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Examining Influence Requirements Part II

Analysis by detail251 - July 28, 2017

Last week I spoke about using the hypergeometric distribution to determine how many influence sources are required to reliably play your cards on curve. I want to take a closer look at a few aspects of this, and provide some useful tables as a quick reference when deckbuilding. Additionally I am going to apply this to a deck I have been tinkering around with on ladder lately.

Play vs Draw Considerations
An interesting point raised by people on last weeks article is that I was assuming you were on the play (going first). This leaves you one card short of what you would have if you were on the draw (going second). This really just equates to one additional turn by the logic on the program but here are some numbers just to be thorough: For the sake of argument, let's consider a theoretical deck that is playing Sandstorm Titan and Obliterate along with 31 total power sources, of which, 24 produce Time and 19 product Fire. The probability of being able to run out a Titan on turn 4 on the play in this deck is 93%. Turn 4 on the draw is 95.2%. The probabilities of Obliterate on turn 5 on the play and draw are 85.3% and 88.7% respectively. Really this doesn't mean a whole lot, we see a 2.2% and 3.4% bump which isn't incredibly significant. What is interesting to see though is that there isn't a uniform increase in percentage for the 2 cards. Is this because turn 4 and turn 5 are different or because the fewer Fire sources in the deck mean extra time leads to a larger percentage gain? Let's consider the same situation but with a nameless double Fire card on 4. The probability of having double Fire on turn 4 on the play in this deck is 81.1% and on the draw it's 85.3% or a 4.2% bump. This seems to indicate that the difference is coming from the influence distribution rather than the increased cost.

A take away here might be that for cards in your secondary or tertiary faction you are gaining a significantly larger percentage chance to be able to cast them each additional card you draw than you are for your primary faction cards (nearly double in the case of this deck). Is this useful information? One conclusion that can be drawn from this is that secondary faction cards that are more expensive are preferable since they come at a point in the game where you are quite likely to have the influence available. I.E. If you have a strange Argenport deck that is basically mono Justice but you are splashing a few Shadow cards, you would much prefer those Shadow cards to be expensive things like Shadowlands Feaster rather than cheaper things like Argenport Instigator. Despite costing more influence than Instigator we are more likely to be able to cast the expensive cards on curve for a heavily Justice leaning power base. It's by a slim margin and likely wouldn't significantly impact gameplay in this instance but the sentiment holds.

Some Tables
I had a few requests for some tables emulating what Karsten posted in the original article I was adapting. These tables represent the number of sources required in a deck to attain the requisite number of sources by turn X with 90% probability. In addition to these tables, the code is now available as a more user friendly tool on EternalWarcry and can be found here for you to use at your own leisure.

1 InfluenceBy Turn 1By Turn 2By Turn 3By Turn 4By Turn 5By Turn 6By Turn 7
45 Card Deck10998776
75 Card Deck17161514131211

2 InfluenceBy Turn 1By Turn 2By Turn 3By Turn 4By Turn 5By Turn 6By Turn 7
45 Card Deck-151413121111
75 Card Deck-252422211918

3 InfluenceBy Turn 1By Turn 2By Turn 3By Turn 4By Turn 5By Turn 6By Turn 7
45 Card Deck--1716161514
75 Card Deck--2827262524

The first obvious thing this shows is that triple influence cards are exceptionally difficult to play on time. Particularly in draft where many decks don't even run 17 power let alone 17 of a single influence. Groundbreaker should only make your deck if you are ok playing it on turn 6+ even in a heavily Fire deck.

Practical Application
Finally, I'd like to talk about this in the context of a deck I have been playing around with a bunch on ladder lately. Lifeforce looked to be an appealing mechanic to me with Katra, the Devoted and Beckoning Lumen being particularly exciting standouts. I was a bit disappointed initially though as I found that most of the cards that trigger lifeforce were underwhelming. Oasis Seeker just doesn't do enough otherwise. Vara's Favor and Extract are both pretty great but they often get played early in the game before many of the lifeforce engine cards come down as a means of surviving. Ayan, the Abductor was the only card in the deck that I found consistently was triggering lifeforce when I wanted to. I came to the conclusion that Xenan simply didn't have the tools to make it work by itself and for this reason I opted to add Justice to the deck. With this addition I got Combrei Healer and Stonepowder Alchemist both of which are cards which trigger lifeforce and are not cards that need to be played early to get maximum value. As an added bonus, I could now play Stand Together which helps with one of the decks major blind spots: Harsh Rule. My list currently looks like this:

I arbitrarily decided that I wanted all my Justice requiring cards active by turn 5. My logic here is that Stand Together must be active on 5 in order to be effective against Harsh Rule. Additionally, Stonepowder Alchemist/ and Combrei Healer need to be active once Katra, the Devoted is in play for maximum effectiveness. Earlier would be fine but I don't deem it necessary for playability. This means I'd like to be 90% to have a single Justice source available by turn 5. This means I should play 13 Justice sources, which I am. Looks like I did my homework!

Moving forward, if you are interested in trying this deck I would strongly consider some changes. As I mentioned previously, Oasis Seeker is pretty unexciting. I'd be interested in testing out Desert Marshal in this slot. This might require 1 or 2 more Justice sources but that isn't a huge deal, and frankly, turn 5 might just be good enough for the marshal anyway. Slimespitter Slug has been a beast for me so far but I could also see him changing to Mystic Ascendant as a compliment for Becky on the card advantage front.

detail251 detail251+4032


konan Eternal Version: 1.22.4
Now that I had a second look at the code itself you are skewing the numbers quite a bit, by not applying the mulligan to 2 power hands. That just seems odd, since I would never keep a 2 power opening hand, when I didn't have the appropriate influence already. On top of that you don't take into account that you can't draw 0 and 7 power opening hands - I guess that is still ok for the sake of simplicity, but you should state that somewhere.
konan Eternal Version: 1.22.4
I think I found the real bug in your code. You got the mulligan rule wrong! You state random.getNextInt(2)+2 which returns 2 or 3 instead of 2-4. It has to be getNextInt(3)+2. I think you have to rerun all the numbers.
konan Eternal Version: 1.22.4
I just ran through a few numbers with the simulator and some things are a bit off. I think you took 29 sources as a baseline for your tables and not 25. I guess that's the typical scenario of 25 Power + 4 Seek, but you should really document that better, because it's not clear from the article itself.

There also seems to be a bug related to the probabilities of non 75 card decks. E.g. I enter 45 cards, 15 influence sources, 15 good sources, turn 1, 1 needed influence - the result should be 100%, since that's a mono deck and your guaranteed 1 source by the mulligan design. Yet the probability I get is 81.593%.