The Beautiful Struggle – The Limits of Math

Today’s Beautiful Struggle sees Mark go Math Crazy! With the aid of some high-level mathematics, Mr Young attempts to cure the Magic populace of one of its cardinal sins… the belief that a winning play is the correct play regardless of the probabilities behind it. We’ve all made mistakes such as these, but Mark suggests we let math be our guide.

A lot of people talk about math in Magic, but the reality is that the game defies many types of easy mathematical analysis. For example, Vintage players often say that you’ll have Force of Will in the opening hand 40% of the time, but when I calculated that same figure myself, I found that I was making a lot of assumptions. For example, my 40% figure included those hands which might have six non-mana cards and a Force, or a Force and no other Blue cards, or which might not be keepers for some other subjective reason. The odds of having a castable Force of Will in the opener, or having an opener with a Force and some action, could be considerably different.

To illustrate what I am talking about, I am going to first refer to a game where the math is pretty clear-cut: poker. Yes, yes, I know you don’t want to hear a bad beat story, but I swear we are getting to a Magic-related point here, if you’ll just bear with me.

The Play

Our Hero (not me) has about $400 in front of him and is in the small blind with the king of spades and the three of clubs. Five players join Hero and limp in the pot for the minimum $3, and the flop comes Ace-4-5 with two spades. Hero is first to act and bets $15 on his inside straight draw. All opponents fold except for The Villain (also not me) in late position, who has about $200 total in front of him. Villain raises another $30, making it $45 total to stay in. Hero ponders for about ten seconds and calls.

The turn is an offsuit 2, giving Hero his straight. He checks, Villain moves all-in, and Hero obviously calls. Villain had Ace-4 of hearts for two pair, and lost when an irrelevant card came on the river. A full analysis of the hand would cause this story to get too long, so I’d like to restrict myself to the math behind Hero’s play.

The Analysis

Let’s focus on the situation after Villain raises. Hero is facing another $30 from a person who would not raise in that situation without at least a medium Ace. To win the hand he will most likely have to hit his straight, so he has four outs out of 46 (the known cards are the flop and his own hand, and he is certain that at least one card of Villain’s hand pairs the board, so 6 cards are taken care of; 52 – 6 = 46). Hero realistically has to hit on the turn, because if he does not hit the turn he will not be able to call the huge bet Villain will likely make. Hero’s odds of winning are thus 11.5 to 1.

The good news is, if Hero hits his straight he is likely to get Villain’s whole stack. However, the maximum Hero can win is $233 ($18 before the flop + Hero’s $15 flop bet + $200 in Villain’s stack). So Hero is faced with the choice of risking $30 to win $233, putting the money risk at less than 8 to 1. Hero would be risking his money out of proportion to his odds of hitting the straight, so he should fold. This is why gamblers going back to the riverboats of the Old West have said that you shouldn’t draw to inside straights – often your opponent’s entire stack isn’t even enough money to justify the odds of hitting.

Now to Magic

The Magic-related point here is something that I’m sure every reader out there has seen before: the winning player used the fact that he won to justify his play, regardless of whether the play was actually right. In response to criticism, Hero simply said over and over, “Look, I played it so that I could get his whole stack, and I did!” as though that should end the argument. He never once acknowledged that even his opponent’s whole stack might not contain enough money to justify drawing to the straight.

This is also why I had to tell you the entire details of the hand, instead of keeping the story short and speaking in generalities: because even number-savvy people will not accept the math of a play if they don’t want to. A lot of players (even some who should know better) develop “eyes bigger than their heads.” That is, they are tempted more by the possibility of a huge win than they are tempered by the odds of the win actually happening. Then in those low-percentage cases when they do win, of course it was because they are great players, regardless of what the universal language of mathematics might have to say about it.

In Magic, you see this happen in a lot of different ways. You’ll see a player keep a hand he probably should not have kept, usually on the reasoning that if he draws a land of a certain type he’s golden. If he needs that land to play a four-drop, it’s not so bad; hitting at least one out of eight or nine outs in four draw steps is about 70% according to my back-of-the-napkin calculations. However, if that certain land would enable a two-drop, it would be a much more dangerous move. Similarly, people will sideboard reactively, such as bringing in Mana Tithe because they have no other answer to a bomb like Triskelavus, not considering what the odds might be of having the Tithe at the same time the opponent has the Trike and exactly seven mana.

Once you start doing math at the level I am doing it now – i.e., the level where you’re aiming toward a Masters degree and a cushy job to better support your Magic Online habit – one of the most important concepts is infinity. I’m not a religious man, but the preachers are right about one thing: the infinite does not lie. Wacky low-probability events are weeded out, and the system tends toward whatever probability dictates it ought to be. If you flip a coin fifty times and wager $1 on each flip, there might be a miser who takes $50 from you. If you were able to flip that coin an infinite amount of times, though, the two of you should just split your money 50/50 and find something better to do with your eternal life.

An interesting example comes from a recent 4322 draft on Magic Online. Yes, I know, there’s almost no value in the 4322 queues. However, I am in a bad slump in TTP draft, battling just to keep my rating above 1700. As with poker, when you face a lengthy slump sometimes you have to move to a lower-limit game.

My draft was a bit of a mess; I saw quality cards in Blue, Red, and White, and just when I thought I was going to commit to two of those three colors, I received a strong pick in the third one that I couldn’t bring myself to pass. I opened a couple of mediocre Blue cards, and then I got passed Firemaw Kavu and Lightning Axe; just when U/R seemed the way to go I opened Serra Avenger and nothing else in Blue or Red; just when W/R might have been the plan, I opened Serra Sphinx in the third pack. I ended up with roughly equal devotion to all three colors, and only 13 creatures.

The first two rounds were not particularly close as I blew people out with Firemaw Kavu + Snapback. In the finals, I faced an opponent whose deck appeared to be G/R with a couple of splashes. I won game 1 when I read him for Tromp the Domains and saved some removal spells to play in response; he took game 2 on the back of Intet, the Dreamer.

The Play

In game 3 I was on the play and kept Temporal Isolation, Prodigal Pyromancer, Dead / Gone, and four lands in all three colors; a mediocre hand that I kept because my opponent had shown me a lot of one-toughness guys and he could well be blown out by the Pyromancer if it stuck around. My opponent mulliganed to five on the draw and the game proceeded…

My turn 1: Mountain, go.
His turn 1: Forest, suspend Search for Tomorrow, go.
My turn 2: Plains, go.
His turn 2: Greenseeker, go (missed his second land drop).
My turn 3: Island, Prodigal Pyromancer, go.
His turn 3: Find a Swamp with Search for Tomorrow, play Premature Burial on the Pyromancer, attack with Greenseeker, go (missed his land drop again).

He surprised me with that Premature Burial; I had seen no Black spells from him in games 1 or 2. During this time I had drawn an Aven Riftwatcher and a Snapback, and on my turn 4 I drew Firemaw Kavu. I played the Riftwatcher and dropped Dead on the Greenseeker.

The Analysis

Games on MTGO tend to move at a fast pace due to all of the extraneous clicking one has to do. Given that, I spent a considerable amount of time (a minute or two) thinking about playing the left half of Dead / Gone on that Greenseeker. Typically, I let Greenseekers live. Yes, the opponent is thinning his deck and straightening out his mana by using the Seeker, but they usually have to throw away some useful cards to do so. Additionally, I really don’t want to waste a card on a 1/1 dude if I can help it. Finally, even if the Seeker lets him curve out, I have Temporal Isolation for his dragon and the Firemaw / Snapback combo for anything smaller.

On the other side of the argument, the opponent is clearly mana-screwed. His deck needs its mana more than, say, a G/R deck that it just trying to thin its lands so that it draws more fat creatures. Also I have almost no offense to speak of; the Riftwatcher will get in for four damage, and perhaps I could also Snapback it for four more, but the remainder of my hand is completely reactive. Killing the Greenseeker is really the only way to put pressure on my opponent, by forcing him to draw land.

As you might imagine, I think that the best way to solve this dispute is with math. My opponent has seen nine cards (mulled to 5, three draw steps, and searched out a Swamp), two of which were lands. If he is running 17 land, that means he has 16 spells and 15 land left in his deck. The odds of his next card being a land are 48%, and if he misses, the odds of his getting there on the following turn are exactly 50/50 (15 land out of 30 cards left).

However, that analysis assumes that every land in his deck gets him there. It may not be so. He might not even have a three-drop; he might have kept his five-card one-lander because it had Forest + Search and he didn’t want to go to four. Even if he has a three-drop in hand, his one-of Urborg and Island may well be blank draws if his three-drop is a Red card or the Yavimaya Dryad he had shown me in game 2. In reality, the above estimates of his odds are very generous, and his real odds of topdecking a playable land are probably lower.

I decided to kill the Greenseeker because the opponent is slightly worse than a coin flip (assuming, at best, that every single land gets him there) next turn, and if he misses he improves to an exact coin flip (again, at best) on the following turn. I have a 25% chance that’s he’ll miss lands on the next two turns, and about a 50% chance that he’ll get there on a third land but won’t make it to four land anyway. Note that it was easy to estimate these odds in my head because Magic Online gives you the count of cards remaining in a player’s library; my internal dialogue was something like, “he has 31 cards left, if he has 17 land then his land odds are 15/31, that’s just less than half…” and so on.

I decided that if we played this position an infinite amount of times and I killed the Greenseeker every time, I would win a majority of those games. Maybe only a slight majority, 52% or something like that, if his hand is full of gas and he just needs one land. That’s still a majority, though, and it means that I am the favorite if I kill the Greenseeker. That’s why I made the decision.

Now, you may be asking yourself, “What happened? Did the decision pay off?” If so, you have missed the entire point of the article. Even if you intend to never play a hand of poker in your entire life, you should take the lesson of the poker example with you into Magic: there is winning and losing, and there is correct play, and one does not always follow the other. In the same way that you can play well and still lose to a lucky topdeck, it is possible that you played quite poorly in a match that you won.

In the Dead / Gone example, one out of these three scenarios definitely happened…

* My opponent topdecked a forest, enabling first a Yavimaya Dryad and then a Penumbra Spider the next turn. I had drawn a Crookclaw Transmuter, so I felt compelled to play Temporal Isolation on his Spider to force through damage. However, he later played a Sporesower Thallid and some more random Green dudes while I drew land, and I was overrun.

* My opponent topdecked a Forest and an Island on consecutive turns, enabling Hunting Wilds into Intet, and when I played Temporal Isolation on the flier he had a sideboarded Krosan Grip. He had boarded in Grip because I had shown him two Isolations, a Prismatic Lens, and a Foriysian Totem in the prior two games, and it paid off here as the Dragon scored from 24.

* My opponent topdecked a Mountain and had no play. I used Snapback on my Riftwatcher to get in quite a bit of damage and lifegain, and followed with a Jedit’s Dragoons that were nigh impossible for him to kill. I Kavu’d his first relevant blocker, and he was simply unable to stabilize.

… but none of them changes the rightness or wrongness of the Dead / Gone play at the time I was considering it. If I think the play makes me a favorite on the numbers, it’s the right play. If not, then some other play is correct. Any other sort of analysis… well, it just doesn’t add up.

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