{"id":8000,"date":"2014-09-28T23:20:26","date_gmt":"2014-09-29T04:20:26","guid":{"rendered":"http:\/\/www.ygorganization.com\/?p=8000"},"modified":"2014-09-29T11:14:30","modified_gmt":"2014-09-29T16:14:30","slug":"dicetheory","status":"publish","type":"post","link":"https:\/\/ygorganization.com\/staging\/?p=8000","title":{"rendered":"Dice Theory and Branch Theory"},"content":{"rendered":"
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\u00a0 \u00a0 By making approximations on a system, it is far simpler to see general rules and logic lines, and their effects, than pre-approximations. \u00a0To this end, let us look at Yu-Gi-Oh! after making some approximations in this manner.<\/div>\n
<\/div>\n
“Dice Theory”<\/div>\n
<\/div>\n

<\/p>\n

—<\/div>\n
<\/div>\n

\u00a0 \u00a0 What does it mean to win? \u00a0To win, it simply means that the combination of your decision making, deck building, and luck were greater than the decision making, deck building, and luck of your opponent. \u00a0In a way, a game can be described or approximated as the two players each pulling out their [heavily modded, sleeved, 1st edition hobby league ready-microwaved] dice, and rolling them. \u00a0Whoever rolls higher would be the winner.<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 In this comparison, the various numbers on the dice would be the results of a formula accounting for the three qualities stated above (decision making, deck building, and luck). \u00a0It should be noted that your opening hand is both a subset of deck building and luck, and thus the variance in your opening hands is shown in the dice\u2019s face values. \u00a0A dice could have various amounts of sides (just like how decks have various amounts of standard plays \u2013 example, +1 Fire Fist vs Dragon Rulers). \u00a0In the following examples, I\u2019m going to set the amount of sides to 10 for all, because it makes it easier to illustrate my point.<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 Player one has a dice with not only holographic, astral pack edges but also an 8 on each face \u2013 so 10 faces with 8 on each. \u00a0If he rolls any amount of times, his average value will be 8. \u00a0This kind of dice, consistent and having decent values, could be alikened to +1 fire fist.<\/div>\n<\/p>\n

\u00a0 \u00a0 Player two is going for maximum yolo. \u00a0He just got back from 360 n0sc0ping some pleb on COD, and is using a minimum scope maximum skill dice, on which 6 sides are 9, and 4 sides are 1. \u00a0The average value of this dice\u2019s faces are 5.8.<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 So, between player one and player two, who is more likely to win? \u00a0The simple answer which new players may mistakenly pick is player one, because on average his roll is an 8, versus player two, who\u2019s rolling not only his swagged out tricycle but also a 5.8. \u00a0However, at no point does player two\u2019s dice land on 5.8 \u2013 it will always land on either a 9, or a 1. \u00a0If it lands on a 9 player two wins, and if it lands on a 1 he loses. \u00a0So in other words, player two has the advantage with a 60% win rate (as games approach infinity).<\/div>\n<\/p>\n

\u00a0 \u00a0 Let\u2019s change player one out for player three. \u00a0Player three is using a consistent dice from a past meta, so it has 6 on each of its 10 sides. \u00a0Player three will have the same win \u2013 loss rate as player one versus player two\u2019s dice, but will have a 0% win rate against player one. \u00a0So, let\u2019s look at this:<\/div>\n<\/p>\n

\u00a0 \u00a0 1. Player one vs Player two: 40-60<\/div>\n
\u00a0 \u00a0 2. Player one vs Player three: 100-0<\/div>\n
\u00a0 \u00a0 3. Player three vs Player two: 40-60<\/div>\n<\/p>\n

\u00a0 \u00a0 Note that player one and player three seem to be equal based on our win rates in (1) and (3).\n<\/p>\n

\n\u00a0However, they are obviously not equal when we look at (2), with a 100% win rate for player one. \u00a0So, what does this mean?<\/p><\/div>\n<\/p>\n

\u00a0 \u00a0 Traditionally, a difference between OCG and TCG deck building has been noted: OCG seems to run builds with more variance between hands (sacrificing consistency for higher power). \u00a0TCG traditionally (not saying it is forever to be static) runs builds with lower variance between hands, but at the cost of some of the higher power (and inconsistent) hands. \u00a0This does not only extend to variance in the consistency sense \u2013 \u2018win-more\u2019 has become a buzzword in the TCG, but with good reason? \u00a0When looking at the game in the approximated dice form above, we see consistency is not the deciding factor \u2013 nor is a card like Dark Hole. \u00a0In the past, Dark Hole was a very dualistic card \u2013 good for baiting your opponent into overcommitting and good when you were in a losing position, but suboptimal when you have an established field. \u00a0The argument here is generally \u2018if I\u2019m winning its fine to have a suboptimal card, as if I start to lose the card becomes optimal\u2019. \u00a0However, is this correct?<\/div>\n<\/p>\n

\u00a0 \u00a0 Let\u2019s say that player two adds a Dark Hole, or a similar card, to his deck. \u00a0What happens here is the values it at suboptimal times and optimal times become averaged over all faces, but are applied differently to different faces of the dice? \u00a0For example, two of the faces with 1\u2019s on them may become 4s, while two of the faces with 9s on them (when he is in a winning position \/ established field) may become 7s (he lost a combo piece for dark hole). \u00a0His dice now has two faces with 1s, two faces with 4s, four faces with 9s, and two faces with 7s. <\/p>\n

\n\u00a0His average dice roll is now a 6, up from 5.8 \u2013 a more consistent deck. \u00a0Let us examine his win rates.<\/p><\/div>\n<\/p>\n

\u00a0 \u00a0 1. Player two vs Player one: 40-60<\/div>\n
\u00a0 \u00a0 2. Player two vs Player three: 60-40<\/div>\n<\/p>\n

\u00a0 \u00a0 While his win rate vs player three is still 60%, his win rate versus player one has dropped 20% (from 60% to 40%) because he sacrificed power for consistency. \u00a0So power is a broad word, how are we defining it? Power is generally used to describe how strong a field can be thrown up (example, Abyss-Teus + Aqua Spirit used to be a common power play), as well as how much damage (both life point and card advantage) something can do (Judgment Dragon is powerful). \u00a0However, speed \/ tempo is built into this definition, and often overlooked. \u00a0There is a winning threshold in this game, where if you are above your opponent by a certain amount (dependent on point in the game and match up) you win.<\/div>\n<\/p>\n

\u00a0 \u00a0 To better implement describe this, let\u2019s give each player three hearts. \u00a0A player loses the difference in rolls in hearts, and players keep rolling until one dies. \u00a0What does this do? \u00a0If one player rolls a 9 and the other a 6, the player whom rolled a 6 will lose right on the first roll. \u00a0However, if one player rolls a 9 and the other an 8, and then the next roll is 4 vs 8, the player rolling the 8s will have one: as the variance based player failed to reach the winning threshold in the allotted time (a better way to describe this would be to have players gaining hearts per turn given by a non-linear function dependent upon the number of turns).<\/div>\n<\/p>\n

\u00a0 \u00a0 At this point we realize that if we are the variance player we have to just forgo the lost hands (because if we roll a 1 or a 4 in Yu-Gi-Oh! it\u2019s not likely to turn into a 9) and find the winning threshold. Once we have done this, we transition into maximizing the likelihood we reach that threshold (as there is no difference between rolling an 11 against an 8 or a 17 against an 8), and then maximize it. \u00a0Of course, due to the sheer number of hands possible in a 37 card deck, our plot will not simply be us seeing the threshold and then having a line for two units at four, for two units at one, for four units at nine, and for two units at seven \u2013 if we were to graph it holistically, with \u2018true\u2019 values assigned to each hand, it would reflect a traditional, continuous graph much more due to the number of points. \u00a0At this point we would graph different builds, and see which has the largest unit distance above the threshold: in effect, which value of deck satisfies this for the largest percentage of hands. \u00a0This would of course be extremely tedious to do by hand, and thus would be done by computer (if we generalize hands to a degree this is actually not hard to code).<\/div>\n<\/p>\n

Returning from this tangent to the concept of assigning different values to different faces, we have four dice, with the following faces:<\/div>\n<\/p>\n

\u00a0 \u00a0 1. Ten 8s (Tier One Consistency)<\/div>\n
\u00a0 \u00a0 2. Ten 6s (Lower Tier Consistency)<\/div>\n
\u00a0 \u00a0 3. Four 1s, Six 9s (Variance)<\/div>\n
\u00a0 \u00a0 4. Two 1s, Two 4s, Two 7s, Four 9s (Variance with Dark Hole type card(s))<\/div>\n<\/p>\n

\u00a0 \u00a0 Our averages are 8, 6, 5.8, and 6, respectively. \u00a0We previously examined the variance vs consistency match up, and saw that it was based on the winning threshold being reached as fast as possible (using the model with non-linear amount of hearts gained per turn to represent grind game). \u00a0In Yu-Gi-Oh!, this is seen how a combo heavy, aggressive deck will generally win with the superior power it generates over a limited amount of turns \u2013 compared to the relatively linear power generation of a deck such as +1 Fire Fist or HAT, or will simply fail to win early and then get out grinded as the average power decreases with following turns. \u00a0This is due to Yu-Gi-oh!’s lack of a restraining mana system \u2013 the only thing limiting the combo deck’s aggression is its ability (how long it takes) to convert hand advantage into field advantage. \u00a0While we are on the Magic related note, it should be noted that dice interactions describe deck interactions between agro, combo, and control relatively well, while allowing us to easier compare this to Yu-Gi-Oh! despite the mana system.<\/div>\n<\/p>\n

\u00a0 \u00a0 Let us examine the consistency mirror, and the variance mirror. \u00a0In the consistency mirror, the win will almost always go to the player with the higher average roll. \u00a0For example, going back to January 2014 we see things like Bujins, Blackwings, and Hunders doing abysmally. \u00a0This represents the concept of fairness that has been used increasingly in recent times \u2013 these decks are consistency based in rolls, and for the most part do not have high power variance rolls. \u00a0Because of this, the winning threshold between these decks and a tier one consistency deck (+1 fire fists, represented by dice (1) above) is very rarely achieved in early game (first couple rolls), and almost without fail the lower tier consistency deck falls. \u00a0From this, we see that there is almost no benefit in playing lower tier consistency based decks.<\/div>\n<\/p>\n

\u00a0 \u00a0 In contrast, the variance mirror includes two dice with large bounds in roll values. \u00a0This is allows for the winning threshold to be achieved extremely easily, as if one person rolls an 8 and the other a 2, the game immediately ends. \u00a0This concept is exemplified in a match up like Quickdraw quasar vs Karakuri. \u00a0Essentially whoever bricks will lose immediately if the other does not brick equivalently (for example rolling a 3 vs a 2). \u00a0However, at this point we notice that with the high variance rolls there is little to be gained by increasing your roll power \u2013 for example, if your opponent rolls a 2 there is no difference between rolling a 6 or rolling a 9. \u00a0So, if the meta is mostly composed of variance based decks or dice, you should seek to restrict your high rolls to the minimum winning threshold, and increase your lower variance rolls enough to increase your win rate slightly (for example, adding dark hole type cards).<\/div>\n<\/p>\n

\u00a0 \u00a0 Referring to a post by Hoban in the ban list thread, we see him note that the format generally goes consistency -> variance in terms of tier one decks \/ dice. \u00a0Of course, over time builds become refined and more consistent at hitting their goal (what they\u2019re winning condition is). \u00a0In terms of a constancy deck, which generally has a stable, non-combo based core, we might see slight adjustments to the monster line up (such as the addition of Card Car Ds to Fire Fist, though Card Car is not a true monster) and standardizations to the main deck, such as fiendish. \u00a0This is because in a consistency-based meta, there is only one \u2018best dice\u2019, and thus this dice changes to combat itself better. \u00a0This represents the first half of January 2014\u2019s format. \u00a0Then variance is introduced in the form of Mermail. \u00a0Contrary to +1 Fire Fist, Mermail did in fact have the ability to brick, which resulted in something resembling dice (3) or dice (4) (this was further accentuated games two and three where the fire fist could draw a macro and win unless Mermails could quickly out it). \u00a0However, as noted in the constancy vs variance section, there is no difference between losing with a difference of 10 cards in card advantage between you and your opponent, and a difference of 1 card \u2013 both mean a loss. \u00a0Because of this, Mermail\u2019s variance based style allowed it to accept the losses to macro \/ difi, and focus on the six or so faces of very high value rolls which were enough to reach the winning threshold over fire fist without progressing to the late game, or at least reach the late game with correct set-up (waters for Tidal in grave, and preferably a controller in hand).<\/div>\n<\/p>\n

<\/div>\n
—<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 The second point that I wished to talk about is branches, in relation to card effects, deck building, and decision making. \u00a0Branches are almost never talked about directly, despite being implicitly (and often unknowingly) referenced all the time.<\/div>\n<\/p>\n

<\/div>\n
“Branch Theory”<\/div>\n
<\/div>\n<\/p>\n

—<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 What do I mean by branches? \u00a0Let\u2019s relive elementary school, and say that a scummy player (Let’s call him player X) wakes up one morning. \u00a0There are three things he can do: top a locals, rip someone\u2019s trade binder on Pojo, or post on the Hearthstone thread. \u00a0Let\u2019s look at this.<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Let\u2019s also assume that you\u2019ve been dutifully stalking him for the last five years, and know that he has a 30% chance of going to locals, a 40% chance of stealing a binder through Pojo and a 30% chance of posting about Hearthstone.<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 You also know that when he goes to locals, he has a 60% chance of topping, and steals binders without getting caught 80% of the time.<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 If he has a 60% chance of topping, he has a 100% – 60% chance \u2013 40% — of not topping, and similarly a 20% chance of getting caught. \u00a0So:<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Now, let\u2019s go back to the start. \u00a0If player X has just woken up, and we want to know how likely he is to go and steal a binder without anyone catching him, we follow the branches to that point: so we\u2019ll get 40% * 80%, which comes out to 32%.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 To try and maintain a logical progression, let\u2019s start with card effects, move to deck building, and then touch on decision making.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Pot of Duality is best described by the use of branches (people have difficulty understanding how it helps consistency because they often lack an understanding of branches). \u00a0For this example, I\u2019m going to leave out Upstart Goblins and other draw cards, because the tree becomes very large and daunting. \u00a0Let\u2019s say you\u2019re trying to figure out how much running a Pot of Duality will help you draw Exodia\u2019s head, as opposed to running a non-draw card in that slot.<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 You\u2019ll notice there are two places where your goal is fulfilled \u2013 hitting Exodia\u2019s head in your opening hand, and getting it off a Pot of Duality. \u00a0So, your probability of getting Exodia turn 1 will simply be a + (1 \u2013 a) * p * h. \u00a0You simply do what we did with the previous example, but add each possible path: so we added the first path and the second one in this case. \u00a0Note: these values may be calculated by using a hypergeometric calculator or one of the programs I released if you\u2019re interested in them.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Now that we have talked about using branches to better describe card effects, let\u2019s move onto deck building. \u00a0A question was posed in a deck thread \u2013 is Forbidden Lance or Mystical Space Typhoon optimal? To answer this question, first examine both cards.\u00a0They share a common overlap in preventing your monster from dying to a trap. \u00a0Now let us compare the differences:<\/div>\n<\/p>\n

1. MST is optimal when there is only one backrow and your opponent went first because it plays around Solemn Warning unlike Lance.<\/div>\n
2. Lance is better when there are multiple backrow due to being able to chain to a trap.<\/div>\n
3. MST kills Vanity’s Emptiness.<\/div>\n
4. Lance lets things beat over El Shaddoll Winda.<\/div>\n
5. And more stuff that I\u2019m not going to list, because you\u2019re getting the point of this at this point.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 What we do now is we divide the branch goals between ones that show Lance is better and ones that show MST is better. \u00a0We then use branches to calculate which is more likely to occur by summing the probabilities, and then get our answer. \u00a0I did this to a limited degree here<\/a> (I did not do the full branch as it was an example).<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 \u201cAnd Miss Click, because this deck does not fear any commonly played flood gates, it’s not too difficult to see where Lance is better than MST. \u00a0All we do is look at the average number of vanities + average number of Solemn Warnings (1 or 0) \/ deck, and compare it to other backrow \/ deck and see which is greater. \u00a0Note that Lance has applications outside of merely protecting from backrow due to the attack reduction. \u00a0For example, it allows Kuick to out Winda.<\/div>\n<\/p>\n

<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Also note another thing, if you play MST over Lance solely as a counter for Vanity’s:<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Your opponent has x % chance of drawing Vanitiy’s, multiplied by 0.5 * y due to the fact they’ll normally have to go first to use it against you (sucks against established field). \u00a0y represents your ability to combo at least one synchro monster. \u00a0I will code this in tonight (your chances of not bricking basically).<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 So: getting skilled on by vanities lock is x * y \/ 2. \u00a0(Note that is never actually 0.5 but because we look at the full match we average the value of going first or second).<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Now, let\u2019s say you run a number of MST. \u00a0You have a z % chance of drawing it. \u00a0In order for you to get the desired value out of your MST, you need to have your chance of MST overlap with them locking you with Vanity’s. \u00a0So, you’ll see that to calculate which is better we have to look at the tree:<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Let g be the % chance of them having multiple backrow, L = % to draw lance.\u201d<\/div>\n

\"\"\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 You can apply this line of thinking and problem solving to most deck building applications in the same fashion, so now let\u2019s talk about decision making in game (as in technical play). \u00a0Before we begin, please note that there is always a correct technical play, and the correct technical play can still lose you the game \u2013 the correct play changes based on the amount of information available (before seeing your opponents face-downs, versus being omniscient and knowing the face-downs). \u00a0You have a spell card that will let you target one monster on the field and destroy it, and if the monster is destroyed (ie nothing happens before spell resolves) your opponent takes 8000 life points. \u00a0However, if the monster is not destroyed during the resolution of your spell, you take 8000 life points instead. \u00a0What are the chances of you killing yourself as opposed to your opponent?<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 So, what do we notice about this? \u00a0Our tree is very simple right now, but unintuitive. \u00a0This is because we have yet to expand the branches. \u00a0Let us begin to do this.<\/div>\n

\"\"\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 The monster being destroyed means that your opponent did not have an out \u2013 so let us clarify this by expanding the branch vertically. \u00a0Expanding a branch vertically does not change the values because there is only one path to take (straight down) meaning the probability will simply be multiplied by 100% (1). \u00a0So now, using information such as \u2018what traps does my opponent\u2019s deck normally run, what s\/t have I seen, what has he used\u2019, you are able to make the optimal choice based on the information available to you. \u00a0I am sure many of you do this implicitly already. \u00a0However, what the branches let you do is calculate the correct play by looking multiple plays down (by branching as in previous examples).<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Let us view one more example, in which we see multiple branches upon the play-path. \u00a0You’re in the U.S national finals, sitting across from a player, lets call him Josh.\u00a0 You control a Machina Fortress, and a face-down Forbidden Lance (one in grave, none in deck). \u00a0Your opponent has two face down cards. \u00a0You enter your battle phase, and Josh activates a Mirror Force. \u00a0Now, generally you would protect the Fortress with lance, but in this case it is game three, and in game two X activated a Magic Cylinder against you. \u00a0You are at 2500 lp, and consider that he may be trying to bait your lance to game you with a Magic Cylinder. \u00a0He asks you if you have a response to Mirror Force, what do you do?<\/div>\n

\"\"<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 In this example, we notice varying gains values depending on the path the duel follows — so in other words, different results can be better to aim for even if they are less likely. \u00a0Let us fill out the probabilities as we have done previously.<\/div>\n

\"\"\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Now to solve this, you simply add up the possibilities from activating Lance, and compare them to the possibilities from not activating Lance, and see which is higher.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Lance: -s * w + (1 – w) * r * s + (1 – s) * (1 – r) * p * -w + (1 – w) * (1 – s) * (1 – r) * w<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Don’t Lance: g * r \u00a0* w + g * (1 – r) * m + (1 – g) * -b<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Note: You could probably expand the tree if you’re feeling skilled, but niche probabilities have extremely low impacts. \u00a0Remember that when comparing possible plays you don’t need to find the probable gains value (probability of it happening * gains) but only figure out which play has higher value (in other words you don’t need to find x, but only if x > y).<\/div>\n<\/p>\n

<\/div>\n
—<\/div>\n
<\/div>\n
\u201cCombining Dice and Branch Theory\u201d<\/div>\n
<\/div>\n
—<\/div>\n
<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 The next question is likely \u2018how do I combine dice theory and branch theory\u2019. \u00a0The main application (not to say that there aren\u2019t others you can link them for) which I use is the match system. \u00a0In the dice theory section, the players had been playing singles \u2013 however, Yu-Gi-Oh! is played in matches obviously. However, are all matches the same? \u00a0No, as you may play two games in a match and go second both times, or may go first, second, first, etc. \u00a0The thing we have to note here is what our dice face values are representing — the faces are a combination of the deck building, luck, and technical play (decision making). \u00a0Note that contained inside the deck building set is the subset of your opening hand — which varies going first or second (five cards versus six), and that the deck itself changes after siding. \u00a0For example, in the variance vs consistency match up that we were representing before, game one player three may have a 60% win rate against player one. \u00a0However, post side floodgates are generally applicable for the variance deck (due to variance being generally more combo oriented and weak to floodgates). \u00a0For example, by adding macro and dimensional fissure to his deck, the fire fist player changes two of his faces from 8s to 11s, making it so that unless the variance \/ Mermail player rolls a high enough number (a number containing an MST), he will simply auto lose. \u00a0This changes the win \/ loss rate from 60%, which is important. \u00a0We are still able to generalize these face values to constant however, by creating different faces for going first \/ second, pre-siding \/ post-siding, etc. \u00a0So let us look at a picture.<\/div>\n

\"\"\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Without any information other than we’re playing a game, we could aliken this to a coin flip — if it heads, we win, if tails, we lose. \u00a0At that point it would be a 50-50 w\/l. \u00a0You will notice that with no information, we assume each branch is a 50-50, and we will end up with 6 red circles (lose the match) and 6 blue (win the match) showing that in fact with no other information it remains a 50-50 through the branching. \u00a0Now, let us apply dice theory. \u00a0By applying dice theory, you can approximate win \/ loss ratios for pre \/ post siding and going first \/ second. \u00a0So in other words, you will end with 8 percentages, shown here.<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Win % Pre siding going first: a<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Win % Pre siding going second: b<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Win % Post siding going first: c<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Win % Post siding going second: d<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Loss % Pre siding going first: 1 – a<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Loss % Pre siding going second: 1 – b<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Loss % Post siding going first: 1 – c<\/div>\n
\u00a0 \u00a0 \u00a0 \u00a0 Loss % Post siding going second: 1 – d<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 If you wish to be extremely precise you may approximate percentages for information gained and such (making differences between game two and game three), as well as the percent of the time that your opponent will opt to let you go first after they lose. \u00a0However, I want to keep the example relatively simple, so I shall stick with this and assume that your opponent chooses to go first each time they can, and that you do the same. \u00a0Viewing music for resulting picture<\/a>.<\/div>\n

\"\"\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 After you’ve taken in the monstrosity that is the resulting jpg, note that your chances of winning the match are:<\/div>\n<\/p>\n

0.5 * a * d + 0.5 * a * (1 – d) * c + 0.5 * (1 – a) * c * d + 0.5 * b * d + 0.5 * b * (1 – d) * c + (1 – b) * c * d<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 and your chances of losing are:<\/div>\n<\/p>\n

0.5 * (1 – a) * (1 – c) + 0.5 * 1 – a * c * (1 – d) + 0.5 * a * (1 – d) * (1 – c) + 0.5 * b * (1 – d) * (1 – c) + 0.5 * (1 – b) * (1 – c) + 0.5 * (1 – b) * c * (1 – d)<\/div>\n<\/p>\n

\u00a0 \u00a0 \u00a0 \u00a0 Now, the skilly thing here is that if you have a 50-50 win \/ loss ratio at a point (lets say for a, so a = 50% or 0.5) then 1 – 0.5 = 0.5, and we see that the chances of winning and losing are equal (when a = b = d = c = 0.5). \u00a0Note that I did not reduce and compact the above lines (for example, have 0.5 * ( a …… * d)) because I wished to better demonstrate the application of branching here. \u00a0Note that you may go one set higher, and generalize your match win rate (from this branch chart, as a function of round number among others) over the possible paths your win \/ loss record could take (this depends on how long the tournament is), and see your chances of winning, topping, etc.<\/div>\n<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u00a0 \u00a0 By making approximations on a system, it is far simpler to see general rules and logic lines, and their effects, than pre-approximations. \u00a0To this end, let us look at Yu-Gi-Oh! after making some approximations in this manner. “Dice Theory”<\/p>\n","protected":false},"author":43,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[5,25,26],"tags":[521,46,520],"class_list":{"0":"post-8000","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-articles","7":"category-deckbuild","8":"category-strategy","9":"tag-decision-trees","10":"tag-deck-building","11":"tag-math"},"acf":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=\/wp\/v2\/posts\/8000","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=\/wp\/v2\/users\/43"}],"replies":[{"embeddable":true,"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8000"}],"version-history":[{"count":0,"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=\/wp\/v2\/posts\/8000\/revisions"}],"wp:attachment":[{"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8000"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8000"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ygorganization.com\/staging\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8000"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}