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By: nmc, Alex KK
Nov 06 2019 12:00pm
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It's been a few months since the London Mulligan rule took effect on Magic the Gathering. As players, we have had enough time to get a feel for it and unanimously agree that the new mulligan rule favors taking mulligans, especially in combo decks. However, we don't need to rely solely on intuition in our evaluation of the new mulligan rule. We can instead actually compute just how much better it makes combo and linear decks by programming a computer simulation that plays out simplified versions of these decks. Additionally, we can decide on an optimal deck list for each one and decide when it's optimal to mulligan. We will focus on the new London rule in comparison to the older Paris rule. The Vancouver rule (scry after the mulligan) adds consistency, but marginally so compared to the new London rule.

Play Strategy

The idea is that we will have a fixed strategy for each deck and simulate goldfishing it millions of times. For simplicity, we will assume that there is no maximum hand size.

The simplified combo deck consists of Island, Mountain, Pestermite and Splinter Twin. We try to play a land on each turn and prioritize getting to single blue and double red mana. Once we have enough lands in play and a Pestermite, we cast it and then try to play Splinter Twin in the following turn to win instantly.

Our simplified linear deck consists of just Mountain and Lightning Bolt. We try to play a land on every turn and then as many bolts as possible. We win the game when we have cast 7 bolts for lethal.

Computing the average win turn

We start by simulating millions of games for each possible starting hand of a deck. We save the turn that the deck goldfished on and  compute the average win turn for each  configuration. This is calculation is irrespective of mulligan rule and we need it to proceed in our evaluation.

We immediately notice that the Splinter Twin deck has 7 and 6 card hands that will always win on turn 4. This is because you just need 1 Pestermite, 1 Splinter Twin, 2 Mountain, 1 Island and any other fourth land in your opening hand to guarantee lethal on turn 4.

But in other cases, the win turn is higher. For example in a deck with 12 Pestermite, 12 Splinter Twin, 13 Island, 23 Mountain then this starting hand

Island Mountain Pestermite Splinter Twin Island Island Pestermite

has an average win turn of 4.76

The Burn deck does not get such broken 7s. It turns out that our best 7 in a deck consisting of 16 Mountain and 44 Lightning Bolt is

Mountain Lightning Bolt Mountain Lightning Bolt Lightning Bolt Lightning Bolt Lightning Bolt

which wins on turn 4.23 on average.

When to mulligan under the Paris Rule

Once the data above is available we can calculate when it's optimal to keep or to mulligan a particular hand under the Paris Rule. The idea is that:

- We always keep 1 card hands, since going down to 0 will never make us win faster.

- We will keep a 2 card hand if its average win turn is faster than the weighted average win turn of all 1 card hands.

What is the weighted average? Notice that because of our decklist choices, not all 1 card hands are equally likely to come up

One Card Hand Average Win Turn Chances Weighted Average Win Turn
1 Pestermite 9.67 20% 1.93
1 Splinter Twin 9.91 20% 1.98
1 Island 9.46 22% 2.08
1 Mountain 9.70 38% 3.68
Sum:     9.67

Since we know that going to 1 card will yield an average win turn of 9.67 we will mulligan

Splinter Twin Splinter Twin

since it has an average win turn of 9.76, but we will keep

Island Mountain

because this hand will win on average in 8.20 turns!

- We keep increasing our hand size and decide whether to keep or mulligan each unique starting hand in the same fashion.

We end up knowing if it's optimal to keep or mulligan every possible hand.

When to mulligan under the London Rule

We will now try to compute when it's optimal to keep or mulligan under the new London Rule. Under this rule we always draw 7 but need to scry some cards to the bottom. So we need to first compute which subgroup of our starting 7 to keep; that is for each 7 card hand and for every hand size from 1 to 6 cards, we need to know which cards to keep in order to optimize our average win turn.

In order to make the calculation we just need to find all the possible keepers for a starting 7 and keep the one that has the lowest average win turn.

So for example when we need to keep 5 from this hand:

Lightning Bolt Lightning Bolt Mountain Lightning Bolt Mountain Lightning Bolt Lightning Bolt

there are 3 different hands we could keep:

Lightning Bolt Mountain Lightning Bolt Lightning Bolt Mountain

Lightning Bolt Mountain Lightning Bolt Lightning Bolt Lightning Bolt

Lightning Bolt Lightning Bolt Lightning Bolt Lightning Bolt Lightning Bolt

but it's optimal to keep

Lightning Bolt Mountain Lightning Bolt Lightning Bolt Lightning Bolt

as it has the fastest average win turn of the 3.

 

We can then proceed with the calculation as we did for the Paris rule:

- Always keep 1 card hands by choosing the best 1 card hand of the 7 that we drew.

- For every 7 card hand, compare the average win turn of the best 2 cards to that of going to a 1 card hand.

- Proceed to higher hand sizes.

Interestingly, in this case we always generate 7 card hands and compute the weight according to the chances of getting this particular 7 card hand.

Playing Burn under the Paris and London rules:

After performing the calculations above, we play millions of games with different decklists and keep some statistics

Paris Rule:

Bolts Mountains Average Win Turn Keep 7 Keep 6 Keep 5
43 17 4.93 82.47% 14.49% 2.44%
44 16 4.90 82.62% 14.24% 2.44%
45 15 4.91 82.43% 14.16% 2.59%

London Rule:

Bolts Mountains Average Win Turn Keep 7 Keep 6 Keep 5
43 17 4.87 82.44% 14.48% 2.76%
44 16 4.856 82.64% 14.33% 2.68%
45 15 4.859 82.38% 15.37% 1.95%

 

Things to note:

- Under both the London and Paris rule, the optimal decklist consists of 44 Lightning Bolt and 16 Mountain

- The average win turn is faster by only about 1%

- We keep the same amount of 7 card hands under both rules, but mulligan a bit more under London with smaller hand sizes.

- The mulligan decision is the same for 7 card hands under both rules, ie. in both cases we only keep a starting 7 if it has 1, 2 or 3 lands and mulligan the rest.

- The program finds that it's generally optimal to drive a starting hand towards a configuration with 2 lands for hand size 6 and mostly towards 2 lands for hand size 5.

AWT per Hand Size HS 1 HS 2 HS 3 HS 4 HS 5 HS 6 HS 7
Paris 10.5 9.5 8.6 7.6 6.6 5.6 4.9
London 10.3 9.1 8.0 7.0 6.1 5.3 4.85

 

Overall this goes to show that a linear deck that requires a high card number to win is only marginally affected by the new rule.

Playing Splinter Twin under the Paris and London rules:

We do the same for the Splinter Twin deck

Paris Rule:

Pestermites Splinter Twins Islands Mountains Average Win Turn Keep 7 Keep 6 Keep 5
12 12 13 23 4.728 62.4% 23.8% 9.4%
11 13 13 23 4.729 60.8% 25.6% 9.5%
12 13 13 22 4.733 63.8% 23.3% 9.3%
12 13 12 23 4.734 60.3% 19.9% 13.3%

London Rule:

Pestermites Splinter Twins Islands Mountains Average Win Turn Keep 7 Keep 6 Keep 5
12 12 13 23 4.34 37.8% 28.3% 21.2%
11 13 13 23 4.35 37.4% 29.3% 20.2%
12 13 13 22 4.352 37.5% 28.2% 20.6%
12 13 12 23 4.354 37.4% 28.5% 21.1%

 

 - The optimal decklist is different depending on the mulligan rule!

- The new rule makes us 1/3 turns faster on average.

- We need to mulligan a whole lot more. It was correct to keep the starting 7 about 62% of the time but now we only keep about 37% of the time.

- Different rules lead to different decisions regarding keeping or mulliganing our starting seven. Namely, these are some of the hands are a keep under the old rule, but a toss under the new one. The common characteristic is that they only have 1 Mountain:

Pestermite Pestermite Pestermite Splinter Twin Splinter Twin Island Mountain

Pestermite Splinter Twin Splinter Twin Splinter Twin Island Island Mountain

Pestermite Splinter Twin Splinter Twin Island Island Island Mountain

- The program generally drives hands towards having at least 1 Pestermite and 1 Splinter Twin

AWT per Hand Size HS 1 HS 2 HS 3 HS 4 HS 5 HS 6 HS 7
Paris 9.6 8.6 7.6 6.7 5.9 5.2 4.74
London 9.5 8.2 6.8 5.6 4.9 4.5 4.36

 

I think the biggest take away from this experiment is that we as players often like to obsess over our decklists and pretend that an extra land over the 8th draw spell is going to make a dramatic difference in our deck's performance. In reality, it is much more important to make informed decisions on the hands we keep and the ones we mulligan.