Will It Hit? – The Math!

HomuMath

HomuMath™

Author’s Note: This article is almost exclusively about one aspect of the math involved in Weiss Schwarz. The charts in this article will be used as references for other articles, such as how to withstand +2 soul/how to best use +2 soul, or when to side attack.

You have three characters on the center stage. Each of them has 1 soul. You have three CXs in hand: a +2 soul, a 1k1, and a 2k1. You’re at 3/6, and your opponent is at 3/2 with no characters. Which play gives you the best chance of winning, assuming that the rest of your deck contains blank triggers and your opponent has 6 CXs left in his or her deck?

Or how about this one?

You have 3 level 0 characters, and your opponent has a level 0, a level 1, and an open slot. Your opponent has 6 CXs in his deck, with 38 total cards remaining. In your hand is a +2 soul CX. What attack configuration and order would allow for the greatest potential damage, and what attack configuration and order offers the highest probability of damage?

But isn’t it all luck?

Yes, insofar as another card game such as blackjack is “all luck”. However, blackjack has a “book”, or a chart of what one should do given a situation. This does not make blackjack a “solved” game, but it does mean that there is an optimal way to play. People play “off the book” all the time and can claim that a “gut feeling” is the way to go, but the player that plays by the book is favored in the long run.

But WS doesn’t have a “book”!

It does now!

1 Soul

Damage Table 1 Soul

Here is the probability table for the chances of a successful attack for a soul value of 1 at any given point for any number of CXs remaining.

By looking at this chart, we can see that most attacks for 1 soul will be very safe. That is, until you hit a point where your probability is below 50%, the attack is “good”. Because luck is a factor, anything at 50% or better (or ideally, just better than 50%), is a chance worth taking; game-winning long shots on the brink of loss notwithstanding.

Short Table

Damage ShortTable 1 Soul

This short table shows the thresholds where attacks for 1 become unfavorable, and is probably not worth memorizing because almost every attack for 1 is going to hit. Even attacking for 1 with 1 card remaining (if it’s a CX) is effectively a hit, because of the refresh penalty. Again, for the sake of these short tables, probabilities of 50% or more are considered ‘favorable’. Anything below 50% is considered ‘unfavorable’ and ranges goes from 49.85% (barely unfavorable) to 0.03% (extremely unlikely).

But people attack for 2 or more!

Fortunately for us, the problem isn’t so much the math as it is inputting it on Microsoft Excel.

2 Soul

Damage Table 2 Soul

Short Table

Damage ShortTable 2 Soul

3 Soul

Damage Table 3 Soul

Short Table

Damage ShortTable 3 Soul

4 Soul

Damage Table 4 Soul

Short Table

Damage ShortTable 4 Soul

5 Soul

Damage Table 5 Soul

Short Table

Damage ShortTable 5 Soul

6 Soul

Damage Table 6 Soul

Short Table

Damage ShortTable 6 Soul

Side story: During a tournament in 2013, our other writer Felix was playing a game to break the top 8 of an event. I showed him the 4 soul table, and his reaction was an expression of sadness deeper than five levels of hell. Why? Turns out he had lost that game to an attack for 4 when he had 7 cards (3 of them CXs) in his deck – a 2.86% chance!

How did you do this?

About the math – these are probability hypergeometric distribution functions. We take the number of successes in a given sample, A (0), followed by the sample size, B, where B is the soul value, the number of successes in the population, C, where C is the number of CXs remaining in the deck, and the population size, D, which is the number from 1-50. We then find out the probability of seeing exactly 0 CXs show up in the sample. This is different from the chances of finding 1 or more in a sample, for a couple of reasons. First, a Brainstorm effect has a certain termination: 4 cards. Damage however, does not, and so calculating the probability of seeing 1, or 1 or more CXs in a given sample will give us an incorrect number.

In Excel, the function is =HYPGEOM.DIST(A,B,C,D,TRUE/FALSE).

What kind of information do these tables give us?

Now that the data have been put out for all to see, it’s a matter of making statements based on the comparisons of situations. Here are some examples of things we can say based on the numbers:

Attacks for 1 are almost always (highly) favored to hit.

The percentages for attacks for 1 don’t weaken until the compression in the opponent’s deck is lower than a 50% chance of hitting. That is, your opponent has to have at least a 1:1 CX/card ratio in their deck to hope to consistently cancel an attack for 1.

Attacks for 3 are usually barely better than coin flips.

If an attack for 3 falls even slightly into the favorable side of the scale, it’s worth making, even if it does cancel.

Attacks for 2 are only slightly less safe than attacks for 1. 1k1 effects are among the best in the game.

1k1 effects help decks before/during level 2 to attack for 2 damage as regularly as possible. Door triggers are almost exclusively found on 1k1 effects, which, aside from the card advantage aspect of triggering said Doormakes the 1k1 Door CX arguably the most powerful CX possible.

An attack for 5 is a high-risk, high-reward play, and is generally ill-advised.

A trap that newer players could fall into is the idea that attacking for as much as possible as often as possible is the way to play WS. If a player disagrees and says “+2 SOUL OR DEATH”, then it becomes a matter of how aggressively against the numbers he or she wants to play. In the long run, a player who insists on attacking for a lot as often as possible will lose more games than he or she will win.

A singleton copy of a +2 soul CX in certain decks, especially those without many effects (e.g. Wooser), is worth having.

In the off chance that an opponent is CX flooded, a single +2 soul CX can capitalize on it much better than a 1k1 effect can. However, because that occasion will come up so rarely, it is better to have fewer copies of the card, because otherwise you will end up skewing the majority of your games.

Now, these are again just a few examples of what one can get from having access to these numbers. Obviously, during a game, we can’t constantly ask how many cards are in our opponent’s deck, and it can be difficult to determine how many CXs remain in their deck, especially if they did not discard any during mulligans. However, the most favorable approach is to assume that if the opponent has not discarded any CXs, that they have drawn at least 1 in their opening hand/turn. Therefore, the viability of attacks, if one is playing to beat favorable odds, should be considered from the beginning as if the opponent has 7 CXs remaining in the deck.

Thanks for reading! Questions? Comments? Want to have the full spreadsheets? Send us an email at theninthcx AT gmail DOT com!

Note: The spreadsheets were made using Microsoft Excel 2013, and may not function properly in older versions.

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