The Role of Luck in Magic

It is a widely known matter that the trading card game Yu-Gi-Oh! originated in Ancient Egypt. What is much less widely known is that it technically isn't the world's oldest CCG. Why not? Well, allow me to explain...

July 2005 - London

Bananas are a great source of
"You hold down 'Control' as you play the card," I explain, looking glumly at my opponent's Uyo, Silent Prophet. He takes two points of mana burn, having failed to copy the Rend Flesh which just brutally killed my powered up Akki Underling. I take a mental inventory of my deck, searching for a card that can get rid of Uyo before the flying damage proves fatal. Besides things already in my graveyard, only two cards will save me. I draw neither and die horribly. My opponent does not draw Uyo for game two. This is something of a relief, since I can accept thanks for my good sportsmanship without it needing to cost me a game. Wait, did I say "game"? I meant "match." Oh, wait, did I say "match"? I meant "5-0 record."

That's the game I play every time a new set comes out online. Only a perfect 5-0 record in the release leagues wins you the special avatar for that set. On paper, I should be easily good enough to have won one by now. In practice...

"You suck?" my cat suggested.

Betrayers of Kamigawa was the closest I ever came. With a 4-0 record I ran into an opponent who played Umezawa's Jitte against me on turn two of game one. His play was OK, but nothing special - more than enough to win that game. Swearing at the screen a bit helped to calm me down enough for game two. Not quite calm enough, as it turned out. He had no turn two play to answer my Nezumi Cutthroat. On turn three he played a Dripping-Tongue Zubera to face my Bloodthirsty Ogre. Turn four, I dropped an Ogre Marauder and was feeling good about my chances this round. He only had a few turns to draw his Jitte and with Ryusei, the Falling Star and a [CARD]Scuttling Death[CARD] in my hand my endgame was looking good too. Then my opponent attacked with his Zubera. I didn't really want to trade the Marauder with it, so I let it through...


They say that to a man with a hammer, everything looks like a nail. Mathematics is my hammer in this respect. As I began the long wait for Ravnica to be released online, I pondered the theoretical aspects of how one achieves a 5-0 record in Sealed. In theory it should be no different from trying to get a good record in any Sealed tournament, because each match is independent. Whenever you play a match, you want to win, so trying to go 5-0 is no different to going 1-0 except for being a bit harder.

As I was immersed in the calculations, a thought occured to me: I was actually working on one of the most interesting theoretical questions in Magic. I mentioned this fact to my cat.

"Sorry to break this to you," my cat responded sarcastically, "but the true extent of your suckage is not an interesting theoretical question."

But it knew what I meant really. Constructed or Limited, the question is always there in the mind of every serious player. Some of your losses are to luck and some are to your own errors. What you want to know is: how often are you unlucky when you lose?

Thanks to Rysorian Badger, my
secret is safe.
I went to the fridge to retrieve two litres of my caffeinated, carbonated beverage of choice. You have to understand that I take great care not to specify exactly what it is because amongst gamers this is a religious issue and I don't wish to lose half my audience by making my preference explicit. Armed with my drink I diligently covered several dozen pages with equations. Then I pushed back my swivel chair and gazed at my work. I wasn't sure I had the right answer.

"Your problem," said my cat, "is that you don't really understand the question. Do you need to borrow my time machine?"

September 1683 - Vienna

The time machine appeared outside an enormous ornate tent. The smell in the air was incredible: a heady mixture of gunpowder, perfume and horse manure. The cat ran into the tent effortlessly, being a cat. For my own part I looked nervously at the huge guards with their polished scimitars. Fortunately, there was a clipboard in the time machine.

"Just taking some readings," I explained vaguely as I walked nonchalantly past the guards with my clipboard. They nodded. One of them muttered something disapproving at my lack of a waxed moustache.

We had arrived just in time (unsurprisingly) for the start of the final of Worlds 1683 between Pasha Kara Mustapha and the Polish national champion Jan Sobieski. Now obviously all my readers will know which way the match went, but it's worth pointing out that the Turks had won Worlds for the last 31 consecutive years, so Kara Mustapha was definitely the favourite.

The format was Extended, best of three. What I was looking out for was the role of randomness in the match.

Now on the face of it, with the two best players in the world facing each other you might expect perfect play on each side. If they were playing perfectly, the element of luck would work as follows:

* Before decks were chosen, each player would have exactly 50% chance to take the match.
* Once deck and sideboarding choices were known, each player would have fixed chances of winning each game.

As it turned out, Jan Sobieski was playing Tradewind-Survival whilst Kara Mustapha was playing a customised five-colour Blue deck. Mustapha's deck had taken down four consecutive Tradewind-Survival decks 2-0 in the earlier rounds, so he was fairly confident.

I conferred with my cat. The empirical statistics suggested that Mustapha was going to win, but did his deck really have the advantage in this matchup, or had he massively outplayed his earlier opponents or even been lucky in earlier rounds?

As a level 2 judge, Sobieski was
entitled to wear DCI barding
in battle.
Game one saw Sobieski play an Ophidian on turn two, which was countered by a pitched Force of Will. He continued to apply pressure, but his first Tradewind Rider met a Mana Leak and the second decided to do a spot of farming. Sobieski still managed to muster some board presence, but he had only one counterspell in hand in the form of Force of Will. When Mustapha was ready, he cast a Morphling with Counterspell backup and the game was his shortly afterwards.

Game two went rather differently. Sobieski now knew that Mustapha's deck contained few to no basic lands. He sideboarded in four copies of Back to Basics. This raised the deck's total of must-counter threats massively. Mustapha's use of resources was flawless, but in the end he was left almost tapped out after a counter war over Wrath of God and Sobieski was able to force through a Back to Basics. Tradewind Rider followed next turn and Mustapha was locked down.

Game three was the one that raised my eyebrows. Of course I knew that Sobieski was going to win. However, as Mustapha drew him into a counter war over Wrath of God it wasn't clear to me how it could happen. Then the game turned. Sobieski had drawn three of his four copies of Back to Basics. Mustapha simply had no options and the decline of the Ottoman Empire began.

My cat and I made ourselves scarce as armed melee broke out amongst the spectators.

"So," asked the cat, "was Sobieski lucky because he drew so many threats, or was that the correct result because his deck was packed with threats?"

"Never mind that," I responded over the sounds of cavalry charges and dying Austrians, "how come they're playing Magic when Richard Garfield didn't invent it until 1993?"

"You have enough on your plate with your questions about luck in Magic," my cat replied as we climbed back into the time machine, "but if you must know, the history of Magic was covered up by Pope Pius VI at the time of the French Revolution."

One the inquisition missed:
A statue at Nantes Cathedral
ponders whether to mulligan.
December 690 - Bingzhou prefecture, China

As we stomped through thickets of bamboo, I began to understand why the cat had shown me the 1683 match. With two perfect players, the result of a match is in fact completely random. Importantly, this means that even if the particular sequence of events which leads to the actual win seems extraordinary, the match was still a close one in every sense that matters.

This is an important idea, because it means that the question "How often are my defeats down to chance?" is really just a rephrasing of the question "How often do I make mistakes?". Because when it isn't a mistake that costs you the game, all else is chance. This is a fact which is often lost when commentators remark that Magic is essentially a game of skill. But of course, nobody does play a perfect game. Because of this, I still had a lot of research to do.

The match we were here to watch was between Emperor Shengshen, the ruler of China, and a nameless monk who had challeneged him to a match. At least that's what I thought. We arrived at the ceremonial duelling mat to discover I was wrong in at least one respect.

"Hey, I never knew Emperor Shengshen was a woman!" I hissed at my cat, "And just typical that the one time you find a woman playing Magic she's in her sixties."

"The Emperor's name was once Wu Meiniang," my cat explained patiently, "She entered the harem of Emperor Taizong at the age of thirteen and was elevated to the fifth rank of the harem within a few years. Or to put it in your terms, she was once 't3h h0tn355.' She learned to play Magic from her father, whose position as a civil servant meant he travelled a lot and was easily able to attend all the PTQs in China."

I was impressed by my cat's ability to use l33t-speak in spoken conversation, but I suppose by the time you're a talking cat it's not much of an extra stretch. The Emperor's opponent was a much more enigmatic character. Nobody knew who he was. He arrived, bowed deeply to the Emperor, then spoke. My cat translated.

"He says that the Emperor's behaviour in leaving the monastery to become the concubine of Emperor Gaozong was against the teachings of Confuscius and..."

"Wait a sec - this guy's insulting the Emperor to her face?!"

"Yep," the cat nodded, "he's basically calling her a tart. He gets executed for it after the match, but first the Emperor wishes to prove that he is wrong. She will do this by defeating him at Magic the Gathering."

The Imperial headgear was a source of
inspiration for Matt Cavotta.
As play began, it was pretty clear that the Emperor was playing Sligh. The monk simply played and used a Wall of Roots and I watched with interest as it blocked a Mogg Fanatic, was pinged by the same Fanatic, and was then Shocked into the graveyard. Card advantage to the monk, I thought to myself. However, Sligh did that thing that Sligh does and eventually the monk was dead. It was still unclear to me what he was playing except that it was something which didn't respond well to having its lands Wastelanded.

Game two saw the Emperor play an early Dwarven Miner and I wondered if the match was going to be a 3-0 whitewash. Then I saw the key card. Restricted to only green mana, the monk still managed to play Survival of the Fittest. After a long game, Recurring Nightmare brought Verdant Force into play, earlier discarded to Survival. The monk won the game. He was playing Rec-Sur!

Game three, the Emperor played a Mountain, then used Wasteland to destroy the monk's Karplusan Forest. I assumed this meant a win for Sligh... but the Emperor never drew a third land and remained on a single Mountain. There was much murmuring from the courtiers. They were a superstitious bunch and the Emperor's ill fortune seemed to them to be an omen.

Game four, the Emperor needed to win to stay in the match. Watching this game, it became clear to be why we were here. Mogg Fanatic, Dwarven Thaumaturgist, Ball Lightning, and Ironclaw Orcs played from one side against Verdant Force, Staunch Defenders, and Spirit of the Night put into play via Recurring Nightmare on the other side. It was all a bit one-sided.

"Wait a minute," I tugged the cat's tail to get its attention, "I'm no history expert, but I thought the Emperor won that match?"

"Oh come on," the cat gave me a despairing look, "Who do you think writes history?"

Realising we had better get back to the time machine before we were put to death as witnesses, we ran for it.

August 1915 - Budapest

The lesson from China seemed simple enough, but it's something all players are prone to forgetting. Most of the time when you lose a match to a player of about the same skill level it's because of the decks involved.

Picking your deck for Constructed formats is a game unto itself. If you really care about winning, the received wisdom is that you should simply play the best deck in the format. In the modern era, it's usually pretty clear which deck that is. Sometimes there isn't a single best deck, but instead a set of so-called tier-one decks. How does this happen? It's the rock-paper-scissors effect. If you have one deck which does well on raw power, but another deck with marginally less power mostly beats it then both are viable. And from there it's quite possible that a third deck will be viable which is somewhere between the power level of the other two but doesn't lose easily to either. In theory more complex structures could arise with four, five or more top decks. I'll believe it when I see it.

This is not the same thing as the metagame. Players get confused about that all the time. The metagame is not the set of decks people could pick, it's the process of deciding what deck to actually pick based on what decks you think everyone else will actually pick. This is very different, because even when there is a clear best deck - such as Trix or Affinity - not everyone will play it. In fact, metagaming for a tournament like States, you expect to see mostly random rubbish instead of any of the good decks until you reach later rounds.

"Here we are," announced the cat.

We were standing in a covered cloister to one side of a grassy courtyard. On the ground sat two schoolboys playing a game of Magic. One of them seemed to be spending some time thinking about his moves, whilst his opponent played with blinding speed, completing each turn in seconds. After a few minutes, the slower player conceded.

"Was that interesting?" I asked the cat.

"Well yes," the cat nodded, "the boy is János Neumann, he has won his last 49 matches. That should be relevant to your assessment of randomness in Magic, no?"

"Seriously, 49? No way is that possible!"

"You're not thinking straight," the cat sighed, "If you look around you, we're in a school. How good do you think his opponents are here?"

At that moment, another boy arrived at sat down to play.

"This match will not be so quick," the cat pointed with a paw, "This boy is Eugene Wigner, a finer player than you or I will ever be."

As the game got underway, it was immediately clear I was watching a Rebels mirror match. Both sides got off to a good start, but János was doing the damage thanks to repeatedly attacking with Steadfast Guard. More importantly, he was the first to cast Lin Sivvi, Defiant Hero, which he then used to summon a Rebel Informer.

"That's game," I said to the cat. After all, they were playing with the old Legend rule that meant Eugene could not cast his own Lin Sivvi if he drew one.

"Nope," the cat replied, "keep watching."

Eugene had Dominate for the Rebel Informer. János was up 20-6 in terms of life total, so he cast all his creatures in quick succession, looking to beat down for the win before the Informer could swing things. Eugene had the answer, however. A Rout cleared the board. János was unfased, he simply laid a Chimeric Idol and used his Rishadan Ports to keep enough tempo to secure the win.

"You can learn a lot from that game," my cat explained, "lesser players might have thrown that game away from either side at a much earlier stage and claimed bad luck. Random events occur in Magic all the time; the question is simply whether they are enough to end the game."

Game two was again a matter of tempo. After both players mulliganned, an early Ramosian Sergeant for János meant plenty of cheap attackers. The details of the match were confusing to me, since without knowing the decklists it wasn't clear whether Eugene was right to counter János' Parallax Wave. He was probably right, since the second Parallax Wave a few turns later took the game.

Game three was interesting. János managed to establish board advantage with an active Mageta the Lion thanks to multiple Rishadan Ports. Choosing his timing carefully, he tapped down the last of Eugene's blue mana, then went for the Armageddon. That was game and match.

"That makes fifty matches," I announced as the players shook hands. The bell rang for the end of the lunch break.

Raphael's portrait of Pope Leo X.
Inset shows a close up of the
cardinal on the right.
"The question is, how should Eugene assess his luck in that match?" the cat asked, "Clearly to assume the match is 50/50 would be to underestimate the role of János' skills. Eugene also had inferior draws in at least two of the matches. On the other hand, his build should have been the stronger of the two in this match. What is your view?"

My cat never asks me my opinion unless it's planning to make me look foolish, so I just shrugged rather cautiously.

"Eugene sideboarded out his copies of Absorb. That left him with a shortage of ways to keep Armageddon from resolving," the cat explained.

"How much did that matter?" I asked, wondering how the cat even knew this fact.

"Don't know," it twitched its whiskers thoughtfully, "but János was not confident he would win before the match and after the match that was his own explanation of why he won. What's interesting is that immediately after the match, Eugene might have had no idea that this was his main problem."

February 1748 BC - Babylon

By now I felt I had a pretty clear picture of the issue of luck in games. What I had learned was this:

Luck is not a directly measurable element of the game because it is what is left when you take everything else away. We measure light, not darkness. We measure sound, not silence. We don't make notches on the bedpost to count the number of cute girls we've failed to sleep with (because quite apart from anything else that would do considerable damage to the bed).

What you need to measure is not your opponent's skill either. Whenever your opponent plays less than perfectly, that's simply a bonus. Forget about it. The one thing it does determine is what proportion of matches we could win if our own play was good enough. That's never going to be lower than 50%. If it's higher, that's simply a bonus.

What we really need to measure is our own skill, or rather our lack of it. What we care about is whether we made a mistake that cost us a game.

In short: we need to measure the extent to which we suck. I realized belatedly that my cat had been teasing me again.

"I think I understand now," I said, "But if Eugene couldn't spot his own error then what chance do I have?"

"That's not the point," the cat responded in its best tone of indifference, "what matters is that you realise that the level of luck involved has little impact on playing Magic, only on the results. What you want to do is get a 5-0 record with your Sealed deck? If each match was 50-50 that would take you thirty-two attempts. At three Expert sets per year, that'll take you ten years. Or I suppose you could have a couple of tries each time, which would still mean an average of five years. Let's suppose you instead win 87% of your matches, then you'll get a 5-0 record almost half the time."

"But it might be impossible to win 87% no matter how well I play!"

"That doesn't matter either; that was just an example," the cat grinned evilly at me, "What matters is that if 87% was possible, you'd still win far less. Maybe 70%, more likely only 60%. Because you suck badly."

At first I thought my cat was just being mean again, but the more I thought about it, the more it made sense. I couldn't be playing perfect Magic, because the rating system told me exactly how good the best players were. There are a few players online who are as much as 200 points above my rating. If I played one of them I'd win only about 1/4 of the time according to the way ratings are calculated.

That's nothing like 50%.

The oldest known copy of
Giant Growth.
At about this point, it suddenly occurred to me to wonder where we were. We had been walking through a dusty wilderness as we walked and we now arrived at a group of crude buildings made from mud bricks. Standing in front of us was a large stone into which were carved hundreds of tiny markings.

"Hammurabi's Comprehensive Rules," my cat declared with a hint of awe, "the first ever time the rules of Magic the Gathering were written down."

As I was examining the stone, a dark skinned man approached and bowed to us, then spoke. The cat translated for him, explaining that he wanted to play me at Magic.

It was too hot to play outside, so we went inside the mud hut. The format was Type 1, since these people had no concept of other formats. My opponent was running some kind of White Weenie deck featuring the unstoppable power of Benalish Hero and Mesa Pegasus. I had an unpowered version of Welder-MUD which I hadn't tuned in almost a year.

After the first 20 games, I was ready to call it a night. My cat, however, pointed out that my opponent's pride would not allow him to concede until he had won a match. So, the next game I carefully arranged to lose, which required quite a bit of ingenuity to prevent my opponent from noticing. Fortunately, the stacking of Smokestack and Tangle Wire activations was a mystery to him, so it was an easy matter to "accidentally" stack them the wrong way and end up sacrificing my own board to my Smokestack having "unfortunately" not drawn a Goblin Welder. He recovered and began to beat down, albeit rather slowly. I was "mana flooded" at this point, so was "forced" to throw my Metalworkers away as chump blockers. I conceded and quickly shuffled my hand back into my deck in case he inspected it and wondered why I was holding back two Powder Kegs.

As we walked back to the time machine, my cat said nothing, whilst I reluctantly admired its ingenuity. It had brought me to an environment where I was the master of the game. What better way to understand how 80% or better records are possible than to achieve one? I wondered somewhat sadly whether my opponent had been cursing his bad luck. From his perspective, my deck had far too many mana sources and too few creatures. Did he understand why I won?

July 2005 - London

Home at last. Somewhat cruelly, my cat had brought me back only a minute after we had left, meaning that I had to set out for work.

As I was heading out of the door, I turned to my cat and asked, "Hey, how come you never use that machine of yours to change the past? Or to visit the future?"

The cat looked guilty and mooched off to find food. I chased after it and picked it up by the tail. "Hey fur-face, I asked you a question!"

"I did," the cat said quietly, "you know how much I loved Dirk Baberowski's Elves deck, right? So I stuck this post-it note on the fridge in Seattle."

I raised by cat to my eye level, still holding it by the tail, "What did it say?"

"Thought Extractor is too good, change the +1/+1 to +1/-1 instead."

My eyes went wide with shock, then I drop-kicked my cat out of the window and headed off to work without giving it breakfast. After all, it wouldn't do to be late.

A Discreet Footnote
The Vatican, following the edicts given by Pope Pius VI, hid the details of the famous matches described in this article as follows:
* Sobieski vs Mustapha was retold as William Jensen vs Jason Zila for the first ever Masters title in 2000.
* Emperor Shengshen vs the nameless monk was retold as Ben Rubin vs Brian Selden in the finals of Worlds 1998.
* János Neumann vs Eugene Wigner was retold as Kai Budde vs Kamiel Cornelissen in the finals of Pro Tour Chicago 2000.

Editing: Goblinboy


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