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Applying game theory to your multiplayer Magic

Theoretically Gaming

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The letter F!rom time to time, this column makes a quick reference to game theory. Game theory is "the study of the ways in which strategic interactions among rational players produce outcomes with respect to the preferences (or utilities) of those players, none of which might have been intended by any of them" (source: Stanford Encyclopedia of Philosophy).

Put another way – game theory helps you figure out how people make decisions, and what comes out of those decisions.

Because Magic is a game where people make decisions, there's a place for game theory in Magic. This is particularly true of multiplayer Magic, where players continually face concepts such as the Prisoner's Dilemma and the Tragedy of the Commons.

Today, I'm going to get some of you up to speed on game theory and how to use it in multiplayer Magic. If folks like what they read, I may revisit it.

Disclaimer: reading this article will not help you pass your next economics or political science class. No complaining if you don't like the level of detail I choose to use (or not use). And I reserve the right to pull out all the academic gobbledygook and sacrifice a little precision for the sake of broader understanding, and not driving away thousands of readers who come to this web site to read something besides a dissertation. This is an entertainment site, folks. I'm treating it that way.

Because Magic is a game where people make decisions, there's a place for game theory in Magic.

For now we'll start with two basic concepts: something called zero-sum games, and then the idea of rational decisions.

C'mon, this'll be fun! I promise to throw a pink dinosaur somewhere in the middle. Bet your teacher never does that.

Zero, My Hero. But Not Your Hero.

Zero-sum situations happen when your gain has to come at someone else's expense.

The definitive zero-sum spell is Consume Spirit. When you resolve that spell properly, someone loses (your target, or its controller) and someone gains (yourself). Your gain comes through someone else's pain. What's more, the down and up sides are exactly the same – if you want to gain six life, you need to deal six damage. If you want to deal twenty damage, you have to gain twenty life.

Black has a lot of similar spells, like Syphon Soul, Syphon Mind, Drain Life, and so on. If you know the flavor of black, you understand why: black believes that resources are finite, and the only way to benefit is to crawl over someone else to get there.

Finite resources. That's important. We'll get back to that.

Sometimes it's easier to define a term by pointing out what it's not. You know when you play Prosperity for 13, and all nine players in the game "have" to draw 13 cards? That's not a zero-sum game. Everybody gains 13 cards (and assumes the risk that comes with drawing a large portion of the library).

You know when you Hull Breach the Winter Orb and Stasis that have been driving five other players nuts? That's not a zero-sum game, either. Even if you're the most glad of the six players to see those cards go, the other players also benefitted (since it was driving them nuts before).

If you're helping someone else as you hurt someone else, that's not zero-sum. That's "win-win". (Or, from the perspective of the guy who got hurt, "win-win-lose".)

With zero-sum, there are only so many marbles in the world. You only get a marble if you manage to take it from someone else. Marbles don't appear out of thin air.

Zeroing In On Magic Applications

Three tips for Magic applications:

1) The zero-based format. If you want to understand zero-sum games in multiplayer Magic better, try the following format (with any or all of the following elements):

  • Every time a player draws a card, that player chooses an opponent. That opponent discards a card.
  • Every time a player or a permanent a player controls deals damage to an opponent, that player gains that much life.
  • Every time a creature goes to the graveyard because of an opponent's permanent, spell, or ability, that opponent may put a creature card from his hand or graveyard into play.

With zero-sum, there are only so many marbles in the world. You only get a marble if you manage to take it from someone else.

Don't play this format for too long – it's easy to build an abusive deck for any one of the three elements. But an evening spent doing this (especially with a limited card pool) should drive home the point that cards like Geth's Grimoire are used often in multiplayer Magic for a reason.

2) The zero-based deck. Once you return to your regular free-for-all format, try a blue-black deck based on the following cards:

ZeroOut.deq
Multiplayer deck fragment

4 Consume Spirit
4 Syphon Mind
4 Control Magic
2 Confiscate
2 Blatant Thievery

3) The zero-based play. The next time you're playing multiplayer Magic, look quietly for situations where a player might face a zero-sum game. Remember, for zero-sum to happen, someone has to gain, while taking something from someone (or a total over several someones) in equal measure.

To get you ready, a quick quiz. Which of the following situations are zero-sum?

  • A Battle-Mad Ronin is the only permanent on the board in a three-player game. It attacks one player instead of the other.
  • In a five player game, I entwine Reap and Sow, destroying your forest and getting an island.
  • In a three-player game where every player controls two creatures, I play Inferno, destroying all creatures on the board. In response to my own spell, I play Congregate, which resolves before the creatures die.

Of the three, I would say that the entwined Reap and Sow is the only clear zero-sum play. A land for a land – it's that simple. In the Battle-Mad Ronin situation, the "gain" for the other two players (the attacker, and the player not attacked) is quite unclear. And in the Inferno example, I may have regained the life lost and simulated a sort of Syphon Soul; but the unequal life movements and total creature loss make this non-zero-sum.

That's enough about zero-sum; I can see your eyes glazing. Let's move on to something scintillating – rational decision-making!

Let's Be Rational For A Moment

An awful lot of analysis in the world depends on assumptions of "rationality". Whether you're looking at stock markets, consumer products sales (including Magic cards), nuclear missile treaties, or even fishing regulations – virtually all of the professionals involved are using analytic tools based on rational behavior.

There are lots of fancy definitions for rational behavior. In plain English, it generally means that when all other things are equal, people will act in their own self-interest. If apples cost $1 in store A and $2 in store B down the street, people will buy them from store A. Those same rational people won't try to shoot someone with a bow and arrow when their target has a howitzer pointed at them. And (most relevant to Magic, perhaps) they won't willingly help someone hurt them.

If you have a choice, with one Terminate in your hand, between Eager Cadet and Grizzly Bears, you will almost certainly hit the Bear. If you are rational.

If you have a choice, with a Hill Giant available to attack, between reducing one player to zero life and another player to 49, you will pick the player with close to zero life. If you are rational.

If you have a choice, with a Jilt in your hand and only a Faceless Butcher and a Royal Assassin under an opponent's control, you will almost certainly bounce the Butcher and smack the Assassin. If you are rational.

We all know intuitively that in Magic games, there are people who act in "crazy" ways. They play Obliterate when their teammates are just starting to dominate the game; or they play Aether Burst when X=3 and pick three creatures that don't appear to be doing anything at all, leaving the Verdant Force untouched.

Those are examples of irrational behavior. Irrational behavior crops up a lot in casual play, because there are little (or no) stakes involved in winning or losing. In tournament play, irrational behavior is punished ruthlessly. (You spent your tournament fee to go nowhere; or you didn't win the $500.) Part of the disdain tournament jockeys have for casual players is based in some casual players' tendency to…

…BE A PINK DINOSAUR SUDDENLY!

See? That kind of crap holds no water for the tournament jockey. They can't stand it. Drives 'em nuts.

When you try to analyze rational behavior, it's easy to gloss over one phrase: "when all other things are equal". I even did it a few paragraphs ago – and all of those quick examples really depend on it! Let's revisit them.

If you have a choice, with one Terminate in your hand, between Eager Cadet and Grizzly Bears, you will almost certainly hit the bear…unless you also have a Slay in hand.

If you have a choice, with a Hill Giant available to attack, between reducing one player to zero life and another player to 49, you will pick the player with almost zero life…unless that player has the only enchantment removal available and the player with higher life has a Test of Endurance on the board.

If you have a choice, with a Jilt in your hand and only a Faceless Butcher and a Royal Assassin under an opponent's control, you will almost certainly bounce the Butcher and smack the Assassin…unless the Butcher removed a Phage the Untouchable under your control.

Another example of an "equal" violation is the player who goes a little nuts. All players must be equally rational for rationality to work.

Let's look at this from both sides, because I've received many emails that involve hurt feelings all around. I've heard from group players who have been playing Magic at the kitchen table for years. They're casual players, but there's one guy in the group who drives them crazy – always making purposely bad plays because they're "funny", always flipping coins that end up destroying teammates' permanents, and so on.

I think what's fundamentally bothering these players is the lack of rationality in their friend's actions. Unpredictability can be fun – but Magic already supplies a healthy amount, and there's no need to intentionally season it with more randomness. At some point, you cross from "fun" into a bunch of coin flips, either figuratively or literally.

Now take it from the other side. I've heard from players who feel like others in the group take everything too darn seriously. Everything has to be efficient, everything has to make sense – why can't they mix it up from time to time?

I think what's fundamentally bothering these players is the presence of too much rationality. Magic is a game, and around the kitchen table things should be less formal. The point isn't to have a bunch of robots all calculating the same percentages. The point is get people smiling and laughing.

Exercises In Rationality And Irrationality

Sometimes, just talking this sort of thing out among your group can go a long way. I hope I've given you some language to do so above. But you can also suggest a few things for your group, if you want to play on the edges of rationality and what it means to Magic:

1) Second best target. There's a concept in game theory called "second price auctions". I'll get into it more in another article, if I make this a series. Suffice it to say, using this sort of auction is an excellent way to get people to see and express how they truly value something.

How does this apply here? Like this: for at least one game, no one may attack another player or target anything without first openly pointing out what the BEST target for an attack or effect is…and then, that player must go after the SECOND BEST target.

This forces everybody to compromise, for a game. First, ultra-irrational players have to go through the exercise of actually analyzing the true threat on the board. But then they get to opt out of it, which ought to be fun. At the same time, ultra-rational players have to watch as the best decisions don't get made, time and time again – though they do at least win the moral victory of having everyone at the table know what the correct play actually is.

There is a slight risk that folks will not agree on what the best/second best target is, every time. In those cases, the group should move on to the third-best target. Don't spend your time arguing – that's not the point.

2) Daring team play. Play your favorite team format, with this twist: you cannot win if your teammates have more than 5 life.

Once again, this forces people to make intentionally irrational plays – but again, they have to acknowledge the rational play before adopting the irrational one. And of course, because of the way the format works, the irrational play is the rational play. So no one can argue. (Or at least not rationally.)

3) You could…BE A PINK DINOSAUR!

See, I bet you didn't see the second one coming. No one ever does. That's why the game warden in Jurassic Park died.

Wrapping Up

I know those of you who are students or experts in game theory will be itching to tell me all of the nuances I've "missed" thus far. No doubt your keen analytic mind will also understand the sound theoretical principle of assuming someone else will send the correction you are thinking of sending. If you're right, I'll hear about it and will take appropriate action – and you won't have to do a thing. If you're wrong, I won't hear about it at all.

Oh, the dilemma! So delicious in its beauty, so deadly in its irony.

Tell you what – since I pay the costs of you erring on the side of sending an email, let me suggest this alternative. Post any errors to the message boards, where others may benefit from your devotion to academic precision. I'll gladly acknowledge corrections there.

Here's further reading for those interested in game theory. I used some of these sites to refresh my memory of some exact terms; but the application of game theory to multiplayer Magic is my own doing. I'm not saying that to take credit for anything spectacular; I'm saying that to assure readers that errors in this article are mine alone.


Anthony has been playing multiple Magic formats for over seven years, and has been writing far longer than that. His new fantasy young adult novel, Jennifer Scales and the Ancient Furnace, was co-written with his wife MaryJanice Davidson, and comes out August 2005 from Berkley Books.

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