Going Deep – The (Re)Draw Rule

azindel_the_wayfinder

Hello Friends! As you should all know, we got a little patch this past week with a couple important additions to the game. RNG did a great guide to his initial impressions of the promos, so I am going to do a run-down of the redraw changes. Before launching into things, I want to commend DWD on spelling out the changes in such an official and clear way. I am a strong believer in acknowledging progress or “things I like” since it is easy to just complain about things I don’t like. Some people have expressed frustration about the communication patterns, and although I don’t think it has bothered me as much as others, I will admit it there was room for improvement. The way they rolled out the changes to this patch both built the Discord hype we are used to, and spelled out things in official channels accessible to all. Good job!

With that, let’s talk redraws!

(Quick note – flash2351 did a some similar work to this, but didn’t put a lot much work into figuring out the effect of Sigil searchers, and put more effort into finding #power vs. #turns. You can also check out his Excel sheet, while mine is a functional disaster.)

Intro

Hopefully you have all looked at the patch notes that explain the changes of the redraw rule, as well as the philosophy of the development team. If not you should check it out. I am going to spell things simply for us all to see:

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Easy enough to understand, right? Well, kinda. The rules make sense, but we are going to need to do some math to determine what the changes actually mean in terms of deckbuilding and redraw decisions. What I am going to break down here is the impact of these changes on the chance of different amounts of power in your opening hand.

Since counting “power” in Eternal is a touch complicated because of the number of power-searching effects, I am going to separate things out a bit. I am counting a “power searcher” as any card that is guaranteed to draw a power that costs 3 or less upon casting it. This includes Seek power, any Favor, Amber Acolyte and Privilege of Rank. Find the Way is also obviously included, but will only be counted as one power searching effect despite being an Echo card. Spire Chaplain or Initiate don’t count since they can die/be silenced before they generate power in any fashion. Monuments, Seats, Banners and Seals are all going to be counted the same, even though there are obvious differences between them all. I understand that cards like Inspire, Quarry or Temple Scribe are part of the powerbase of some decks, but given that their influence on the powerbase is difficult to capture, I am ignoring them entirely. I am also totally ignoring influence requirements, even though that clearly matters in games. Just focusing on total power numbers gives us enough to talk about.

For our little research project, I did some research into the typical decks that are around. I looked at the stock decks from our site, as well as ETS results. 25 power with no power seeking effects was pretty common among low-curve aggro decks, 25 power with 4 searching effects was common for a deck with intermediate power needs, and a power deck with intense power needs often had about 27 power and 6 sigil searchers. These will be our 3 case study decks moving forward.

I’m going to assume the average reader is not very interested in my mathematical process, so I am going to move all that to the end of the article. If you want to learn a little about that, I encourage you to check out the end! I also explain my assumptions and such, so if there are any math geeks out there with suggestions on how to improve please include that in the comments. For the average player – you just want the numbers, so lets dive in!

1st Hand – Everyone

The first table and graph here shows the chances of getting any number of power in your 1st hand under the old rule and the new rule. Here, Sigil searchers and power are considered the same. This adds a slight wrinkle I will talk about in a moment.

Note – there are a number of boxes in this table and the next that say 0 but are not actually truly 0. Any number below .1% was considered too small to matter, so they are functionally 0. I will denote all of those with 0* as oppose to 0.

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Oh, can you not see the dashed lines clearly in the graph? That is because they are all right on top of the solid lines! What this data essentially means is that there is almost no difference between the previous rule and the new rule for 1st hands. Now, to be fair, there are some minor differences specifically for 25 power decks. About 5% of you hands would be 0 power, and that has been eliminated. That being said, 5% is not a lot. Given that 0 power hands are very rare for either 29 or 33 power decks, and that 7 power hands are extremely rare in all cases, we see no important differences between the old rule and the new rule for first hands.

Now I will admit this is cheating a smidge. I am counting Sigil searchers as power, and if you have no power in your opening hand, your Sigils searchers are not exactly effective. A 75-card deck with 25 power has a 5% chance of drawing 0 power in its opening hand no matter how matter how many searchers you play. Given the fact that almost all decks play close to 25 actual power, it seems that the rules change probably has very little impact on deck/strategy balance. In that case it looks like the change is neutral across the field. What it really does is limit the “feel bad” moments of drawing 0 or 7 power opening hands. Although this is a rare event, they still suck.

CONCLUSION: Given that most decks play close to the minimum possible power, this change probably has almost no impact on balance. The quality of 1st hands has increased marginally, but not enough to impact deckbuilding decisions. This might represent a small net buff to 25 power aggro decks, but it the benefit is small.

2nd Hand – 25 Power Aggro

Okay, lets take a look at the new rule for the 2nd hand. Here the difference between true power and power searchers is more important, so those have been accounted for in the calculations. The table below summarizes my results, while the chart that follows shows these results visually.

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Ok, now we are seeing some serious shifts. Let’s talk about these each individually. For the 25 power + 0 Sigil searcher deck we see a fairly big shifting of probability. The probability of a 2-power hand has gone down by about 10%, the chance of a 3-power hand has dropped by about 2.5%, and the chance of a 4 or 5 power hand has increased by about 12.5%. Given that we are talking about a hyper-aggro deck, the average 2 or 3 power hand is better than the average 4 or 5 power hand, so this is probably an overall nerf to aggro decks. There was 4.4% chance of a 5-power hand before. I could see someone making the argument that as an aggro player may accept an increase in the number of 4 power hands you draw just so that you never have 5 power in the opener. The price they are paying here seems steep, but at least there is a silver lining.

A good way to summarize the impact of this change is to just focus on the average power in your opening hand. Under the current system every deck without Sigil searchers has a forced 3 power average opening hand (33.3% chance 2 power, 33.3% chance 3 power and 33.3% chance 4 power). How much power would we need to run in our deck to get an average of 3 power in our opening hand if we were just drawing randomly? About 33. That’s right – the redraw on the most power-light decks is currently similar to the 1st draw of the most power hungry decks. That is a big change.

CONCLUSION: This is a nerf for 25 power aggressive decks, and even outweighs any buff they got from first hand changes. Playing this style of deck, you are more incentivized than ever to play 25 power, but you should be more willing to keep 1st hands, given that the 2nd hand is more susceptible to flood.

2nd Hand – 25 Power + 4 Sigil Searchers

For decks with 25 power + 4 Sigil searchers we have some important differences. The chances of drawing only 2 power has significantly dropped (by 12%) and the chance of drawing 3 power has dropped a touch (2%). This probability has shifted to the chance of drawing 4 power, which now has a 36% chance up from 21.8%. The chance of drawing 5, 6 or 7 power remains about the same.

To grossly oversimplify things, we can say they “gave” a bunch of the probability from 2-power hands to 4-power hands. This greatly limits the number of games that you stall on 2 or 3 power, but adds a little bit to the chance of flood. What is strange here is that the decision to use 4 Seek Power over 4 Sigils seems very important. A deck with no Seek Power draws about .3 less power in their opening hand than a deck running 4 Seek Power. These two decks have, functionally, have the same probability to draw power for the rest of the game, but the opening hand will see 1 less power about a third of the time.

I have been kinda shocked by the gravity of these changes when looking at the numbers. A 12% shift in probability is a lot, and the shift is in a very specific way. To me, it appears as if any deck with 25 power + 4 Seek power is very likely to hit it’s 4th power on turn 4. According to my math, under the old rules, a 25+4 deck would miss their 4th power just over 30% of the time. Now? About 22% of the time. That is a big drop!

Although this is a big shift in probability, I can’t tell what it means in terms of deckbuilding. DWD always tells us they think we play too little power in our decks. In reality, players tend to fill the deck with Sigil searchers, while playing the bare minimum of actual power. I feel like this change will only magnify this. Given that hitting power in the early game is most important, players may build decks in such a way to maximize the number of power in your opener, and minimize your chance of drawing power afterwards. A deck light on actual power but heavy in Sigil searchers achieves this under the current rules.

CONCLUSION: decks playing 25 power + 4 Sigil searchers are going to cast their 4 drops on time way more often than before. Very few games well be decided by stalling on 3 power. Although few decks play more than 25 actual power, it seems that there is no reason to change. This is probably a buff to midrange decks.

2nd Hand – 27 Power + 6 Sigil Searchers

For 27 + 6 deck, it appears as if the same shifts seen in the 25 + 4 deck shows up here, though they are slightly dulled. The chance of a 2-power hand is reduced by only 9%, which has similarly been shifted to 4 power hands. The difference in opening hands between these two builds is going to be almost unnoticeable in the early game, which leads to some interesting possibilities.

As of right now decks like Hooru or Feln Control tend to play 4 Seek Power in their deck as a form of fixing. In addition to acting as a depleted power of any faction, it also thins your deck of Sigils, so your deck is slightly leaner as you move through the game. Obviously, the impact of playing an additional Seek Power versus and additional Sigil is extremely minor for every draw step, but over the course of a long game, a deck with 4 Seek Power is probably going to draw 1 more non-power card than the opponent. That being said, with the redraw rule change, it might be possible to game the system a little. Let’s say we play a deck with 35 power, including some number of monuments, but absolutely 0 Sigil seekers. Our opening hand would have an average of 4 non-power cards. We would have a good chance of drawing into the power that we need for our deck to function, but we would be guaranteed to have something to do in the early game. I’m not quite sure this is “right” yet, but it is certainly an interesting consideration. Although I have the numbers sitting in front of me, it is hard to process them in terms of deckbuilding.

CONCLUSIONS: the chance of 2 power hands has been decreased, while the chance of 4 or 5 power hands has increased. This may imply a shift in deckbuilding behaviour, although more work is needed to figure things out.

Draft

Although I have been focusing on ranked (as I usually do) the application of this rule change is probably most important in draft. A ton of draft games are decided by flood/screw, which is probably not a surprise given that draft has fewer tools for fixing power or smoothing draws. The number of 2 power, mono 4-drop hands is somewhat discouraging, so now we will have a bit less of that, which is always a plus! I’m not going to get into the math here since there is very little to actually analyze, but if I had to guess you can probably shave about half a power from your draft decks. If you are playing a 17-power deck you should expect your opening hands to draw more powerful than before. Doing a quick check, I think that if you are playing only 15 power in your draft deck you are still slightly favored to hit your 4th power on turn 4 even if you are on the play. We are going to need to jam a lot of games to work out the intuition here, but it seems like people might want to try and get a little greedy, given that draft is often determined by flood.

Method Behind the Mathness

In doing the calculations for probabilities I used a hypergeometric calculator a lot. If you don’t know what that is, I actually did a podcast on the subject of math and Eternal, where I give an introduction to hypergeometric calculations. I’m going to just assume that you understand how that works.

For the 1st hands via old rules question I simply did the hypergeometric on 0-7 power for 25, 29 and 33 power. Very simple. Changing things to the rule to 1-6 rule was easy, as I simply pretended that the 0 power hands didn’t exist. For example, in the 25-power deck I just divided all the probabilities by .95, which boosted them slightly. The .05 that was “lost” was the 5% of hands that are supposed to be 0 power.

This same logic was used for the “2-5 Rule”. Since it is possible for decks with Sigils searchers to draw more than 5 “power” in the opening hand, I only removed the probability of 0 and 1 power hands. This misrepresents the odds slightly, but not enough to really matter, given the odds of drawing a opening hand with 6+ actual power is very unlikely.

Figuring out the odds for the new redraw hands was actually fairly easy. There is a 33% chance of 2, 3 or 4 power no matter what, so you just need to figure out the chance of drawing power searching effects with the rest of the hand. For example, when calculating the chance of drawing 4 power with a 25 + 4 powerbase you start with a 33% chance of starting with 2, 3, or 4 power. You then figure out the odds of drawing 0, 1 or 2 searchers for each hand. For example, you have a 26.4% chance of drawing a searcher in your opener if you draw 3 real power. This accounts for just under 9% of hands. You then add up all these possibilities and find the probabilities of all possible paths to difference kinds of hands.

I did some quick calculations a few times in the piece on “chance you have your 4th power on time”. This was determined by taking the distribution of starting hands, followed by determining the chance of drawing 0, 1 or 2 power in the next 2 draw steps.

Bonus Decklist!

For those who stuck around for the end of the article, I have a little decklist I wanted to share with ya’ll. Of the promos, Azindel seemed to be the weakest, but I noticed a neat interaction that might make him playable! Whispers of the Void “makes” the Spitelings in the void, and kinda pushes them onto the field. Since the little gremlins start in the void they count towards Azindel’s max power bonus. That’s nice and all, but what are we doing this for? Why do we need this much power? Well, what is another card in Shadow that benefits from Whispers of the Void? WITCHING HOUR! This list is something that has been buzzing around my head that I have not had a chance to test yet since I have been busy over the weekend, but maybe someone out there is able to take it to the next level. Check it out! (Picture says 1 Azindel, but should be 4)

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