Magically Applied Game Theory

How is your strategic decision making? Mike Flores talks about how game theory can be applied to Magic in order to improve your in-game play with illuminating examples.

How is your strategic decision making?

A lot of Magic players—especially Magic players who fancy themselves thinkers—talk about being involved in game theory… Perhaps without taking into consideration what that is under a greater umbrella than having a theory or two about one particular game (even if it is a game all of us love).

One definition of game theory is the study of mathematical models of conflict and cooperation between intelligent rational decision makers. As Magic players, it can be useful to think about not just what seems to be the “right” go-forward decision based on the information you have but also the right decision given an assumption of what your opponent might know or think— or what you could have him believe.

One particularly tense moment for me was being on the cusp of making day 2 of a team Pro Tour. My teammate Brian Kowal had gotten his opponent below three life and had a Magma Burst in hand—which he seemingly refused to cast. Turn after turn went by with Kowal’s position slipping slowly away…and that Magma Burst would not leave his hand. Nor would he tap mana. So the opponent would do stuff, and Kowal could neither reinforce his position nor move forward on some other vector. Back then teammates were not allowed to talk to each other during games; I don’t know what my actual demeanor was—how easily I might have been read by the boys across the table—but I remember almost bursting out of my chinos. WHY DON’T YOU BLAST HIM TO DEATH ALREADY?

Finally, Kowal had enough mana to make some proactive move; something that might win the game. The opponent Absorbed it, threatening to go up to six—only then did Kowal respond with the lethal Magma Burst.

The obvious go-forward strategy was to burn the opponent with Magma Burst. This would have proven a disaster! Not only would he have not closed the gap on those last three life points but the opponent would have gone up to six. Kowal used a combination of a [correct] read and a thimble full of patience to vary his play from the obvious go-forward strategy to essentially win a game of Chicken.


Two drivers drive towards each other on a collision course; one must swerve or both will die in the resulting crash. If one driver swerves and the other does not, the one who swerved will be called a “chicken” (a coward). In game theory, each player prefers not to yield to the other (though the worst result occurs when neither player yields). In our example from the team Pro Tour, Kowal’s first and most important move was to realize he was playing a game of Chicken (rather than Ultimatum Game or some other game) and act accordingly. In this game [of Magic], whoever acted first was going to cede massive percentage to the other; only by identifying he was forced into playing Chicken was Brian able to induce the opponent to act first.

Sudden Shock became legal in Extended at the exact right time to be played at the World Championship. Sudden Shock’s virtues given the format at the time were endless. It was the nth burn spell to give Boros a critical mass of great creatures and efficient burn; it was a perfect burn spell for purposes of murdering Wild Mongrels, Arcbound Ravagers, and Psychatogs. But as good as it was for killing creatures that had traditionally torn their way past red burn spells, pro players wanted nothing more than to get their opponents to one or two, play lame duck for a while, and then Split Second slow roll for the win. As soon as you had priority, you could insta-win no matter how bad things had gotten (or how deeply down the rabbit hole you had let the opponent go)…

But what about when two players had the same idea and both had Sudden Shock?

I found myself in what amounted to a game of Chicken at a [relatively] recent SCG Standard Open in New Jersey. My opponent was playing The Aristocrats, and I was playing Bant Hexproof. My opponent had won game 1. I looked at my game 2 draw; all other things held equal, the hand was not so weak that I would want to throw it back. It had lands, Auras, and an Aura beneficiary.

The problem?

That one guy was a lowly Avacyn’s Pilgrim.

I kept, took two, and played my Pilgrim.

My opponent immediately paid two life for a Godless Shrineand played Champion of the Parish.

Realizing a moment of relative safety, I slammed down Spectral Flight and got in for three.

We were in a rollicking race at that point, which I won overwhelmingly with Unflinching Courage and Ethereal Armor.

The game of Chicken was a short one. I actually lost it; I acted first. Luckily for me, my opponent handed it right back. He had the opportunity on that Godless Shrine to Tragic Slip me into a position where I would never draw another creature in time. But instead he played his default go-forward strategy, which was not in this case good enough.

This is the true essence of Game Theory in Magic; not what you should pick as a default, but what you should pick factoring in the information your opponent has and gives you as his decisions can and should alter yours as you jockey for supremacy in a conflict of intelligent actors.

Nash Equilibrium

Equilibrium is an important concept for non-cooperative games where each player is informed of the possible decisions that other players can make. Players are said to be at equilibrium if they are all making the best decisions they can while taking into account the potential decisions of the others.

A simple example of a game at equilibrium is Rock/Paper/Scissors. In Rock/Paper/Scissors, Rock smashes Scissors, Scissors cuts Paper, and Paper covers Rock; each of these beats one of the other choices but loses to the remaining option. A game of Rock/Paper/Scissors would be at equilibrium if each player is .33 likely to choose Rock, Paper, or Scissors.

Poor predictable Bart.

The thing is that in Magic formats are very rarely actually at equilibrium. One reason for this is that players have a tendency to copy the decks that just did well. Let’s say Rock just won the last Standard Open. Rock is very exciting, and lots of people now want to play Rock. The format at equilibrium would have 33% Rock, 33% Paper, and 33% Scissors, but because of the big Open win it’s now 50% Rock.

What deck is most likely to win at a proportion of Rock, Rock, Paper, Scissors?

More importantly, what deck is the right choice to play?

If we imagine a four-player event of Rock, Rock, Paper, and Scissors, there are only three possible configurations:

Rock1 v. Rock2 (Rock) + Paper v. Scissors (Scissors)

Rock v. Scissors (Rock)

Rock1 v. Paper (Paper) + Rock2 v. Scissors (Rock2)

Paper v. Rock2 (Paper)

Rock1 v. Scissors (Rock) + Rock2 v. Paper (Paper)

Rock1 v. Paper (Paper)

Paper wins two-thirds of the time! Rock, which makes up 50% of this small field, wins only one-third of the time.

Obviously, we are making a few assumptions here. There are three options and all options win at an equal rate, which is not 100% transferrable to deck choices in the Magic metagame, but I think the principle stands. You don’t necessarily want to play the Deck To Beat; all other things held equal, the most populous deck is not even favored to win.

Matching Pennies

Let us call it Previous Level Rock/Paper/Scissors.

Matching Pennies is the two-strategy equivalent of R/P/S.

Player A and Player B each have a penny and must secretly turn it heads or tails; they reveal their pennies simultaneously. If the pennies match (both heads or both tails), Player A keeps both pennies. If the pennies do not match (one heads and one tails), Player B keeps both pennies.

Imagine you are Player A. Do you reveal heads or tails? If you think Player B will reveal heads, you should reveal heads; if you think player B will reveal tails, you should reveal tails.

Imagine you are Player B. Do you reveal heads or tails? If you think Player A will reveal heads, you should reveal tails; if you think player B will reveal tails, you should reveal heads.

Does this remind you at all of that scene from The Princess Bride?

Here’s the thing about matching pennies: there is no obvious strategy that gives you edge.


What if human beings don’t play at equilibrium? What if you can simply predict (like Lisa in that Rock/Paper/Scissors video) what the opponent is going to do? How do you apply this to Magic?

World Championship some years ago. Two teammates are paired against one another and know that the best strategy is Mono-Blue Tinker. Three days later, two [different] all-time greats will pair off in a Tinker mirror for the World Championship.

Player A is a future Hall of Famer. He has a Tinker deck in front of him.

Player B—a multiple PT Top 8 competitor and Grand Prix winner—has brought two decks with him and registered two forms. One is the team’s Tinker deck. The other is a Mono-Blue Control that he thinks has the advantage against Tinker.

Player B brought two decks for a reason. Player B realizes this is a game of Matching Pennies.

Player B chose to give himself some free percentage in round 1. He played the anti-team deck in order to play with edge in round 1 even though it might have a weaker expectation against the field (to be fair, had he made Top 8, he might have been in a Paper versus Rock/Rock scenario later on).

Player A got the Metalworker nut draw and beat him anyway. Hey! Two-time PT winner and future Hall of Famer… Doesn’t mean Player B made the wrong Matching Pennies choice.

The conceit of a regular game of Matching Pennies is that there is no better strategy than to simply play at equilibrium. However, if you can predict what the opponent will do, you can gain value (or win unilaterally) by adjusting your strategy accordingly.


Blotto is an interesting game about resource deployment. Not only is it highly applicable to Magic, but I know of a forum of high-level Magic players where a Blotto win (again, over a bunch of Magic DIs) was used as the primary resume centerpiece to initiate a career in trading!

There are many variations on games of Blotto, but they generally involve two players deploying an equal number of resources over a designated number of battlefields. Here is a simple example:

  • There are three Towers, each of which is worth one point.
  • Each player has six soldiers.
  • Each player must deploy at least one soldier to each tower and must deploy them in non-decreasing order (i.e., you can’t put three soldiers on Tower 1 and two soldiers on Tower 2).
  • A Tower is won by the player with more soldiers on that Tower; equal numbers of soldiers draw the Tower.

How would you play this game of Blotto?

There are only three possible solutions:

1) 1,1,4

2) 1,2,3

3) 2,2,2

Of these one is superior to the other two and ergo the optimal strategy.

1,1,4 draws with 1,2,3 (they draw Tower 1 and each win one other Tower).

1,1,4 is defeated by 2,2,2 (2,2,2 wins the first two Towers and 1,1,4 wins Tower 3).

1,2,3 draws with 2,2,2 (1,2,3 wins Tower 3 and 2,2,2 wins Tower 1).

The best strategy is 2,2,2 since it is even with one other possible strategy and beats the other.

Here is where we can get interesting with regards to valuation. Jon Finkel defines a mistake in Magic as anything but the optimal play. 1,2,3 is better than 1,1,4 but is still not optimal. Given how limited our choices are, it is difficult to justify any choice but 2,2,2; on the other hand, if we know that everyone knows that 2,2,2 is the best, there is also no down-side to picking 1,2,3 simply because no rational actor will pick 1,1,4.

A more complicated Blotto game might allow you to deploy 100 troops any way you like across 10 Towers [as long as you play at least one soldier per Tower] but with a twist: Tower 1 is worth 1 point, Tower 2 is worth 2 points all the way to Tower 10 being worth 10 points.

How might you play that game?

There are many different possible solutions that get quite heavy into trying to figure out what an opponent will do. When presented with this problem, I put only one soldier on Tower 10 but fought heavily for Towers 8 and 9. My thinking was that my opponents would put tons of ultimately wasted soldiers on Tower 10 and that Towers 8 and 9 would be worth nearly as many points but that I could pick them up more cheaply.

The idea of your opponent committing heavy resources to a seemingly important part of the game—but your actual strategy being focused somewhere else—is an important tool to acquire in Magic, especially for hybrid decks or decks with disparate haymakers.

My favorite look at this might have been KarstenBot BabyKiller; against the dominant B/U/W decks of the time, the strategy was to get out an Ohran Viper and use your resources to keep getting in with it and /or drawing with Scrying Sheets. The opponent would divert resources to stemming your Ohran Viper and try to stay even with your card draw. The real plan was to just Demonfire them to death! The problem was that they could not actually ignore your card advantage game, so the balance was putting enough attention to that so they wouldn’t get blown out but putting enough resources into fighting you that they could race an uncounterable X spell.

Today you have decks like Four-Color Reanimator and to a lesser extent The Aristocrats: Act 2.

These decks play multiple plans, including Boros Reckoner + Harvest Pyre and / or Blasphemous Act. The opponent must measure resources against just losing to one of the best three-drops of all-time, let alone a combo kill. He must be wary because a mistap will result in bad news via Unburial Rites or Falkenrath Aristocrat. Both decks are capable of overwhelming advantage even outside of a “take thirteen” situation.

How the opponent approaches resource deployment against your three or four unique lines of proactive play are very much like a game of Blotto. Remember: you probably have to deploy resources in response to his proactive attempts for the veritable Towers.

One possible solution might be to maintain an air of deception as you move from game 1 to game 2. Let’s say you can play conservatively in such a way that you never reveal your Boros Reckoner but instead use Faithless Looting, Mulch, and Grisly Salvage to play a Lingering Souls / Angel of Serenity game only. You may induce your opponent into sideboarding heavily for Reanimator, either bringing in tons and tons of Ground Seals and Cremates or perhaps going with or reinforcing a Blasphemous Act plan of their own.

How devastating is it for an opponent who has gone Blasphemous Act when you reveal your own Boros Reckoner? Their soldiers are in terrible position, fighting for the wrong towers.

Ultimatum Game

I used to hold fairness as a core value.

Then I read this blog post by onetime DailyMTG and StarCityGames.com Premium columnist (and Harvard MBA and Pro Tour Top 8 competitor) Chad Ellis.

I threw fairness largely out of my decision-making algorithm.

From Chad’s blog:

“Imagine you’re playing the following game. You and another person have been chosen at random to divide $100 between you, but there’s a catch. The other person gets to make the division—that is, they decide how much they get and how much you get. You then get to decide if you accept their split. If you accept, the split happens, and if you don’t accept, then neither of you gets anything. You’re not allowed to talk beforehand, and the game is anonymous—no one will ever know you were chosen to play unless you decide to tell them about it.

Now suppose the split is revealed, and it’s $90 to the other person and $10 to you. Do you accept? Remember, your only alternative to accepting is to get nothing—there are no second chances or renegotiations.

From a purely rational perspective, you should take the $10. Refusing is equivalent to finding a $10 bill on the street and throwing it in the trash just to make sure that some stranger out there doesn’t get $90. And yet a very large number of people will refuse “unfair” offers when this game is played with real money.”

I have applied several looks at this game to various situations in work and Magic since. If you propose splitting $10 $9 to $1, you will very often get a rejection. $1 is unfair! Well, what if you split $1,000,000,000 $999,900,000 to $100,000? That is way more unfair (proportionally) to $1 v. $9, but I can’t imagine very many people wanting to turn down $100,000.

The gut reaction of annihilating a $90 to $10 split feels natural. It certainly does feel unfair—until you think about the fact that given the parameters of the game you did nothing to earn the $10. You are simply rejecting it because someone else (who admittedly also did nothing to earn their $90) got more.

This gut move based on a fuzzy idea of fairness is both anti-utility and inherently self-destructive. Forget about the fact that it is also value destroying (you also cost the other player $90).

It’s not that fairness in and of itself is so horrible, but I came to realize after being exposed to Ultimatum Game the first time that decisions predicated on fairness were not particularly likely to give us the best possible results.

What is the solution? I think a good player takes any amount of money no matter how unfair the split seems. After all, we are playing in an imaginary game theory universe, and we are ultimately talking about free money. In fact, the more sophisticated a player I have (in my admittedly limited n) polled on this, the more immediately willing they were to take smaller amounts, recognizing the value of value over what might otherwise seem like a natural revulsion at an unfair split. “I would take a penny,” said one VP and mother.

For me, in Magic there are at least two lessons here:

1) Strive to pick up on any value you can: unasked for, found, incidental, or just on the bonus. If your opponent taps the wrong land or searches up the wrong creature, think about how adjusting your strategy might uncover a greater likelihood of success. Consider the opposite (you probably don’t play this way, but in my first couple of PTQs I did). If I made a grievous mistake I would sometimes concede; I didn’t feel like I deserved to win, and the Pro Tour was for the best player in the room. The lesson of Gabriel Nassif is to play super hard out of your mistakes, not give into some fuzzy idea or despair when we make one.

2) Fairness in and of itself is something we want to avoid strategically. Why do we read Magic: The Gathering articles? Why do we strive for broken, powerhouse, decklists? Because we want to play with an edge. If you approach this consciously—the way Kowal realized he was playing a game of Chicken—I think you will have better results in the long term.

So… How is your strategic decision making?


[author name=