There is No Such Thing as “Randomness” in Trading Card Games
Randomness Does Not Exist
An event is random only if it is unknown (in its totality). A state is random if it is unknown. Randomness is thus a synonym for unknown.
—William Briggs, Uncertainty: The Soul of Modeling, Probability & Statistics
“Randomness” does not exist. It does not exist anywhere. It certainly, then, does not exist in trading card games (TCGs).
What we call “randomness” is shorthand for “unknown” or “uncertain”. We invoke “randomness” to describe states or processes that are too complex or too hidden for us to make statements about with certainty.
The Classic Problem of the Urn
In statistics, we frequently use drawing unknown-colored lots from an urn. (This is the classic problem in statistics, dating back to Jacob Bernoulli in the 18th century.)
Consider the problem anew.
We have an urn with two colors of balls inside it: some black and some white. We’re not sure how many balls are inside the urn nor the proportion of each color among the balls.
We choose a ball, unseen, “at random.” What color should we expect?
Consider Figure 1.
It is reasonable, given the situation, to say that we think the probability of a black ball will be 0.5000 and that the probability of a white ball will be 0.5000. Our probability assignments are conditional on our state of knowledge.
But is the choice really “random”? The ball that we pick is a physical object in physical space. The balls are either black or they are white. Each ball occupies a position inside the confines of the urn. We are free to pick, unseen, any ball from inside the urn.
In what sense is this “random”?
The word random, in this description, is operating as unknown. When the balls where placed in the urn (before we had a chance to pick from among them), their colors were not determined “randomly”. They were given definite colors by some manufacturing process that was decided upon by other agents. The positions of the balls themselves, even assuming that they were dumped into the urn without much care or consideration, follow the laws of physics, and are expected to fall in positions consistent with those laws of physics.
The colors of the balls and their positions in the urn are definite. We just don’t know what they are.
Now consider this: what if the urn is transparent? See Figure 2.
If we can see the interior contents of the urn, the idea of “randomness” disappears entirely.
We now know for certain the colors and positions of the balls inside the urn. Randomness is just our ignorance of the state of the world.
“Randomness” in TCGs
In trading card games, we like to think of the shuffling of cards in a deck, the order in which cards are drawn, and the amount of player skill affected by “luck” (good or bad) as random.
Just like with the problem of the urn, what we call random is really just shorthand for unknown.
If the cards were transparent, we’d know which ones would be drawn next. If we had better knowledge and modeling techniques, we might get closer and closer to certainty about the outcome of a match. (Consider that, however, this would be an insurmountable amount of required knowledge: to know not only the order in which cards will turn up in players’ decks, but what each player will do in response to each available card and each card as played by his or her opponent.)
In the Pokémon TCG, each player sets aside six “prize” cards at the start of the game, face down and unseen, off to the side of the play field (see Figure 3). As players defeat one another’s Pokémon, they draw these prize cards (again, unseen); when a player draws his or her last prize card, he or she wins.
Prizes are commonly understood to be placed “at random” at the start of the game and, when drawn by the players, are thought to be at likewise “at random”. But as we’ve seen, random is just a stand in for unknown.
The expected pay off of taking Prize Card 1 instead of Prize Card 4, for instance, isn’t random; it’s unknown. The prize cards are definite already. Their identities have been set.
But what if, like the audience in a tournament stream, we can see the prize cards, like in Figure 4? How random is the selection of a prize card now?
This is the exact same issue as that of the urn.
Randomness does not exist. We just have an inferior state of knowledge.
Implications for Modeling TCG Probabilities
In the every day practice of playing TCGs, this insight that randomness does not exist may bear little consequence.
The big changes come in when he model probabilities and form hypotheses about various outcomes. If randomness doesn’t exist, applying it can’t really help us model probability.
How Much Does “Card X” Affect Win Probability?
Consider a common technique in statistics: the randomized trial. In a randomized trial, we set out to test some effect—say the inclusion of a new card called Card X and its effect on a player’s win probability. We take 100 players and “randomly” assign them to two groups: Group 1, consisting of 50 players, uses 4 copies of Card X in their decks, while Group 5, consisting of 50 players, does not use any copies of Card X in their decks.
We then set off the two groups, Group 1 and Group 2, to a tournament and track their win loss record. Suppose that Group 1 wins 68.75% of their matches and Group 2 wins 43.75% of their matches. Since we “randomized” our groups, we believe we are safe to conclude that Card X does in fact have an influence on player win probability and it is recommended to use it.
Focusing only on the use of “randomization” (and ignoring some other big problems with this inference), we should recognize that the randomness has done nothing for us. In fact, it may have made our understanding of what Card X is doing, if anything, even worse.
Using “randomization” has potentially introduced new, unknown variables into our trial. If random is unknown, then the random process has selected a slew of unknown qualities possessed by the members of either Group 1 or Group 2. Do, perhaps, the members of Group 1 have a certain deck strategy that Group 2 does not that is better suited to win given the current format? Are the members of Group 2 younger than those of Group 1? We don’t know. We didn’t control for that. We assumed “randomizing” would remove all bias from our results.
We’re now in a worse state than when we started; worse than not having relied on “randomization” at all.
A Better Way: Controlling Variables
What is needed instead, is the precise control of variables.
Real science, including statistical science, should focus on controlled experiments that leave as little room for doubt about cause and effect as possible.
Consider the trials of Card X. Suppose, instead of using randomization, we select just 10 players our of 100 participating in a tournament (call them Group X). We track all the variables we can and that we believe might have an influence on win probability: age, gender, deck strategy, contents of decks, previous win/loss record, previous number of tournaments attended, etc. in each match (for both those in Group X and those not in it). We then track the win/loss records over the course of the tournament.
After this is done, we invite the same 100 players back and form the same 10 players in Group X. We have Group X use 4 copies of Card X and repeat the tournament, including the same matchups against the same opponents and track the win/loss records.
We know have a baseline: one without Card X and one with Card X.
Now comes the harder part: controlling variables. Ideally, we’d want to test each applicable variable: how does Group X (with Card X) fare when one player is above 18 and one player is below 18? How about when both are above 18? How about if both are below 18? Etc., etc., etc. for every possible combination of variables. (This could be done with modeling.)
We can then draw inferences about the effect of Card X (and probably many more effects, besides, thanks to such a robust dataset).
Then the experiment can be repeated with 10 (or however many) new players. Do these new experiments show the same effect for Card X as the previous ones? If so, we are finding evidence for the Card X effect; if not, we must return to our original experiment and results and see if we haven’t made a mistaken inference.
Real (Statistical) Science
What we’re doing now is finding out how much Card X really makes a difference, which is what we wanted all along.
This is science.
“Randomization” isn’t required because, at best, it does nothing but lull us into believing our results are unbiased, and at worst, introduces new unknown variables that skew our results and lull us into believing in effects that either aren’t really there (false positives) or discrediting effects that are there but we can’t see due to this confusion (false negatives).
We must first admit to ourselves that randomness isn’t real, it’s just a stand in for what we do not know. Then we can move on to finding out the things we really want to know by precisely controlling variables of interest to find out the effects each has on the outcomes we care about.
This is hard work, for sure, which I think is why statisticians prefer to throw “randomness” and “random trials” into the mix to cloak their findings in an aura of mystique. Doing so doesn’t lend to their credibility; it detracts from it.
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