Dreaming of Happy Draws

Warning: This article contains little strategic value for the advanced competitive player.

In the fall of 2000, I drove through the night to my parents' house to visit for Thanksgiving. Arriving on Thanksgiving morning, I decided to take a nap. My mother comes to wake me up for Thanksgiving dinner, and while I began to wake up I was simultaneously conscious and dreaming, which is why I can remember the latter part of this dream.

It was an odd dream since there was very little visually, and my dreams are almost strictly pictorial. Not odd for me, my dream was from the first person but I was not myself. In this case I was a deck of Magic cards, except that it wasn't a deck. It was an abstract system. And it did not include cards, only abstract components which could not be described individually. None had any significance alone.

When my mother told me it was time to go upstairs, I was going to explain to her that I'd be up as soon as the artifact-based, mana-producing combo centered in my legs began functioning fully, but I had to wait for the untap step so a small, catalystic component would be available to re-ignite the process. Not fully aware of reality, I was also lacking in eloquence, and she couldn't understand what I was trying to say. She asked me to repeat, but just then I snapped back into reality and realized I had better shut up before my mother becomes even further convinced that I'm addicted to Magic.

So now you think I'm a lunatic. Maybe you're right, but the fact that I have very strange dreams isn't the point of this article.

When retelling this story to a friend, I commented on how strange it was that it wasn't even a deck. What a weird way to think of a deck! And how did I know that's what it was, anyway?

A day or two later, I started thinking about that silly way of visualizing a deck, and realized it's not entirely wrong. The more I thought about it, the more I realized I had been wrong. Prior to this, I always saw a deck as a pile of cards. Deck X was defined solely by the 60 component cards. The more I thought about it, the more I began to theorize. I came to the conclusion that it isn't the 60 cards that defines the deck, but instead the interactions between those cards. It wasn't a pile of cards any more, it's a system; a machine if you will. I started doing something really strange. I would lay out all the cards of a deck face-up, and try to see the deck as a whole, not considering the individual cards. Then I would make a change, and see the new whole. Previous to this I had compared the card I was considering removing with the one I was going to replace it with. Sure, I would think about interactions a little, but I didn't really ever see the big picture.

What about interactions with your opponents' cards? Sure there is something to be said for pure utility, synergy aside! I knew this to be true, both logically and intuitively, but how was I to reconcile this with my new understanding?

The answer came fairly quickly and it was simple. The deck itself isn't the system of interest, it was the combination of the two decks that formed a larger system that was the concern. Concentrating solely on the inter-workings of a deck is folly, for an absolute answer to which play and which card choice is correct one can only look at the system known as the matchup. The idea of two systems coming together to form one larger one was fairly easy for me, as I was a physics major at the time. The matchup is a closed system, comprised of two closed systems.

Magic strategy had always been about guessing for me, but I wanted my educated guesses to be better. I longed for a deductive path where I could see objectively which decisions were correct. I began trying to apply my new concepts to strategic thought. I soon realized that guessing would have to continue, as deductive reasoning through millions of possibilities simply takes too long. I still had hope, though, since knowing what made a decision correct and where the balances must be struck should give me insight and make my guesses more educated.

It was about this time that I saw a Magic site was accepting unsolicited submissions. I stayed up one night, and began organizing my thoughts. I found it more difficult that I expected, since every time I wrote something, something else seemed to pop into my head. I also began to see places where my theories were lacking, or simply incorrect due to things I hadn't considered.

I submitted it anyway. The site never responded. I made a revision or two, as things kept jumping into my head. I submitted it to another site. No response. Finally I submit it to MagicAddict.com, who gleefully accepts it, but they won't post it because the Greek letter Sigma (which was a part of some of the equations I used, used to mean "sum of all") doesn't quite fit into HTML.

What I had come up with was the Happy Draw model of deckbuilding, so named because it intended to maximize a deck's potential to have Happy Draws (drawing the most helpful cards in the given situation) given a particular field of competitors. I've included that article at the end of this one, for humor purposes.

Several months have passed since then, and I decided to give it another shot. This time, I've realized one key thing, though. Simplicity is of the utmost importance when speaking in generalities. So here are the general tenets of the Turpish School:

1) All gameplay decisions can be broken into binary (yes or no) questions. If you don't see it email me at EmpSchao@aol.com and I will try to explain it. This is done for simplicity of explanation.

2) To determine the value of making a particular decision during gameplay: take the probability that you will have a favorable outcome if your answer is yes, minus the probability that you will win if the answer is no. Compare that difference to the values of all mutually-exclusive decisions that could be made, and answer yes to the one with the highest value. This is more difficult than it sounds, because for the complete thought process one must consider every bit of information one has at one's disposal, including each of the 60 cards in your deck, what zone they are in, any of the opponent's cards that have been revealed to you and their zones, life totals, etc. There is also extrapolation to be done about information not granted, such as the identity of the unrevealed cards in your opponent's library and/or hand. Favorable outcome typically means winning, but if you've won the first game and in the second game there is an Opalescence in play with no other creatures, playing a Dance of Many (to cause the game to draw) is favorable since it causes you to win the match.

3) When considering an individual card, consider what options it grants you in what situations. For instance, Call of the Herd grants me the option to play it if it is in my hand and I have 2G available, as well as granting me the option to flash it back if it is in my graveyard and I have 3G available. Then consider every situation possible in a given matchup. At this point you should see why I say guessing is still necessary. For each situation, how likely is that situation to come up? Do you have an option due to Call of the Herd in that situation? If so, what is the value of the decision it is facing you with? You can compile the answers to these questions to see how useful Call of the Herd is in the average situation in this matchup. Consider how changing that card to another will affect the usefulness of other cards in your deck in this matchup. With these considerations you can see which cards are useful in this context and which are not.

4) How likely is this matchup to occur, in other words how many people are playing that deck? What are your odds of winning? Trying to keep the highest average odds of winning based on the entire fields is admirable, but also consider that the standard deviation is a consideration. Being able to brutally massacre 60% of the decks in the field while folding to the other 40% is a bad plan, since two losses will usually keep you out of the top 8. It would be much better to have 60% odds of beating every deck in the field.

5) Compiling 3 and 4, consider the alteration of a card choice for each matchup, then weight the value of that alteration in a given matchup by that matchup's probability of occurring, compile it to see the value of altering that card choice in the overall field.

And that is how I tune decks. Doggone it! I forgot about the sideboard again. Well, it's the same basic process, so I won't insult your intelligence by getting into it now.

Lucky Draw Magic
The Rack is banned in this format, so don't get any silly ideas while I describe it.

There is no need to shuffle your deck before the game begins. Players begin the game with no cards in hand. Instead of having a natural draw, that is the draw that occurs as a result of the game rules during your draw step, you have a natural Demonic Tutor. This means that at the beginning of your draw step, instead of drawing a card, you search your library for a card and put that card into your hand, then shuffle your library. If you do not put a card into your hand this way (usually because there are no cards left in your library) you lose the game.

Deckbuilding is considerably different, too. Instead of having a minimum deck size, there is a maximum. You don't want people to come with a shoebox full of cards and claim it to be their deck. The maximum is not necessarily a number of cards though. Let me explain.

The typical restriction of cards, that one may not have more than one copy of a card in one's deck, is not very helpful in this format since your chances of getting that card into your hand are still fairly high. Instead, when the play group feels a card is too strong, they increase its point value. Every card not restricted as a result of its power has a point value of 1, even basic land. Then you total the point values of all of the cards in your deck. This total may not exceed 100.

No sideboarding, that would get ridiculous.

This is a major break from your typical Magic experience. For one, luck is significantly reduced. Mana screw is optional, unless your opponent's strategy is to encourage it (e.g. land destruction decks). Strategies that would never work in a normal game are just fine here and vice versa. Mana curves tend to be tighter, and keeping the contents of your deck secret is more important. If I think you'll be playing a white weenie deck I may include a single Anarchy in my deck and simply choose not to draw it against my other opponents.

Overall, I'd say it's a great format. If you prefer constructed over limited I think you'll like this one.

May you find what you seek.
John B. Turpish

Addendum: Happy Draw
Below you'll find what I submitted to MagicAddict.com several months ago, except that I've changed the Greek letter sigma to &. Keep that in mind whenever you see an &.

Happy Draw - by JohnB Turpish
I. Introduction
The Happy Draw model is one of deck construction. Any deck construction model that is worth its salt will delve into play strategy, and this is no exception, but Happy Draw is concentrating on card choices for deck construction. I propose this not as error-free, but in full knowledge that it probably has some errors, which I hope the reader will be kind enough to show me.

II. Definitions
A. Decision - any option a player has between two or more alternatives, where the choice may affect the outcome of the game.
B. Round - one complete set of turns, a period of time which includes one entire turn of each player.
C. Relative round - a round which begins with the turn to which the round is respective.
D. Play(n) - a set of choices made, inclusive of all responses by one player to decisions presented to them during the course of a relative round, relative to them.
E. Absolute round - a round which begins with the turn of the player who had the first turn of the game
F. Good play - a play which results in a higher probability of that player winning than any other play that player could've made given the situation
G. Happy draw - a draw which, assuming a good play is made, will increase that player's chance of winning the game
H. Happiest draw - a draw which, assuming a good play is made, will increase that player's chance of winning the game more than drawing any other card in the given format would
I. Situation - combination of all current objective factors in a game, known and not. This includes which cards are involved, which zone each is in, the order of the libraries and graveyards, states of all cards in play (tapped/not, counters, local enchantments' targets, permanent changes to attributes, et c.), life totals, number of poison counters, what step of what phase of whose turn is it, what's on the stack, how many absolute rounds have passed, and anything else I might be forgetting.
J. No-play ability - an ability which doesn't require playing the card or interaction with any other card you own to take effect. This includes cycling, Gaea's Blessing's whole-graveyard shuffling, 1996 World Champion's topdecking, et al..
III. Postulates
I hold these truths to be self-evident that:
A. The game is based on decisions.
B. All decisions can be broken down into binary decisions, yes or no questions. This has little to do with this model, but it's interesting to think about and important for those desiring to write AI for the game.
C. All playing skill can be reduced to choices made for decisions presented to that player.
D. Card choices will increase the odds of winning a game if they are made to maximize the Happy draw potential

IV. Card Choices
To decide if a card should be used in a particular deck, forsaking all others available, only one thing need be considered - by how much does playing this card increase my probability of winning the average matchup. To decide this, two things need to be considered: interaction with each other card in the deck, and each other card the opponent might play, with respect to the probability of each such interaction occurring.
A. Raw probability increase
Raw probability increase occurs when playing the card brings one closer to a win condition than an opponent to hers. For example, some cards that have raw probability increase are Brush With Death, Heroes' Reunion, and Lava Axe. This is difficult to measure exactly, except in reference to your win condition (life swing, poison, et c.)
B. Adjusted probability increase
This is the same as rpi, except that it accounts for choices that will be likely be made, and is measured in turns. API=&(P(p')tp')/n' - &(P(p)tp)/n, where p' is any possible set of good plays (made by both players) that could occur surrounding and after the playing of this card extending to the end of the game, P(p') is the probability of the conditions necessary for that set of plays to occur existing, tp' is the number of rounds that would pass before the game ends as a result of the given set of plays (and is negative if an opponent wins the game), and p is a set of plays that could occur if the card was not played, and all subscript references are the equivalent of what was mentioned for p'. API is, as one might have guessed, the adjusted probability increase. I forgot, n and n', which are the number of possible good plays that could possibly occur if the card was not played and if it was, respectively.
Some things should be noted here:
1. Unless one is omniscient and superhumanly intelligent, one should never be capable of precisely calculating the API. Guesses, however, can be made, and the better the guesses, generally, the better the player.
2. The API is situation-specific. It should, and does, change for any given card depending on one's opponent, the point in the game, and random factors such as shuffling.
3. If the API is negative, do not play the card. If the API is positive, but less than the API of another card and the playing of the cards is mutually exclusive, then play the one with the higher API. Otherwise, play any card with a positive API.
4. The API could be applied to any decision, however, since this will ultimately be a model of deck construction, it's the API for playing of a card that I am interested in. One should also notice I've neglected no-play abilities. This was a mistake, but I shan't change it because it would complicate things. Just remember to consider them.
5. The API takes into account the entire situation. This includes how it will affect things that will happen later in the game. This is what gives good API's to basic lands, and that is why the mana curve is entirely contained herein.
6. The API assumes all players will always use good plays. This is clearly not true, and so the formula would have to be amended for gameplay use. However, it is my contention that when constructing one's deck one should not count on the weaknesses of opponents and one should strive to work out all of one's own weaknesses, and since this model is based on my opinions, that's how it'll stay.

V. Making card choices for one's deck based on Happy Draw.
A. The first thing that must be decided upon is the field. The field consists of all situations that could occur in the format. To calculate, not estimate, one must find all possible combinations that are legal in the format that could be called "Decks". Since this is would necessitate limits, because there are an infinite number of unique decks in any given format (I could use 90,000 basic land) I recommend even for those attempting to get "exact" numbers to put a maximum for deck sizes that they would consider. Then decide on the probability of actually playing against each deck. The probability of many of these decks will be essentially 0 (decks like 20 Forest, 1 Shock, 4 Terminal Moraine, and 35 Swamp are highly unlikely to be faced) so estimation gets the luxury of neglecting most of the possible decks. The probability of a given deck being faced is P(d). Then consider how each possible game would most likely occur. Consider here the likelihood of draws and the likelihood of a player choosing a certain play. Now calculate the probability of each possible situation occurring, based on simple probability (multiply the chance of it happening in each possible combination of random factors such as deck order (this being determined by the player's likelihood of choosing certain plays) by the chance of those random factors being such and OR these all together). This is P(s).
B. Now for each deck in the field, find the average AVI for each card in it (different copies count separately). The AVI, you may remember is calculated for one given situation. AVIA = &(AVIsP(s)P(d))/n, where AVIs is the AVI of that card in a given situation, P(s) is the probability of that situation happening in that matchup, P(d) is the probability of facing that deck, and n is the number of possible situations in the format. Now calculate deck power. DP=&(AVIAC)/n, where AVIAC is the AVIA of each card, and n is the number of cards in the deck. If one indeed calculates this for every possible deck, then he should simply choose the deck with the highest DP.
It should be noted that:
1. This seems to be entirely metagamed, but if fact it is not. Decks with greater internal synergy have higher DP's, given equal interaction with the field.
2. If followed strictly, all advantages will be idealized and balanced, as well as the mana.
3. The probability of how your opponent will play her deck, and which she will play, is guesswork. If it is known for fact, then this model should yield VERY good results. However, would one really need to go to this extent in that case?
C. Sideboarding
1. Test your deck's DP, only slanting the field this time. Change the P(d)'s, increasing if your decks DP, relative to that deck, is lower than your overall DP, and vice versa.
2. Replace any one card in your deck with any one card that would be legal. Test its DP in the slanted field again. Note the change in DP. Repeat for each card in your deck being replaced by each card that would be legal to do so with.
3. Choose the 15 cards that had the greatest improvement of your DP in the slanted field.
Things to note:
a. If this were done, be sure to take special note of which cards you're replacing. Which ones improve it the most?
b. Be sure, also, to take note as to which replacements work against which decks.
c. This isn't ideal. Actually, sideboarding was entirely an afterthought on my part, and I honestly don't care enough to get it right. If you want to, then what you must correct is my disregard for interaction of multiple cards sideboarded simultaneously and your opponent's sideboarding. What I've written here would work to some extent, I'm sure, but it'll never bring you a "surprise" sideboard, you'll never side in a combo, you won't reverse sideboard, et c..