Sullivan Library – Math, Chance, and Winning

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Today’s Sullivan Library looks to quantify that most elusive of commodities – Luck. In fact, Adrian is here to help us understand the true role of luck, and math, in our glorious game. We all know what it means to roll the die and take our chances… but do we understand the true nature of such things? How can we grant ourselves the best chances of success?

“Some people — pros even — won’t play no-limit. They can’t handle the swings.”
People Magazine’s “Sexiest Man Alive,” Matt Damon, in Rounders

This is not a poker article. This is an article about math. But math impacts poker, it impacts Magic, and it impacts any game in which strategy is employed, even football.

What is the meaning, though, behind that quote?

If you know much about poker, you know that there are plays that you should make (they have the highest Expected Value — EV) that put you in for every chip you have. The proper bet, the one that will, reliably made, net you the best EV, is to go all in.

A proper bet is not necessarily a winning bet. But, if you were to make that bet 10 times, a thousand times, a million-million times, it would be the play that would result in the most return. That does not mean it will give you the best return this time. The best poker players are not only those players who can calculate odds and know the value of a hand, of their position and any number of other factors, but it is also measurable in their ability to actually play to this EV. One of the reasons that I quit playing in regular poker games is that I can’t do that. When I’m sitting at a poker table, the money is real money to me, and I can’t treat a dollar or a chip like a weapon at the table. At the moment of truth, when I look at the EV (if I’m even calculating it correctly), I don’t have the resolve to just put all the money in there. To me, it isn’t a weapon. It’s money. So, I’ll make plays that are going to be sub-optimal, and over time, I’m going to lose more.

But… um, Magic? Right?

A game of Magic on any level, even a team draft, or even a Pro Tour, doesn’t intrinsically have money built into it. It’s an aside. It doesn’t matter how much money might be won or lost. The game itself only cares about the cards on the table. It’s amazing, then, how often Magic players — truly dedicated Magic players — are unable to handle the swings in the game.

What is the most common “swing”…? It’s at the heart of every bad beat story out there in Magic. It’s the loss that defies chance. I’ve been on both sides of this. I always think about a set of games versus Dave Peterson in the Top 8 of yet another PTQ that season. I was playing a deck that had a very good, but not overwhelming, matchup against Sligh. He was with, of course, Sligh. We went to three games, and he came out on top, ever so barely. A playtest partner asked me why I wasn’t furious. “That’s a great matchup you just lost! He got so lucky! And you played it so well! *expletive deleted* this game!”

“We’re shuffling cards,” I said. “If we’re going to bother playing this game, we have to be able to handle the randomness that comes with that.”

It doesn’t matter how good your matchup is — you can lose it.

Take the example of my match at Grand Prix: Columbus versus my co-columnist Nick Eisel. My deck had a good matchup, though not overwhelming, versus the “typical” build of Flash-combo, the Kiki-versions. I worked hard to get it to that point. But the Disciple kill of the deck was nearly unbeatable. I had tried to get some kind of win out of it, but warping my deck to even make the attempt cost me massive percentages everywhere else, and didn’t give much in the way of return. In addition, I was pretty confident that I wouldn’t see much of the Disciple kill, as my own preliminary testing showed that the Kiki-versions were far more resilient and self-sufficient (your mileage may vary). I knew that my win versus Disciple versions approached 0% in a fair game, but the EV was in just forgetting about it. And besides, Magic isn’t a fair game.

Nick had the Disciple version. My chance at winning if things went normally was very, very bad. Game 1, they went normally, and he just blew me out. Game 2, he triple mulligans. Game 3, he doubles.

The thing about Magic is that this can happen. If, in a fair game, you have no chance, remember that you still can win if they get screwed. My reckoning puts Disciple-Flash as a deck that gets screwed, in some way or other, a larger percent of the time. If my game one chance is “0” it is probably closer to 5 or 10%, depending on the nature of the deck, just based on drawing poorly, unlucky mulligans, or any other crap happening. Even assuming I don’t actually help out my chances with sideboarding and the like, that still puts me at about a 1 in 35 chance of winning the match. If my sideboard is at all relevant, that can go up even more. Personally, I felt like my sideboarded chances (including screw) went up to a better, though still wretched 20 to 25% in that matchup. That puts the matchup around 1 in 10 to 1 in 14. That’s bad, to be sure. But I also know that I’ve rolled a 19 or 20 on a 20-sided die.

Wizards has maintained on and on that luck is an essential ingredient to the success of the game at a larger level. Who would want to play a game, for example, where they had literally no chance at winning (unless they simply wanted to say they played against someone)? There is a lot to be said for consistency, of course. But I know I would want to play a game of Magic versus Jon Finkel or Kai Budde or Kenji or anyone of high caliber, and to know that I could win. I absolutely could not say that about any chess master that you could name. This randomness can provide a lot of encouragement to players who are beginning to build themselves up. Before Paul Cheon was Paul Cheon, contender for Player of the Year, he was Paul Cheon, PTQer. Before that, he was probably Paul Cheon, new guy at the game store, or new guy at the Magic game with his friends. If he simply lost every game that he played at the beginning again and again, he almost certainly would not have continued and progressed to where he is now.

So, we live with luck. We need to remember that, and move on, with that in our minds.

Take a typical good matchup. You have a deck, and you know that it beats another deck. Your playtesting wasn’t great, perhaps, but your estimate of the matchup is that it is maybe a 75% matchup for you in the first game, and it gets slightly worse in the second game, going down to 65%. Wow! Pretty great, right?

But you do lose. Perhaps your playtesting was wrong (after all, it wasn’t great, right?). Perhaps you misplayed. Perhaps your opponent was lucky, or you were unlucky. But remember, a 75/65 matchup only wins just over 75% of the time over a full match. They really did have a decent shot (1 in 4). Even if your matchup was better, say 80/80 (a truly incredible matchup), they still have just over a 10% chance of beating you. They can roll a 10 on a d10. It happens.

Really, when it boils down to it, there is a complex interweaving of percentages and math that are going on from the beginning of a tournament, to its completion. Imagine a small tournament of eight people. They each have a deck, and their matchups vary all over the place. There are some 50/50s, some 70/30s, some 40/60s, etc. With only eight people, there are 28 different matchup pairings.

Add onto this the varying play skill of players. Maybe a player could be abstractly attached a “rating” of +10% or more, or a poor player might have a —30%. But it might not even be that simple. In trying to capture the abstract nature of the skill of a player, we can only guess at what it means to capture, say, a Zvi Mowshowitz. Is Zvi best captured by a +X%, or is there some kind of crazily complicated formula that goes on in which deck choice and player skill can interact with results that can spike wildly?

Take an oldie but a goodie. Classically, playing Fires against the best Fact or Fiction based Blue-X control decks would initial favor the Fires player heavily at the low levels of the skill spectrum (i.e., both players were weak to moderate players). Then, at slightly higher levels, the Blue player would begin to come out on top. Towards the top end of the play spectrum, Zvi discovered a strategic choice that would wildly shift the matchup back towards Fires: “stumbling” on your threat tempo. Essentially, with some hands, rather than cast a threat in the previous turn (only to have it most likely countered), he would wait a turn, and cast the spell when the Blue player had out four mana, and force them to have to choose between their answer (the counter) or Fact or Fiction. This could be repeated until the opponent had six mana, when they could do both. At the very top level of play, this again reversed the direction of the matchup, and perhaps a strategic answer might be found that would reverse it again. But modeling for this mathematically is pretty high on the difficulty scale. For the sake of this model, a +X% is going to have to suffice.

In our 8 man tournament, the best player in the room, who happens to be fairly better than the rest of his opponents, might have a +X%. If you run a simulation of the event, where each win percentage is accurately represented, over the course of the event it is likely that there will be upsets. Still, though, the deck/player with the highest chance will tend to float to the top.

But they won’t always.

In a larger tournament, you could simulate the event as well. If you ran that simulation a large number of times, what you’d be likely to find is that certain players and certain decks would float to the top, on average, but that they would reside in a certain “area” of finishes. Jeroen Remie, playing the Rock, might most commonly expect a result in some range of the field, say the 5th to the 15th percentile, with a couple of outliers, one of which might even include a tournament win. Another player, much less experienced, but with an incredible deck, might have a huge range of results that are likely, and this may, in fact, include a larger chance of winning the whole tournament than Jeroen, but also a fair number of finishes that are in the bottom third of the field.

It’s often said that it is rare that the best deck in a field wins the tournament, and while I don’t know that I agree that it is necessarily rare, I definitely agree that it is common that the best deck doesn’t win. Was Wafo-Tapa’s deck or Fortier’s deck the best deck for their respective tournaments? It’s entirely possible. I know that I believe that a computer model of Valencia and Yokohama would result in more Wafo-Tapa wins than Fortier wins, largely because of the nature of the formats themselves. Wafo-Tapa’s deck was definitely an exemplary deck… a +10%, if you will, in a roomful of largely 0 to +2%, whereas Fortier’s, if it was as good, was in a roomful of +10%s.

But, how practical is it thinking about these things?

First of all, there is the poker concept of tilt. To be “on tilt” is to be emotionally and intellectually off your game. Usually, it is linked to being shaken somehow. When you lose a matchup that is, in your estimation, good, it is very easy for a player to become incredibly angry, or to experience a sudden lack of confidence.

You’ll hear again and again from players that some level of confidence is incredibly important to winning. Maybe a part of this can be explained by making plays that would lead you to win out of difficult positions that one might not otherwise make, simply because one expects to win. Take a wonderful example of this in game 1 of round 9 from Tiago Chan tournament report from Krakow. Here he discusses a play which will make him have a shot at winning a game:

So, one play is unlikely to happen, but will keep me alive… but then it’s almost guaranteed I’ll lose a couple of turns later. The other play is extremely unlikely to happen, but if it happens, it can win me the game.


I now needed three things:

1- Draw a second Rack
2- Hit with Tarmogoyf
3- Force the discard of his remaining card with Thoughtseize.

I drew and saw a second Rack staring at me. Good, two to go. I attacked, he did not sacrifice the War Marshal to Greater Gargadon to make a token, and so the Tarmogoyf successfully hit and put him at 5 life. I played Thoughtseize, having only three life remaining and needing to see a non land, and he responds by Pongifying his Goblin token. I played a second Rack and passed. He had a Riftwing Cloudskate coming into play on his upkeep, but there was no way out against two Rack. He’s the active player his triggers go to the stack first, meaning my two Rack will resolve first.

As, Richard Feldman brilliantly describes in his article “One Game,” there is a potential alternate read to a game. “He got lucky,” for example, or from Richard’s article, “I got flooded.”

Chan’s story might have ended differently. He might not have drawn the right card. The odds were against him. But he looked at his play, and went for something that might increase his EV. He recognized chance, and played for it. Had he lost, I have a feeling that he would have shrugged it off, and moved onto the next game.

If you can recognize the power of luck, you can make plays that might require some luck to come to a win, but they at least give you chances for your outs. If you can recognize the power of luck, you can keep your cool more successfully, and be more mentally and emotionally prepared for the next game.

Another thing to recognize when it comes to dealing with luck are the ways that you can improve the range in which luck impacts your game. The game is clearly not merely luck. Working on our own game matters. Take the example of a near-mirror match of Guile-Blue against Guile-Blue. I don’t know about you, but I wouldn’t want to be playing in that mirror against Guilliame Wafo-Tapa. On the other hand, I’d love to be playing against a typical FNM player in the mirror. Even if the matchup is, we’ll say, a 50/50, knowing your deck or knowing how to play in the mirror can entirely shift the matchup one way or another. Certainly, a skill or strategic edge would lean it one way or another, the question might be how much? But, you want to be the one that has the edge, not the person with the edge pressing on your throat.

This also matters, too, in exploring solutions to problems. Sometimes the answer in a matchup, can be completely surprising. One of my favorite examples of this from my own history was a discovery I made in the development of Chevy Blue, a mono-Blue descendant of Alan Comer’s Turbo-Xerox versus Counter-Rebels, back in 1999 or 2000. Fact or Fiction, clearly an incredible card, existed in the earlier version of the deck. That version would die horribly to Counter-Rebels. But, in shifting a card out of the deck, the ubiquitous Fact or Fiction, and transforming it into Thieving Magpie, the matchup went from a blowout against me to a blowout against Counter-Rebels.

Take on moment from a match I had versus Dave Humpherys at U.S. Nationals. From the coverage: “Adrian had more to cast though and a Temporal Adept entered the fray. Dave searched up a Lin Sivvi and things were still looking pretty good for the YMG star.” From my end, I remember thinking that the match was over a few turns earlier when the Magpie had resolved. Dave Humpherys, at his height, was one of the best players that the game has ever seen. But exploring the matchup had given me a strategic edge. Sometimes there exists no decent answer to a matchup, and the correct answer is to ignore the matchup or abandon your archetype of choice, depending on the EV of either choice. The key is to do the work. If you do it well, it will provide you with rewards.

One other important thing to remember about the “range of results” that you can get with a deck, is that there is variance. If you want to qualify for the Pro Tour, that means one thing: maximize your chances by going to as many events as you can! If you want to qualify, you can’t just go to the one PTQ near you and expect that you’ll get a good return. If you’re really hungry for it, you should go to that PTQ that is a little farther. You should attend that expensive PT. You should try to make it to that GP that’s a little bit off the beaten path. More chances are more trials to reroll the dice and maybe make that Q happen. The only real limit is how much you can invest in it. Maybe you have a family or work or school that is competing for your time. Maybe you have a limited budget. Within those constraints, give the time and the money that you rationally can. No matter how good you are, you’re unlikely to just qualify on one try.

I’ve walked into so many PTQs, with a greater or lesser degree of preparedness. I’ve qualified for a lot of Pro Tours, and even done pretty well at a few. I didn’t get there by rolling the dice once. I got there by rolling the dice a bunch of times and recognizing that I am rolling dice. I’m not a Finkel. Most of us aren’t. If Wizards is to be believed, the Finkels of the world represent less than .00002% of the Magic populace at large. If, like me, you’re a part of that 99.99998%, give yourself as many chances that you can.

See you at the next PTQ…

Adrian Sullivan