Articles

# Mulligan and Combinatorics Force of Will implements the Mulligan as a game mechanic. This is a huge advantage towards consistency in the early game, allowing to quantifiably reach combos to a larger extent.

In the ongoing article I want to highlight the positive effects of the mulligan, solidifying my arguments by combinatorics.
The Mulligan enables a better access to certain cards (eg. Hate-Cards) and Combos as well. Many of the following thoughts are easy to understand with a certain level of experience at any card- or boardgame. The applied combinatorics is a useful tool for card players, especially if considered prior to deckbuilding, or to evaluate combos.

Let’s start at the basic level and delve in with a simple example on combinatorics:
The first important word to learn is factorial (at least for us as card players): n!= n*(n-1)*…1

If you take 5 different numbers and try to arrange them in a given order, you may do that by multiplying all possibilities for the given position. If you draw 5 numbers out of a bowl a this translates to:

5*4*3*2*1= 5!

Lotto would offer more numbers than positions to the drawn numbers. A smaller version of Lotto – take for example 40 ‘numbers’ – with 5 drawn numbers offers 40 possibilities for the first position, 39 possibilities for the second position and so forth:

40*39*…*36 = 40!/35!

Figure 1 –  5 out of 5 numbers in given order; 5 out of 40 numbers in given order

If we now consider the order indifferent, the amount of possibilities reduces drastically, because the combination of all variation converges. Mathematically the possibilities to arrange the same numbers in a different order is called Permutation. How many Permutations may be omitted? For 5 distinguishable numbers it is 5!, like we discussed earlier. This means the possibilities reduce by the faction of 5!, because those options converge.

If the order won’t matter the possibilities are:

Figure 2 – The binominal coefficient directly translates to a start hand of k cards out of n deck cards

This expression (called binominal coefficient) sums up all possibilities for 40 different numbers to be drawn without a given order. I guess you caught that this simplified Lotto version directly translates to a 5 card hand out of a 40 card deck. Luckily, we may use playsets of cards to guarantee more consistency for a starting hand, thus varying the model to access cards in our favour.

The probability for a certain scenario is defined by the possibilities under the scenario’s parameters divided by all possible scenarios. With few tricks one might build models for Force of Will decks that will help to re-evaluate card choices and offer the chance to take a closer look upon deck construction and optimisation.

The first pesky topic is deck size. Let us assume that a card A is available as playset (k = 4) in a 40-card deck (n = 40) and we want to draw card A into our starting hand.

Figure 3 – Possible different start hands from n deck cards and k hand cards

We may consider the following scenarios:

1. Card A is not in hand
2. Card A is at least 1 time in hand (1,2,3 or 4 times)

This simplifies our deck to 36 cards we may ignore and 4 cards we want to access in our start hand.
The overall possibilities are simply the options multiplied:

Notice: the added positions in the upper row sum up to n (=40) and the bottom row adds up to k (=5).

Figure 4 – the probability to draw into card A

How does probability vary due to mulligan? We may look at a deck with n cards a consider the increasing hand size k due to mulligan and its effect on our probability to draw a card.

Figure 5 – Probability to hit A at least 1 times

Figure 5 shows all possible events to hit card A at least one time (don’t be confused by the probability normed to 1: you might as well multiply the result with 100 and end up in percent). The rest may be calculated the same way (with increasing hand size). However, that might be a bit too much information, so I decided to plot the probabilities with increasing Mulligan.

Figure 6 – Correlation of deck size and Mulligan to the probability to find a certain card A

What do we learn from the graphic? Well, one additional deck card results in 0,9 % loss of probability to start the game with the wanted card A. Be careful: it isn’t always disadvantageous to play more than 40 cards, especially if you try to avoid a certain card in the starting hand (like Satan, God of the Fallen (DBV-099), which may only to be in the deck to be removed via Invitation to Purgatory (DBV-079) and is rendered useless if drawn to hand). But most of the time it is crucial to optimise the deck as far as possible.

The model for Figure No. 6 is based on views the Mulligan as an extended hand. Looking at Mulligan 1 as a 6 card starting hand, etc. The Mulligan 5 (so to speak a 10 card starting hand!) gives a probability to draw into card A of about 70 % compared to the 42% without Mulligan, thus increasing consistency dramatically. This has real implications to the game, especially considering Hate-Cards to counter certain strategies (like: Mourning Angel (AO3-013) = Grave-Hate; Regalia Break (AO1-032) = Regalia-Hate; Pier, the Godspeed Archer (AO3-014) = Addition-Hate, etc.).

For example, if you play Arla (AO3-BaB-1) // Arla [J-ruler] (AO3-BaB-1J) against Rezzard (AO3-BaB-3) // Rezzard [J-ruler] (AO3-BaB-3J) with good Grave Recursion you might want to start with a Mourning Angel (AO3-013) at the beginning of the game.
As we learned the probability to start with Mourning Angel (AO3-013) is about 70%, if we consider to Mulligan 5 to find the card. Making Arla able to start with the Grave-Hate in approximately 2 out of 3 games. Compared to the Mulligan 0 scenario giving the Arla player the opportunity to start with at least 1 Mourning Angel (AO3-013) in 2 out of 5 games.

Combinatorics helps to understand other phenomena in deck construction and gameplay (i.e. building a 3-colour Magic Stone deck: how high is the probability to hit a certain colour until turn Nr. X), but more importantly combo decks may calculate and optimise due to consideration upon combinatorics. Let’s take a look.

We want to pull a combo from card A and card B. If a 40 card Deck with a 5 card starting hand is used to pull the combo all possible starting hands may be calculated with:

A and B are in the deck as playset and the rest of the deck consists 32 cards irrelevant to the combo:

X times card A out of 4 cards

Y times card B out of 4 cards

Z times card C out of 32 cards

The sum out of x,y,z should be 5 and we need to find all possible combinations. The math is the same, since the amount of possibilities is given via multiplication:

Figure 7 – Possibilites to create a starting hand with cards A,B,C

Figure 8 – Scenarios with A and B in a 5 card starting hand

How do we determine the influence of the Mulligan to this problem? The best way to start is to implement a simple rule: if we find neither A nor B we will take a Mulligan 5 out of a 35 card deck. This would be a deck consisting of 4 times A and B and 27 times card C (which is irrelevant to the combo).

Figure 9 – Implemented Mulligan rules to find the Combo

The probabilities are calculated by the overall possibilities for the given scenario: The possibilities for the combo with 5 out of 35 card and 4 out of 35 card with the given result of respectively

Figure 10 – No combo piece is in hand: the probability to draw in to the combo via a 5 card Mulligan

The sum of all possibilities reflects the scenario in which the combo is not in the start hand. So, the overall probability to draw into the combo can be calculated by the multiplication of the probability without the combo and the probabilities after Mulligan and summed up for all scenarios.

Figure 11 – One combo piece in hand: the probability to draw into the combo with a 4 card Mulligan

The resulting probability is simply added to the 16% we calculated before (without Mulligan). This results in additional 27,83% to start with the 2-card combo!

Figure 12 – Additional Probability gained due to the applied Mulligan rule

Overall, there is a 43% probability to gain A and B to the starting hand. This means 2 out of 5 games, considering a 40 card deck.

Figure 13 – Decksize dependence to draw into the combo

How does the deck size effect the chance of starting with the combo? We lose about 1,6% per card. A 43 Card deck loses 4,7%, a significant loss in combo-potential.

Sending