Hi I'm Buckwheat and this is the first installment of my recurring column, Gleaning Grain.
Gleaning Grain means to gather the grains, of knowledge in this case, left over by less careful havesters. It is also a pun on my name, which is a grain. In particular, this column will discuss situations in Eternal that people don't usually think about, that hardly matter, but have a non-zero chance of affecting a game. Often, I will analyze a single "what's the play" situation. Other times, I will investigate a specific, recurring pattern of play that has a minimal, but real, effect.
I'm not sure who exactly I'm writing this column for, but fresh beginners should avoid going too deep down this rabbit hole. I write about these topics mostly out of curiosity, because these topics are interesting in their own right, not because I expect them to noticeably improve anyone's play. Most people, therefore, should treat this content as entertainment. If you are good enough at the game to treat it as genuine advice, I don't have to tell you that, because you know who you are.
Today's topic is on casting Seek Power in the late game for deck thinning. Imagine playing a slow matchup where you have enough power and influence for the rest of the game. You cast Seek Power, and can choose to draw your only Time Sigil or one of your two Primal Sigils. If you think this decision doesn't matter, you're wrong. I will use the remainder of this article to show you why.
0. Contents
1. Simplifying Assumptions
2. Analysis
2.1) With only good colors
2.2) With no good colors
2.3) With favors
2.4) Systematic errors
3. Other Factors
3.1) Multiple searches
3.2) Strategize
3.3) Depleted sigils
3.4) Premium sigils
4. Conclusions
5. Closing Remarks
6. Acknowledgements
1. Simplyfing Assumptions
To keep this article to a reasonable length, and to keep the math simple enough for me to do, I need to ignore some factors that might matter in a real game. I will assume the following:
-Your opponent's deck and plays will not affect your decision.
-You do not benefit from playing more power cards this game.
-You want to minimize the chances of drawing power.
-The game will not go long enough for you to naturally draw cards you've previously put on the bottom of your deck.
2. Analysis
I define the following terms:
-The failure rate of a color is the odds you will draw a sigil you've put on the bottom if you search for a sigil of that color.
-A good color is a color with a failure rate of zero.
-To spoil a color means that there's now at least one sigil of the specific color on the bottom of your deck
-An active card is a card you have not put on the bottom.
2.1) With only good colors
First, I analyze the case where you have at least two good colors and no favors. When there is exactly one good color, you should obviously Seek that one. When there are no good colors, there is usually one with the lowest failure rate among the colors, which is the best color to Seek. I will examine the case where there are no good colors and a tie for the lowest failure rate later. Favors complicate the calculations, so I will leave those for later as well.
You want to minimize the chances of all your colors becoming spoiled by the time you Seek Power again. A color can be spoiled by:
-Scouting a sigil of the color to the bottom, or
-Strategizing a sigil of the color to the bottom, or
-Drawing all sigils of the color.
To scout a sigil to the bottom, you must play a scout effect, which must reveal a sigil. The chance this happens is approximately the density of scout effects multipled by the density of sigils in your active deck, per draw. Multiply that by the average number of cards you expect to see before your next Seek Power, and you get the chance of scout effects spoiling the color.
To Strategize a sigil to the bottom, you must play a Strategize and not have anything better than a sigil to put back. You should put back a non-sigil power card or a dead card in the matchup over a sigil. You have some control over the odds of Strategize spoiling a color, and can even wait to do it until you know it has no chance to spoil any color. For these reasons, I assume you can't spoil any colors with Strategize.
You draw all the sigils of a color before your next Seek Power if all such sigils are above your next Seek Power.
Given these approximations, I can write a formula for the odds of a color being spoiled before drawing the next Seek Power.
Sp = (Sc * Si) / (D * Se) + Si! * Se! / (Si + Se)!
Where:
Sp is the chance the color becomes spoiled before the next Seek Power;
Sc is the number of active scouting effects in the deck;
Si is the number of sigils of the color in the deck;
D is the total number of active cards in the deck;
Se is the number of Seek Powers in the deck;
! is the factorial operator.
This formula is not only approximate, it is also too complicated to apply multiple times for every Seek Power. I studied the equation using a graphing calculator and found some patterns, but they were hard to generalize. You can find the graphical calculation here, which I also have shown below. A key is of the terms from the link are include as well.
Red line is the combined chances of spoiling the color, before your next seek
Here is what I know:
-The more sigils a color has, the more often it gets spoiled by scouting.
-The fewer sigils a color has, the more often it gets spoiled by drawing.
-The graph starts at 100% chance of spoilage with 0 sigils, has a steep initial decrease, reaches a local minimum, then climbs nearly linearly as the spoilage from scouting becomes the dominant factor.
-The local minimum occurs at around 2 to 3 sigils. This is where the total chance of spoilage is lowest.
-The steady linear climb occurs beyond about 5 sigils.
-The most important sigil is going from 0 sigils to 1 sigil in almost every case.
-The shape of the graph, especially the steepness of the linear portion, depends strongly on how many active scout effects you have left and weakly on how many active Seek Powers you have left, compared to the size of your active deck. The linear portion of the graph gets steeper with a higher density of scouting or a lower density of Seek Powers.
-If the linear portion of the graph becomes flat enough for the sweet spot to move outside the 2-3 sigil range, it also becomes flat enough that the linear portion essentially does not increase.
You want to minimize the product of the chances your remaining colors will spoil before your next Seek Power. This is because you will only end up without a good color for your next Seek Power if all of your remaining colors become spoiled. The following is how to do this:
-If up to one color has 4 or more sigils, choose the color with the most sigils.
-If more than one color has 4 or more sigils, choose the color with the fewest sigils among those.
This is not always correct, but should be close enough in practice. It would take a far more complex procedure to determine the correct choice for every case without error. That is infeasible because the resulting heuristic would be too complex to quickly learn and apply.
2.2) With no good colors
It is rare, but possible, for two or more colors to be tied for the lowest failure rate when there are no good colors. If the tied colors have identical numbers of active and inactive sigils, the decision doesn't matter as all choices are the same. It, therefore, makes sense to rank the colors in order of the number of sigils if such a ranking helps.
The current Seek Power does not change the failure rate of the color you choose. You might get a sigil that is not on the bottom, which raises the failure rate, or get one that is on the bottom and lower the failure rate. You can't tell which case actually happened, so you average out every possible case, where your overall failure rate for the color remains unchanged.
Unlike with good colors, you only need to draw a single sigil to make its color worse. Like with good colors, you need to scout a single sigil to make its color worse. Also like the case with good colors, your next Seek Power only becomes worse if all of your tied colors become worse in the meantime.
You cannot control the total number of sigils among your tied colors remaining after Seeking, but you can control how they are split. You want to minimize the product of the number of active sigils in your best colors. The caveat is you don't want to eliminate a color entirely. If that happens, only one thing instead of more than one thing would have to happen to worsen your next Seek. Based on this analysis, you should:
-If at least one color has at least two active sigils, choose the color among those with the least sigils.
-If no color has at least two active sigils, choose the color with the most sigils.
Note that the number of active sigils might not be a whole number. For example, Seeking a color with 3 active sigils and 1 inactive sigil results in 2.25 active sigils and 0.75 inactive sigils remaining. Fractions of a sigil, of course, do not exist, but you can't tell whether you now have 3 active and 0 inactive sigils, which happens 25% of the time, or with 2 active and 1 inactive sigil, which happens 75% of the time. All you can do is to assume the average case, even if it's impossible.
2.3) With favors
Until now, I have ignored favors. Favors do not change the method the math is done, but they skew some of the numbers. They act like Seek Powers for their color, and reduce the effective number of sigils of their color.
A deck with, for example, 2 sigils and 2 active favors will draw through its sigils at an accelerated rate, as fast as if it had 4 sigils. This means the sigils will run out as fast as if the deck only had 1 sigil. In general, a color with S sigils and F active favors will run out of sigils as fast as a deck with S^2/(S+F) sigils and no favors. This number is useful, so I call it the color's effective sigil count. This only applies to drawing. The odds of scouting sigils are not affected by favors.
All the odds calculated so far are in terms of "until the next Seek Power". For a color with favors, this becomes "until the next favor or Seek Power". A color with favors can add its favors to its Seek Power count.
Since favors do not invalidate previous methods of analysis, I don't have to repeat the entire analysis for either of the previous cases when favors are involved. I only need to revise the results I've already reached.
With good colors, I need to replace each instance of "sigil" with "effective sigil". With lots of sigils, the color with the biggest proportion of favors is the least likely to spoil, because it has fewer cards to go until the next Seek Power, except for extremely lopsided and unrealistic sigil counts. The following are the revised heuristics for Seeking with favors when there is more than one good color:
-If up to one color has 4 or more effective sigils, choose the color with the most sigils.
-If more than one color has 4 or more effective sigils, choose the color with the most favors per sigil among these.
For the case without good colors, I only need to replace each instance of "sigil" with "effective sigil":
-If at least one color has at least two active effective sigils, choose the color among those with the least sigils.
-If no color has at least two active effective sigils, choose the color with the most sigils.
2.4) Systematic errors
My method for determining the optimal Seek Power target is approximate. In particular, I calculated everything as sampling with replacement, while drawing cards in reality is sampling without replacement. I did this to simplify the math to a point where I could actually do it. More accurate numbers probably need to come from simulations and statistical methods.
Sampling with replacement approximates sampling without replacement reasonably well at large deck sizes, but my method breaks down below deck sizes of about 20. I think there is no easy heuristic available for Seeking Power correctly at very small deck sizes.
It is sometimes impossible to spoil every color by scouting because there are fewer scouting effects than good colors left in the deck. I omitted this possibility from my calculations because the resulting error is minuscule.
I rounded the cutoffs for my heuristics up to whole numbers. The actual cutoff for good colors is closer to 3.7, not 4, and for no good colors it is closer to 1.5 than 2. I did this to make the systems easier to learn and less prone to serious error. In general, the odds change slower the more sigils there are, so I'd rather make my numbers too big than too small to err on the side of less costly mistakes.
My math on favors was sloppy, but I don't know enough math to calculate it any better than I did. I think the errors due to this are acceptable, and are no more serious than the errors resulting from my other methods.
I only considered how to optimize the very next Seek Power, not all Seek Powers remaining in the game. Doing that would require knowing how long the game is expected to last, which I lack the information to predict.
3. Other Factors
3.1) Multiple searches
You might cast multiple sigil search effects in the same turn. With the way my heuristics are set up, you can treat them as multiple separate searches.
3.2) Strategize
Bottom dead cards in the matchup and non-sigil power cards before sigils. If you must bottom a sigil, and have more than one option, choose the one from your worst color. Your worst color is either your color with the highest failure rate, or the highest chance of being spoiled if all your colors are good. Good colors with exactly 1 sigil spoil the most easily, followed by those with 4, 5, 6, ..., with 2 and 3 close to tied for least likely to spoil. If you must choose between a 2 and a 3, choose the 2 to spoil the least number of sigils.
3.3) Depleted sigils
Depleted sigils can be put on the bottom of the deck without spoiling their color, unless you have favors in the color. You should treat them as "non-sigil power" when you don't have favors of that color, and as regular sigils when you do, for the purpose of Strategize.
3.4) Premium sigils
Premium sigils let you split one color into two effective colors, making your Seek Powers better. You can, for example, run 3 premium and 2 non-premium Fire Sigils instead of 5 premium Fire Sigils, to get two separate colors instead of one. This is good because with more colors, you're more likely to have good ones remaining at any stage in the game. With fewer sigils in each color, correct decisions have bigger positive effects on failure or spoilage rates. There is very little cost to building your deck this way, if you own premium sigils.
The correct split of premium and non-premium sigils is as evenly as possible, up to 9 sigils. With 10 or more sigils, it's better to split them such that one side has exactly 4 sigils, such as 4-8 with 12 total sigils.
4. Conclusions
Here's a summary of the heuristics from the article.
-Your choice doesn't matter if you have no active favors or Seek Powers left. Simply choose any color with or tied for the lowest failure rate.
With at least two good colors and no favors:
-If up to one color has 4 or more sigils, choose the color with the most sigils.
-If more than one color has 4 or more sigils, choose the color with the fewest sigils among those.
With no good colors and no favors:
-If at least one color has at least two active sigils, choose the color among those with the least sigils.
-If no color has at least two active sigils, choose the color with the most sigils.
With at least two good colors and favors:
-If up to one color has 4 or more effective sigils, choose the color with the most sigils.
-If more than one color has 4 or more effective sigils, choose the color with the most favors per sigil among these.
With no good colors and favors:
-If at least one color has at least two active effective sigils, choose the color among those with the least sigils.
-If no color has at least two active effective sigils, choose the color with the most sigils.
Where the number of effective sigils is given by S^2/(S+F), S being the number of sigils and F being the number of active favors.
An important question is how much any of this matters. My answer is it doesn't matter unless you have the luxury of untimed turns, such as in a tournament. A prerequisite to applying any method presented in this article is to track every sigil, Seek, and favor on the bottom of your deck. Most players would do better if they ignore this information to focus on more important things.
5. Closing Remarks
I believe my method, albeit inexact, approximates reality closely enough to be useful. "Useful" is a generous term here. I don't expect anyone to use these findings in their actual games. It just doesn't seem worth the effort.
At the same time, this investigation illustrates the complexity of Eternal and how difficult it would be to play it to perfection. This mathematical curiosity doesn't need to affect anyone's play to be interesting on its own.
I hope you enjoyed the first installment of Gleaning Grain. I will return later to pick up more grains of knowledge from the vast fields of the unknown, and you can come with me.
