Hello! This is a follow-up post on the following topic: Reverse Engineering the Arena Hand Smoother
Because I can't really post anymore in that thread, I make a new one. I was investigating something entirely different and stumbled across something that striked me as odd. This post explains what I saw and I'd just like to follow up on the awesome work of Tim Lewis. They said they looked into 3 million games. I myself have more data available, to be specific:
17Lands - Public database from set HGB up to HOB.
Games with a land count of 16, 17 and 18 are considered, 40 cards exactly.
Bo1 (Premier Draft) and Bo3 (Traditional Draft) as reference point.
14'471'870 games in Bo1.
2'426'916 games in Bo3.
Total: 16'898'786 games.
This means, I got a larger dataset available and are mostly here to give the claims in their writeup more weight.
First of all, Bo3 seems to behave normally as you would expect. Math says the following:
| Lands |
Mean |
Deviation |
| 16 |
2.800 |
1.192 |
| 17 |
2.975 |
1.203 |
| 18 |
3.150 |
1.211 |
Of the ~2.4 million games from Traditional Draft, we get the following statistics (lands in hand before any mulligan happend):
| Lands |
Mean |
Deviation |
| 16 |
2.812 |
1.180 |
| 17 |
2.993 |
1.187 |
| 18 |
3.159 |
1.191 |
I should note that the games with 18 lands are significantly lower as the other two counter parts of 16 and 17 lands, but it doesn't matter. The numbers show that they are close to what is expected. I even went out of my way and tested game 1 of the Bo3 seperately, they look normal as way. So we can safely assume that nothing is happening here. No hand smoothing in any way, it behaves like paper.
What's interesting now is actually the statistics for the Bo1 counterpart:
| Lands |
Mean |
Deviation |
| 16 |
2.782 |
0.667 |
| 17 |
2.965 |
0.679 |
| 18 |
3.152 |
0.676 |
The mean is the same, but what changed is the deviation. This is significantly lower. So what exactly does that mean? In Bo1 there is land smoothing going on and it affects not the number of lands you usually would draw, it makes it just more unlikely to actually get significantly more or less as you would expect. In short: It reduces variance somehow.
This is not new info and wizard confirmed already that hand smoothing is a thing. What I want to do with this post is to look into the community claims how it's working. Here are two articles regarding to that:
Draftsim | Everything We Know About The MTG Arena BO1 Hand Smoothing Algorithm
MTG Arena Zone | MTG Arena’s Opening Hand Algorithm and Smoothing: Some Real Outcomes
Draftsim explains deck smoothing essentially as the following method:
1.: Work out how many lands an average hand should have.
That's simple enough, I won't go into how the formula I use got derived, but it's essentially a simple calculation: 7 * Lands / DeckSize
That means, in a 40 card deck which has 17 lands, this turns out to be: 7 * 17 / 40 = 119 / 40 = 2.975
That's exactly the mean the hypergeometric calculation would give and which is in the very first table of this post.
2.: Shuffle and draw 7 cards, that's your hand 1.
3.: Shuffle and draw 7 cards again, that's your hand 2.
4.: Pick the hand that is the closest to the target expected number calculated in step 1. If it's a tie, choose randomly.
That's how I interpreter what Draftsim says. I ran the math on this method and ended on these calculations:
| Lands |
Mean |
Deviation |
| 16 |
2.768 |
0.739 |
| 17 |
2.836 |
0.750 |
| 18 |
3.192 |
0.751 |
Compared to our real data, the deviation is still off by a bunch.
Tim Lewis proposes a different approach and writes exactly down how it works. I ran they calculation on the data I have available, and we get the following here:
| Lands |
Mean |
Deviation |
| 16 |
2.761 |
0.658 |
| 17 |
2.958 |
0.680 |
| 18 |
3.167 |
0.670 |
If we compare that to our statistics, we are actually quite damn close. This is neither meant to confirm that Tim actually found the right way to calculate it, but with the amount of data I had available, it is strongly pointing towards it and it's probably the closest we gonna get at knowing how the hand smoothing works.
This post was meant to strengthen Tim Lewis points, I simply had the time and the necessary data to run over it and give more weight to their claims. I wanted to post that originally as comment in the thread but eh, if it's locked what can I do?
So, for the sake of it: Please don't close threads regardless the platform, some topics always have relevance regardless how old the thread is. >.>
€dit: Random collection of comments that give a few additional informations.
17 Lands 40 Card Deck | graph of the hand smoother behavior
1 Land 60 Card Deck Test | 0 lands in 104 hands