comprehensive pieces (written in Japanese) which cover the multiple options:

And if you are not a Japanese reader, my fellow WCC teammate Zabuton has written his Quick Guide on some of the basic lines.

While all these were helpful with listing out the available lines on bringing out Alices, I was in search for some more in-depth answers. Is it better to do the 9 card play or the 4 card play with 4 stock? How much am I improving my odds if I spend one card to go from a 10 card deck to a 9 card deck? Is it worthwhile to make one stock with Leafa to get that one extra brainstorm?

Well if I couldn’t find readily available answers, I suppose the only option is to work it out myself.

Methods, Assumptions, and Disclaimers

The numbers you will see in the following sections are statistical sample means generated from simulations run on GNU Octave. While it’s possible to calculate the true mathematical probabilities, it’s much less hassle to write a few lines of codes than to account for all the branches of scenarios. I’ve somewhat arbitrarily chosen n = 100,000 runs for each simulation, a sample size that I felt was large enough to have the sample means converge on the true values while not needing too much runtime. I have also calculated the true mathematical probabilities on selected cases to verify that the simulations are coded correctly.

The Basic: 3n+1

The questions to start with: how much does each additional brainstorm improve the odds, and how important is getting to that 3n+1 deck size (compared to 3n+2 and 3n+3)?

*Improvement from additional brainstorms*

The left three columns indicate the starting conditions, while P(X) = probability of hitting exactly X times. The simulations are coded to stop brainstorming once we hit 3 Alices, allowing to calculate E(Cost) which is the expected value of stock spent. I will talk more about E(Cost) later, but for now, the focus is the column of P(3).

It shouldn’t be surprising to see that there is a diminishing return as the number of brainstorms increases. The main takeaway from this set of data is to note the significant improvements in the range 3S/10D → 4S/13D → 5S/16D on P(3), and to also note the unimpressive increases when you move into 6S/19D and 7S/22D.

What we can see from here is that, in the 3-5 stock range, every additional card in deck above the desired 3n+1 minimum equates to about 8-9% decrease in P(3).

3 Stock: 51.5% → 40.8% → 32.7%
4 Stock: 71.9% → 62.2% → 53.6%
5 Stock: 81.6% → 73.4% → 66.4%

With the numbers available it’s now not difficult to conclude that reaching 3n+1 is not only desirable, but rather crucial to the success rate of bringing out 3 Alices. To point out a few examples, brainstorming on 4S/15D is only marginally better than 3S/10D, and 5S/18D is in fact worse than 4S/13D. If conditions allow, I am more inclined to drop an Asuna millling off those two extra cards before I start making the brainstorm plays, even if it means one less card in hand from doing so.

The Alternative: Re-Draw (再抽選)

By now you are most likely thinking all that stuff I said above contradicts with the title. Well, I did refer 3n+1 as The Basic in the heading. It’s now time to look the other line of play where, rather than spending all stock on brainstorming, one stock is instead used for the purpose of returning 2 cards in the middle of the process. (For technicality, you don’t have to use one stock if you have three copies of Brainstorm to work with, but it’s much more realistic to consider the case of using 1 Brainstorm Alice and 1 Sortiliena.)

Zabuton has covered the detailed steps on starting with 4 stock and 8 card deck containing 4 Alices. 梨王’s article also has a nice chart showing all branches if you are nihongo jouzu.

(A small correction on Zabuton’s steps: you only need 1 Brainstorm Alice + 1 Sortiliena, as you can begin with Sortiliena + 1st copy of Brainstorm returning 4 cards to make the 8 card deck, and have Soriliena salvage 2nd copy of Brainstorm which will carry out the 3rd time of returning 2 cards).

We can now generalise this line of play for other numbers of stock. Starting with m+1 stock, 1 Brainstorm Alice + 1 Sortiliena:

Spend 1 stock on Sortiliena to salvage a second copy of Brainstorm Alice. Return 4 Alices with Sortiliena and one copy of Brainstorm Alice such that there are 4 Alices in a deck of 3m-1
Brainstorm until 2 copies of Alice in waiting room (extra copies from brainstorm mills) or 2 cards remaining in deck, whichever happens first.
Return 2 cards from second copy of Brainstorm Alice, putting back Alices where applicable, and finish the remaining brainstorms. The total number of cards milled by brainstorm is 3m+1

The first thing to notice is that this Re-Draw line of play completely eliminates the possibility of hitting brainstorm only once, meaning at least two Alices are guaranteed to be on the field. With the 3n+1 method, while the values of P(1) are generally very small, the fact they are non-zero means you will inevitably hit that low roll once in a while.

Now I can finally answer the very first question I asked in the beginning section: Is it better to do the 9 card play (3n+1 line) or the 4 card play (re-draw line) with 4 stock?

We can see that there is a ~15% increase in P(3) using the Re-Draw method for the 4 stock scenario, and similar increases with 5 stock and 6 stock. It’s understandable that the increase is smaller with 7 stock, since P(3) in Re-Draw is quickly approaching the 100% upper bound as the number of stock increases.

So that’s it? We should just go the Re-Draw method whenever we can?

The answer is (sadly) not something I can give you, but rather requires your own judgement (and it may or may not be the same conclusion for everyone). There is one more factor to consider here, and it’s all about this E(Cost) which I haven’t talked about thus far.

Let’s consider 4 stock scenario again. When you go for the Re-Draw method, you use 1 stock for Sortiliena and 3 stock for brainstorming. This happens 100% of the time. You will always be spending 4 stock for this 86.9% chance of getting 3 Alices. On the other hand, while 3n+1 can only give you P(3) = 71.9%, you can get lucky and simply hit 3 out of 3 brainstorms, saving yourself one stock (and also leaving 4 cards in deck which means your combo CX goes back on Refresh). The natural question is how often this happens.

On the 4 stock 3n+1 line, E(Cost) = 3.7353. This means on average we will spend this much stock when making this play. Since in reality we can only spend whole numbers of stock in game, a better way to phrase this is we will spend 4 stock in ~73.5% of our games, and save one stock in the other ~26.5%. It is now up to you to decide whether the 15% boost in P(3) is worth the value of saving one stock approx. 1 in 4 games, because it’s definitely not something we can easily compare quantitatively, and attempting to do so will most likely only produce meaningless results.

Before moving on, I do want to point out that, for 5+ stock scenarios, the difference in E(Cost) is small enough that it’s reasonably safe to say Re-Draw method is simply better in this range.

The Prodigy: Guaranteed Triple

In his Triple Alice World Map on yuyu-tei, K named the 5 stock 2 card setup as the Prodigy Node (神童ノード). After all, none of the above has P(3) = 100%. Again, Zabuton has covered this line in his Quick Guide under Method 3.

So far I’ve been making comparisons without any mentions to the hand/resource required to get to the desired deck size, almost as if we were making a naive assumption that we can get to any deck size with equal ease. This is of course not the case. The intention of these comparisons to know which method to choose when we can reach the deck size for either, and to give ourselves a basis on deciding whether we should spend additional resource to go for one method over another.

Although there was no need to run any simulations on a line with 100% success rate, we can make some observations by comparing with the other two method on the 5 stock scenario.

The comparison between 3n+1 and Re-Draw has already been covered previously. If milling to 7 cards would cost one extra hand than milling to 12 cards, I am probably okay with spending this one extra hand for the 15% boost in P(3). Now, if milling to 2 cards would cost one extra hand than milling to 7 cards, the answer to this decision making is probably “It depends”. Milling to 2 instead of 7 only gives a ~4% boost in P(3), which may or may not be worthwhile depending many other factors. If your opponent already has 3 Alices on board, you might be more inclined to take the Prodigy line so you know you won’t be behind with only getting out 2 Alices. If you are the one to make Alice plays first, saving this one card in hand as well as the 45% chance of saving one stock might look more favourable.

The Odd One: 1 Alice Left In Deck

Here’s a not-so-uncommon scenario: you have 4 stock, 2 copies of Alice Brainstorm in hand, no Sortiliena, so you start milling towards the 9 card deck since 3n+1 is the only line of play here. As you reach the 9 card deck, you notice there are only 3 Alices in the bin. The fourth copies is still in the deck. You are now presented with 2 options: you can return 3 Alices + 1 other card using both copies of Alice Brainstorm (and make the usual 3n+1 play), or you can return only 2 Alices with one Brainstorm, start brainstorming, and aim to return 2 Alices with the second copy of Brainstorm.

Once again the first thing to notice is that by “saving” that second copy of Alice Brainstorm, P(1) becomes 0. This shouldn’t be surprising as the second Alice Brainstorm here has a very similar function to the Re-Draw line of play. We can see the odds are significantly better than the usual 3n+1 line, and given there would be no difference in resource spent on milling, this is a simple case of one option being objectively better than the other. Of course we have no control on whether that 4th Alice stays in the deck during milling, but it’s important to know what to do when the situation does arise.

The Accidents: Alice Stuck In Stock/Clock

Speaking of things we have little to no control, if having Alices left in deck after milling can be considered some form of high-roll, it’s time to look at the other side when low-rolls happen. During first two turns, our Alices can, and will, make themselves inaccessible by going to stock or clock. Let’s break down our options in different scenarios.

Alice at the top of stock:

This one is probably the easiest to deal with. The answer is to go for the Re-Draw line, since you spend one stock before returning any cards to deck, meaning this Alice is effectively the same as being in the bin.

Alice at the 2nd/3rd card of stock:

The best solution is the Prodigy line if conditions allow us. The first step only requires two Alices in the bin. You spend one stock for the first instance of brainstorm. Then the next two stock are spent on Sortiliena returning cards, meaning this Alice in stock will become accessible in the process.

If Prodigy line is not an option, the next best option would be some slightly modified form of the Re-Draw line. Keep in mind that the core concept of the Re-Draw line is having one instance of return 2 in the middle of brainstorming. Using this to put back the Alice once it is paid out from stock (along with other extra Alice milled where applicable) generally will give the best results.

Alice in the deep part of stock or in clock:

JUST GIVE UP AND ACCEPT IT.

If the Alice in buried deep in stock, it’s likely not possible or not worth the resource to make it accessible. If it’s in clock, there is generally very little options besides some fringe cases of clocking up to LV2. Obviously it’s not ideal, but we will just have to resign to the fact that we only have 3 Alices to work with.

I did think about running numbers on scenarios when only 3 Alices are accessible, but in the end I’ve decided they are not going to be too meaningful. The conclusions we’ve made from the comparisons above will generally still apply. While you might be interested in the drop in P(3) from 4 Alice case to 3 Alice case, the reality is, if we cannot control whether this 4th Alice is accessible, the knowledge of P(3) being lowered by 15% or 20% isn’t going to change our plays.

Some Closing Words

While having a handle name like 博士 and showing all these numbers in tables probably give off an impression that I like crunching numbers in WS, this is in fact the first time I’ve done something like this. I hold the opinion that the complexity in WS as a game makes most pre-researched calculations a little lacking in generality, and I would prefer to make case specific estimates on the spot during games if there is a need. However, SAO is a deck consistent enough to reach those desired deck sizes before carrying out the steps of bringing out Alices, which makes it worthwhile to do a quantitative analysis to know my lines of play.

The important takeaway from all these tables is not to memorise these numbers. It’s not for the purpose of being salty about “missing that 87%” when only 2 Alices landed on your board. It’s about having knowledge of “If I can afford to, I should mill to 4 cards instead of 9 cards when I have 4 stock”. It’s about “If I have 5 stock, milling to 12/7/2 cards are all valid options and I should plan my milling with this in mind”. Playing Alice should never be “Mill down my deck to a somewhat thin size and hope for the best”. Admittedly, it can still work out to have the same result due to the probabilistic nature in brainstorm (and you won’t feel the difference once your 3 Alices are on the board), but I hope you will agree with me that a systematic approach is preferred if we want consistent results.

博士notes