This means that a metagame with 8 decks with perfectly even meta share would have a slightly lower uncertainty than this metagame, while a metagame with 9 perfectly even decks would have a slightly higher uncertainty. In Mythic Championship 6 where the meta share looked like this:
Let’s do some examples and unpack what this really means. So in the realm of magic metagames this number tells you how many decks with exactly equal metagame share another metagame would need in order to have the same “diversity” as the meta you computed the entropy of. If you take your entropy and raise 2 to that power the number that you get is how many equally likely events you need to produce the same uncertainty. This corresponds to the two binary bits it would take to catalog the 4 possible outcomes.Īnother way of thinking about what entropy means is to construct a diversity index. The maximum is still at p=½ and the E(½)=2.
This should make sense, we are considering more events, so there are more possible outcomes that could occur and thus our uncertainty is higher. If we graph this on the same plot as the single coin, we can see that for all values of p the uncertainty is higher than before.
There are now 4 outcomes with the following probabilities To make things a bit more interesting we can think of two flips of this (potentially) unfair coin. bits) it would take to represent the two different outcomes (we could use 0 for heads and 1 for tails). It’s no coincidence that this is the number of binary digits (i.e. We cannot count on one outcome happening more frequently than the other. But when p=½ the two outcomes are equally likely. When p=0 or 1 there is no randomness, the outcome is always determined. This should correspond to your intuition. We can see the entropy is maximized when p=½ and is minimized at p=0 and p=1 (1). Which we can graph with p on the horizontal axis and E on the vertical entropy) there is in a coin flip that comes up heads with probability p and tails with probability 1-p.
This gives us a negative number and so we multiply this result by -1, so that larger entropy corresponds to more format variance.Īs a simple example we can think about how much uncertainty (i.e. To compute a metagame’s entropy first take the share of a deck and multiply that by the base 2 logarithm of that share, do that for every single deck and add up the results. One way to quantify such uncertainty is a concept from information theory known as (Shannon’s) entropy. To some extent the value of format diversity comes from the uncertainty of an opponent’s deck choice or the novelty of playing against different decks. These aren’t easy questions to answer, but they are important ones for play designers to answer. Which draft format was more diverse: Adventures in the Forgotten Realms or Innistrad, Midnight Hunt? When was standard at its most diverse? How do different constructed formats compare?
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