Variance and Entropy

One of the most compelling qualities of Magic is its tremendous replay value. If you sit down two players with two sealed decks to play ten games, every one of those games will be different. Why? Because of the randomness that drives Magic.

However, the way we talk about randomness in games is often imprecise or even misleading. When we say that a card is "unpredictable", we usually conflate two similar but distinct concepts. I'd like to define and separate these ideas, so that we can say what we mean to say. (And maybe even design what we mean to design!)

Variance: the lack of predictability of a card's contribution towards winning the game.
Entropy: the lack of predictability of the player's experience of playing a card.

The former definition is one you've probably seen before in some form. It's how poker players talk about variance. Tom LaPille wrote about it in this excellent Latest Developments column.

But the latter definition is entirely different: it's talking about the variety of ways in which a card can be experienced, independent of that card's likelihood of winning the game for you.

Variance

The most obvious types of high-variance cards are those with explicit randomness: Enlisted Wurm, Friendly Fire, Haunted Fengraf, etc. However, the more common and less obtrusive way for a card to be high-variance is situational usefulness. Some cards are only good early (Carnophage), only good late (Boulderfall), only good alongside other cards (Boggart Sprite-Chaser), only good as answers to other cards (Eyes of the Wisent), only good if you’re winning (Ruthless Cullblade), only good if you’re losing (Gavony Ironwright), only good in very particular situations (Hidetsugu’s Second Rite), etc. Linear mechanics such as Constellation, Slivers, and Affinity also fall into this category, as do threshold mechanics. Whenever you have a card that’s potentially awesome or terrible depending on the game state, that’s a variance-creating card.

Conversely, low-variance cards tend to have more evenly spread out utility. They may be the kind of spell like Compulsive Research that you’ll be happy to cast in almost any circumstances. Another way to reduce variance is to give a card modes. This isn’t solely about Charms and Commands, though; many cards have non-explicit modality, like Searing Spear (removal or dome) and Unsummon (tempo play or removal dodge). Also, there are oodles of keyword mechanics that create modality and thereby reduce variance: Cycling, Kicker, Multikicker, Entwine, Fuse, Replicate, Overload, Bestow, Evoke, Reinforce, Retrace, Suspend, and Transmute, to name a few.

Some cards reduce variance not by virtue of their own power, but because they allow you to make a choice between situationally useful cards. Scry and its many non-keyworded cousins fall into this category. Lastly, any card that lessens the problem of drawing too many or too few lands is also a variance-reducer, because it mitigates the randomness of screw and flood. This includes Landcycling and Scry on the low end, and Landfall and Spellshapers on the high end.

Which direction do we turn these knobs in? Too little variance, and the game turns into chess; too much, and it’s Mario Party. Ultimately, the exact degree depends on the needs of the set. For example, Theros had Heroic, an inherently high-variance mechanic that relied on having both the Heroic creature and an aura or combat trick in hand. Therefore, it needed Scry to smooth things out and increase the likelihood of getting appropriately Heroic behavior.

Entropy

A low-entropy card is one that only gives the player a single kind of experience. Such cards do the same thing any time you play them: Lava Axe always burns face. Elvish Mystic always taps for {G}. But even simple cards can provide players with some variety of experiences: Divination may draw you a game-winning Frost Titan, or just a pair of Islands. Mind Rot might catch nothing, or it could clear out the Doom Blade they’ve been sandbagging.

High-entropy cards create a larger diversity of experiences. As before, explicit randomness is one way to achieve this effect, either through coin flips (Molten Birth), random discard (Hanabi Blast), or using library order (Goblin Guide, Bloodbraid Elf). Entropy also comes from spell modality (Gnarlid Pack) and on-board modality (Wooly Loxodon). Threshold mechanics increase entropy; attacking with a Sabertooth Outrider is a completely different experience if you’ve got Formidable.

While tuning variance is a mostly mechanical problem, managing entropy is about the players’ emotional state. Low entropy comforts players with familiar patterns; high entropy excites them with novelty. Every game needs some of both, although the balance point depends on each player’s experience level and personality.

These two types of randomness are not perfectly correlated. As shown here, a card can be high or low variance and high or low entropy:

	low variance	high variance
low entropy
high entropy

Unpredictability and the Player

The diversity of Magic’s audience requires different kinds of cards for different people. Let’s take a careful look at how variance and entropy are received by players.

Appreciation for variance is negatively correlated with Spikiness. In part, this is because Spike wants to win by demonstrating her mastery of the game; success via random chance feels hollow to her. However, it’s also for a more practical reason: variance-reducing cards create more reliable decks, which are simply more likely to win consistently throughout a tournament. Spike will still play the most powerful cards if they’re high-variance, but she won’t be happy about it. And she’ll be even more unhappy about losing to high-variance cards. Remember the amount of vitriol directed at Bloodbraid Elf and the “Cascade lottery” during Jund standard?

On a different axis, appreciation for entropy is positively correlated not with any of the psychographics, but with experience. The more games a player has under his belt, the more he has seen all the common situations, and craves variety in card effects and board states. Many of the highest-entropy formats such as Cube and Commander are extremely popular with veteran players, who can process the giant smorgasbord of weird old cards and mechanics. Conversely, new players don’t need quite as much entropy. They can derive enjoyment from analyzing much more conventional game states, because they haven’t seen them all so many times before.

Wrapping up

I hope this article has changed your mind a bit about what “randomness” means in Magic. More importantly, I hope it informs your decisions on how to use these tools in design! Sound off in the comments and let me know what you think.